from Lessons Nanoscience: A Lecture Note Series
Volume 1: Lessons from Nanoelectronics: A New Perspective on Transport Lectures 1-9 Supriyo Datta Purdue University [e-mail]
August 22, 2011
For more information about the lecture note series, see http://nanohub.org/topics/LessonsfromNanoscience.
This DRAFT copy is provided by the author to students enrolled in ECE 453/595 at Purdue University IT SHOULD NOT BE DISTRIBUTED.
Copyright World Scientific Publishing Company, 2011
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Preface
Preface
Everyone is familiar with the amazing performance of a modern laptop, powered by a billion-plus nanotransistors, each having an active region that is barely a few hundred atoms long. These lectures, however, are about a less-appreciated by-product of the microelectronics revolution, namely the deeper understanding of current flow, energy exchange and device operation that it has enabled, which forms the basis for what we call the bottom-up approach. I believe these lessons from nanoelectronics should be of broad relevance to the general problems of non-equilibrium statistical mechanics which pervade many different fields. To make these lectures accessible to anyone in any branch of science or engineering, we assume very little background beyond linear algebra and differential equations. We hope to reach those who are not experts in device physics or transport theory and would like to keep it that way. For dedicated graduate students and the experts, I have written extensively in the past. But they too may enjoy these notes taking a fresh look at a familiar subject, emphasizing the insights from mesoscopic physics and nanoelectronics that are of general interest and relevance. I should stress that these are "lecture notes" in unfinished form, and suggestions from readers are very welcome. Readers can view the actual lectures online http://nanohub.org/topics/LessonsfromNanoscience.
July 2011
Contents
Preface I. The New Ohm's Law 1. The Bottom-Up Approach 2. Why Electrons Flow 3. The Elastic Resistor 4. How Many Channels? 5. Conductivity 6. Diffusion Equation for Ballistic Transport 7. What about Drift? 8. Role of Electrostatics 9. Smart Contacts II. Old Topics in New Light 10. Thermoelectricity 11. Heat Flow 12. Measuring Electrochemical Potential 13. Hall Effect 14. Spin Valve 15. Kubo Formula 16. Second Law 17. Fuel Value of Information III. Contact-ing Schrodinger 18. The Model 19. Non-Equilibrium Green's Functions (NEGF) 20. Can Two Offer Less Resistance than One? 21. Quantum of Conductance 22. How to Rotate an Electron 23. Beyond the Elastic Resistor 24. Does NEGF Include "Everything"? 25. Spins and Magnets: Quantum and the Classical References Appendices
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11 15 23 30 35 50 62 73 84 92 101 112 126 139 153 160 175 186 205 226 235 244 261 268 277 283 284
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Detailed Contents
Detailed Contents
Preface Useful Constants I. The New Ohm's Law 1. The Bottom-Up Approach 2. Why Electrons Flow 2.1. Two Key Pieces 2.2. Fermi Function 2.3. Non-equilibrium: Two Fermi Functions 2.4. Linear Response 2.5. Difference in "Agenda" Drives Flow 3. The Elastic Resistor 3.1. Conductance of an Elastic Resistor 3.2. How an Elastic Resistor Dissipates Heat 3.3. Why an Elastic Resistor is Relevant 4. How Many Channels? 4.1. Ballistic and Diffusive Transport 4.2. Channels of Conduction 5. Conductivity 5.1. E(p) or E(k) Relations 5.2. Counting States 5.3. Drude Formula 5.4. Is Conductivity proportional to Electron Density? 5.5. Quantized Conductance 6. Diffusion Equation for Ballistic Transport 6.1. Flow Dynamics on Electronic Highway 6.2. Electrochemical Potentials Out of Equilibrium 6.3. From Currents to Potentials 6.4. The New Boundary Condition
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50
Detailed Contents
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7. What about Drift? 7.1. Boltzmann Transport Equation, BTE 7.2. Diffusion Equation from BTE 7.3. The Two Potentials 8. Role of Electrostatics 8.1. The Nanotransistor 8.2. Why the Current Saturates 8.3. Role of Charging 8.4. Extended Channel Model 8.4. Rectifier Based on Electrostatics 9. Smart Contacts 9.1. Why P-n Junctions are Different 9.2. Contacts Are Fundamental II. Old Topics in New Light 10. Thermoelectricity 10.1. Linear Response 10.2. Optimizing Thermoelectric Effects 11. Heat Flow 11.1. Electron Heat Current 11.2. Phonon Heat Current 11.3. ZT Product 12. Measuring Electrochemical Potentials 12.1. The Landauer Formulas 12.2. Buttiker Formula 13. Hall Effect 13.1. Why N- and P- Conductors Are Different 13.2. Spatial Profile of Electrochemical Potential 13.3. Measuring the Potential 13.4. Non-reciprocal Circuits 14. Spin Valve 14.1. Conductivity Mismatch or Mode Mismatch? 14.2. Interface Resistance Due to Mode Mismatch 14.3. Spin Potentials 14.4. Polarizers and Analyzers 14.5. Topological Insulators
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Detailed Contents
15. Kubo Formula 15.1. Kubo Formula for an Elastic Resistor 15.2. Onsager Relations 16. Second Law 16.1. Absorption and Emission Rates 16.2. Law of Equilibrium 16.3. Entropy 16.4. Free Energy 17. Fuel Value of Information 17.1. Extracting Energy from a Metastable State 17.2. Fuel Value Comes From Knowledge 17.3. Landauer's Principle 17.4. Maxwell's Demon III. Contact-ing Schrodinger 18. The Model 18.1. Schrodinger Equation 18.2. Electron-electron Interactions 18.3. Differential to Matrix Equation 19. Non-Equilibrium Green's Functions (NEGF) 19.1. One-level Resistor 19.2. Do Multiple Sources Interfere? 19.2. Multi-level Resistors 19.3. Dephasing 20. Can Two Offer Less Resistance Than One? 20.1. Quantum Resistors in Series 20.2. Potential Profiles 20.3. NEGF-Based Numerical Model 21. Quantum of Conductance 21.1. Point Contact 21.2. Hall Effect 21.3. NEGF-Based Numerical Model
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Detailed Contents
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22. Rotating an Electron 22.1. Quantum Transport with Spinors 22.2. Spin Precession 22.3. Rotation Matrices 22.4. Matrix Potentials 22.5. Spin-Hall Effect 23. Beyond the Elastic Resistor 23.1. Two-Level Heat Engine 24. Does NEGF Include Everything? 24.1. Coulomb Blockade 24.2. Fock Space Description 24.3. Entangled States 25. Spins and Magnets: Quantum and the Classical References Appendices A. Relating Fermi Function Derivatives B. Angular Averaging C. Hamiltonian with E- and B-Fields D. Transmission Line Parameters from BTE Equations E. NEGF Equations F. MATLAB Codes for Text Figures
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263 270
279 285 286
Some Symbols Used
Constants 1.6e19 coul. 1 eV + 1.6e19 Joules Planck's constant h 6.626e 34 Joules 1.055e 34 Joulesec ! = h / 2! Boltzmann constant k 1.38e23 Joule / K kT ~ 25 meV / 300K Free electron mass m0 9.11e31 Kg Electronic charge Other Symbols V Voltage volts (V) U Electrostatic Potential eV Electrochemical Potential eV (also called Fermi level, quasiFermi level) I Electron Current amperes (A) (See Fig.3.2) R G G(E) D ! ! A W L Resistance Conductance Conductance at 0K with =E Diffusivity Mobility Ohms (V/A) Siemens (A/V) Siemens (A/V) m2 /sec m2 /Vsec
q
Resistivity Conductivity Area Width Length
Ohmm (3D), Ohm (2D) S/m (3D), S (2D) m2 m m
Some Symbols Used
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E f (E) FT(E) D(E) N(E) N n M(E)
Energy eV Fermi Function Dimensionless Thermal Broadening Function /eV (See Section 2.2) Density of States /eV Number of States with Energy < E Dimensionless Same as Number of Electrons at 0K with =E Number of Electrons Dimensionless Electron Density /m3 Number of Channels Dimensionless (also called transverse modes)
T t T, t ! ! ! !"
Temperature Transfer Time Transmission Probability Transfer Rate Energy Broadening
degrees Kelvin (K) seconds Dimensionless /second eV
[X]+ Complex conjugate of transpose of matrix [X] R G (E) Retarded Green's function /eV GR(E) = [GA(E)]+ Advanced Green's function /eV n (E) / 2! (Matrix) Electron Density /eV, per gridpoint G (Matrix) Density of States /eV, per gridpoint A(E) / 2! (Matrix) Energy Broadening eV !(E)
A New Perspective on Transport
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Lecture 1
The bottom-up approach
"Everyone" has a computer these days, and each computer has more than a billion transistors, making transistors more numerous than anything else we could think of. Even the proverbial ants I am told have been vastly outnumbered. There are many types of transistors, but the most common one in use today is the Field Effect Transistor (FET), which is essentially a resistor consisting of a "channel" with two large contacts called the "source" and the "drain" (Fig. 1.1a).
Fig.1.1a. The Field Effect Transistor (FET) is essentially a resistor consisting of a "channel" with two large contacts called the "source" and the "drain", across which we attach the two terminals of a battery.
The resistance R = Voltage (V) / Current (I) can be switched by several orders of magnitude through the voltage VG applied to a third terminal called the "gate" (Fig.1.1b) typically from an "OFF" state of ~100 Megohms to an "ON" state of ~10 Kilohms.
Fig.1.1b. The resistance R = V/I can be changed by several orders of magnitude through the gate voltage VG.
Actually, the microelectronics industry uses a complementary pair of transistors such that when one changes from 100M to 10K, the other changes from 10K to 100M. Together they form an inverter whose output is the "inverse" of the input: A low input voltage creates a high output voltage while a high input voltage creates a low output voltage as shown in Fig.1.2. A billion such switches switching at GHz speeds (that is, once every nanosecond) enable a computer to perform all the amazing feats that we have come to take for granted. Twenty years ago computers were far less powerful, because there were "only" a million of them, switching at a slower rate as well.
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Lessons from Nanoelectronics:
Fig.1.2. A complementary pair of FET's form an inverter switch.
Both the increasing number and the speed of transistors are consequences of their ever-shrinking size and it is this continuing miniaturization that has driven the industry from the first four-function calculators of the 1970's to the modern laptops. For example, if each transistor takes up a space of say 10 m x 10 m, then we could fit 3000 x 3000 = 9 million of them into a chip of size 3cm x 3cm, since
3 cm /10 m = 3000
That is where things stood back in the ancient 1990's. But now that a transistor takes up an area of ~ 1 m x 1 m, we can fit 900 million (nearly a billion) of them into the same 3cm x 3cm chip. Where things will go from here remains unclear, since there are major roadblocks to continued miniaturization, the most obvious of which is the difficulty of dissipating the heat that is generated. Any laptop user knows how hot it gets when it is working hard, and it seems like a formidable problem to increase either the number of switches and/or their speed any further. These Lectures, however, are not about the amazing feats of microelectronics or where the field might be headed. They are about a less-appreciated by-product of the microelectronics revolution, namely the deeper understanding of current flow, energy exchange and device operation that it has enabled, based on which we have proposed what we call the bottom-up approach. Let me explain what we mean. According to Ohm's law, the resistance R is related to the cross-sectional area A and the length L by the relation
R"
V I
=
#L A
(1.1a)
" being a geometry-independent property of the material that ! channel is made of. The reciprocal of the the
resistance is the conductance
!
G = " A /L
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(1.1b)
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A New Perspective on Transport
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which is written in terms of the reciprocal of the resistivity " = 1/ # called the conductivity. Eq.(1.1a) says that if we reduce the length, L of the channel (Fig.1.1) by half, the ! resistance should go down by half. But what if we reduce the length to very small dimensions? Will the resistance approach zero? Our conventional view of electronic motion through a solid is that it is "diffusive," which means that the electron takes a random walk from the source to the drain, traveling in one direction for some length of time before getting scattered into some random direction as sketched in Fig.1.1. The mean free path, ! that an electron travels before getting scattered is typically less than a micrometer (also called a micron = 10-3 mm, denoted m) in common semiconductors, but it varies widely with temperature and from one material to another.
Fig.1.3. The length of the channel of an FET has progressively shrunk with every new generation of devices ("Moore's Law") and stands today (2010) at ~ 50 nm, which amounts to a few hundred atoms!
It seems reasonable to ask what would happen if a resistor is shorter than a mean free path so that an electron travels ballistically ("like a bullet") through the channel. Would the resistance still obey Ohm's law? What if we reduced the length of the channel down to a few atoms? Would it still make sense to talk about its resistance? These questions have intrigued scientists for a long time, but even twenty five years ago one could only speculate about the answers. Today the answers are quite clear and experimentally well established. Even the transistors in commercial laptops now have channel lengths L ~ 50 nm, corresponding to a few hundred atoms in length! And in research laboratories people have even measured the resistance of a hydrogen molecule. Interestingly, however, this progress in our understanding has not yet influenced the way we think, teach and discuss the concept of resistance or analyze and design electronic devices. For historical reasons, the subject is always approached top-down, from large complicated conductors down to hydrogen molecules. As long as there
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Lessons from Nanoelectronics:
was no experimental evidence for what the resistance of a small conductor might be, it made good sense to start from large conductors where the answers were clear. But now that the answers are clear at both ends, a bottom-up view starting from the hydrogen molecule seems called for, at least to complement the top-down view. After all that is how we learn most things, from the simple to the complex: quantum mechanics, for example, starts with the hydrogen atom, not with bulk solids. But there is a deeper reason why the bottom-up approach can be particularly useful in transport theory and this is the "new perspective" we are seeking to convey in these lectures. Transport involves two distinct branches of physics. One of these is pure frictionless mechanics of the type described by Newton's laws or the Schrodinger equation. The other involves the generation of heat described by the science of thermodynamics. The first is reversible: when run in reverse, the process is still a legitimate one. The second is irreversible: when run in reverse it looks absurd, like heat flowing spontaneously from a cold to a hot body. What makes transport conceptually complicated is that these two distinct aspects are inseparably intertwined. In a nanoelectronic device the two processes are spatially separated: electrons transfer elastically (that is, without exchange of energy) through the channel and lose energy to heat in the contacts. Of course this "elastic resistor" is only an idealization, but there is experimental evidence that it is met at least approximately in practice in ultrasmall electronic devices. More importantly as we will show, it is extremely useful in gaining a unified view of disparate transport phenomena in a wide variety of electronic devices, both small and large. Indeed the concept of an "elastic resistor" was introduced by Landauer in 1957 (see for example, Landauer 1988) as a conceptual tool long before the developments in nanoelectronics made it experimentally relevant. The separation of the mechanics from the thermodynamics is the key factor that we believe makes the elastic resistor and the bottom-up approach based on it, such a powerful conceptual tool. These lectures are intended to encourage both beginners and experts to try this new approach, by showing how insightful and rewarding it can be, not only in terms of innovating nanoscale electronic devices but also in terms of clarifying fundamental concepts. It is our belief that the most exciting vistas of the future will involve far from equilibrium transport on an atomistic scale and we hope the bottom-up approach presented here will allow students and professionals from different disciplines to learn about the insights from modern nanoelectronics and use them creatively.
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A New Perspective on Transport
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Lecture 2
Why electrons flow
2.1. Two Key Pieces 2.2. Fermi Function 2.3. Non-equilibrium: Two Fermi Functions 2.4. Linear Response 2.5. Difference in "Agenda" Drives the Flow
The central phenomenon we wish to discuss in these Lectures is a well-known and well-established fact, namely that when the two terminals of a battery are connected across a conductor, it gives rise to a current due to the flow of electrons across the channel from the source to the drain.
If you ask anyone, novice or expert, what causes electrons to flow, by far the most common answer you will receive is that it is the electric field. However, this answer is incomplete at best. After all even before we connect a battery, there are enormous electric fields around every atom due to the postive nucleus whose effects on the atomic spectra are well-documented. Why is it that these electric fields do not cause electrons to flow, and yet a far smaller field from an external battery does? The standard answer is that microscopic fields do not cause current to flow, a macroscopic field is needed. This too is not satisfactory, for two reasons. Firstly, there are well-known inhomogeneous conductors like p-n junctions which have large macroscopic fields extending over many micrometers that do not cause any flow of electrons till an external battery is connected. Secondly, experimentalists are now measuring current flow through conductors that are only a few atoms long with no clear distinction between the microscopic and the macroscopic. This is a result of our progress in nanoelectronics, and it forces us to search for a better answer to the question, "why electrons flow."
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Lessons from Nanoelectronics:
2.1 Two key pieces To answer this question, we need two key pieces of information. First is the density of states per unit energy D(E) available for electrons to occupy inside the channel (Fig.2.1). For the benefit of experts, I should note that we are adopting what we will call a "point channel model" represented by a single density of states D(E). More generally one needs to consider the spatial variation of D(E), as we will see in Lecture 8, but there is much that can be understood just from our point channel model.
Fig.2.1. The first step in understanding the operation of any electronic device is to draw the available density of states D(E) as a function of energy E, inside the channel and to locate the equilibrium electrochemical potential 0 separating the filled from the empty states.
The second key input is the location of the electrochemical potential, 0 which at equilibrium is the same everywhere, in the source, the drain and the channel. Roughly speaking (we will make this statement more precise shortly) it is the energy that separates the filled states from the empty ones. All states with energy E < 0 are filled while all states with E > 0 are empty. For convenience I might occasionally refer to the electrochemical potential as just the "potential".
Fig.2.2. When a voltage is applied across the contacts, it lowers all energy levels at the positive contact (drain in the picture). As a result the electrochemical potentials in the two contacts separate: 1 - 2 = qV.
When a battery is connected across the two contacts creating a potential difference V between them, it lowers all energies at the positive terminal (drain) by an amount qV, - q being the charge of an electron (q = 1.6 x 10-19 coulombs) making the two electrochemical potentials separate by qV as shown in Fig.2.2:
1 " 2 = qV
(2.1)
!
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Just as a temperature difference causes heat to flow and a difference in water levels makes water flow, a difference in electrochemical potentials causes electrons to flow. Interestingly, only the states in and around an energy window around 1 and 2 contribute to the current flow, all the states far above and well below that window playing no part at all. Let us explain why. 2.1.1 Energy Window for Current Flow
Each contact seeks to bring the channel into equilibrium with itself, which roughly means filling up all the states with energies E less than its electrochemical potential and emptying all states with energies greater than . Consider the states with energy E that are less than 1 but greater than 2. Contact 1 wants to fill them up since E < 1, but contact 2 wants to empty them since E > 2. And so contact 1 keeps filling them up and contact 2 keeps emptying them causing electrons to flow continually from contact 1 to contact 2. Consider now the states with E greater than both 1 and 2. Both contacts want these states to remain empty and they simply remain empty with no flow of electrons. Similarly the states with E less than both 1 and 2 do not cause any flow either. Both contacts like to keep them filled and they just remain filled. There is no flow of electrons outside the window between 1 and 2, or more correctly outside a few kT of this window, as we will discuss shortly. This last point may seem obvious, but often causes much debate because of the common belief we alluded to earlier, namely that electron flow is caused by the electric field in the channel. If that were true, all the electrons should flow and not just the ones in any specific window determined by the contacts. 2.2 Fermi function Let us now make the above statements more precise. We stated that roughly speaking, at equilibrium, all states with energies E below the electrochemical potential 0 are filled while all states with E > 0 are empty. This is precisely true only at absolute zero temperature. More generally, the transition from completely full to completely empty occurs over an energy range ~ 2 kT around E = 0 where k is the Boltzmann constant ( ~ 80 eV / K) and T is the absolute temperature. Mathematically this transition is described by the Fermi function :
f 0 (E) =
1 # E " 0 % exp +1 $ kT &
(2.2)
!
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Lessons from Nanoelectronics:
Fig.2.3. Fermi function and the thermal broadening function.
This function is plotted in Fig.2.3 (left panel), though in an unconventional form with the energy axis vertical rather than horizontal. This will allow us to place it alongside the density of states, when trying to understand current flow (see Fig.2.4). For readers unfamiliar with the Fermi function, let me note that an extended discussion is needed to do justice to this deep but standard result, and we will discuss it a little further in Lecture 16 when we talk about the key principles of equilibrium statistical mechanics. At this stage it may help to note that what this function (Fig.2.3) basically tells us is that states with low energies are always occupied (f=1), while states with high energies are are always empty (f=0), something that seems reasonable since we have heard often enough that (1) everything goes to its lowest energy, and (2) electrons obey an exclusion principle that stops them from all getting into the same state. The additional fact that the Fermi function tells us is that the transition from f=1 to f=0 occurs over an energy range of ~ 2kT around 0. 2.2.1. Thermal Broadening Function Also shown in Fig.2.3 is the derivative of the Fermi function (multiplied by 4kT to make it dimensionless). This "thermal broadening function" appears often enough that we will give it a name:
FT (E " 0 ) # "
$f 0 $E
=
E # 0 1 ex where x " kT kT (e x + 1) 2
(2.3)
=
!
f 0 (E) (1" f 0 (E)) kT
!
!
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Note: (1) FT (x) = FT (" x) , so that FT (E " 0 ) = FT ( 0 " E) . (2) Also, if we integrate the thermal broadening function the total area is one,
!
+# "#
!
+#
!
$ dE FT (0 " E) %
"
"#
$ dE
&f 0 = &E
f 0 (" #) " f 0 (+ #) = 1 (2.4)
!
so that we can approximately visualize FT(E) as a rectangular "pulse" centered around E=0, with a peak value of (1/4kT) and a width of ~ 4kT.
2.3 Non-equilibrium: Two Fermi Functions When a system is in equilibrium the electrons are distributed among the available states according to the Fermi function. But when a system is driven out-ofequilibrium there is no simple rule for determining the distribution of electrons. It depends on the specific problem at hand making non-equilibrium statistical mechanics far richer and less understood than its equilibrium counterpart. For our specific non-equilibrium problem, we argue that the two contacts are such large systems that they cannot be driven out-of-equilibrium. And so each remains locally in equilibrium with its own electrochemical potential giving rise to two different Fermi functions (Fig.2.4):
f1 (E) =
1 # E " 1 % exp +1 $ kT & 1 # E " 2 % exp +1 $ kT &
(2.5)
and f 2 (E) =
!
The "little" channel in between does not quite know which Fermi function to follow ! and as we discussed earlier, the source keeps filling it up while the drain keeps emptying it, resulting in a continuous flow of current.
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Lessons from Nanoelectronics:
Fig.2.4. Electrons in the contacts occupy the available states with a probability described by a Fermi function f(E) with the appropriate electrochemical potential .
In summary, what makes electrons flow is the difference in the "agenda" of the two contacts as reflected in their respective Fermi functions, f1(E) and f2(E). This is qualitatively true for all conductors, short or long. But for short conductors, the current at any given energy E is quantitatively proportional to
I(E) ~
f1 (E) " f 2 (E)
representing the difference in the probabilities in the two contacts. This quantity goes ! to zero when E lies way above 1, 2 since f1 and f2 are both zero. It also goes to zero when E lies way below 1, 2 since f1 and f2 are both one. Current flow occurs only in the intermediate energy window, as we had argued earlier. 2.4 Linear response Current-voltage relations are typically not linear, but there is a common approximation that we will frequently use throughout these lectures to extract the "linear response" which refers to the low bias conductance, dI / dV, as V 0. The basic idea can be appreciated by plotting the difference between two Fermi functions, normalized to the applied voltage
F(E) !
where
f1(E) " f2 (E) qV / kT
(2.6)
1 = 0 + qV / 2
2 = 0 ! qV / 2
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Fig.2.5. F(E) from Eq.(2.6) versus (E-0)/kT for different values of x=qV/kT.
Fig.2.5 shows that the difference function F gets narrower as the voltage is reduced relative to kT, and it can be checked that the total integrated area stays the same. The interesting point is that as qV is reduced below kT, the function F becomes proportional to the thermal broadening function we just discussed in Section 2.2:
F(E ! 0 ) " kT FT (E ! 0 ) , as qV/kT 0
so that from Eq.(2.6)
f1(E) ! f2 (E) " qV FT (E ! 0 )
(2.7)
if the applied voltage 1 - 2 = qV is much less than kT. For those who like to see a mathematical derivation, Eq.(2.7) can be obtained using the Taylor series expansion described in Appendix A to write
( f ! f0 ) = FT (E ! 0 ) ( ! 0 )
(2.8)
Eq.(2.8) and Eq.(2.7) which follows from it, will be frequently used in these lectures. 2.5. Difference in "Agenda" Drives the Flow Before moving on, let me quickly reiterate the point we are trying to make. It is wellknown that the conductivity varies widely changes by a factor of ~1020 going from copper to glass, to mention two materials that are near two ends of the spectrum. The common explanation for this difference in conductivity is that it results from the difference in the number of "free electrons." But this begs the question of which electrons are free and which are not, a question that becomes more confusing for atomic scale conductors. The viewpoint we have presented suggests a simple answer. Conductivity depends on the available density of states in an energy window a few kT wide around the
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Lessons from Nanoelectronics:
equilibrium chemical potential 0 defined by the function FT, which is like a "window function" (see Fig.2.3) that is non-zero only over a small energy window, a few kT wide around the equilibrium chemical potential. What matters is not the total number of electrons which is of the same order of magnitude for all materials from copper to glass. The key point is the availability of states around 0 which can be widely different from one material to another. Furthermore this is a quantity that can be and is routinely measured in independent experiments like photoelectron spectroscopy. This answer itself is not a new one and is well-known to experts in the field. However, discussions of conductivity typically start from the Drude formula (discussed in Lecture 5) which played a major historical role in our understanding of current flow. Unfortunately, it also gives rise to the two common misconceptions that we are trying to dispel, namely that (1) current is driven by electric fields, and (2) current depends on the total number of electrons. One could argue that the two misconceptions are related, since if current were driven by fields, all electrons should be affected. This is commonly explained away by saying that there are mysterious quantum mechanical forces that prevent electrons in full bands from moving. The thermal broadening function FT(E) in Fig.2.3 arising from the difference in the "agenda" of the two contacts
f1(E) ! f2 (E)
could be called a mysterious force since the Fermi function (Eq.(2.2)) reflects the exclusion principle. But this is the only mystery, and our objective is to show that we can obtain quantitatively correct results without invoking any further mysteries. As we have seen, the function (f1 f2) is non-zero only in an energy window around the equilibrium electrochemical potential, 0. And so whether a conductor is good or bad is determined by the availability of states in this window., a fact that is wellknown to experts. But this answer typically appears much later in any discussion, after advanced concepts like the Boltzmann (Lecture 7) or the Kubo formalism (Lecture 15) have been introduced. By contrast, our bottom-up approach allows us to start our discussion with the correct picture, thus getting us off "on the right foot."
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Lecture 3
The Elastic Resistor
3.1. Conductance of an Elastic Resistor 3.2. How an Elastic Resistor Dissipates Heat 3.3. Why an Elastic Resistor is Relevant
We saw in the last Lecture that the flow of electrons is driven by the difference in the "agenda" of the two contacts as reflected in their respective Fermi functions, f1(E) and f2(E). The negative contact with its larger f(E) would like to see more electrons in the channel than the positive contact. And so the positive contact keeps withdrawing electrons from the channel while the negative contact keeps pushing them in. This is true of all conductors, big and small. But it is generally difficult to express the current as a simple function of f1(E) and f2(E), because electrons jump around from one energy to another and the current flow at different energies is all mixed up. In this Lecture I want to introduce the concept of an elastic "resistor" (Fig.3.1), which is a very useful idealization that describes short devices very well and provides insights into the operation of long devices as well. In this idealized device electrons squeeze through the channel from the source to the drain elastically, meaning without any loss or gain of energy.
Fig.3.1. An elastic resistor: Electrons travel along fixed energy channels.
In an elastic resistor, the current in an energy range from E to E+dE is decoupled from that in any other energy range, allowing us to write it in the form
dI
= dE G(E) ( f1(E) ! f2 (E))
and integrating it to obtain the total current I. Making use of Eq.(2.7), this leads to an expression for the low bias conductance
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Lessons from Nanoelectronics:
I G! V
+#
=
"#
$
dE FT (E " 0 ) G(E)
(3.1)
where FT is the thermal broadening function (see Eq.(2.3)) which can be visualized as a rectangular pulse of area equal to one, with a width of ~ 2kT. Eq.(3.1) tells us that for an elastic resistor, we can define a conductance function G(E) whose average over an energy range ~ 2kT around the electrochemical potential 0 gives the experimentally measured conductance. At low temperatures, we can simply use the value of G(E) at E = 0. The concept of an elastic resistor may sound like an oxymoron since we traditionally associate current flow (I) through a resistor (R) with a Joule heat I2R. How can we talk of a resistor if the electrons do not lose energy? The answer is that while the electron does not lose any energy in the channel of an elastic resistor, it does lose energy both in the source and the drain and that is where the Joule heat gets dissipated. In short, an elastic resistor has a resistance R determined by the channel, but the corresponding heat I2R is entirely dissipated outside the channel. There is experimental evidence that this is approximately true of many nanoscale conductors which would have burnt up if all the heat were dissipated inside the conductor and it is believed that they do not burn up because most of this heat is generated inside the contacts which are large regions capable of dissipating it. Note that by elastic we do not just mean "ballistic" which implies that the electron goes straight from source to drain, "like a bullet." We also include the possibility that an electron takes a more traditional diffusive path as long as it changes only its momentum and not its energy along the way. We are introducing the concept of an elastic resistor not just because it is useful in understanding nanoscale devices, but because it also helps understand transport properties like conductivity of large resistors. Indeed it is the key concept that makes the bottom-up approach so powerful in clarifying transport problems in general. We will elaborate on this a little more at the end of this section. For the moment, let us proceed to obtain an expression for the conductance of an elastic resistor.
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3.1 Conductance of an Elastic Resistor In the spirit of our "bottom-up" approach, consider first the simplest elastic resistor having just one level with energy " in the energy range of interest (Fig.3.2), through which electrons can squeeze through from the source to the drain. We can write the resulting current as
! Ione level =
q t
( f1(" )
# f 2 (" ))
(3.2)
where t is the time it takes for an electron to transfer from the source to the drain.
!
Fig.3.2. The "bottom-up" approach starts by considering the smallest channel having only one energy level in the energy range of interest. Note that because an electron carries negative charge, the direction of the electron current is always opposite to that of the conventional current.
Let me briefly comment on a general point that often causes confusion regarding the direction of the current. As I noted in Lecture 2, because the electronic charge is negative (an unfortunate choice, but something we cannot do anything about) the side with the higher voltage has a lower electrochemical potential. Inside the channel, electrons flow from the higher to the lower electrochemical potential, so that the electron current flows from the source to the drain. The conventional current on the other hand flows from the higher to the lower voltage. Since our discussions will usually involve electron energy levels and the electrochemical potentials describing their occupation, it is also convenient for us to use the electron current instead of the conventional current. For example, in Fig.3.2 it seems natural to say that the current flows from the source to the drain and not the other way around. And that is what I will try to do consistently throughout these Lectures. In short, we will use the current, I, to mean electron current. We can extend Eq.(3.2) for the current through a one-level resistor to any elastic conductor (Fig.3.1) with an arbitrary density of states D(E), noting that all energy channels conduct independently in parallel. We could first write the current in an energy channel between E and E+dE
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Lessons from Nanoelectronics:
dI
= dE
D(E) q ( f1(E) ! f2 (E)) 2 t
since an energy channel between E and E+dE contains D(E)dE states, half of which contribute to carrying current from source to drain. Integrating we obtain an expression for the current through an elastic resistor:
+" !"
I
=
1 q
#
dE G(E) ( f1(E) ! f2 (E) )
(3.3)
where G(E)
= q 2 D(E) / 2t(E)
(3.4)
If the applied voltage 1 - 2 = qV is much less than kT, we can use Eq.(2.7) to write
+"
I
= V
!"
#
dE FT (E ! 0 ) G(E)
which yields Eq.(3.1) stated earlier. We will generally write
G = q 2 D / 2t , R = 2t / q 2 D
(3.5)
with the understanding that the quantities D and t are in general energy-dependent and have to be averaged over an energy range of 2kT around the electrochemical potential 0. Eq.(3.5) seems quite intuitive: it says that the conductance is proportional to the product of two factors, namely the availability of states (D) and the ease with which electrons can transport through them (1/t). This is the key result that we will use in subsequent Lectures. But before moving on let us talk about a few conceptual issues that may be bothering the reader. 3.2 How an Elastic Resistor Dissipates Heat Let us now get back to the question we mentioned at the start of this section: Our entire discussion leading to the expression for resistance (Eq.(3.5)) is based on the
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premise of an elastic resistor, whose resistance R is determined by the channel, but the corresponding heat I2R is entirely dissipated outside the channel. To see how this happens, consider the one level elastic resistor having one sharp level with energy " . Every time an electron crosses over through the channel, it appears as a "hot electron" on the drain side with an energy " in excess of the local electrochemical potential 2 (Fig.3.3). Energy dissipating processes in the contact quickly make the electron get rid of the excess energy ( " -2). Similarly at the source ! end an empty spot (a "hole") is left behind with an energy " that is much less than the ! local electrochemical potential 1, which gets quickly filled up by electrons dissipating the excess energy (1- " ) .
!
! !
Fig.3.3. (a) Everytime an electron crosses from the source to the drain through the energy level it appears as a "hot electron" in the drain with energy in excess of the local electrochemical potential 2, leaving behind an empty spot in the source way below the local electrochemical potential 1. (b) Energy relaxation processes quickly return the structure to the initial state.
In effect, every time an electron crosses over from the source to the drain, an energy 1 " # is dissipated in the source and an energy " # 2 is dissipated in the drain.
The total energy dissipated is 1 ! 2 = qV , so that if N electrons cross over in a ! time t (Current, I = qN / t)
!
Dissipated power = qV
N t
= V *I
exactly as we would expect. Note that V*I is the same as I2R or V2G. The heat dissipated by an! "elastic resistor" thus occurs in the contacts and as we mentioned earlier, there is experimental evidence that this is approximately true of many short conductors. The reason it is such a powerful conceptual tool is that in an elastic resistor, the mechanics and the thermodynamics are spatially separated:
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Lessons from Nanoelectronics:
Electrons transfer elastically through the channel and lose energy to heat in the contacts. The latter is a complicated process whose details are completely bypassed simply by legislating that the contacts are always maintained in equilibrium with a fixed electrochemical potential. 3.3 Why an Elastic Resistor is Relevant The elastic resistor model is clearly of great value in understanding nanoscale conductors, but the reader may well wonder how an elastic resistor can capture the physics of real conductors which are surely far from elastic? In long conductors inelastic processes are distributed continuously through the channel, inextricably mixed up with all the elastic processes (Fig.3.4). Doesn't that affect the conductance and other properties we are discussing? One way to justify the success of the elastic resistor model is to say that a large conductor with distributed inelastic processes can be broken up conceptually into a sequence of elastic resistors (Fig.3.5), each much shorter than the physical length L, having a voltage that is only a fraction of the total voltage V. We could then argue that the total resistance is the sum of the individual resistances.
Fig.3.4 Real conductors have inelastic scatterers distributed throughout the channel.
Fig.3.5 A hypothetical series of elastic resistors: Can it model a real resistor with distributed inelastic scattering?
This splitting of a long resistor into little sections of length shorter than Lin (Lin: length an electron travels on the average before getting inelastically scattered) also helps answer another question one may raise about the elastic resistor model. We obtained the linear conductance by resorting to a Taylor's series expansion (see Eq.(2.6)). But keeping the first term in the Taylor's series can be justified only for voltages V < kT/q, which at room temperature equals 25 mV. But everyday resistors are linear for voltages that are much larger. How do we explain that? The answer is that the elastic resistor model should only be applied to a short length < Lin and as
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long as the voltage dropped over a length Lin is less than kT/q we expect the current to be linear with voltage. The terminal voltage can be much larger. 3.3.1 Connecting Elastic Resistors in Series
However, this splitting into short resistors needs to be done carefully. A key result we will discuss in the next Lecture is that Ohm's law should be modified
from R =
!L A
to R =
! (L + " ) A
(Eq.(4.2))
(Eq.(1.1))
to include an extra fixed resistance !" / A that is independent of the length and can be viewed as an interface resistance associated with the channel- contact interfaces. Here ! is a length of the order of a mean free path, so that this modification is primarily important for near ballistic conductors (L ~ ! ) and is negligible for conductors that are many mean free paths long (L >> ! ). Conceptually, however, this additional resistance is very important if we wish to use the hypothetical structure in Fig.3.5 to understand the real structure in Fig.3.4. The structure in Fig.3.5 has too many interfaces that are not present in the real structure of Fig.3.4 and we have to remember to exclude the resistance coming from these conceptual interfaces. For example, if each section in Fig.3.5 is of length L having a resistance of
R =
! (L + " ) A
then the correct resistance of the real structure in Fig.3.4 of length 3L is given by
R =
! (3L + " ) ! (L + " ) and NOT by R = 3 A A
So if we want to obtain the correct expression for the conductivity of long conductors from the elastic resistor model for a short conductor, we have to be careful to separate the interface resistance from the length dependent part. This is what we will be doing next.
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Lessons from Nanoelectronics:
Lecture 4
How Many Channels?
4.1. Ballistic and Diffusive Transport 4.2. Channels of Conduction
We saw in the last Lecture that the resistance of an elastic resistor can be written as
R = 2t / q 2 D
(see Eq.(3.5))
In this Lecture I will first argue that the transfer time "t" across a resistor of length L for diffusive transport with a mean free path ! can be related to the time tB for ballistic transport by the relation (Section 4.1)
! t = t B #1 + "
L$ & !%
(4.1)
Combining with Eq.(3.5) we obtain the new Ohm's law
R=
! (L + ") A
(4.2)
where the additional interface resistance is given by
!" A
=
q2 D
2 tB
(4.3)
The standard Ohm's law predicts that the resistance will approach zero as the length L is reduced to zero. Of course no one expects it to become zero, but the common
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belief is that it will approach a value determined by the interface resistance which can be made arbitrarily small with improved contacting technology. What is now well established experimentally is that even with the best possible contacts, there is a minimum interface resistance equal to !" / A determined by the properties of the channel, independent of the contact. The modified Ohm's law in Eq.(4.2) reflects this fact: Even a channel of zero length with perfect contacts has a resistance equal to that of a hypothetical section of length " . But what does it mean to talk about the mean free path " of a channel of zero length? The answer is that neither " nor " mean anything for a short conductor, but their ! product does. The ballistic resistance "# / A has a simple meaning that has become clear in the light of modern experiments. As we will see in Section 4.2, it is inversely ! proportional to the number of channels, M(E) available for conduction, which is ! ! proportional to, but not the same as, the density of states, D(E). The concept of density of states has been with us since the earliest days of solid state physics. By contrast, the number of channels (or transverse modes) M(E) is a more recent concept whose significance was appreciated only after the seminal experiments in the 1980's on ballistic conductors showing conductance quantization. 4.1 Ballistic and Diffusive Transport
2 Consider how the two quantities in G = q D /2t , namely the density of states, D and the transfer time (t) scale with channel dimensions for large conductors. The first of these is relatively easy to see since we expect the number of states to be additive. A channel twice as big should have twice as many states, so that the density of states ! D(E) for large conductors should be proportional to the volume (A*L).
!
Regarding the transfer time, t, broadly speaking there are two transport regimes:
Ballistic regime: Transfer time t ~ L
G = q 2 D /2t ~ A
Diffusive regime: Transfer time t ~ L2
!
G = q 2 D /2t ~ A /L
The ballistic conductance is proportional to the area (note that D ~ A*L as discussed ! above), but independent of the length. This "non-Ohmic" behavior has indeed been observed in short conductors. It is only diffusive conductors that show the "ohmic" behavior G ~ A/L.
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Lessons from Nanoelectronics:
These two regimes can be understood as follows. In the ballistic regime electrons travel straight from the source to the drain "like a bullet," taking a time
tB
= L /u
(4.4)
where u is the average velocity of the electrons in the z-direction: u = < vz > . But conductors are typically not short enough for electrons to travel "like bullets." Instead they stumble along, getting scattered randomly by various defects along the way taking much longer than the ballistic time L / u . We could write
t
=
L u
+
L2 2D
(4.5)
viewing it as a sort of "polynomial expansion" of the transfer time t in powers of L. However, it is well-known from the treatment of random walks (see for example, Berg, 1983) that coefficient D can be identified as the diffusion constant 2 D = < vz ! > , " being the mean free time. In defining the two constants D , u appearing in Eq.(4.5), we have used the symbol "" to denote an average over the angular distribution of velocities which yields a ! different numerical factor depending on the dimensionality of the conductor (see Appendix B), d = {1, 2, 3} dimensions :
!
u
where and
=
K! =
< vz > = K v v(E)
(4.6a)
{1, 2 / " , 1 / 2}
(4.6b)
2 D = < vz ! > = v 2 (E) ! (E) / d
We could use Eqs.(4.5) and (4.4) to obtain the result stated earlier (Eq.(4.1))
! t = t B #1 + "
where
L$ & !%
(same as (4.1)) (4.7)
!
= 2D / u
= (2 / K v d) "#
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The quantity " is basically the mean free path " # that an electron travels before getting scattered, apart from a numerical factor
!
2 Kvd
! 4 = {2, , } 2 3
!
for d = {1, 2, 3} dimensions.
Some readers may not find this "derivation" of Eq.(4.1) relating the ballistic and diffusive transfer times rigorous enough, but this approach has the advantage of getting us to the new Ohm's law very quickly using simple algebra. In Lecture 6 we will re-derive Eq.(4.2) more rigorously by solving a differential equation, without invoking Eq.(4.1). 4.2 Channels for Conduction The modified Ohm's law in Eq.(4.2) tells us that even a channel of zero length with perfect contacts has a resistance equal to that of a hypothetical section of length " . This ballistic resistance "# / A has a simple meaning that has become clear in the light of modern experiments.
! Numerous experiments since the 1980's have shown that for small conductors, the ballistic conductance does not go down linearly with the area A. Rather it goes down ! 2 in integer multiples of the conductance quantum, q /h , suggesting that we make the replacement
A GB = !"
=
~ 38 S
q2 h !
!
Integer
M !
(4.8)
so that using Eqs.(4.3) and (4.4) we have
Du 2L
=
M h
(4.9)
How can we understand this relation and what does the integer M represent? This result cannot come out of our elementary treatment of electrons in classical particlelike terms, since it involves Planck's constant "h". Some input from quantum mechanics is clearly essential and this will come in Lecture 5 when we evaluate D(E).
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Lessons from Nanoelectronics:
For the moment we note that heuristically Eq.(4.8) suggests that we visualize the real conductor as "M" independent channels in parallel whose conductances add up to give Eq.(4.8) for the ballistic conductance. Each channel contributes
Du 2L
=
1 h
(For each channel)
Eq.(4.9) then suggests that we introduce an important new concept, namely that of conduction channels (or transverse modes) defined by turning Eq.(4.9) around a little and using Eq.(4.6a):
M (E) =
h Du 2L
= Kv
h D(E)! (E) 2L
(4.10)
To estimate M(E) theoretically we would need a model for the density of states D(E) and the velocity v(E), which we will take up in Lecture 5. We could, however, use experimental data to estimate M(E) at energies E around the equilibrium electrochemical potential, 0. For example, copper has
! = 17 e ! 9 " ! m
! = 40 nm
so that from Eq.(4.9)
M = A
h
1
q 2 !"
=
1 (0.16 nm) 2
showing that copper has approximately one mode per 0.16 nm * 0.16 nm which is of the order of the area occupied by a single copper atom. It is as if every atom contributes a channel for electrons to conduct. This is about as large as it ever gets in any material and is typical of metals. Semiconductors typically have far fewer modes, typically one for every 104 to 106 atoms, while insulators have hardly any modes.
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Lecture 5
Conductivity
5.1. E(p) or E(k) Relations 5.2. Counting States 5.3. Drude Formula 5.4. Is Conductivity proportional to Electron Density? 5.5. Quantized Conductance
A well-known expression for conductivity is the Drude formula relating the conductivity to the electron density (n), the effective mass (m) and the mean free time
"#
1 = $
q2 n % m
(5.1a)
This is a very well-known expression since even freshmen physics texts start by ! deriving it. It also leads to the widely used concept of mobility
= q! / m
such that
(5.1b) (5.1c)
!
= qn
On the other hand, we can easily come up with a different expression for conductivity based on what we have done so far. All we have to do is to take the transfer time for the diffusive regime
tD
= L2 / 2D
(see Eq.(4.5))
use it in in our expression for conductance
G = q 2 D / 2t
(see Eq.(3.5))
and combine it with the definition of conductivity ( G = ! A / L ) to write
G!
!A L
=
q2 D D L2
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Lessons from Nanoelectronics:
so that using Eq.(4.6b) for the diffusion coefficient, D we have:
!!
1 "
" D% 2 = q2 $ ! " /d # AL ' &
(
)
(5.2)
Note that just like the conductance (see Eq.(3.1)), we have to average the energydependent conductivity from Eq.(5.2) over an energy range of a few kT's around E=0, using the thermal broadening function,
+)
!
=
!)
*
# "f & dE % ! 0 ( ! (E) $ "E '
(5.3)
Eq.(5.2) does not look like the Drude formula (Eq.(5.1)), but it is also a standard one that is derived in many textbooks. However, the usual derivation of Eq.(5.2) requires advanced concepts like the Boltzmann or the Kubo formalism and so usually appears much later in any discussion. For example, the classic solid-state physics text by Ashcroft & Mermin, introduces Eq.(5.1) in Chapter 1 and Eq.(5.2) in Chapter 13. Not surprisingly, most people remember Eq.(5.1) and not Eq.(5.2). But the point we wish to stress is that while Eq.(5.1) is often very useful, it is a result of limited validity that can be obtained from Eq.(5.2) by making suitable approximations based on a specific model. But when these approximations are not appropriate, we can still use Eq.(5.2) which is far more generally applicable. For example, Eq.(5.2) gives sensible answers even for materials like graphene whose nonparabolic bands make the meaning of mass somewhat unclear, causing considerable confusion when using Eq.(5.1). In general we should really use Eq.(5.2), and not Eq.(5.1), to shape our thinking about conductivity. There is a fundamental difference between Eq.(5.2) and (5.1). The averaging implied in Eq.(5.3) makes the conductivity a "Fermi surface property", that is one that depends only on the energy levels close to E=0. By contrast, Eq.(5.1) depends on the total electron density (n) integrated over all energy. But this dependence on the total number is true only in a limited sense. Experts know that "n" only represents the density of "free" electrons and have an instinctive feeling for what it means to be free. They know that there are p-type semiconductors which conduct better when they have fewer electrons, but in that case they know that "n" should be interpreted to mean the number of "holes". For
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beginners, all this appears confusing and much of this confusion can be avoided by using Eq.(5.2) instead of (5.1). Interestingly, Eq.(5.2) was used in a seminal paper to work "backwards" to Eq.(3.5) which is essentially the same as Eq.(1) of Thouless (1977). By contrast our use of the bottom-up approach based on the elastic resistor allowed us to use elementary arguments to obtain Eq.(3.5), and then easily work "forwards" to Eq.(5.2). Eq.(5.2) stresses that the essential factor determining the conductivity is the density of states around E=0. Materials are known to have conductivities ranging over many orders of magnitude from glass to copper. And the basic fact remains that they all have approximately the same number of electrons. Glass is not an insulator because it is lacking in electrons. It is an insulator because it has a very low density of states or number of modes around E=0. So when does Eq.(5.2) reduce to (5.1)? Answer: If the electrons are described by a "single band effective mass model" as I will try to show in this Lecture. So far we have kept our discussion general in terms of the density of states, D(E) and the velocity, v(E) without adopting any specific models. These concepts are generally applicable even to amorphous materials and molecular conductors. A vast amount of literature both in condensed matter physics and in solid state devices, however, is devoted to crystalline solids with a periodic arrangement of atoms because of the major role they have played from both basic and applied points of view. For such materials, energy levels over a limited range of energies are described by a E(p) relation and we will show in this Lecture that irrespective of the specific E(p) relation, at any energy E the density of states D(E), velocity v(E) and momentum p(E) are related to the total number of states N(E) with energy less than E by the relation
D(E)" (E) p(E) = N(E) . d
which can be combined with Eq.(5.2) to yield
(5.4)
!
! (E) = q 2
N(E) " (E) A L m(E) p(E) / v(E)
(5.5)
where we have defined mass as
m(E) =
(5.6)
For parabolic E(p) relations, the mass is independent of energy, but in general it could be energy-dependent.
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Lessons from Nanoelectronics:
Eq.(5.5) indeed looks like Drude expression (Eq.(5.1)) if we identify N/AL as the electron density, n. At low temperatures, this is easy to justify since the energy averaging in Eq.(5.3) amounts to looking at the value at E = 0:
!
N "$ ! = # q2 & " AL m % E= 0
= q 2 n" / m
since N(E) at E = 0 represents the total number of electrons (Fig.5.1). At non-zero temperatures one needs a longer discussion which we will get into later in the Lecture. Indeed as will see, some subtleties are involved even at zero temperature when dealing with differently shaped density of states.
Fig.5.1. Equilibrium Fermi function f0(E), Density of states D(E) and integrated density of states N(E).
Note, however, that the key to reducing our conductivity expression (Eq.(5.2)) to the Drude-like expression (Eq.(5.5)) is Eq.(5.4) which is an interesting relation for it relates D(E), v(E) and p(E) at a given energy E, to the total number of states N(E) obtained by integrating D(E)
E
N(E) =
"#
$ dE D(E)
How can the integrated value of D(E) be uniquely related to the value of quantities like D(E), v(E) and p(E) at a single energy ? The answer is that this relation holds ! only as long as the energy levels are given by a single E(p) relation. It may not hold in an energy range with multiple bands of energies or in an amorphous solid not described by an E(p) relation. Eq.(5.2) is then not equivalent to Eq.(5.5), and it is Eq.(5.2) that can be trusted. With that long introduction let us now look at how single bands described by an E(p) relation leads to Eq.(5.4) and helps us connect our conductivity expression (Eq.(5.2)) to the Drude formula (Eq.(5.1)). This will also lead to a different interpretation of the quantity M(E) introduced in the last Lecture, that will help understand why it is an integer representing the number of channels.
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5.1 E(p) or E(k) relations for crystalline solids The general principle for calculating D(E) is to start from the Schrodinger equation treating the electron as a wave confined to the solid. Confined waves (like a guitar string) have resonant "frequencies" and these are basically the allowed energy levels. By counting the number of energy levels in a range E to E+dE, we obtain the density of states D(E). Although the principle is simple, a first principles implementation is fairly complicated since one needs to start from a Schrodinger equation including the nuclear potential that the electrons feel inside the solid. One of the seminal concepts in solid state physics is the realization that in crystalline solids electrons behave as if they are in vacuum, but with an effective mass different from their natural mass, so that the energy-momentum relation can be written as
p2 E( p) = E c + 2m
(5.7a)
!
where Ec is a constant. The momentum p is equated to !k , providing the link between the energy-momentum relation E(p) associated with the particle viewpoint and the dispersion relation E(k) associated with the wave ! viewpoint. Here we will write everything in terms of "p", but they are easily translated in terms of k = p /! . Eq.(5.7a) is generally referred to as a parabolic dispersion relation and is commonly used in a wide variety of materials from metals like copper to semiconductors like ! silicon, because it often approximates the actual E(p) relation fairly well in the energy range of interest. But it is by no means the only possibility. Graphene, a material of great current interest, is described by a linear relation:
E = Ec + v 0 p
(5.7b)
!
!
where v 0 is a constant. Note that "p" denotes the magnitude of the momentum and we will assume that the E(p) relation is isotropic, which means that it is the same regardless of which direction the momentum vector points. For any given isotropic E(p) relation, the velocity points in the same direction as the momentum, while its magnitude is given by
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Lessons from Nanoelectronics:
v"
dE dp
(5.8)
This ! a general relation applicable to arbitrary energy-momentum relations for is classical particles. On the other hand, in wave mechanics it is justified as the group velocity for a given dispersion relation E(k). 5.2 Counting states One great advantage of this principle is that it reduces the complicated problem of electron waves in a solid to that of waves in vacuum, where the allowed energy levels can be determined the same way we find the resonant frequencies of a guitar string: simply by requiring that an integer number of wavelengths fit into the solid. Noting that the de Broglie principle relates the electron wavelength to the Planck's constant divided by its momentum, h / p, we can write
L h/ p
= Integer
!
" h% p = Integer * $ ' # L&
where L is the length of the box. This means that the allowed states are uniformly distributed in "p" with each state occupying a "space" of
"p =
h L
(5.9)
Let us define a function N(p) that tells us the total number of states that have a ! momentum less than a given value "p". In one dimension this function is written down by dividing the total range of 2p (from -p to +p) by the spacing h/L:
N( p) =
2p h /L
= 2L
" p$ # h%
1D
!
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In two dimensions we divide the area of a circle of radius "p" by the spacing (h/L)*(h/W), L and W being the dimensions of the two dimensional box.
N( p) =
2 " p2 # p% = " WL 2D $ h& (h /L)(h /W )
!
In three dimensions we divide the volume of a sphere of radius "p" by the spacing (h/L)*(h/W1) *(h/W2), L, W1 and W2 being the dimensions of the three dimensional box. Writing A = W1 * W2 we have
N( p) =
(4" /3) p 3 (h /L)(h 2 / A)
=
3 4" # p% AL $ h& 3
3D
We can combine all three results into a single expression
!
" p$ N( p) = K N # h%
d
(5.10)
where K N
! # = $2L, "WL, %
4" & AL' for d = 3 (
{1,
2, 3} dimensions
!
We could use a given E(p) relation ! turn this function N(p) into a function of energy to N(E) that tells us the total number of states with energy less than E. 5.2.1. Density of states, D(E)
This function N(E) that we have just obtained must equal the density of states D(E) integrated up to an energy E, so that D(E) can be obtained from the derivative of N(E):
E
N(E) =
Hence from Eq.(5.10),
"#
$ dE D(E)
!
" D(E) =
dN dE
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Lessons from Nanoelectronics:
D(E) =
dN dp dp dE
dp p d!1 d = KN dE h d
Making use of Eqs.(5.8) and (5.10), we obtain the relation stated earlier
D(E)" (E) p(E) = N(E) . d
(same as Eq.(5.4))
which is completely general independent of the actual E(p) relation. ! 5.3 Drude formula As noted earlier, using this relation we can rewrite Eq.(5.2) in the form
! (E) = q 2
N(E) " (E) A L m(E)
(same as Eq.(5.5))
with an energy-dependent mass m(E) defined in Eq.(5.6)). As we have seen, It is straightforward to connect this relation to the Drude formula (Eq.(5.1)) at low temperatures where the energy averaging in Eq.(5.3) amounts to looking at the value at a single energy E=0. What about non-zero temperatures? 5.3.1. n-type Conductors Using Eq.(5.5) and assuming "m" and " ! " to be energy-independent we have
!
=
q 2" 1 m AL
+) !)
*
# "f & dE % ! 0 ( N(E) $ "E '
(5.11)
The integral can be carried out "by parts" to yield
+) !) +)
*
# "f & dE % ! 0 ( N(E) = $ "E '
[
+) ! N(E) f0 (E) ! )
]
+
!)
*
dE
dN f0 (E) dE
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+"
=
!"
#
dE D(E) f0 (E)
= Total number of electrons, N
since dE D(E) f0(E) tells us the number of electrons in the energy range from E to E+dE. When integrated it gives us the total number of electrons. Eq.(5.11) then reduces to
!
=
q 2" N m AL
which is the Drude formula, noting that n = N / AL. 5.3.2. p-type conductors
An interesting subtlety is involved when we consider a p-type conductor for which the E(p) relation extends downwards, for example, something like
p2 E( p) = E v " 2m
!
Fig.5.2: Equilibrium Fermi function f0(E), Density of states D(E) and integrated density of states N(E): p-type conductor.
E
Instead of
N(E) =
!"
#
dE D(E)
we now have (see Fig.5.2)
+#
N(E) =
E
$
dE D(E) ! D(E) = "
dN dE
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Lessons from Nanoelectronics:
This is because we defined the function N(E) from N(p) which represents the total number of states with momenta less than p, which means energies greater than E for a p-type dispersion relation. Now if we carry out the integration by parts as before
+) !)
*
# "f & dE % ! 0 ( N(E) = $ "E '
[
+) ! N(E) f0 (E) ! )
]
+)
+
!)
*
dE
dN f0 (E) dE
we run into a problem because the first term does not vanish at the lower limit where both N(E) and f0(E) are both non-zero. We can get around this problem by writing the derivative in terms of (1-f0) instead of f 0:
+) ") +)
*
# !(1 " f0 ) & dE % N(E) = $ !E ( '
+"
[
+) N(E)(1 " f0 (E)) " )
]
"
")
*
dE
dN (1 " f0 (E)) dE
=
!"
#
dE D(E) (1 ! f0 (E))
= Total number of "holes", P
What this means is that with p-type conductors we can use the Drude formula
" = q 2 n# /m
but the "n" now represents the number of empty states or holes. A larger "n" really means fewer electrons.
!
5.3.3. "Double-ended" density of states How would we count "n" for a density of states D(E) that extends in both directions as shown in Fig.5.3 (left panel). This is representative of graphene, a material of great interest (recognized by the 2010 Nobel prize in physics), whose E(p) relation is commonly approximated by E = v 0 p . People usually come up with clever ways to handle such "double-ended" density of states so that the Drude formula can be used. For example they divide the total density of states into an n-type and a p-type component !
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D(E) = Dn (E) + D p (E)
as shown in Fig.5.3 and the two ! components are then handled separately.
Fig.5.3. A "double-ended" density of states can be visualized as a sum of an "n-type component" and a "p-type component."
But the point we would like to stress is that there is really no particular reason to insist on using a Drude formula. One might just as well use Eq.(5.2) which reflects the correct physics of conduction, namely that it takes place in a narrow band of energies around 0. 5.4. Is conductivity proportional to electron density? Experimental conductivity measurements are often performed as a function of the electron density and the common expectation based on the Drude formula (Eq.(5.1)) is that conductivity should be proportional to the electron density and any nonlinearity must be a consequence of the energy-dependence of the mean free time. What is not often recognized is that for non-parabolic dispersion relations, the mass itself defined as p/v can be energy-dependent and this will affect the conductivityelectron density relation. First we note that from Eq.(5.10)
!2p # n( p) = g " , # h $
! p2
h2
,
4! p3 % # & 3 h3 # '
(5.12)
where we have defined n as N/L or N/WL or N/AL in 1, 2 and 3 dimensions respectively. We have also included the degeneracy factor "g" to denote the number of equivalent states. For example all non-magnetic materials have two spin states with identical energies, which would make g=2. Certain materials also have equivalent "valleys" having identical energy momenta relations so that the N we calculate for
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Lessons from Nanoelectronics:
one valley has to be multiplied by g when relating to the experimentally measured electron densities. For graphene, g = 2*2 = 4.
Writing
n( p) = K p d
n( p)" ( p) m( p)
(d = dimensions, K = constant)
we have
! = q2
= q 2 K p d!1 v( p)" ( p)
(5.13)
If we know how the velocity and the mean free time vary with E (and hence with p) we could eliminate "p" from the expressions for the conductivity, ! and the electron density, n to obtain a direct relation between them. For example, with graphene, E = v 0 p , so that the velocity dE/dp is a constant (v0), independent of p. If we assume an energy independent mean free time ! , we obtain
!
!
=
q2 4 gn # h "
(5.14)
from Eq.(5.13) after a little algebra, making use of Eq.(4.7) for the mean free path and noting that grapheme is two-dimensional (d=2) with K = ! g / h 2 . Note that Eq.(5.14) says that the conductivity is ~ n and not ~ n, as is commonly assumed, with a constant (energy-independent) mean free time. The calculated results from Eq.(5.14) with ! = 2 m and with ! = 300 nm compares well with the experimental data on graphene reported in Bolotin et al. (2008). Note that the values of the mean free path indicated in the paper are half the values we have used. This is because our definition of mean free path differs from the standard one by the factor of 2 /K v d (see Eq.(4.7)).
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5.5. Quantized Conductance Before moving on, let me also relate the conductivity to the number of channels M(E) introduced in the last Lecture and use the concepts developed in this Lecture to interpret it in a very different way that helps us understand one of the seminal results of mesoscopic physics, namely the quantized conductance in low dimensional ballistic structures. Firstly we note that our starting expression for conductivity (Eq.(5.2)) can be written in terms of M as
!
! D$ 2 = q2 # " # /d " AL & %
(
)
=
q2 M $ h A
(5.15)
making use of Eq.(4.10) for M and Eq.(4.7) for ! . I noted in the last Lecture that the ballistic conductance given is by
GB = ! A / "
= (q 2 / h) M
(same as Eq.(4.8))
and that experimentally M is found to be an integer in low dimensional conductors at low temperatures. However, M was introduced in the last Lecture as the product of the density of states and the velocity and it is not at all clear why it should be an integer. Using the E(p) relations discussed in this Lecture we will now show that we can interpret M(p) in a very different way that helps see its integer nature. Earlier in this Lecture, we introduced N(p) as the total number of states with a momentum that is less than p and we have seen that it is equal to the number of wavelengths that fit into the solid (see Eq.(5.10)).
! L WL # N( p) = g "2 , ! , (h / p)2 # h/ p $
4! AL % # 3& 3 (h / p) # '
for d={1,2,3} dimensions, where we have also included the degeneracy factor `g' that was explained following Eq.(5.12).
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Lessons from Nanoelectronics:
We can start from the definition of M given in Eq.(4.10) and make use of the general result in Eq.(5.4) with appropriate values for d and Kv from Eq.(4.6) to write
! W A % # # M ( p) = g "1, 2 , ! 2& (h / p) (h / p) ' # # $
(5.16)
Just as N(p) tells us the number of wavelengths that fit into the volume, M(p) tells us the number that fit into the cross-section and this result is independent of the actual E(p) relation, since we have not made use of any specific relationship. Now we are ready to look at the origin of conductance quantization. If we evaluate our expressions for N(p) and M(p) for a given a sample we will in general get a fractional number. However, since these quantities represent the number of states, we would expect them to be integers and if we obtain say 201.59, we should take the lower integer 201. This point is commonly ignored in large conductors at high temperatures, where experiments do not show this quantization because of the energy averaging over 0 2kT associated with experimental measurements. For example, if over this energy range, M(E) varies from say 201.59 to 311.67, then it seems acceptable to ignore the fact that it really varies from 201 to 311. But in small structures where one or more dimensions is small enough to fit only a few wavelengths the integer nature of M is observable and shows up in the quantization of the ballistic conductance. We should then rewrite Eq.(5.16) as
! W A % # # M ( p) = g * Int "1, 2 , ! 2& h/ p (h / p) ' # # $
(5.17)
where Int (x) represents the largest integer less than x. For one dimensional conductors the number of modes is equal to g, which is the number of spins times the number of valleys. Ballistic conductors have a resistance of ~ h / q2M, so that the resistance of a 1D ballistic conductor is approximately equal to 25 K ! divided by g. This has indeed been observed experimentally. Most metals and semiconductors like GaAs have g=2, and the 1D ballistic resistance ~ 12 K ! . But carbon nanotubes have two valleys as well making g=4 and exhibit a ballistic resistance of 6 K ! .
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For two- and three-dimensional conductors, Eq.(5.17) is not quite right, because it is based on the heuristic idea of counting modes by counting the number of wavelengths that fit into the solid (see Eq.(5.5)). Mathematically it can be justified only if we assume periodic boundary conditions, that is if we assume that the cross-section is in the form of a ring rather than a flat sheet for a 2D conductor. For a 3D conductor it is hard to visualize what periodic boundary conditions might look like though it is easy to impose it mathematically as we have been doing. Most real conductors do not come in the form of rings, yet periodic boundary conditions are widely used because it is mathematically convenient and people believe that the actual boundary conditions do not really matter. But this is true only if the cross-section is large. For small area conductors the actual boundary conditions do matter and we cannot use Eq.(5.10). Interestingly a conductor of great current interest has actually been studied in both forms: a ring-shaped form called a carbon nanotube and a flat form called graphene. If the circumference or width is tens of nanometers they have much the same properties, but if it is a few nanometers their properties are observably different including their ballistic resistances.
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Lessons from Nanoelectronics:
Lecture 6
Diffusion Equation for Ballistic Transport
6.1. Flow Dynamics on Electronic Highway 6.2. Electrochemical Potentials Out of Equilibrium 6.3. From Currents to Potentials 6.4. The New Boundary Condition
The title of this Lecture may sound contradictory, like the elastic resistor. Doesn't the diffusion equation describe diffusive transport? How can one use it for ballistic transport? An important idea we are trying to get across with our bottom-up approach is the essential unity of these two regimes of transport and hopefully this lecture will help. Most people when they hear the diffusion equation, think of the following equation
I A
= !D
dn dz
where D is the diffusion coefficient defined in Lecture 4 (see Eq.(4.6b)). However, this is appropriate only for homogeneous materials. For example if we have two materials one with a low density of states and one with a high density of states as shown, at equilibrium, both should have the same electrochemical potential 0 making the number of electrons n1 on the left less than the number n2 on the right. If current were proportional to dn/dz, we would expect a current to flow, which is not true since the two are in equilibrium. More generally, the current is proportional not to the slope of the electron density, but to the slope of the electrochemical potential (z)
I ! d = ! A q dz
(6.1a)
where ! is the conductivity (Eq.(5.2)) from the last Lecture. It takes a little algebra to show that this does reduce to the equation in terms of dn/dz if the density of states is spatially constant.
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For one-dimensional structures (see Fig.6.1), under steady-state conditions, the current must be the same at all points z:
dI dz
= 0
(6.1b)
The standard approach is to solve Eqs.(6.1a,b) with the boundary conditions
(z = 0) = 1 (z = L) = 2
(6.2a) (6.2b)
Fig.6.1. Solution to Eqs.(6.1a,b) with the boundary conditions in Eq.(6.2). Note that we are using I to represent the electron current as explained earlier (see Fig.3.2).
It is easy to see that the linear solution sketched in Fig.6.1 meets the boundary conditions in Fig.6.2 and at the same time satisfies both Eqs.(6.1a,b) since a linear (z) has a constant d/dz, so that with a constant conductivity, dI/dz = 0. Could we use the same equation (Eqs.(6.1a,b)) to model a ballistic channel? Many would say that a whole new approach is needed since quantities like the conductivity or the diffusion coefficient mean nothing for a ballistic channel. Now the point I wish to make in this Lecture is that we can still use Eqs.(6.1a,b) provided we modify the boundary conditions in Eq.(6.2) to reflect the interface resistance
(z = 0) = 1 ! qI (RB / 2)
(6.3a) (6.3b)
(z = L) = 2 + qI (RB / 2)
RB being the inverse of the ballistic conductance GB defined in Eq.(4.8)
RB
=
! "A
=
h q2 M
(6.4)
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Lessons from Nanoelectronics:
Fig.6.2. Eqs.(6.1a,b) can be used to model both ballistic and diffusive transport provided we modify the boundary conditions in Eq.(6.2) to reflect the two interface resistances, each equal to RB/2.
The new boundary conditions in Eqs.(6.3a,b) can be visualized in terms of lumped resistors RB/2 at the interfaces as shown in Fig.6.2. leading to additional potential drops as shown.
It is straightforward to see that this approach applied to a uniform resistor leads to the new Ohm's law discussed in Lecture 4. Since (z) varies linearly from z=0 to z=L, the current is obtained from Eq.(6.1a)
I
=
=
! A (0) ! (L) q L
! A " 1 ! 2 qI RB % ! $ ' q # L L &
I
so that using Eqs.(6.3a,b)
and noting that ! A RB = " ,
! !$ I #1 + & " L%
=
" A ! 1 ' 2 $ # & q " L %
Since 1 - 2 = qV (Eq.(2.1)),
I
=
!A V "+L
(6.5)
which leads to the new Ohm's law obtained earlier in Lecture 4 (Eq.(4.2)). Note, however, that our prior argument leading to Eq.(4.2) was based on a semiheuristic argument for the transfer time (see Eq.(4.5)).
t
=
L u
+
L2 2D
(see Eq.(4.5))
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By contrast, in this Lecture we will make no use of this relation, thus providing a different derivation and what some might consider a firmer basis for the new Ohm's law. However, it is mathematically a little more complicated since it involves a differential equation. Basically what we will do in the rest of this Lecture is to "derive" Eqs.(6.1a,b) along with the "new" boundary conditions in Eqs.(6.3a,b). Once we have these, the new Ohm's law follows, as we have just seen. 6.1 Flow Dynamics on Electronic Highway We can visualize the source and drain as two cities connected by a highway with a set of lanes running from west to east, and an equal number from east to west. For ballistic conduction, crossing over from the eastbound to the westbound lanes (or the other way) is forbidden so that at steady-state, both the current I+(z) on the eastbound lanes and the current I-(z) on the westbound lanes are fixed:
d I+ dz
= 0 =
dI ! dz
Fig.6.3. Source and drain are analogous to two cities connected by a highway with a set of lanes running from west to east, and an equal number from east to west. I+(z) denotes the flux on the eastbound lanes and I-(z) denotes the flux on the westbound lanes.
But for diffusive transport electrons are continually crossing over from the eastbound to the westbound lanes at a rate characterized by the mean free path ! . We will show next that the correct equation describing the electronic flow on a diffusive highway is described by
d I+ dz
= !
I +! I ! !
=
dI ! dz
(6.6)
To obtain this equation we define the mean free path ! as a quantity such that the fraction of electrons that cross over from the westbound to the eastbound lanes in a short length ! z is given by ! z / ! .
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Lessons from Nanoelectronics:
We can then write (Fig.6.3)
I + (z + !z) = I + (z)
Fraction NOT turned around
# !z & %1 " ( $ ' ! "!$ # #
+ I ! (z + !z)
Fraction turned around
! !z $ # & " !% !
which is rearranged to give
I + (z + !z) " I + (z) !z
I + (z) " I " (z + !z) = " !
Letting the length ! z tend to zero, we have half of Eq.(6.6)
d I + (z) dz
= !
I + (z) ! I ! (z) !
(6.7a)
To obtain the other half, we write similarly
I !(z) = I ! (z + ! z)
Fraction NOT turned around
! "z $ #1 ! & " % ! "!$ # #
+ I + (z)
Fraction turned around
! !z $ # & " !% !
rearrange
I ! (z + "z) ! I ! (z) "z
= !
I +(z) ! I ! (z + "z) !
and let the length ! z tend to zero:
d I ! (z) dz
= !
I +(z) ! I ! (z) !
(6.7b)
Combining Eqs.(6.7a, b) we have Eq.(6.6).
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6.1.1.
Brief Note Regarding Energy Dependence of the Mean Free Path
Before moving on, let me note that the above discussion could be applied to an energy range between E and E + dE using I to denote the current per unit energy, so that we can write Eq.(6.6) in the form
! I + (z; E) !z
= "
I +(z; E) " I " (z; E) ! (E)
=
! I "(z; E) !z
(6.8)
The appeal of the elastic resistor lies in the decoupling of energy channels that allows us to analyze one energy at a time and simply add up (that is, integrate) the results at the end. For example we could find the net current per unit energy
I(z; E) = I + (z; E) ! I ! (z; E)
and integrate it to obtain the current:
+"
(6.9)
I(z) =
!"
#
dE I(z; E)
(6.10)
Can we use Eq.(6.6) directly for the integrated currents, instead of using Eq.(6.8) for one energy at a time? If the mean free path ! were independent of energy we could do that without losing anything. We could then simply integrate Eq.(6.8) to obtain Eq.(6.6), thus demonstrating their exact equivalence. But if the mean free path is energy-dependent then we cannot exactly replace
+" !"
#
I +(z; E) ! I ! (z; E) dE ! (E)
with
I +(z) ! I ! (z) <!>
where < ! > is a constant, without knowing what the function I(z;E) looks like. In this case, we could use Eq.(6.6) only in an approximate sense assuming some reasonable average value for < ! >. And that is what we will do in this Lecture.
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Lessons from Nanoelectronics:
6.2. Electrochemical Potentials Out of Equilibrium Eq.(6.6) is expressed in terms of currents I(z). How do we get to the diffusion equation (Eq.(6.1)) which is in terms of the electrochemical potential (z)? So far we have only talked about electrochemical potentials inside the contacts which are large regions that always remain close to equilibrium and hence are described by Fermi functions (see Eq.(2.5)) with well-defined electrochemical potentials. By contrast we are now using (z) to represent quantities inside the out-ofequilibrium channel, where it is at best an approximate concept since the electron distribution among the available states need not follow a Fermi function. In general one has to solve a full-fledged transport equation like the semiclassical Boltzmann equation to be introduced in the next Lecture which allows us to calculate the full occupation factors f(z;E). More generally for quantum transport one can use the nonequilibrium Green's function (NEGF) formalism to be introduced in Part three to solve for the quantum version of f(z;E). But even though (z) is an approximate concept it is a remarkably useful one, which is why device engineers routinely use it as an aid to visualizing the physics in large devices. However, for ballistic or near ballistic devices a single (z) misses much of the physics and it is important to recognize that we need two separate potentials +(z) and -(z) to capture the physics of near-ballistic transport. So how do we determine these potentials? Interestingly for a perfectly ballistic channel with good contacts, the correct answer is deceptively simple. All drainbound (or "eastbound") electrons are distributed according to the source (or "West") contact with + = 1
f + (z; E) =
f1(E) !
1 #E" & 1 1 + exp % ( kT ' $
(6.11a)
while all sourcebound (or "westbound") electrons are distributed according to the drain (or "East") contact with - = 2:
f ! (z; E) =
f2 (E) "
1 #E! & 2 1 + exp % ( kT ' $
(6.11b)
This is justified by noting that the drainbound (or "eastbound") channels from the source are filled only with electrons originating in the source and so these channels remain in equilibrium with the source with a distribution function f1 (E) . Similarly
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the sourcebound (or "westbound") channels from the drain are in equilibrium with the drain with a distribution function f 2 (E) . Suppose at some energy f1(E) =1, and f2(E) = 0, so that there are lots of electrons waiting to get out of the source, but none in the drain. We would then expect the drainbound lanes of the highway to be completely full ("bumper-to-bumper traffic), ! while the sourcebound lanes would all be empty. Of course this assumes that electrons do not turn around either along the way or at the ends. In electronic terms, this means ballistic channels with good contacts where there are so many channels available that electrons can exit smoothly with a very low probability of turning around. If we either have bad contacts or diffusive channels, the solution in Eq.(6.11a,b) wouldn't work. In Lecture 15 on spin valves we will see some consequences of bad contacts, but for the moment let us talk about diffusive channels with good contacts.
Fig.6.4. Spatial profile of electrochemical potentials +, across a ballistic channel.
Eqs.(6.11a,b) suggest a reasonable guess for what we might expect the distributions to look like in a diffusive channel. We assume the same Fermi-like function but with spatially varying electrochemical potentials reflecting the fact that electrons from the drainbound channels continually transfer over to the sourcebound lanes:
f + (z; E) =
1 " E ! +(z) % 1 + exp $ ' kT # & 1 " E ! !(z) % 1 + exp $ ' kT # &
(6.12a)
f ! (z; E) =
(6.12b)
Note that the potentials are in general energy-dependent and could be written as (z;E). In an elastic resistor, every energy is independent and in general each one could exhibit a different spatial variation in the potential if the mean free path is energy-dependent. But as noted earlier, for simplicity, we will ignore this point assuming some average energy-independent mean free path.
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Lessons from Nanoelectronics:
6.3. From Currents to Potentials Our next objective is to translate our earlier equation for currents I (Eq.(6.6)) into an equation for potentials . The currents appearing in Eq.(6.6) can be written in terms of the occupation functions in Eqs.(6.11a,b) by noting first that I+ equals the amount of charge exiting from the right per unit time. In a time ! t , all the charge in a length ! z ! t exits, so that
I+
= q * Electrons per unit length * ! z
The number of electrons per unit length is equal to half the density of states (since only half the states carry current to the right) per unit length, D(E) / 2L, times the fraction f+ of occupied states, so that
I + (z; E) = q
D(E) u(E) f + (z; E) 2L"# !# $
M (E)/h
Here u is the average vz as defined in Eq.(4.6a) and making use of the definition of the number of channels M from Eq.(4.9) we have
I + (z; E) = (q / h) M (E) f + (z; E)
Similarly
(6.13a) (6.13b)
I ! (z; E) = (q / h) M (E) f ! (z; E)
To get from distribution functions f to electrochemical potentials , we make use of the low bias result (Eq.(2.8)) using a Taylor series expansion to relate the deviation from the reference value of f0(E) to the deviation of the corresponding electrochemical potential (z) from the reference value of 0 (see Eq.(A.4), Appendix A)
f +(z; E) ! f0 (E) = f !(z; E) ! f0 (E) =
( +(z) ! 0 ) FT (E ! 0 ) (6.14a) ( !(z) ! 0 ) FT (E ! 0 ) (6.14b)
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This allows us to write the currents in the form
I +(z; E) ! I 0 (E) = I !(z; E) ! I 0 (E) =
( +(z) ! 0 ) q M (E) FT (E ! 0 ) (6.15a) h ( !(z) ! 0 ) q M (E) FT (E ! 0 ) (6.15b) h
where the equilibrium current I0 must be the same for both the "+" and the "-" streams since they must cancel out giving zero net current at equilibrium. We can now rewrite Eq.(6.8) in the form
!(I + " I 0 ) !z
(I + " I 0 ) " (I " " I 0 ) = " !
=
!(I " " I 0 ) !z
and make use of Eqs.(6.15a,b) to obtain an equation for the potentials:
d+ dz
= !
+ !! !
=
d! dz
(6.16)
We have thus translated the equation for currents (Eq.(6.6)) into an equation for potentials. To get to our main result in Eq.(6.1), we first take the sum and difference of the two equations in (6.16):
d " + + ! % $ ' dz # 2 &
= !
+ ! ! !
(6.17a)
d + ! ! dz
(
)
= 0
(6.17b)
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Lessons from Nanoelectronics:
We now need a relation connecting the current to the electrochemical potentials. Using Eqs.(6.15a,b) in Eq.(6.9)
I (z; E) =
Using Eq.(5.15),
( +(z) ! ! (z)) q M (E) FT (E ! 0 ) h
I(z; E) =
( +(z) ! ! (z)) ! (E) A FT (E ! 0 ) q" ( )
Integrating over all energy and making use of Eqs.(6.10) and Eq.(5.3), we have
I(z) =
!A + (z) ! ! (z) q!
(6.18)
Combining Eq.(6.18) with Eqs.(6.17a,b) and defining
(z) !
+(z) + " (z) 2
(6.19)
we obtain what we were looking for, namely, Eqs.(6.1a,b). 6.4. The New Boundary Condition And that brings us to Eq.(6.3a,b), the new boundary condition that replaces the standard Eqs.(6.2a,b). To obtain Eqs.(6.3a,b) we first note that the correct boundary conditions apply to + on the left and - on the right
+ (z = 0) = 1, ! (z = L)
= 2
(6.20)
This is because we expect the distribution f+ to look like f1 at the source end (z=0) and the distribution f- to look like f2 at the drain end (z=L). In the ballistic case these values stay fixed throughout the channel (see Fig.6.4), while in the diffusive case a solution of Eq.(6.1a) leads to linearly decreasing solutions for both as sketched in Fig.6.5, such that their difference is constant and Eq.(6.18) gives a constant current as required by Eq.(6.1b).
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Fig.6.5. Spatial profile of electrochemical potentials +, - across a diffusive channel obtained by solving Eq.(6.17).
But the point to note is that the boundary conditions in Eq.(6.20) do not apply to (z) which is defined as the average of (see Eq.(6.19)). The boundary conditions for (z) are obtained by starting from Eq.(6.19)
" +! ! % (z = 0) = $ + ! ' 2 # & z=0
and making use of Eqs.(6.18) and (6.20) to write
(z = 0)
= 1 !
q! I 2" A
= 1 ! (q I RB / 2)
making use of Eq.(6.4) for RB. Similarly,
" +! ! % (z = L) = $ ! + ' 2 # & z=L
= 2
+ (q I RB / 2)
These are the new boundary conditions (Eqs.(6.4a,b)) we mentioned at the beginning of the Lecture.
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Lecture 7
What about Drift?
7.1. Boltzmann Transport Equation, BTE 7.2. Diffusion Equation from BTE 7.3. Equilibrium Fields Do Matter 7.4. The Two Potentials
Interestingly in our Lectures so far we have hardly ever mentioned the electric field, in contrast to most treatments of electronic transport which start by considering the electric field induced force as the driving term. It may seem paradoxical that we could obtain the conductivity without ever mentioning the electric field! Electric fields are typically visualized as the gradient of an electrostatic potential (=U/q). By contrast, we have been using the electrochemical potential as the basis for our discussions. It is important to recognize the difference between the two "potentials":
Electrochemical
!
=
Chemical
( ! U) !"# # $
+
Electrostatic
U !
(7.1)
Fig.7.1. The two potentials: Electrostatic (U/q) and electrochemical (/q). D(z;E) denotes the spatially varying density of states.
is a measure of the energy upto which the states are filled, while U determines the energy shift of the available states, so that ( - U) is a measure of the degree to which the states are filled and hence the number of electrons. In the last chapter we obtained the equation
I
A = !!
d ( / q) dz
(7.2)
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But what we really showed was that
I
A = !!
d ( !U) / q dz
(7.3)
assuming zero electric field, dU/dz = 0. So how do we know what the correct equation is, when we include U? It would seem that we needed to solve a whole new problem including the effect of the field (= d(U / q) / dz ) on electrons. However, this is unnecessary because the basic principles of equilibrium statistical mechanics require the current to be zero for a constant , just as there can be no heat current if the temperature is constant. Hence the current expression must have the form given in Eq.(7.2) which can be written as the sum of a drift term and a diffusion term
I
d ( !U) / q dU / q = !! !! A dz dz !## ## " $ !# # " $
Diffusion Drift
(7.4)
both of which must be described by the same coefficient ! , a requirement that leads to the Einstein relation between drift and diffusion. And that is why we can find ! considering only the diffusion of electrons with U = 0, obtain Eq.(7.3) and just replace it with Eq.(7.2) which correctly accounts for "everything." There is really no need work out the drift problem separately. What we called the diffusion equation is really the drift-diffusion equation even though we did not consider drift explicitly. Couldn't we instead have neglected diffusion completely and just gone with the drift term? That way we could stick to the view that current is driven by electric fields and not have to bother with electrochemical potentials. The problem is that if we take this view then one has to invoke mysterious quantum mechanical forces to explain why all electrons are not affected by the field. In our discussion the energy window for transport (FT, see Fig.2.3) arises naturally from the difference in the "agenda" of the two contacts
f1(E) ! f2 (E) " FT (E ! 0 ) ( 1 ! 2 )
as discussed in Lecture 3. The point is that regardless of which potential we choose to work with, it finally affects transport through the occupation factor, f. In this Lecture we will justify our neglect of drift more explicitly by introducing the Boltzmann Transport Equation (BTE) which is the standard starting point for all
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Lessons from Nanoelectronics:
discussions of the transport of particles. We too could have used it as the starting point for but we did not do so because it is harder to digest with its multiple independent variables, compared to the ordinary differential equation in Lecture 6, which follows from relatively elementary arguments. Even in this Lecture we will not really do justice to the BTE. We will introduce it briefly and use it to show that for low bias, the current indeed depends only on d/dz and not on dU/dz, thus putting our discussion of steady-state, low bias transport without electric fields on a firmer footing and identifying possible issues with it. Note the two qualifying phrases, namely "steady-state" and "low bias." We will show later in this lecture that for time varying transport, the neglect of electric fields can lead to errors, but we will not discuss it further in these lectures. However, even under steady-state conditions, electric fields can play an important role in determining the full current-voltage characteristics, once we go beyond low bias, as we will discuss in the next Lecture. 7.1 Boltzmann Transport Equation, BTE In Lecture 6 we introduced electron distribution functions f and electrochemical potentials describing the drainbound and sourcebound currents I . Both the drainbound and sourcebound current, however, is composed of electrons traveling at different angles having different z-momemtum pz, even though they all have the same energy (we are still talking about an elastic resistor) and hence the same total momentum. To include the effect of the electric field we need "momentum-resolved" distribution functions f (z, pz , t) The BTE describes the evolution of such "momentum-resolved" distribution functions f(z,pz,t) that tell us the occupation of states with a given momentum pz and velocity vz at a location z at time t :
!f !f !f + ! z + Fz !t !z !pz
= Sop f
(7.5)
where Fz is the force on the electrons, and Sopf symbolically represents the complex scattering processes that continually redistribute electrons among the available velocity states. The BTE with the right hand side set to zero (that is without scattering processes)
!f !f !f + ! z + Fz !t !z !pz
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is completely equivalent to describing a set of particles each with position z(t) and momenta pz (t) that evolve according to the semiclassical laws of motion:
!z !
Fz !
dz = dt
"E "pz
(7.7a)
dpz #E = " dt #z
(7.7b)
where E(z,pz,t) is the total energy. Eqs.(7.7a,b) describe semiclassical dynamics in single particle terms where the position z(t) and momenta pz (t) for each of the electrons is a dependent variable evolving in time. By contrast, the BTE provides a collective description with all three independent variables z, pz , t on an equal footing. To get from Eqs.(7.7) to (7.6) we start by noting that in the absence of scattering,
f (z, pz , t) =
f (z ! ! z "t, pz ! Fz "t, t ! "t)
reflecting the fact that any electron with a momentum
pz at z at time t ,
must have had a momentum of
pz ! Fz "t at z ! ! z "t a little earlier at t ! "t .
Next we expand the right hand side to the first term in a Taylor series to write
f (z, pz , t) =
f (z, pz , t) !
"f "f "f ! z #t ! Fz #t ! #t "z "pz "t
Eq.(7.6) follows readily on canceling out the common terms. The left hand side of the BTE thus represents an alternative way of expressing the laws of motion. What makes it different from mere mechanics, however, is the stochastic scattering term on the right which makes the distribution function f approach the equilibrium Fermi function when external driving terms are absent. This last point of course is not meant to be obvious. It requires an extended discussion of the scattering operator Sop that we talk a little more about in Lecture 16 when we discuss the second law.
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For our purpose it suffices to note that a common approximation for the scattering term is the relaxation time approximation (RTA)
Sop f
! "
f " f0 !
(7.8)
which assumes that the effect of the scattering processes is proportional to the degree to which a given distribution "f" differs from the equilibrium distribution f0. One comment about why we call this approach semiclassical. The BTE is classical in the sense that it is based on a particle view of electrons. But it is not fully classical, since it typically includes quantum input both in the scattering operator Sop and in the form of the energy-momentum relation. For example, graphene is often described by a linear energy-momentum relation
! ! E = !0 p
a result that is usually justified in terms of the bandstructure of the graphene lattice requiring quantum mechanics that Boltzmann did not live to see. But once we accept that, many transport properties of graphene can be understood in classical particulate terms using the BTE that Boltzmann taught us to use. 7.2 Diffusion equation from BTE We start by combining the RTA (Eq.(7.8)) with the full BTE (Eq.(7.5)) to obtain for steady-state ( ! / !t = 0 ),
!z
!f !f + Fz !z ! pz
= "
f " f0 "
(7.9)
In the presence of an electric field we can write the total energy as
E(z, pz ) = ! ( pz ) + U(z)
(7.10)
where ! ( pz ) denotes the energy-momentum relation with U=0 and this gets shifted locally by U(z) as sketched in Fig.7.2.
Fig.7.2. The energy momentum relation with U=0 is shifted locally by U(z). At equilibrium the electrochemical potential 0 is spatially constant.
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The first point to note is that the equilibrium distribution with a constant electrochemical potential 0
f0 (z, pz ) =
1 " E(z, pz ) ! 0 % exp $ ' +1 # & kT
(7.11)
satisfies the BTE in Eq.(7.9). The right hand side of Eq.(7.9) is obviously zero, but it takes a little differential calculus to see that the left hand side is zero too. Defining
X0
! E(z, pz ) " 0
= ! ( pz ) + U(z) " 0
(7.12)
we have
!z
! f0 !f + Fz 0 !z ! pz
" ! f % " ! X0 ! X0 % = $ 0 ' $! z + Fz !z ! pz ' # ! X0 & # & = 0
" ! f % " !E !E % = $ 0 ' $! z + Fz ! pz ' # ! X0 & # ! z &
making use of Eqa.(7.7a,b).
Fig.7.3. Same as Fig.7.2, but the electrochemical potential (z) varies spatially reflecting a non-equilibrium state.
Out of equilibrium, we assume the distribution function f(z,pz) to have the same form as Eq.(7.11) but with a spatially varying electrochemical potential (z):
f (z, pz ) =
1 " E(z, pz ) ! (z) % exp $ ' +1 # & kT
(7.13)
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Using Eq.(7.13), the left hand side of BTE (see Eq.(7.9)) reduces to
" ! f % " !X !X % !z + Fz $ !X ' $ !z ! pz ' # &# &
=
"!f%" d % $ ! X ' $ ( !z d z ' # &# &
X
where
! E(z, pz ) " (z)
(7.14)
=
X0 (z, pz ) + 0 " (z)
We now assume small deviations in (z) from the equilibrium value so that we can write the left hand side as
"!f% $ !X' # & X =X 0
" d % $ ( !z d z ' # &
and use a Taylor series expansion (as discussed in Lecture 2) to write the right hand side of BTE as
!
f ! f0 !
$#f' (z) ! 0 " & ) ! % # X ( X =X 0
Combining the two sides
!z
d dz
= !
(z) ! 0 "
(7.15)
We now introduce two separate electrochemical potentials + and - for the rightmoving ( ! z > 0 ) and left-moving ( ! z < 0 ) electrons to write
d + dz
= !
+ ! 0 ! z"
,
d ! dz
=
! ! 0 ! z"
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+ ! Assuming 0 = ( + ) / 2 , we obtain the same equations (see Eqs.(6.16))
d+ dz
= !
+ !! !
=
d! dz
(7.16)
as in the last lecture (with ! = 2" z# ), except that now we have done it "right", by including electric fields explicitly and showing that their effect cancels out. 7.3. Equilibrium Fields Do Matter However, there is an important subtlety we would like to point out. Although the externally applied electric field does not affect the low bias conductance, any inbuilt fields that exist within the conductor under equilibrium conditions can affect its low bias conductance. Let me explain. Note that in our treatment above we assumed that under non-equilibrium conditions, the electrochemical potential is a function of z (Eq.(7.13)) and the resulting linearized equation (Eq.(7.15)) does not involve the field Fz = dU/dz. However, the field term would not have dropped out so nicely if we were to assume that the electrochemical potential is not just a function of z, but of both z and pz. Instead of Eq.(7.15) we would then obtain
!z
! ! + Fz !z ! pz
= "
(z, pz ) " 0 "
(7.17)
However, the additional term involving the field Fz does not play a role in determining linear conductivity because it is ~ V2, V being the applied voltage. At equilibrium with V=0, = 0, so that both derivatives appearing on the left are zero. Under bias, in principle, both could be non-zero and to first order ~ V. But the point is that while vz is a constant, the applied field Fz is also ~ V. So while the first term on the left is ~ V, the second term is ~ V2. But this argument would not hold if Fz were not the applied field, but internal inbuilt fields independent of V that are present even at equilibrium. Equilibrium requires a constant and NOT a constant U. The equilibrium condition depicted in Fig.7.2 (also shown here for ease of reference) is quite common in real conductors, with varying U(z) corresponding to non-zero fields Fz. Indeed this picture could also represent an interface between dissimilar materials (called "heterostructures") where the discontinuity in band edges is often modeled with effective fields.
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The point is that such equilibrium fields can and do affect the low bias conductance. For an ideal homogeneous conductor we do not have such fields. But even then we need to make two contacts in order to measure the resistance. Each such contact represents a heterostructure qualitatively similar to that shown in Fig.7.2 with inbuilt effective (if not real) fields. I believe it is these fields that give rise to the interface resistance that distinguishes the new Ohm's law from the standard one. 7.4. The Two Potentials In these Lectures we will generally focus on steady-state transport involving the injection of electrons from a source and their collection by a drain (Fig.7.4). We have seen that the low bias conductance can be understood in terms of the electrochemical potential , without worrying about the electrostatic potential U.
Fig.7.4. So far we have talked of steady-state transport involving the injection of electrons by a source and their collection by a drain contact.
However, we would like to briefly consider ac transport through a nanowire far from any contacts where we have a local voltage V(z,t) and current I(z,t) (Fig.7.5), because this provides a contrasting example where it is important to pay attention to the difference between the two potentials even for low bias, in order to obtain the correct inductance and capacitance.
Fig.7.5. Ac or time varying transport along a nanowire can be described in terms of a voltage V(z,t) and a current I(z,t).
For this problem too we start from the BTE with the RTA approximation as in the last section, but we do not set ! / !t = 0 ,
!f !f !f + !z + Fz !t !z ! pz
= "
f " f0 "
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and linearize it assuming a distribution of the form (compare Eq.(7.13))
f (z, pz , t) =
1 " E(z, pz , t) ! (z, t) % exp $ ' +1 # & kT
(7.18)
Compared to the steady-state problem (Eq.(7.15)) we now have two extra terms involving the time derivatives of E and :
! ! !E + !z " !t ! z !t
= "
(z, t) " 0 "
(7.19)
As we did in the last Section with Eq.(7.15), we can separate Eq.(7.19) into two equations for + and -, whose sum and difference are identified with voltage and current (Eqs.(6.18), (6.19)) to obtain a set of equations
!( / q) !z
= " (LK + LM )
!I !t
"
I !A
(7.20a)
!( / q) !t
# 1 1 & !I = "% + ( $ CQ CE ' ! z
(7.20b)
that look just like the transmission line equations with a distributed series inductance and resistance and a shunt capacitance. The algebra getting from Eq.(7.19) to Eqs.(7.20a,b) is a little long-winded and since time-varying transport is only incidental to our main message we have relegated the details to Appendix D. Those who are really interested can look at the original paper on which this discussion is based (Salahuddin et al., 2005). But note the two inductors and the two capacitors in series. The kinetic inductance LK and the quantum capacitance CQ per unit length, arise from transport-related effects
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LK =
1 , CQ = q 2 2Mvz
h
q 2 2M h vz
(7.21)
while the LM and the CE are just the normal magnetic inductance and the electrostatic capacitance CE from the equations of magnetostatics and electrostatics. The point I wish to make is that the fields enter the expression for the energy E(z,pz,t) and if we ignore the fields we would miss the !E / !t term in Eq.(7.19) to obtain
! ! + !z !t !z
= "
(z, t) " 0 "
and after working through the algebra obtain instead of Eqs.(7.20a,b)
!( / q) !z
= " LK
!I !t
"
I !A
(7.21a)
!( / q) !t
# 1 & !I = "% ( $ CQ ' ! z
(7.21b)
Do these equations approximately capture the physics? Not unless we are considering wires with very small cross-sections so that M is a small number making LK >> LM and CQ << CE. We could recover the correct answer from Eqs.(7.21a,b) by replacing the in with (- U) and then using the laws of electromagnetics to replace
!U !t
with
1 !I CE ! z
and
!U !z
with LM
!I !t
But these replacements may not be obvious and it is more straightforward to go from Eq.(7.19) to (7.20) as spelt out in Appndix D. Note that if we specialize to steady-state ( ! / !t = 0 ), both Eqs.(7.20) and (7.21) give us back our old diffusion equation (Eq.(6.1)). As we argued earlier, for low bias steady-state transport, the applied electric field can be treated as incidental.
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Lecture 8
Electrostatics is Important
8.1. The Nanotransistor 8.2. Why the Current Saturates 8.3. Role of Charging 8.4. Rectifier Based on Electrostatics 8.5. Extended Channel Model
In the last Lecture we tried to justify our "field-less" approach to conductivity which comes as a surprise to many since it is commonly believed that currents are driven by electric fields. However, we hasten to add that the field can and does play an important role once we go beyond low bias and our purpose in this lecture is to discuss the role of the electrostatic potential and the corresponding electric field on the current-voltage characteristics beyond low bias. To illustrate these issues, I will use the nanotransistor, an important device that is at the heart of microelectronics. As we noted at the outset the nanotransistor is essentially a voltage-controlled resistor whose length has shrunk over the years and is now down to a few hundred atoms. But as any expert will tell you, it is not just the low bias resistance, but the entire shape of the current-voltage characteristics of a nanotransistor that determines its utility. And this shape is controlled largely by its electrostatics, making it a perfect example for our purpose. I should add, however, that this Lecture does not do justice to the nanotransistor as a device. This will be discussed in a separate volume in this series written by Lundstrom, whose model is widely used in the field and forms the basis of our discussion here. We will simply use the nanotransistor to illustrate the role of electrostatics in determining current flow. We have seen that the elastic transport model characterized by the current formula
I
=
1 q
+" !"
#
dE G(E) ( f1(E) ! f2 (E) )
(see Eq.(3.3))
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In this Lecture I will use the nanotransistor to illustrate some of the issues that need to be considered at high bias, some of which can be modeled with a simple extension of Eq.(3.3)
I
=
1 q
+" !"
#
dE G(E !U) ( f1(E) ! f2 (E) )
(8.1)
to include an appropriate choice of the potential U in the channel which is treated as a single point. We call this the point channel model to distinguish it from the standard and more elaborate extended channel model which we will introduce at the end of the Lecture. 8.1 The nanotransistor The nanotransistor is a three-terminal device (Fig.8.1), though ideally no current should flow at the gate terminal whose role is just to control the current. In other words, the current-drain voltage, I- VD, characteristics are controlled by the gate voltage, VG (see Fig.8.2). The low bias current and conductance can be understood based on the principles we have already discussed. But currents at high VD involve important new principles.
Fig.8.1. Sketch of a field effect transistor (FET): Channel length, L; Transverse width, W (Perpendicular to page).
Fig.8.2. Typical current-voltage, I-VD characteristic and its variation with VG for an FET.
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The basic principle underlying an FET is straightforward (see Fig.8.3). A positive gate voltage VG changes the potential in the channel, lowering all the states down in energy, which can be included by replacing Eq.(8.1) with Eq.(8.2) and setting U = qVG .
Fig.8.3. A positive gate voltage VG increases the current in an FET by moving the states down in energy.
For an n-type conductor this increases the number of available states in the energy window of interest around 1 and 2 as shown. Of course for a p-type conductor (see Fig.7.2) the reverse would be true leading to a complementary FET (see Fig.0.2) whose conductance variation is just the opposite of what we are discussing. But we will focus here on n-type FET's. We will not discuss the low bias conductance since these involve no new principles. Instead we will focus on the current at high bias, specifically on why the currentvoltage, I- VD characteristic is (1) non-linear, and (2) "rectifying," that is different for positive and negative VD.
8.2 Why the current saturates
Fig.8.4. The current saturates once 2 drops below the band-edge.
Fig.8.2 shows that as the voltage VD is increased the current does not continue to increase linearly. Instead it levels off tending to saturate. Why? The reason seems easy enough. Once the electrochemical potential in the drain has been
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Lessons from Nanoelectronics:
lowered below the band edge the current does not increase any more (Fig.8.4). The saturation current can be written from Eq.(8.1)
I sat
=
1 q
+" !"
#
dE G(E !U) f1(E)
(8.2)
by dropping the second term f2(E) assuming 2 is low enough that f2(E) is zero for all energies where the conductance function is non-zero. In the simplest approximation
U (1) = qVG
The superscript "1" is included to denote that this expression is a little too simple, representing a first step that we will try to improve.
Fig.8.5. The current does not saturate completely because the states in the channel are also lowered by the drain voltage.
If this were the full story the current would have saturated completely as soon as 2 dropped a few kT below the band edge. In practice the current continues to increase with drain voltage as sketched in Fig.8.6. The reason is that when we increase the drain voltage we do not just lower 2, but also lower the energy levels inside the channel (Fig.8.5) similar to the way a gate voltage would. The result is that the current keeps increasing as the conductance function G(E) slides down in energy by a fraction " (< 1) of the drain voltage VD, which we could include in our model by choosing
!
U (2) = (1 ! ! ) qVG + ! qVD
" UL
(8.3)
Fig.8.6. Current in an FET would saturate perfectly if the channel potential were unaffected by the drain voltage.
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Indeed the challenge of designing a great transistor is to make ! as small as possible so that the channel potential is hardly affected by the drain voltage. If ! were zero the current would saturate perfectly as shown in Fig.8.6 and that is really the ideal: a device whose current is determined entirely by VG and not at all by VD or in technical terms, a high transconductance but low output conductance. For reasons we will not go into, this makes designing circuits much easier. To ensure that VG has far greater control over the channel than VD it is necessary to make the insulator thickness a small fraction of the channel length. This means that for a channel length of a few hundred atoms we need an insulator that is only a few atoms thick in order to ensure a small " . This thickness has to be precisely controlled since thinner insulators would leak unacceptably. We mentioned earlier that today's laptops have a billion transistors. What is even more amazing is that each has an insulator whose thickness is! precisely controlled down to a few atoms! 8.3 Role of charging There is a second effect that leads to an increase in the saturation current over what we get using Eq.(8.3) in (8.1). Under bias, the occupation of the channel states is less than what it is at equilibrium. This is because at equilibrium both contacts are trying to fill up the channel states, while under bias only the source is trying to fill up the states while the drain is trying to empty it. Since there are fewer electrons in the channel, it tends to become positively charged and this will lower the states in the channel as shown in Fig.8.5 even for perfect electrostatics ( " = 0) resulting in an increase in the current. This effect can be captured within the point channel model (Eq.(8.1)) by writing the ! channel potential as
U = U L + U0 (N ! N 0 )
(8.4)
where UL is given by our previous expression in Eq.(8.3). The extra term represents the change in the channel potential due to the change in the number of electrons in the channel, N under non-equilibrium conditions relative to the equilibrium number N0, U0 being the change in the channel potential energy per electron. To use Eq.(8.4), we need expressions for N0, N. N0 is the equilibrium number of channel electrons, which can be calculated simply by filling up the density of states, D(E) according to the equilibrium Fermi function f0(E).
+#
N0
=
"#
$ dE D(E " U) f 0 (E)
(8.5a)
!
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while the number of electrons, N in the channel under non-equilibrium conditions is given by
+#
N =
"#
$ dE D(E " U)
f1 (E) + f 2 (E) 2
(8.5b)
assuming that the channel is "equally" connected to both contacts. Note that the ! calculation is now a little more intricate than what it would be if U0 were zero. We now have to obtain a solution for U and N that satisfy both Eqs.(8.4) and (8.5) simultaneously through an iterative procedure as shown schematically in Fig.8.7.
Fig.8.7. Self-consistent procedure for calculating the channel potential U in point channel model.
Once a self-consistent U has been obtained, the current is calculated from Eq.(8.1), or an equivalent version
I
=
1 q
+" !"
#
dE G(E) ( f1(E + U) ! f2 (E + U))
(Eq.(8.1), modified version)
that is sometimes more convenient numerically and conceptually. This simple point channel model often provides good agreement with far more sophisticated models as discussed in Rahman et al. (2003).
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8.4 "Rectifier" Based on Electrostatics Let us now look at an example that can be handled using the point channel model just discussed. However, it does not illustrate any issues affecting the design of nanotransistors, which we could not do justice to anyway. Instead I have chosen this example to illustrate a fundamental point that is often not appreciated, namely that an otherwise symmetric structure could exhibit asymmetric current-voltage characteristics (which we are loosely calling a "rectifier"). In other words, we could have
I(+VD ) ! I("VD )
for a symmetric structure, simply because of electrostatic asymmetry. Consider a nanotransistor having perfect electrostatics represented by " = 0 (Eq.(8.3)), connected (a) in the standard configuration (Fig.8.8a) and (b) with the gate left floating (Fig.8.8b). The basic device is assumed physically symmetric, so that one ! could not tell the difference between the source and drain contacts. This may not be true of real transistors, but that is not important, since we are only trying to make a conceptual point. The configuration in (a) has electrostatic asymmetry, since the gate is held at a fixed potential with respect to the source, but not with respect to the drain. But configuration (b) is symmetric in this respect too, since the gate floats to a potential halfway between the source and the drain.
Fig.8.8. (a) Standard FET assuming perfect electrostatics. (b) Floating gate FET
1 = 0 , 2 = 0 " qVD
! ! ! !
1 = 0 + qVD /2 2 = 0 " qVD /2
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Fig.8.9 shows the current-voltage characteristics calculated using the model summarized in Fig.8.7 (MATLAB code in Appendix F), for each of the structures shown in (a) and (b). The difference is not very significant, but the point I am trying to make is that for the floating gate structure,
I(+VD ) = I(!VD )
but not for the standard configuration where the magnitude of the current is less for positive voltages and more for negative voltages relative to the floating gate case.
Fig.8.9. Current-voltage characteristics obtained from the point channel model corresponding to the confgurations shown in Fig.8.8.
Numerically the difference comes from the electrostatic factor ! which is zero for the standard configuration, but equal to 0.5 for the floating configuration, since the gate floats to + VD/2, corresponding to U = ! qVD / 2 . Physically one could understand the current-voltage characteristics by noting that for the standard FET (Fig.8.8a) we move 2 while holding 1 fixed:
1 = 0 , 2 = 0 " qVD
but for the floating gate FET (Fig.8.8b), we could mentally hold the channel potential fixed and write
!
!
1 = 0 + qVD /2 , 2 = 0 " qVD /2
!
so that we do not have to redraw the channel density of states as we change the bias. It is then straightforward to see that the I-VD characteristic is symmetric, since replacing +VD with -VD only serves to interchange 1 and 2. ! The point is that it is not necessary to design an asymmetric channel to get asymmetric I-V characteristics. Even the simplest symmetric channel can exhibit nonsymmetric I(VD) characteristic if the electrostatics is asymmetric.
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8.5 Extended Channel Model The point channel elastic model that we have described (Eqs.(8.1), (8.2)) integrates our elastic resistor with a simple electrostatic model for the channel potential (U/q), allowing it to capture some of the high bias physics that the pure elastic resistor misses. Let me end this Lecture by noting some of the things it misses. The point channel model ignores the electric field in the channel and assumes that the density of states D(E) stays the same from source to drain. In the real structure, however, the electric field lowers the states at the drain end relative to the source as sketched in Fig.8.10. Doesn't this change the current?
Fig.8.10. Our discussion of current flow in an elastic conductor has assumed the (a) zero-field spatial profile shown, while (b) in the real structure the non-zero electric field lowers the states at the drain end.
For an elastic resistor one could argue that the additional states with the slanted (rather than horizontal) shading are not really available for conduction since (in an elastic resistor) every energy represents an independent energy channel and can only conduct if it connects all the way from the source to the drain. But even for an elastic resistor there should be an increase in current because at a given energy E, the number of modes at the drain end is larger than the number of modes at the source end. This is because the number of modes at an energy E depends on how far this energy is from the bottom of the band determined by U(z) which is lower at the drain than at the source.
Fig.8.11. The number of channels M(E) is larger at the drain end than at the source because of the lower U(z).
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The structure almost looks as if it were "wider" at the drain than at the source. For a ballistic conductor this makes no difference since the conductance function cannot exceed the maximum set by the "narrowest" point. But for a conductor that is many mean free paths long, the broadening at the drain could increase the conductance relative to that of an un-broadened channel. In general we could write
q 2 M1! h L+!
! G(E) !
q2 M1 h
(8.6)
This effect is not very important for near ballistic elastic channels, since the minimum and maximum values of the conductance function in Eq.(8.7) are then essentially equal. Indeed this increase in conductance could be ascribed to a field-dependent mean free path which can be ignored in the low bias limit as we have done so far. How do we include it in a quantitative model? We could simply take our "driftdiffusion" equation from Lecture 6 and modify it to include a spatially varying conductivity:
d I= 0 dz I A = !
! (z) d q dz
(8.7)
What do we use for the conductivity, ! (z) ? Our old expression
+"
!
=
!"
#
dE FT (E ! 0 ) ! (E)
(same as Eq.(5.3))
involved an energy average of ! (E) over an energy window of a few kT around E = 0. The spatially varying U(z) shifts the available energy states in energy, so that one now has to look at the energy window around
E
= (z) !U(z)
from the bottom of the band, suggesting that we replace Eq.(5.3) with
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+"
! (z) =
!"
#
dE FT (E + U(z) ! (z)) ! (E)
(8.8)
For low bias, we could replace (z) with 0 to obtain our earlier result in Eq,(7.12) from obtained by directly linearizing the BTE. Note that to use Eqs.(8.7), (8.8) we have to determine ((z) U(z)) from a selfconsistent solution the Poisson equation ( ! : Permittivity, n0, n: electron density per unit volume at equilibrium and out of equilibrium )
d ! dU $ #! & dz " dz %
= q 2 (n ' n0 )
(8.9)
The electron density per unit length entering the Poisson equation is calculated by filling up the density of states (per unit length) shifted by the local potential U(z), according to the local electrochemical potential
+#
n(z) !
"#
$
dE
D(E "U(z)) 1 E " (z) L 1 + exp kT
+"
=
!"
#
dE
D(E) L
1 E ! (z) !U(z) 1 + exp kT
(8.10)
Solving Eq.(8.10)) self-consistently with the Poisson equation (Eq.(8.9)) is indeed the standard approach to obtaining the correct (z), U(z), which can then be used to find the conductivity ! (z) from Eq.(8.8) and hence the current from Eq.(8.7). Note that this whole approach is based on the assumption of local electrochemical potentials (z) describing right and left-moving electrons whose average is the (z) appearing in Eq.(9.1). In general, electron distributions can deviate so badly from Fermi functions that an electrochemical potential may not be adequate and one needs the full semiclassical formalism based on the Boltzmann Transport Equation (BTE) and much progress has been made in this direction. However, full-fledged BTE-based simulation is time-consuming and the drift-diffusion equation based on the concept of a local potential (z) continues to be the "bread and butter" of device modeling. What our bottom-up approach adds is that Eq.(8.7) can be used even to model ballistic channels if the boundary conditions are modified appropriately (Eq.(6.3)) to include the interface resistance, a result that was obtained by carefully accounting for the distinction between +(z) and -(z) (Lecture 6).
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Lecture 9
Smart Contacts
9.1. Why P-n Junctions are Different 9.2. Smart Contacts
We are now ready to finish up with part one of these lectures, which I entitled "the new Ohm's law" referring to
R=
! (L + ") A
(same as Eq.(4.2))
which includes an extra contact resistance !" / A that depends solely on the properties of the channel and cannot be eliminated by better contacting procedures. As we saw in Lecture 6, the key concept in identifying this interface resistance was the recognition that when a current flows, the electrochemical potentials + and - for the drainbound and sourcebound states are different (Fig.6.5, also reproduced below for convenience). From Eqs.(6.5) and (6.18) we could write (Note: 1 - 2 = qV)
! ! + " "
=
1 " 2 1+ L / "
(9.1)
The contacts held at different potentials 1 and 2 drive the two groups of states (drainbound and sourcebound) out of equilibrium, while backscattering processes described by the mean free path ! try to restore equilibrium. Eq.(9.1) describes the result of these competing forces. Normally we do not like to deal with multiple electrochemical potentials. The diffusion equation for example (see Eq.(6.1)),
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I A
= !
! (z) d q dz
(9.2)
works in terms of a single potential (z) and what we saw in Lecture 6 was how we could sweep the two potentials +(z) and -(z) under the proverbial rug, by defining (z) as the average of the two and including interface resistances into the boundary conditions by replacing Eq.(6.2) with Eq.(6.3). The point I wish to make in this Lecture is that this separation of the electrochemical potentials for different groups of states is really far more ubiquitous and cannot always be swept under the rug. Indeed I would like to go further and argue that the most interesting devices of the future will be the ones where multiple electrochemical potentials will represent the essential physics and cannot be swept under the rug. This is not really as exotic as it may sound. For example, all semiconductor device texts start with the p-n junction for which the need for two separate electrochemical potentials is well-recognized. Let me elaborate. 9.1. Why p-n Junctions are Different
Fig.9.1 shows a grayscale plot of the density of states D(z,E). The white band indicates the bandgap with a non-zero DOS both above and below it on each side which are shifted in energy with respect to each other. A positive voltage is applied to the right with respect to the left, so that 2 is lower than 1 as shown.
Fig.9.1. Simplified grayscale plot of the spatially varying density of states D(z,E) across a p-n junction.
If we look at a narrow range of energies around 1 (see shaded area on the left) it communicates primarily with contact 1. If we look at a narrow range of energies around 2 (see shaded area on the right) it communicates primarily with contact 2. We could draw an idealized diagram with each of these two groups communicating just with one contact and cut off from the other as shown in Fig.9.2. In reality of
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course neither group is completely cutoff from either contact, and people who design real devices often go to great lengths to achieve better isolation, but let us not worry about such details.
Fig.9.2. An idealized version of the pn junction in Fig.9.1.
Would the idealized device in Fig.9.2 allow any current to flow? None at all, if we it were an elastic resistor. There is no energy channel that will let an electron get all the way from left to right. The ones connected to the left are disconnected from the right and those connected to the right are disconnected from the left. But current can and does flow because of inelastic processes that allow electrons to change energies along the channel. Electrons can then come in from the left, change energy and then exit to the right as sketched in Fig.9.3.
Fig.9.3. Current flow in the idealized device of Fig.9.2 is facilitated by distributed inelastic processes.
Indeed this is exactly how currents flow in p-n junctions, by transferring from the upper group of states down to the lower group by inelastic processes, which are generally referred to as recombination-generation (R-G) processes, since people like to think in terms of electrons in the upper group recombining with a "hole" in the lower group. But as we mentioned in Lecture 5, this is really an unnecessary complication and one could simply think purely in terms of electrons transferring inelastically from one group of states to another. The point to note is that this class of devices cannot be described with one electrochemical potential and to capture the correct physics, it is essential to treat the two groups of states separately, introducing two different electrochemical potentials, labeled with the index "n"
In
! d n = ! n q dz
(9.3)
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These currents are all coupled together by inelastic processes generally called "RG processes" in the context of p-n junctions
dI n dz
=
! [ RG ] m"n
m
#
[ RG ] n"m
(9.4)
that take electrons from one group of states "m" to the other "n". This is indeed the way p-n junctions are modeled. It is well-known that the current in a p-n junction is given by an expression of the form
I
= I 0 (eqV /! kT !1)
(9.5)
where the number " ! " as well as the coefficient I0 are determined by the nature of the inelastic or RG processes. The conductivities ! n of either of the two groups of states plays hardly any role in determining this current. The physical reason for this is clear. The rate-determining step in current flow is the inelastic process transferring electrons from one group of states to the other. Transport within any of these groups only adds an unimportant resistance in series with the basic device. Everything we have talked about in these lectures has been about the conductivities ! n of the homogeneous p-type or n-type materials. And this is exactly the physics that is relevant to the operation of the most popular electronic device today, namely the Field Effect Transistor (FET) whose conductivity is controlled by a gate electrode through the electrostatic potential U. But the p-n junction is a totally different device from the FET both in terms of its current-voltage characteristics and the physics that underlies it. It is the basic device structure used to construct solar cells and the principle it embodies is key to a broad class of energy conversion devices. So let me take a short detour to elaborate on this principle.
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9.1.1.
Current-Voltage Characteristics
Consider for example the device in Fig.9.4 assuming that the upper group of states (labeled A) is clustered around an energy ! A while the lower group (labeled B) is clustered around ! B .
Fig.9.4. Same as Fig.9.3 with the two groups of states labeled A and B. Electronic transitions between A and B are facilitated by inelastic interactions.
The essential physics of such p-n junction like devices is contained not in Eq.(9.3), but in Eq.(9.4) which for two levels A and B can be written as
I
A#B !#### "##### # $ ~ DB!A f A (! A ) (1 " fB (! B ))
B#A !#### "##### # $ ! DA"B fB (! B ) (1 ! f A (! A ))
(9.6)
where the coefficients DBA and DAB denote the strength of the inelastic processes inducing the transitions from A to B and from B to A respectively (note that the transition occurs from the second subscript to the first). Interestingly these two rates DAB and DBA are generally NOT equal. DAB involves absorbing an amount of energy
!!
= "A ! "B
(9.7)
from the surroundings, while DBA involves giving up the same amount of energy. A fundamental principle of equilibrium statistical mechanics (see Lecture 16) is that if the entity causing the inelastic scattering is at equilibrium with a temperature T0, then it is always harder to absorb energy from it than it is give up energy to it and the ratio of the two processes is given by
DAB DBA
" !! % = exp $ ! # kT0 ' &
(9.7)
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We can write the current from Eq.(9.4) in the form
I
~ DAB fB (! B ) (1 ! f A (! A )) ( X !1)
(9.8)
where
X
!
DBA f A (! A ) 1 " fB (! B ) DAB 1 " f A (! A ) fB (! B )
(9.9)
Making use of Eq.(9.8), Eq.(9.9) and the following property of Fermi functions (Eq.(2.2))
1 ! f0 (! ) f0 (! )
we can rewrite Eq.(9.9) as
" ! ! 0 % = exp $ # kT ' &
(9.10)
X
" !! !! % " A ! B % = exp $ ! ' exp $ kT ' # & # kT0 kT &
(9.11)
Since Level A is connected to contact 1 and Level B to contact 2, if the inelastic processes taking electrons from A to B are not too strong, level A is almost in equilibrium with contact 1 and level B with contact 2 , so that
A ! B
" 1 ! 2
= qV
If T0 = T, we can write the current from Eq.(9.8) as
I
~
( X !1)
~ eqV /kT !1
which is the standard I-V relation for p-n junctions stated earlier (see Eq.(9.5)) with ! =1. Other values of " ! " would be obtained if we consider more elaborate RG processes rather than the direct "band-to-band" processes considered here.
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But the more important point I want to stress is that this device can be used for energy conversion. If the scatterers are at a temperature different from that of the device (T0 T) then one can have a current flowing even without any applied voltage. This short circuit current is given by
I sc
! I(V = 0) ~ exp
!! # 1 1 & " " 1 k % T0 T ( $ '
(9.12)
One could in principle use a device like this to convert a temperature difference (T0T) into an electrical current. The short circuit current has the opposite sign for T0>T and for T>T0. Readers familiar with Feynman's classic ratchet and pawl lecture (Feynman 1962) may notice the similarity. The ratchet reverses direction depending on whether its temperature is lower or higher than the ambient. One could view more practical devices like solar cells as embodiments of the same principle, the light from the sun having a temperature T0 ~ 60000C characteristic of the surface of the sun, much larger than the ambient temperature. From Eq.(9.8) it is easy to see that under open circuit conditions (I=0), we must have X=1, so that from Eq.(9.11) we have
qVoc !!
= 1!
T T0
The left hand side represents the energy extracted per photon under very low current (near open circuit) conditions, so that this could be called the Carnot efficiency of a solar cell viewed as a "heat engine". However, since T0 >> T, this Carnot efficiency is very close to 100% and my colleague Alam often points out that other factors related to the small angular spectrum of solar energy are important in lowering the ideal efficiency to much lower values. 9.2. Contacts Are Fundamental The point I want to make is how important the discriminating contacts are in the design of this class of devices which we could generally refer to as "solar cells" (Fig.9.5a). The external source raises electrons from the B states to the A states from where they exit through the left contact, while the empty state left behind in B is filled up by an electron that comes in through the right contact. Every electron raised from B to A thus causes an electron to flow in the external circuit.
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But if the contacts are connected "normally" injecting and extracting equally from either group (Figure 9.5b) then we cannot expect any current to flow in the external circuit, from the sheer symmetry of the arrangement. After all, why should electrons flow from left to right any more that they would flow from right to left?
Fig.9.5. (a) Asymmetric contacts are central to the operation of the "solar cell". (b) If contacted symmetrically no electrical output is obtained.
It is this asymmetric contacting that makes p-n junctions fundamentally different from the Field Effect Transistor (FET) that we started our lectures with, both in terms of the current-voltage characteristics and the physics underlying it. It is of course well recognized that the physics of p-n junctions demands two different electrochemical potentials. What is not as well recognized is the generic nature of this phenomenon. Let me explain. For most of these lectures we have discussed how the contacts in an ordinary device drive drainbound and sourcebound states out of equilibrium faster than backscattering processes can restore equilibrium. In p-n junctions we just saw how the contacts drive the two bands out of equilibrium, faster than R-G processes can restore equilibrium. In Lecture 14 we will talk about spin valve devices where magnetic contacts drive upspin and downspin states out of equilibrium faster than spin-flip processes can restore equilibrium. In every case there are groups of states A, B etc that are driven out of equilibrium by smart contacts that can discriminate between them. More and more of such examples can be expected in the coming years, as we learn to control current flow not just with gate electrodes that control the electrostatic potential, but with subtle contacting schemes that engineer the electrochemical potential(s). Many believe that nature does just that in designing many biological `devices', but that is a different story. In the context of man-made devices there are many possibilities. Perhaps we will figure out how to contact s-orbitals differently from p-orbitals, or one valley differently from another valley, leading to fundamentally different devices.
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But this requires a basic change in approach. Traditionally the work of device design has been divided neatly between two groups of specialists: physicists and material scientists who innovate new materials using atomistic theory and device engineers who worry about contacts and related issues using macroscopic theory. Future "solar cells" that seek to function effectively at the microscopic level may well require an approach that integrates materials and contacts at the atomistic level. Perhaps then we will be able to create devices that rival the marvels of nature like photosynthesis.
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