3 Chapter Carrier Transport in Semiconductors
Carrier transport in semiconductors is associated with externally applied electric field, magnetic field, as well as temperature gradients. The subjects that we are going to study include charge and energy transport accompanying the carrier transport. It carries very practical significance because any useful semiconductor devices all involve carrier transport. In this chapter, we limit our discussion to the carrier transport under the condition of weak electric field and magnetic field. We also assume spherical constant energy surface with a parabolic relationship between E and k, i.e. the effective mass is isotropic. 3.1 The Drift of Carriers According to the result of quantum mechanics, the motion of electrons should not be changed moving in a periodic potential field ( k is a constant), i.e. an ideal lattice does not scatter the electrons. If so, the electrons should be able to maintain their status indefinitely. Equilibrium between lattice and electrons can never be achieved. However, any practical lattice has defects, in addition, the atoms of a periodic lattice vibrates about their equilibrium positions, impurities (neutral and ionized) are introduced for doping. All these crystal imperfections cause deviation from a periodic potential. The interaction between the deviation and electrons changes the motion of electrons and results in momentum and energy exchange carrier scattering. The scattering results in the random distribution of electron velocities. Therefore, under thermal equilibrium, the total momentum of carriers is zero, hence the current is zero. In fact, scattering causes the electron lose its original momentum momentum relaxation. Under the condition of applied electric field, carriers acquire momentum along the direction of electric field at a rate of dp / dt = -eE . However, their momentum cannot increase indefinitely since they will have to lose momentum when collided with imperfections (such as defects). They eventually will reach a steady state at which they have a fixed amount of momentum on average. The average velocity gained by carriers under an applied electric field drift velocity v d = E where is the mobility. The current density j = nevd = neE = E where = ne is the conductivity. The mobility can vary in a large range, but for commonly used semiconductors it is typically within 10 2 ~ 10 4 cm2/Vs. Mobility
The drift velocity is actually a statistical average of the electron velocity, Let's use holes as example vi mh v i vd = i = i where the summation is over all holes in a unit volume. The total p pmh
i
momentum per unit volume pmh v d = mv i is affected by two factors: electric field and scattering. Isotropic assumption for scattering If the probability if carrier being scattered to all directions are equal, we can consider carriers lose their momentum completely. Let's define the following quantity: The scattering probability P : number of collisions occurred for one carrier in a unit time. pP : number of collisions in a unit volume in a unit time. pPmh v d : the rate of loss of momentum due to scattering in a unit volume.
d ( pmh v d ) = peE - pPmv d (for holes) dt Introduce = 1 / P - average time traveled for a carrier between two collisions - mean free time, then we have d ( mh v d ) mv = eE - h d dt At steady state, we should have d ( mh v d ) / dt = 0 , then e e vd = E, = mh mh This is equivalent to say that the drift velocity is the additional velocity acquired by the carriers due to the electric field within the period of mean free time. The conductivities are then pe 2 ne 2 n = * for n-type, and p = * for p-type. me mh
It can be shown that the mean free time is also the momentum relaxation time under the condition of isotropic scattering. Let's now remove the electric field at t = 0 moment, we will have d ( mv d ) mv =- d dt
Its solution has the form
mvd (t ) = mvd (0)e - t / Hence, is also the momentum relaxation time. 3.2 The Hall Effect
Consider a rectangular bar of a uniformly doped n-type or p-type semiconductor of length l, width w, and thickness d. Let an electric field E x = Vx / l be applied along the xdirection and a magnetic field Bz along the z-direction.
Fig. Hall effect measurement
When a particle of charge q moves in a magnetic field, the field exerts a force q( v B) on the particle which causes it to move in a direction deflected away from its original direction of motion. When a steady current is flowing along the x-direction, electrons (ntype) or holes (p-type) move to one side of the bar perpendicular to y-axis. As a result, an electric field E y is developed between the two faces opposes the motion of electrons or
holes in the y-direction. A Hall voltage VH can be measured between the two faces. Given a Hall voltage VH , E y is fixed. Thus, the force exerted by E y on each carrier is the same qE y . But the Lorentz force q( v B) is different because the velocity for each carrier can be different. Thus, for each individual carrier qE + q( v B) 0 . In fact, they perform cyclotron movement in the plane perpendicular to Bz and are scattered by various crystal imperfections and other carriers. Instead of treating the movement of individual carriers, we study the momentum conservation of the carrier system under the electric and magnetic field. Since the movement of carriers in z-direction (parallel to B) is not affected, we need only to consider the following quantities in the x-y plane, jx , j y : current density in x- or y-direction
E x , E y : electric field in x- or y-direction vdx , vdy : average velocity in x-or y-direction
Assuming isotropic scattering, and p-type
peE + pe( v d B) =
pmh v d
(steady state)
Writing it in x-direction and y-direction pmh vdy pmh vdx and peE y - pevdx Bz = peE x + pevdy Bz = For n-type, we only need to change e - e , p n . Under the condition of Hall measurement, vd = 0 , thus E y = vdx Bz . 1 1 jx Bz = RH jx Bz , where RH = is Hall coefficient. pe pe 1 For n-type, we have RH = - . ne Use j x = pevdx , then E y = Hall voltage VH = E y w = 1 wd 1 I x Bz jx Bz = for p-type and pe d pe d
VH = -
1 I x Bz for n-type. Since E y 0 , there is an angle between E and j . pe d
Fig. E - j Relationship in Hall effect.
Using E y =
tan p = 1 1 1 pe 2 jx Bz = Bz E x , we obtain the Hall angles p and n as Bz E x = pe pe pe mh
E y eBz eB eB = for p-type and tan n = - z for n-type. If we define = z E x mh mh m cyclotron frequency, then the Hall angle tan p ,n = .
For << 1, then p ,n which implies that the angle for which the electric field has rotated is also the angle that carriers have rotated within the mean free time .
3.3 Anisotropic Scattering
In the above analysis, we have assumed isotropic scattering. This is an oversimplified model to treat the scattering processes of carriers. In reality, most scatterings are anisotropic.
Fig. A spherical polar coordinate system
Consider a carrier traveling in z-direction originally is scattered into a solid angle d at ( , ) with the probability dP = P( , )d where the scattering angle is between incident and scattering direction. The ratio of momentum loss for a scattering process with is (1 - cos ) . In the previous expression for the rate of momentum loss due to scattering, pPmh v d , should now be modified to pPm mh v d where Pm describes the rate of momentum loss which should be evaluated as a weighted integral Pm = P( , )(1 - cos )d and m = 1 / Pm gives the momentum relaxation time. However, the scattering probability should be calculated Pc = P( , )d And c = 1 / Pc is the mean free time.
Only when the scatterings are isotropic, P( , ) = constant, we have cos d = 0 , then Pm = Pc and m = c . But in general, m c , m is more useful in describing the momentum loss (relaxation).
3.4 with Energy Dependence
Since carriers are distributed in a energy range, and carriers with different energies move in different velocities, they are scattered at different rates. With dn being the number of carriers in the energy interval E ~ E + dE , we can calculate the conductivity as follows e 2 ( E ) e 2 ( E ) d = dn = N ( E ) f ( E )dE me* me*
e 2 ( E ) e2 ne 2 < > dn = * ( E )N ( E ) f ( E )dE = me* me me* Where the average of momentum relaxation time ( E )N ( E ) f ( E )dE = ( E )dn < >= N ( E ) f ( E )dE dn
=
The mobilities can be written now as
n =
e < > e < > and p = . * * me mh
For Hall effect, in order to take into account the energy distribution of carriers, using eB = z , we can rewrite m pmh vdx peE x + pevdy Bz =
peE y - pevdx Bz =
pmh vdy
As
2 pe 2 E x + E y pevdx = mh 1 + 2 2
pevdy =
2 pe 2 E y - E x mh 1 + 2 2
Averaging these over energy, we will have < /(1 + 2 2 ) > and < 2 /(1 + 2 2 ) > . We should conclude in general that the Hall coefficient is related to the intensity of magnetic field through the cyclotron frequency. But under the condition of weak magnetic field, eB m << 1 or z = Bz << 1 , Hall coefficient is independent of the intensity of m e magnetic field. (For high mobility material such as InSb, B ~ 103 Gauss can no longer be treated as weak field.) Consider << 1 ,
jx = pe < vdx >=
j y = pe < vdy >=
pe 2 < > pe 2 < 2 > Ex + E y mh mh
pe 2 < > pe 2 < 2 > Ey - E x mh mh
where
< >= <
2
( E )N ( E ) f ( E )dE = ( E )dn N ( E ) f ( E )dE dn ( E )N ( E ) f ( E )dE = ( E )dn >= N ( E ) f ( E )dE dn
2 2
The second term on the right side of the current expressions is called the Hall rotation current. Since j y = 0 in Hall measurement,
Ey < 2 > p = Ex < >
In comparison with the previous result, has been replaced by < 2 > / < > . Finally, the Hall coefficient is given by 1 < 2 > rH R= = , pe < > 2 pe where the Hall factor rH =< 2 > / < > 2 = 1 when is independent of energy.
Through the measurement of conductivity and Hall effect, some important parameters of semiconductors can be determined. Based on the sign of Hall coefficient, one can determine the type of carriers (n-type or p-type). In addition, the carrier concentration can also be determined by the Hall effect. For complete ionization, this gives the impurity doping density. The combined measurement of conductivity and Hall effect can lead to determination of carrier's mobility. RH = rH = H (Hall mobility). This mobility differs from the actual mobility by a Hall factor. Still, mobilities of materials are commonly obtained in this way.
3.5 Carrier Scattering
For a crystal, any imperfections in a periodic potential introduce scattering effect on carriers. There are several scattering mechanisms: (1) Impurities (either ionized or neutral) (2) Defects (point or line) (3) Lattice vibrations All these mechanisms can introduce additional potential in the range of electron wavelength (~100) which scatters electrons or holes.
Carrier Scattering by Ionized Impurities
At room temperature, both shallow donor and acceptor impurities are ionized. The interaction between carriers and ionized impurity centers cause a change in the direction of carrier movement. The calculation of the relaxation time for impurity scattering is based upon the theory of scattering of charged particles by the Coulomb potential of nuclei which was originally developed by Rutherford to explain the scattering of -particles.
Fig. The Coulomb scattering geometry.
For both cases
tan
Ze2 2 4 o s Smv 2 where Ze is the amount of charge on the ionized impurity atom, v is the initial and final velocity, S is the impact parameter (perpendicular distance between the scattering center and the projection of the initial approach). =
Fig. Illustration of the scattering process.
Particles within S ~ S + dS will be scattered to ~ + d . Within period t , the probability for a particle to be at S ~ S + dS around an impurity is N i ( vt )( 2SdS ) , where N i is the impurity concentration. The differential scattering probability P( , )d = 2P( , ) sin d = 2N i vSdS Then N vS dS P( , ) = i sin d 2 Ze Use tan = , we obtain 2 4 o s Smv 2
Ze2 P( , ) = N i v 8 mv 2 sin 2 ( / 2) o s The screening effect due to the presence of other carriers and impurities should be taken into account. For a rough estimate, we can consider for carriers with S greater than half of the average distance between two impurities the interaction is completely screened, S > d = 1 / 2 N i1/ 3 , corresponding to this, there exists a minimum scattering angle
tan
2
min
2
=
Ze2 N i1/ 3 . The rate of momentum loss for a carrier 2 o s mv 2
Pm =
min
P( , )(1 - cos )(2 )d
-1
Carry out this integral, and the momentum relaxation rate m = 1 / Pm 8 2 o s2m 2 v 3 4 o s2 m 2v 4 m = ln1 + Z 2e4 N i N i e 4 Z 2
Following observations can be made if we ignore the weak dependence within the ln function, m 1 / N i and m v 3 E 3 / 2 T 3 / 2 . For carriers with higher kinetic energy, the scattering effect is weak because carriers spend less time in the neighborhood of impurities due to high velocities. Since m T 3 / 2 , as temperature increases, the mobility actually improves for the case in which the ionized impurity scattering is dominant. The scattering by ionized impurities can be a dominant effect at relatively low temperatures (but high enough for most impurities to be ionized).
Phonon Scattering
At a given temperature, all lattice atoms vibrate around their equilibrium lattice sites. These vibrations cause carriers to be scattered. The interaction between carriers and lattice vibrations are usually referred to as phonon scattering. Phonons are a quantum of mechanical energy of lattice vibrations. It obeys the Bose-Einstein statistics. Given a crystal with N primitive cells, the phonon wave vector q can have N different values which are all in the region of the 1st Brillouin zone. This is based on the periodic boundary condition of the crystal. Given a primitive cell with n atoms, there are totally 3n different vibration mode branches corresponding to each wave vector q . The 3 branches are for acoustic waves (1 LA and 2 TAs) and 3n-3 branches are for optical waves (LOs and TOs). For a crystal with N primitive cells with each primitive cell having n atoms, there are totally 3nN different vibration modes. The actual vibration of atoms is the superposition of these 3nN modes. See examples for Si, Ge, GaAs. For optical branch, q o , and for acoustic branch, q vs q at long wavelength limit 1 2 and E q = hq . Each mode can be regarded as n particles with energy hq and momentum hq - phonons. During the interaction between carriers and photons, the energy and momentum are conserved h2 (k '2 -k 2 ) = hq and k'-k = q + K * 2m k, k' : initial and final carrier wave vectors q : phonon wave vector K : reciprocal lattice vector +: phonon absorption -: phonon emission
. For a given vibration mode, the vibration energy is quantized E q = n + hq
A phonon scattering process that involves K 0 is called Umklump process. The average number of phonons in a given mode is
n=
Neglecting
1 [exp (h q / kT ) - 1]
1 hq , the average vibration energy for a given mode 2 hq Eq = n hq = [exp(hq / kT ) - 1]
For acoustic phonons of long wavelengths hq << kT , Eq = n hq kT . Longitudinal Acoustic Phonon Scattering It has been shown by quantum mechanics the carriers in bulk semiconductors can only be scattered by phonons of longitudinal modes. This type of vibrations leads to a periodic change in the inter-atomic distance. This resulted in a deformation potential that interacts with carriers
Fig. Illustration of LA phonons and related deformation potential. Longitudinal Optical Phonon Scattering The two atoms in a primitive cell will move in an opposite direction which leads to one type of atoms have high distribution density at one location where the other type have low density. This is a very important scattering mechanism to the polar semiconductors where two types of atoms have different electro negativities. Since the positive and negative charges will be spacially separated and meanwhile oscillating, we effectively have oscillating electric dipole.
Fig. Illustration of LO phonons and the oscillating electric dipole.
This is so called polar optical phonon. The scattering by these phonons are particular strong for polar compound semiconductors, e.g. GaAs, ZnSe. Phonons participating in carrier scatterings are mostly of long wavelengths (small q ) since carriers are mostly located within a few kT around the band edge (small k or small k ) near equilibrium. The calculation of carrier scattering by acoustic phonons with long wavelengths has shown that the scattering is isotropic and the scattering probability is proportional to vibration energy Eq = n hq kT and to the density of final states E 1/ 2 , P kTE 1/ 2 , or
m = E -1/ 2 / kT T -3/ 2 . This leads to the temperature dependence of the mobility T -3 / 2
This mobility increases as the temperature decreases , opposite to that of impurity scattering. For relatively pure materials, the phonon scattering is a dominant mechanism, especially in the high temperature range. The higher the temperature, the more phonon, and thus stronger the scattering. Multiple Scattering Mechanisms Consider the case in which more than one scattering mechanisms are in the same order of magnitude. The total scattering probability should be the summation of that of each scattering process. For scattering by simultaneously ionized impurities and phonons, we shall have 1 1 1 = +
i
p
: effective relaxation time i : impurity scattering relaxation time p : longitudinal phonon scattering relaxation time.
The effective mobility
1 1 1
=
i
+
p
i : mobility when ionized impurity scattering is dominant p : mobility when longitudinal phonon scattering is dominant.
At higher temperatures: i >> p , p , p T -3 / 2 At lower temperatures: i << p , i , i T 3 / 2 Thus, from the measurement of temperature dependence of mobility we can tell which scattering mechanism is dominant.
3.6 Energy Dependence Theory of Conductivity The previous diccussion on conductivities provides a simple picture on describing the carrier transport behavior in a material. Even though we have considered the energy dependence of carrier momentum relaxation time, the treatment was not highly rigorous. In fact carriers acquire kinetic energy in between their two collisions, which indicates that the kinetic energy of a carrier varies and should not be treated as a time independent variable. Strictly speaking, there is no fixed relaxation time associated with carriers since their energies are uncertain. A rigorous treatment would involve the solution of distribution function of carriers under the externally applied electric and/or magnetic field. All the transport properties and parameters can then be derived through the nonequilibrium distribution function. Consider a general situation in which a nonuniformly doped semiconductor has an equilibrium carrier distribution f o ( k , r ) indicating the probability of state k being occupied by a carrier at position r . f o ( k , r ) is symmetric respect to k = 0 in k -space when an electric field is applied, carriers reach to a new nonequilibrium f ( k , r ) which has no symmetry in k -space.
Fig. Illustration of carrier distributions in equilibrium and nonequilibrium. The change in distribution function (k, r ) = f (k, r ) - f o (k, r ) The number of electrons in the element of volume d 3k is 2 dn = f ( k , r ) d 3k 3 ( 2 ) The current density contributed by electrons 2e j= - vf ( k , r ) d 3k 3 ( 2 ) Since j = 0 under equilibrium f o ( k , r ) , and f ( k , r ) = f o ( k , r ) + ( k , r ) , 2e j= - v ( k , r ) d 3k ( 2 )3 Once we can obtain the nonequilibrium distribution function under the electric field, we can easily obtain the conductivity. (1) Approximation of Relaxation Time
Assume f = f ( k , r, t ) , at instant t, the number of electrons occupying k ~ k + dk and r ~ r + dr is 2 dn( k , r, t ) = f ( k , r, t )d 3kd 3r 3 ( 2 ) Three factors can affect the change of the distribution function f a) The movement of carriers in real space t r f b) The movement of carriers in k -space t k f c) Carrier scattering t s The total rate of change f f f f = + + t t r t k t s Now let's examine each one of these terms. f Term t r The increase of number of electrons in the volume dxdydz is equal to that entering at y subtracting that leaving at y+dy 2 f 3 3 2 [ f (k, x, y, z, t ) - f (k, x, y + dy, z, t )]v y dxdzdzd 3kdt d kd rdt = 3 3 ( 2 ) t r , y ( 2 ) = 2 f - v y dydxdzd 3kdt 3 ( 2 ) y
2 f v y d 3rd 3kdt 3 ( 2 ) y Similar result can be obtained for x and z direction, then we shall have f f f f = - v x - v y - vz = - r f v x y z t r A similar analysis in k -space leads to dk F f = - k f = - k f dt h t k where F is external force. Thus F f f = - r f v - k f + t h t s f = 0 , we arrive For steady state t F f r f v + k f = h t s =-
r , k are the gradient operators in real and k -space respectively. This equation establishes the balance between the movement of carriers in real and k -space and the scattering processes. The change in distribution function due to scattering process can be written as V f {w(k ' , k ) f (k ' )[1 - f (k )] - w(k, k ' ) f (k )[1 - f (k ' )]}d 3k ' = 3 t s ( 2 ) w(k ' , k ) : probability of carriers being scattered from k ' to k . w(k , k ' ) : probability of carriers being scattered from k to k ' . The number of carriers being scattered should also be dependent on the occupation probability of the initial state and the empty probability of the final state. Finally, at steady state V F {w(k ' , k ) f (k ' )[1 - f (k )] - w(k, k ' ) f (k )[1 - f (k ' )]}d 3k ' r f v + k f = 3 h ( 2 ) - Boltzmann integro-differential equation. Generally, this equation has no analytical solution. However, under the approximation of relaxation time, we can obtain an analytical solution. The approximation of relaxation time gives f - fo f =- t s Assuming that the change rate of the distribution function is proportional to the deviation of distribution the function from its equilibrium function f - f o . With application of external field: f o f Removal of external field: f f o with a time constant of . For a uniformly doped semiconductor, r f = 0 when at t = 0 the external field is removed F = 0 , then f f f - fo = =- t t s The solution f - f o = ( f - f o )t =0 e - t / : time constant that describes the relaxation process of the distribution function. With such an approximation, the Boltzmann equation reduces to F f - fo r f v + k f = - h From which we can derive a solution for the distribution function at steady state. For uniform materials r f = 0 under electric field F = -eE and = f - f 0 , we obtain
e k ( f o + ) E (*) h Under the condition of weak electric field, << f o , small deviation, e (k ) = k f o E h f o 1 hk , where is the energy carrier (to avoid k , and v = k = And k f o = me h confusion with the electric field E). Then eh f o f ( k ) = e o v E = m k E e st This is the 1 order expression (E), if substituted into the right hand side of (*), we obtain the second order term, and so on.
(k ) =
For nondegenerate semiconductor, f o exp(- / kT ) , f o / = - f o / kT , then
Consider the electric field is along x-direction, then eh f o (k ) = m k x E e
( k ) = -
eE : order of drift velocity vd me kT : order of thermal velocity vt hk x
eE / me fo kT / hk x
eE / me For weak field, vd << vt , the higher order terms contain higher power of kT / hk and x should be neglected. At room temperature, vt ~ 107 cm / s . The equation eh f - f o = = - m kT f o E k x e Suggests that the steady state distribution function does not have spherical symmetry any more in k -space, but it maintains its symmetry with respect to k x - axis.
Given a constant energy, the change of the occupation probability for the k -state along the E field ( k x > 0 ) is opposite to that for the k -state against the E field ( k x < 0 ), but the magnitude of change are the same, < 0 for k x > 0 along E direction > 0 for k x < 0 against E direction
Fig. Illustration of carrier distribution in k -space under electric field. Therefore, the energy distribution of carriers is not changed by the application of external field, but on a constant energy surface more electrons are distributed in those states with k x < 0 which is against the direction of electric field. f o 2e v E into j = - v ( k , r )d 3k , we obtain ( 2 )3 2e 2 f j= - o v( v E)d 3k 3 ( 2 ) which can be written in tensor form Substituting ( k ) = e
ji = ij E j , i = 1,2,3
j =1
3
Where
2e2 f v i v j o d 3k 3 ( 2 ) When i j , the integral function is an odd function of vi and v j , thus ij = 0 (i j ) .
ij = -
For spherical constant energy surface, = xx = yy = zz
=- =- 2e 2 f v x2 o d 3k 3 ( 2 ) 2e 2 f v 2 o d 3k 3 3( 2 )
2 2 Where we have used v 2 = v x2 + v y + v z3 = 3v x . Since n =
2 ( 2 )
3
f o d 3k ,
2 3 ne2 2 ( me v / 2)(f o / )d k ne2 - = = < > 3 me 3 me fod k
me v 2 Use = , d 3k 1/ 2d , we can write the average relaxation time as 2
The above expressions for current density j , conductivity , and average relaxation time < > are valid for both degenerate and nondegenerate semiconductors. The mobility e < > = has the same form, but < > is different from what we had before which me
3/ 2 2 (f o / )d . < >= - 1/ 2 3 f o d
was < >=
1/ 2
f o d f o d
1/ 2
since N ( ) 1 / 2 .
Nondegenerate semiconductors, f o exp(- / kT ) , f o / = - f o / kT , and
0 0
3/ 2
f o d = f o d
1/ 2
3 kT , then 2
Compared to the previous result, the additional weight is factored into the averaging. For different scattering mechanisms, the carrier energy compositions in a current is different. Take the energy dependence for relaxation time = a and use
x - / kT x +1 x - x +1 e d = (kT ) e d = (kT ) ( x + 1) 0 0
< >=
3/ 2
f o d f o d
3/ 2
.
Where the -function has the following properties, (n + 1) = n! , ( x + 1) = x( x ) , and (1 / 2) = , thus ( + 5 / 2) . (5 / 2 ) For acoustic phonon scattering = -1 / 2 , and for ionized impurity scattering, = 3 / 2 . < >= a (kT ) Degenerate Semiconductors
f + 3 / 2 f +3 / 2 +3/ 2 o d = a o d = -a fod + a f 0 0 0 Since f o = 0 for . Thus
3/ 2
0
= - a f o ( + 3 / 2) +1/ 2 d
0
< >= a
Let = / kT , ( EF - Ec ) / kT = F , we can evaluate
+1 / 2 f o d + 3/ 2 . 3 / 2 1 / 2 f o d
< >= a
+ 3/ 2
3/ 2
( kT )
0
[1 + exp( - )]
+1 / 2
F
-1
d
[1 + exp( - )]
1/ 2
-1
F
d
0
using the Fermi integral Fj (F ) = we obtain 1 -1 j [1 + exp( - F )] d ( j + 1) 0
( + 5 / 2) F +1/ 2 ( F ) . (5 / 2) F1/ 2 ( F ) For the case of strongly degenerate semiconductors, E F is deep into the band, only those energy levels near E F ( ~ kT ) are partially filled. When field is applied, the distribution function has noticeable change for these energy levels only since f o / = 0 for levels < >= a ( kT )
well above and below E F . In this case, 3/ 2 (f o / )d can be evaluated in the neighborhood of E F and - f o / ( - F ) .
2 a With = E - Ec and F = EF - Ec , < >= - 3
+3/ 2
(f o / )d
1/ 2 f o d
can be evaluated by
realizing
0
1/ 2
2 2 2 f 2 f f o d = f o d 3 / 2 = 1/ 2 f o - 3 / 2 o d = - 3 / 2 o d , then 30 3 30 30 0
a < >=
+3/ 2
3/ 2
( - F )d
( - F )d
= a F = ( F ) .
The conductivity for strongly degenerate semiconductors ne 2 = ( F ) . me 3.7 Energy Dependence Theory of Hall Effect In this section we treat the distribution function under the application of weak electric field and magnetic field. The force exerted on electrons due to the magnetic field without electric field is d ( hk ) = - ev B dt i.e. the movement of electrons in k-space in perpendicular to both the velocity and the 1 magnetic field. Since the velocity of electrons v = k is perpendicular to the constant h
energy surface, the electrons circles in a constant energy surface about the B-axis ( k z direction). Apparently, the application of magnetic field only will not change the electron distribution function in k-space.
Fig. Illustration of electron motion in k-space. In fact, if we apply F = -ev B to the Boltzmann equation F f - fo k f = - h We obtain e ( k ) = k f o ( v B) = 0 . h f f Since k f o = k = ( hv ) ( v B) , as a matter of fact, any function that depends on energy Q = Q ( ) , we have k Q ( ) ( v B) = 0 . When both electric field and magnetic field are applied, the electric field will disturb the symmetry of the distribution function and the magnetic field will further disturb the distribution by rotating it about B-axis for an angle of . The distribution function will not be symmetric with respect to E-direction, but to a direction which has rotated angle from E-direction. F = - e - ev B The Boltmann equation under the approximation of relaxation time gives e e ( k ) = k f E + k f ( v B) h h With weak electric field, (k ) needs only to include the first order term of E, in the first term on the right f can be substituted by f o . For the second term, the contribution from f o is zero since k f o ( v B) = 0 . f e (k ) = e o v E + k (k ) ( v B) h f We can see the zeroth order (k ) under the magnetic field B, ( 0 ) ( k ) = e o v E for the second term on the right to obtain the first order The force exerted on an electron
e 2 f o f o ( k ) = e vE+ k v E ( v B) h f f f f Where k o v E = k o (v E) + o k (v E) , but k o ( v B) since h h f o is a function of and k Q ( ) ( v B) = 0 . Also k (v E) = k (k E) = E me me and E ( v B) = - v ( E B) . Finally,
( 1)
( 1) ( k ) = e
Comparing to ( k ) = e
f o e v E - E B me
f o v E for only electric field is applied, the axis to which the distribution function maintains its symmetry has to be changed to direction of e E - E B from E direction. When E B , this axis has rotated me E ( e / me ) B e e tan = = B , for weak magnetic field B. E me me
Fig. Orientations of axes about which the distribution function is symmetric.
Due to the rotation change of the distribution function, there will be current in the direction perpendicular to both the E field and B field Hall rotation current. For this part of the current, (1) ( k ) 2 . If = a with > 0 , then carriers with higher energies will rotate more. Obviously, the carrier energy composition in the Hall rotation current is different from that in the drift current where (1) ( k ) . Under the condition of weak effect, only (1) ( k ) is needed to describe the Hall effect.
Hall effect due to single type of carriers
For n-type semiconductor with electron carriers j = - 2 ( 2 )
3
ev
( 1)
( k , r )d 3k and
( 1) ( k ) = e
f o e v E - E B , thus me
f 2 2 e3 2 f o v[v (E B )]d 3k e 2 o v ( v E)d 3k + 3 3 ( 2 ) ( 2 ) me The first term is identical to that obtained when only electric field E is applied. Comparing to the first term, the second term has replaced with 2 , and replaced E with e - E B . The current density can then be written as me j = o E + E B
j= -
For the case of spherical constant energy surface, o = where < >=
ne 2 ne 3 < > and = - 2 < 2 > , me me
be written in tensor form ji = ij E j , i = 1,2,3 , where the tensor
- By Bx o The presence of magnetic field only contributes to the nondiagonal terms in the conducitivity tensor, which can be proven as follows ^ E B = ( E y Bz - E z By )^ + ( E z Bx - E x Bz )^ + ( E x By - E y Bx )k i j o = - Bz B y
3/ 2
f o d f o d
3/ 2
and < 2 >=
2
3/ 2
f o d
3/ 2
f o d
, the relationship between j and E can
Bz o - Bx
v (E B) = v x ( E y Bz - E z By ) + v y ( E z Bx - E x Bz ) + vz ( E x By - E y Bx )
e3 2 f o When we integrate v x [v (E B )]d 3k , the only term survives is me e3 2 f o 2 3 me vx ( E y Bz - Ez By )d k , the other two contain v x v y and vx vz which result an odd integral function producing zero integral. For y and z directions, it is similar. The second term in the current density expression 2 e3 2 f o 2 E B = ^ vx ( Bz E y - By E z )d 3k i ( 2 )3 me
+^ j
2 e3 2 f o 2 v x (- Bz E x ( 2 )3 me
- Bx E z )d 3k
2 e3 2 f o 2 )d 3k v x (By E x - Bx E y ( 2 )3 me ^ i With all diagonal term equal to zero. Taking E = E x ^ , B = Bz k , then j x = o E x and j = -E B . In Hall measurement, j has to be balanced out by E ^ field built up by j ^ +k
y x z y y
change accumulation j y = -E x Bz + o E y = 0 , therefore,
Ey =
jx E x Bz = Bz = 2 jx Bz . o o o o
The Hall coefficient is then
R=
( ne3 / m 2 ) < 2 > 1 = - 2 4 e2 = - rH (for n-type) 2 2 ( n e / me ) < > ne o 3 2 2 ( pe / m ) < > 1 (for p-type) R = 2 4 h2 = rH 2 ( p e / mh ) < > ne
Where the Hall factor rH =
<2 > and the Hall angle < >2 B ( ne 3 / me2 ) < 2 > e <2 > e < > =- z = Bz = Bz = Bz rH 2 ( ne / me ) < > me < > me o
For nondegenerate semiconductors and = a , < >= a (kT )
( 2 + 5 / 2) , then (5 / 2 ) < 2 > ( 2 + 5 / 2) (5 / 2) rH = = < >2 2 ( + 5 / 2) For = 0 (energy independent), rH = 1 For = -1 / 2 (acoustic phonon scattering), rH = 1.18 For = 3 / 2 (ionized impurity scattering), rH = 1.93 . < 2 >= a 2 ( kT ) 2
( + 5 / 2) and (5 / 2 )
For strongly degenerate semiconductors, we have < >= ( F ) , and < 2 >= 2 ( F ) , obviously rH = 1 . In general, for degenerate semiconductors we have
< >= a ( kT ) ( + 5 / 2) F +1/ 2 ( F ) ( 2 + 5 / 2) F2 +1/ 2 ( F ) , and < 2 >= a 2 ( kT ) 2 , then (5 / 2) F1/ 2 ( F ) (5 / 2 ) F1/ 2 ( F ) < 2 > (5 / 2) (2 + 5 / 2) F1/ 2 (F ) F2 +1/ 2 (F ) = . rH = < >2 2 ( + 5 / 2) F2+1/ 2 (F )
For all above cases, the Hall factor rH is always on the oder of 1. Thus 1 / eR and R should give good approximations for carrier concentrations and mobilities.
Hall Effect due to Both Electrons and Holes j = jn + j p
= nE + nE B + pE + pE B = ( n + p )E + ( n + p )( E B)
In y-direction, j y = ( n + p ) E y - ( n + p ) E x Bz = 0 , we obtain
Ey =
n + p + p n + p jx E x Bz = n Bz = jx Bz . n + p n + p n + p ( n + p ) 2
R=
Therefore,
n + p ( n + p ) 2
Where n = -
ne 3 pe3 ne 2 pe 2 2 < n2 > , p = 2 < p > , n = < n > , p = < p > . Using me2 mh me mh e e 2 n = < n > and p = < p > , we can write n = - nerHn n2 , p = perHp p , me mh n = nen , p = pe p , thus n > p
2 prHp p - nrHn n2 . e( pn + n p ) 2 Assume that the relaxation time has the same energy dependence for both electrons and holes, then rHp = rHn = rH ,
R=
2 2 rH p p - nn rH p - nb2 R= = e ( pn + n p ) 2 e ( p + nb) 2
Where b = n / p . For intrinsic semiconductor, n = p = ni , then
2 2 rH p p - nn r 1- b R= = H 2 e ( pn + n p ) eni 1 + b
For intrinsic semiconductors, typically n > p , b > 1 , thus R < 0 ; For n-type semiconductor, p < nb 2 , thus, R < 0 ; For p-type semiconductor, p > nb2 , thus, R > 0 ; but as temperature increases, p-type material will make a transition to intrinsic situation, the sign of Hall coefficient R will change sign from positive to negative.
Fig. Temperature dep...