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ECE432
Excess Carriers in Semiconductors
Chapter 2 & Patrick Roblin T.H.E OHIO
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UNIVERSITY 0 The Ohio State University % ' $
Lecture # 4 & Patrick Roblin T.H.E OHIO
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UNIVERSITY 1 The Ohio State University % ' $ Carriers Concentrations in Equilibrium o T ( K) In equilibrium the carrier concentrations verify:
9
EF ;Ei (x) >
>
=
n0 (x) = ni exp
kT
! n0p0 = n2i (T )
E (x);EF >
>
p0 (x) = ni exp i kT
In equilibrium the Fermi level EF is constant through out
the device. & Patrick Roblin 2 T.H.E OHIO
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UNIVERSITY The Ohio State University % ' $ Departure from Equilibrium −+
Equilibrium is perturbed when we inject energy in the system:
light (Chapter 4)
applied voltage (Chapter 5)
mechanical stress (piezzo electric e ect) & Patrick Roblin 3 T.H.E OHIO
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UNIVERSITY The Ohio State University % ' $
Excess Carriers Away from equilibrium the carrier concentrations depart from their
equilibrium values:
9
F (x);E (x) >
>
=
n = n0 + n = ni exp n kT i
! np 6= n2i (T )
Ei (x);Fp (x) >
>
p = n0 + p = ni exp
kT
The Fermi level EF is not constant but we can introduce a spatially
varying quasiFermi levels Fn(x) and Fp(x) for electrons and holes. & Patrick Roblin T.H.E OHIO
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Excess Carriers = n0 +
p = n0 + n 9
>
=
n>
>
p> ! np 6= n2i (T ) What happens to excess carriers generated? excess electrons and holes can recombine
they can di use if there is a gradient of excess carrier & Patrick Roblin T.H.E OHIO
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4.1 Optical Absorption
collision with lattice
Ec absorption
h ω > Eg recombination
Ev The light intensity is: intensity (Power) = I (W ) = h! nphoton = photon energy # of photons
sec
The absorption of light is proportional to the intensity:
dI (x) = I
;
)
I = I e; x & dx Patrick Roblin 0 T.H.E OHIO
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Frequency Dependence of Absorption Coe cient α InSb GaAs
GaP
Si
CdSe
CdS SiC Ge ZnS Eg (eV)
0 Eg E =h ω 1 2 3 4 λ (μ m)
5 Infrared 1 0.5 Visible 0.35 Ultraviolet Infrared: GaAs, Si, Ge and InSb
Visible: GaP, Cds, SiC (transparent) & Ultraviolet: ZnS Patrick Roblin T.H.E OHIO
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4.2 Luminescense
9
>
>
>
>
>
= 8
> Photon: ( uorescense, phosphorescense)
Photo>
>
>
<
Cathodo > Luminescence ) > Electron: (CRT: cathode ray tube)
>
>
>
>
>
: LED (light emitting diode)
>
Electro & Patrick Roblin T.H.E OHIO
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UNIVERSITY 8 The Ohio State University % ' $ Fluorescence and Phosphorescence
collision with lattice collision with lattice
Ec absorption
h ω > Eg absorption h ω = Eg Ec
trap h ω > Eg Et
h ω = E t − Ev Ev & Ev Fluorescence Phosphorescence uorescence: excitation ! light emitted (direct bandgap only)
phosphorescence: excitation ! delay ! light emitted Patrick Roblin 9 T.H.E OHIO
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UNIVERSITY The Ohio State University % ' & Patrick Roblin $ Alternating Current Thin Film Electroluminescence T.H.E OHIO
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UNIVERSITY 10 The Ohio State University % ' $ 4.3 Carrier Lifetime & Photoconductivity low high low Ω meter high Ω meter Light on
+ & Light o
+ Excess carriers generated Recombination of excess carriers + + T.H.E conductivity (t) increases conductivity (t) decreases gradually OHIO
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Direct Recombination of Electrons & Holes Rate of change in electron concentration:
dn(t)
= generation rate ;recombination rate
{z
}

dt
optical+thermal
= gopt + r {z} ; r n(t)p(t)
n2
i
n0 p0 In equilibrium: gopt = 0 and np = n0p0 = n2.
i
In equilibrium the total rate is zero: dn(t) = 0
dt & Patrick Roblin T.H.E OHIO
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UNIVERSITY 12 The Ohio State University % ' $ Direct Recombination of Electrons & Holes Introduction excess carriers n and p:
n(t) = n0 + n(t)
p(t) = p0 + p(t)
n= p
(created by optical generation)
in the rate equation we have:
d n(t) = g + n2 ; dt n(t)p(t)
= gopt + r n2 ; r (n0 + n)(p0 + p)
i
= gopt ; r (n0 + p0) n ;  r{zn2 (for n small)
}
opt ri r !0 For low injection level (
d n(t)
n
= gopt ; & dt Patrick Roblin n n small):
with 1
n=
r (n0 + p0 ) T.H.E OHIO
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Steady State and Relaxation Toward Equilibrium For low injection level ( n small):
d n(t) = g ; n
with dt opt n= n 1
r (n0 + p0 ) n
In steady state ( d dt(t) = 0) under continuous illumination (gopt 6= 0):
d n(t)
n
= 0 = gopt ;
) n = ngopt
dt n After the illumination is switched o (gopt(t 0) = 0), the
semiconductor relaxes toward equilibrium:
n
d n(t)
=;
) n(t) = n(0)e;t= n & dt Patrick Roblin n T.H.E OHIO
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Indirect Recombination & Trapping Impureties which introduce deep energy levels (Et ) can act as recombination
centers:
EC EC EC EF
Et EF
Et EF
Et EV EV EV step 1 step 2 EHP recombined Example: Gold (Au) impureties in silicon & The capture time for electron and holes is di erent Patrick Roblin 6
n= p T.H.E OHIO
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UNIVERSITY 15 The Ohio State University % ' $ Rate Equations for Indirect Recombination
EC step 2 τn step 1 τp EF
Et EV For low injection level ( n and
d n(t)
=g ;n & dt
d p(t)
dt Patrick Roblin opt = gopt ; p small): n p p with n 6= p T.H.E OHIO
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UNIVERSITY 16 The Ohio State University % ' $ Photoconductivity low high low Ω meter high Ω meter Light on Light o + + n = ngopt and p = pgopt & + = q n(n0 + n) + q p (p0 + p)
=q n n+q p p Patrick Roblin n(t) = n e;t= n and p(t) = p e;t= p
(t) = q + n n0 + n(t)] + q p p0 + p(t)] T.H.E OHIO
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QuasiFermi Levels Ec
Fn
Ei
Fp
Ev
Away from equilibrium, the Fermi level EF is not constant but we can introduce a
spatially varying quasiFermi levels Fn (x) and Fp (x) for electrons and holes. & n = n0 + n = ni exp
p = n0 + p = ni exp Patrick Roblin Fn (x);Ei (x)
kT
Ei (x);Fp (x)
kT 9
= ! np 6= n2 (T )
i
T.H.E OHIO
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Excess Carrier & Di usion
δn Diffusion Diffusion gopt τ
x gopt τ /2 L= D τ −d/2 d d/2 x d & Patrick Roblin T.H.E OHIO
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Lecture # 5 & Patrick Roblin T.H.E OHIO
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UNIVERSITY 20 The Ohio State University % ' $ Di usion of Carriers Concentration l & 0 t 2t 3t time x T.H.E OHIO Mean free time between collision: t Patrick Roblin 4t Mean free path between collision:S `
T
ATE
UNIVERSITY 21 The Ohio State University % ' $
Di usion Applet http://www.geocities.com/SiliconValley/Campus/2811/browni/Difus.html & Patrick Roblin T.H.E OHIO
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UNIVERSITY 22 The Ohio State University % ' $ Di usion Equation Flux for di usion process:
n=
!;
= n!v! ; n v
= 1 n(x) ` ; 1 n(x + `) `
2
t2
t
`2 n(x + `) ; n(x)
=;
2t
`
`2 dn(x)
=;
2t dx
Electron & Hole di usion uxes: & n
p = ;Dn dn(x)
dx
= ;Dp dp(x)
dx Patrick Roblin and
and `2
Dn = ; 2tn
n
2
`
Dp = ; 2tp
p (electron di usion constant)
(hole di usion constant) OHIO
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UNIVERSITY 23 The Ohio State University % ' $ Di usion Fluxes and Currents Di usion Fluxes:
dn(x)
n = ;Dn
dx
= ;D dp(x)
p p dx Di usion currents:
Jn = ;q n = Dn dn(x)
dx
J = q = ;D dp(x)
p p p dx What are the directions of the electron and hole uxes & currents ?
n(x) & Patrick Roblin p(x)
φn (diff) φp (diff) Jn (diff) Jp (diff)
x 24 T.H.E x OHIO
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UNIVERSITY The Ohio State University % ' $ Di usion Fluxes and Currents Di usion Fluxes:
= ;Dn dn(x)
n
dx
= ;D dp(x)
p p dx Di usion currents:
Jn = ;q n = Dn dn(x)
dx
J = q = ;D dp(x)
p p p dx n(x) & Patrick Roblin p(x)
φn (diff) φp (diff) Jn (diff) Jp (diff)
x 25 T.H.E x OHIO
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UNIVERSITY The Ohio State University % ' $
Di usion and Drift of Carriers Total electron and hole current:
dn(x)
Jn = qn n E + Dn
dx
dp(x)
J = qp E ; D
p p  {z }
drift p dx  {z }
diffusion These equations will be our bread and butter. & Patrick Roblin T.H.E OHIO
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UNIVERSITY 26 The Ohio State University % ' $ Drift, Di usion and Ballistic Transport Consider the simple pastoral scene of a lake with a waterfall on one side and a small creek on the
other. There is clearly a continuous ow of water from the waterfall, through the lake and into
the creek. However if the lake is very wide, the water drift might not even be perceptible to a
sherman on its bank shing for trouts. However a y sherman shing in the creek will see his
dry y quickly drift away and will need to recast his line often. If a red dye or some kind of
liquid trout food is poured in the lake at one spot we can expect it to slowly di use and spread
throughout the lake. A drift di usion model applies therefore well to the lake area. On the .other
T
H.E
OHIO
hand, no dye is expected to be able to di use up the waterfall into its upper reservoir. Clearly
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the waterfall is operating under a ballistic mode.
UNIVERSITY & Patrick Roblin 27 The Ohio State University % ' $
QuasiFermi Level Derivation Nonequilibrium thermodynamic: Prigogine (Nobel prize) demonstrated that the
entropy production rate is minimized:
J = qn E ! J = n dFn(x)
n n n n dx
p(
Jp = qp pE ! Jp = p p dFdxx)
where n(x) and p(x) are given by (assuming local equilibrium):
!
Fn(x) ; Ec(x)
n(x) = Nc exp
kT
!
;
p(x) = Nv exp Ev (x)kT Fp(x)
expressed in terms of the quasiFermi levels Fn(x) and Fp (x) and with
Ec(x) = Ec(0) ; qV (x) and Ev (x) = Ev (0) ; qV (x) & Patrick Roblin T.H.E OHIO
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UNIVERSITY 28 The Ohio State University % ' Flash Back on Potentiel Energy and Electrostatic Potential
Ec (x) = Ec(0) ; qV (x) and ENERGY Ec duct Vale n Ba nce Patrick Roblin Vibrations (heat) and idde Vibrations (heat) = Ev (0) ; qV (x) ion B Forb Ev Ev (x) B A
Con & $ E − qV(x) nd c Band E − qV(x)
v x x A B x
position T.H.E OHIO
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UNIVERSITY 29 The Ohio State University % ' $ DriftDi usion Equation Derived from QuasiFermi Level From the carrier statistics we get:
!
n(x) = N exp Fn(x) ; Ec(x) ! (
Fn(x) = Ec(x) + kT Ln nNx)
kT
c
Using Ec (x) = Ec (0) ; qV (x) the quasi Fermi level (electrochemical potential) is
!
n(x)
+ kT Ln
F n (x) =
Ec(0) ; qV (x)
{z
}

N

{z c }
electrostatic potential energy
chemical potential
The total electron current is given by
n(
Jn = n(x) n dFdxx)
"
!#
dV (x) + n(x)kT d Ln n(x)
= ;qn(x) n
n
dx
dx
Nc
= qn(x) nE + kT n dn(x)
dx
= qn nE + qDn dn using the Einstein relation: Dn = kT
OHIO
dx
q
c & Patrick Roblin ) T.H.E S ATE
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UNIVERSITY 30 The Ohio State University % ' Electron and Hole Currents & Equilibrium Jn = nn Jp = pp In equilibrium:
dFn
dx
dFp =0 dFn (x)
dx
dFp (x)
dx )
) dn(x) E + Dn dx
= qp pE ; Dp dp(x)
dx
= qn Jn n =0 =0
Jp = 0
the quasiFermi levels are constant.
When electrons and holes are also in equilibrium together
EF = Fn = Fp .
dx & $ Patrick Roblin T.H.E OHIO
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UNIVERSITY 31 The Ohio State University % ' $
Di usion Coe cients & Einstein Relation For Instrinsic Semiconductors at room temperature:
Dp
Dn
Dp
Dn
p
n
n
p
(cm2 /sec) (cm2 /V/sec)
(cm2/sec) (cm2 /V/sec)
Si
35
1350
0.0259
12.5
480
0.026
Ge
100
3900
0.0256
50
1900
0.0263
GaAs
220
8500
0.0259
10
400
0.025 & Patrick Roblin T.H.E OHIO
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UNIVERSITY 32 The Ohio State University % ' $ 4.4.3 The Continuity Equation
φ (x) φ (x+ Δx) p(x)
x x+ Δx Δx
Conservation of particles using p(x) = p0 (x) + p(x t):
@p = @ p = ; p + p (x) ; (x + x) = ; p ; @ p(x) @t @t
x
p
p
Continuity Equations for electrons and holes:
@ p = ; p ; 1 @Jp(x) using J = q
p
p
@t
q @x
p
@ n = ; n + 1 @Jn(x) using J = ;q
n
@t
q @x & Patrick Roblin n @x (drift+di usion)
n (drift+di usion) OHIO
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UNIVERSITY 33 The Ohio State University % ' $ Di usion Equation when Electric Field is Neglegible in Uniform Material When the builtin electric eld is negligible the current is carried by di usion only:
(x
(x
Jp = ;Dp @[email protected] t) =;Dp @ [email protected] t) for uniform material: p(x t) = p0 + p(x t)
J = D @n(x) = D @ n(x) for uniform material: n(x t) = n + n(x t)
n
@x
@x
Substituting the di usion currents in the continuity equations:
@ p = ; p ; 1 @Jp(x)
@t
q @x
p
@ n = ; n + 1 @Jn(x)
@t
q @x
n n 0 n we obtain the Di usion Equations:
@ p = D @ 2 p(x) ; p & p
@t
@x2
p
@ n = D @ 2 n(x) ; n
n
@t
@x2
n Patrick Roblin T.H.E OHIO
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UNIVERSITY 34 The Ohio State University % ' $
4.4.4 Steady State Carrier Di usion Length Assumptions:
current is carried strictly by di usion (assumes builtin electric eld is
negligible):
doping is uniform
steady state @p
@t = @n = 0
@t The Di usion Equations reduce to:
@ p = 0 = D @ 2 p(x) ; p ) @ 2 p(x) = p = p with L = qD
p
p
pp
@t
@x2
@x2
Dp p L2
p
p
@ n = 0 = D @ 2 n(x) ; n ) @ 2 n(x) = n = n with L = qD
nn
n
n
@t
@x2
@x2
Dn n L2
n
n
q
p
where Lp = Dp p and Ln = Dn n are the di usion lengths. & Patrick Roblin T.H.E OHIO
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UNIVERSITY 35 The Ohio State University % ' $ Di usion of Injected Carrier
p(x)
diffusion Δp Solving the 2nd order equation
@ 2 p(x) = p p0 @x2
L2
p
we obtain the solution: recombination Qp x 0
Lp p(x) = C1ex=Lp + C2e;x=Lp with 8
< p(0) = p
: p(1) = 0 ) p(x) = pe;x=Lp The di usion current at position x is: Jp(x) = qDp d p(x) = qDp p(x) Lp dx
Lp
The current at the interface is then: Jp (0) = qDp p = Qp = total excess hole charge
OHIO
Lp
recombination time
p & Patrick Roblin T.H.E S ATE
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UNIVERSITY 36 The Ohio State University % ' $ Example δ n =δ p Diffusion Diffusion gopt τ
x gopt τ /2
L= D τ Recombination
−d/2 d d/2 x d See derivation in 2001 midterm # 1 posted in class webpage.
Assumption: negligible electric eld which is only rigorously valid if Ln = Lp = L
) n(x) = p(x) & T.H.E What is the total current? Patrick Roblin OHIO
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UNIVERSITY 37 The Ohio State University % ' $ HaynesShockley Experiment First performed by HaynesShockley Experiment in 1951 at Bell Labs
Permits to measure the minority carrier mobility: for a ntype material
v
L
p = Ed = td E with L the sample length and td the drift time. Permits to measure the minority carrier di usion constant Dp by measuring the
carrier spread
δ p(x,t)
drift time
t d/4 + − Light pulse
at t=0 t d/2 3t /4
d td n−type &
+− Patrick Roblin x
L 0 σ = 2 Dp t L T.H.E OHIO
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UNIVERSITY 38 The Ohio State University x % ' $
Theory for HaynesShockley Experiment Starting from the Di usion Equations (no drift) and assuming for simplicity that
p is larger than td (not necessarily veri ed since d is typically on the order of s
and td ms):
@ p = D @ 2 p(x) ; p ) @ p = D @ 2 p(x)
p
p
@t
@x2
@t
@x2
p
admits for solution the following timedependent Gaussian distribution:
p(x t) = q P e;x2 =(4Dpt)
2 Dpt
with P the number of holes per area created at t = 0. q The width the pulse 2Dp t increases with time & In the presence of an electric eld the pulse will also drift Patrick Roblin T.H.E OHIO
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UNIVERSITY 39 The Ohio State University % ' $
Java Applet for HaynesShockley Experiment http://jas.eng.buffalo.edu/education/semicon/diffusion/diffusion.html & Patrick Roblin T.H.E OHIO
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UNIVERSITY 40 The Ohio State University % ' $ Ballistic Transport
Energy
Ballistic Electrons
Ec
a)
E
xo N P v Position Normalized electron
distribution at x o
f(v )
x
1 Ballistic peak b) & Patrick Roblin vx
1 0 41 1 Normalized
velocity T.H.E OHIO
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UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode
Energy
Position
OFF EF1
EC1 EF2
E C2 a) ON EF1
EC1 qV
EF2
E b) & Patrick Roblin EF1
EC1 C2 OFF qV c) EF2
E C2 T.H.E OHIO
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UNIVERSITY 42 The Ohio State University % ' $ Resonant Tunneling Diode: Potential
Conduction band edge
1.5 1 Energy (eV) 0.5 0
VD=0 V
−0.5 & −1 VD=0.8 V
T.H.E −1.5 Patrick Roblin 0 20 40 60 80
100
120
Position (monolayer) 43 140 160 180 200 OHIO
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UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: Transmission Coe cient
Transmission coefficient for various biases 0 10 −2 VD=0.8 V 10 VD=0 V
Transmission coefficient −4 10 −6 VD=0.8 V 10 −8 10 −10 VD=0 V 10 & Patrick Roblin T.H.E 0 0.1 0.2 0.3
Energy (eV) 44 0.4 0.5 0.6 OHIO
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UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: IV Characteristic
9 3.5 I−V characteristic x 10 300 K
Maximum charge in well
77 K
3 Current density (A/m2 2.5 2 1.5 1 0.5 & 0
T.H.E −0.5 Patrick Roblin 0 0.1 0.2 0.3 0.4
0.5
Voltage (V) 45 0.6 0.7 0.8 0.9 OHIO
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UNIVERSITY The Ohio State University % ' $ Resonant Tunneling Diode: Charge Distribution
5 16 Charge distribution x 10 14 3 Charge density (C/m ) 12 VD=0.8 V 10 8 6 VD=0 V 4 & Patrick Roblin 2
T.H.E 0 0 20 40 60 80
100
120
Position (monolayer) 46 140 160 180 200 OHIO
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UNIVERSITY The Ohio State University % ' $ Inte Material 2 r aye el
rfac Interface Material 1 Interface Roughness Scattering Z Y NIR NIR X
(A) Side view & Patrick Roblin Energy E c (x) VB E c (x) NIR
(B) 47 X=na/2
For <100> T.H.E OHIO
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UNIVERSITY The Ohio State University % ' $
Scattering Assisted Tunneling
Energy
E0x and E1x Interface
roughness E1x(max)=E0 Incident
wave Transmitted
incident wave E0x Scattering assisted
resonant tunneling
Continuum of
scattered waves Quasi boundstate & Patrick Roblin Position
T.H.E OHIO
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UNIVERSITY 48 The Ohio State University % 209 6.10 Results for resonant tunneling structures Fig. 6.4. Equilibrium (no bias) transmission coefﬁcient TF ( E 0x , E 0⊥ = 0) versus E 0x in the
˚
presence of polar scattering for 50/50/50 A diode at the lattice temperature of 4.2 K (dashed line),
100 K (dasheddotted line), and 300 K (dotted line). Also shown is the transmission coefﬁcient in
the absence of polar scattering (full line). (P. Roblin and W. R. Liou, Physics Review B, Vol. 47, No.
4, Pt II, pp. 2146–2161, January 15 1993. Copyright 1993 by the American Physical Society.) shown in Figures 6.5(a) and (b) is seen to induce both a decrease in the peak current
and an increase in the valley current. Polar scattering therefore contributes to the
reduction of the peaktovalley current ratio. At 4.2 K the phonon emission peak of
the transmission coefﬁcient has introduced a secondary peak in the I –V characteristic
at around VD = 0.6 V. At higher temperature (here 100 K) the variation of the
Fermi–Dirac occupation is more gradual around the Fermi energy, and a phonon peak
is usually not resolved because its small contribution is smoothed out in the current
integration. Note that the detection of a phonon peak in the I –V characteristic is
facilitated when plotting higher order derivatives of the I –V characteristic.
The reader is referred to [7] for results on acoustic phonon scattering which usually
induces a weak scatteringassisted tunneling component to the diode current.
The next phononscattering process considered is – X intervalley scattering induced by LOX phonons. The transmission coefﬁcient shown in Figure 6.6(a) is seen
to be subjected to a selfenergy shift. 5% of the current remains carried by the valley.
Intervalley scattering is seen to leave the valley current unchanged but does effectively
increase the classical diode leakage current at high voltages. 210 Scatteringassisted tunneling Fig. 6.5. I –V characteristic in the presence (dashed line) and absence (full line) of polar scattering
˚
˚
calculated for (a) a 50/50/50 A diode at 4.2 K and (b) a 34/34/34 A diode at 100 K. (P. Roblin and
W. R. Liou, Physics Review B, Vol. 47, No. 4, Pt II, pp. 2146–2161, January 15 1993. Copyright
1993 by the American Physical Society.) Next we consider interface roughness scattering. We show in Figure 6.7 the
equilibrium (no bias applied) transmission coefﬁcient TF ( E 0x , E 0⊥ ) plotted versus ' $
IV Characteristics in the Presence of Scattering
8 8 x 10 7 Current density (Amps/m/m) 6 5 4 3 2 1 & Patrick Roblin 0 0 0.1 0.2 0.3 0.4
0.5
Applied voltage (Volts) 0.6 0.7 0.8 0.9 T.H.E OHIO
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UNIVERSITY 50 The Ohio State University % ' $ InterBand Resonant Tunneling Diode
EC
e2DEG
p+ InAlAs
EF
EF
n+ InAlAs InGaAs InGaAs h2DHG
InAlAs EV EF GaSb AlSb n+InAs AlSb EF
EC
EV a) b) n+InAs EC
Sb spike
doped
EC
EV & Patrick Roblin c) B spike
doped
EV T.H.E Si n+ Si
(or Si0.5Ge 0.5 ) Si p+ OHIO
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UNIVERSITY 51 The Ohio State University % ...
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 Fall '08
 LU
 Electron, Ohio State University, Columbus, Ohio

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