Material parameter modeling

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Transcript of Material parameter modeling
Material Parameter Modeling
Dielectric Constant vs. Composition
1
221
11
122
111
211211
21122121
cc
cca
If ε1 and ε2 are the dielectric constants for semiconductors 1 and 2, the dielectric constant εa for an alloy that has mole fraction c of material 1 is given by
(1.1)
Bandstructure Parameters vs. Composition
(i) Bandgap vs. composition
22
gbbgd EcbcaE
Material Direct gap Indirect gap
AlcGalcAs 0.468c2 + 1.042c +1.439 0.29c + 1.87
GaAslcPc 0.21c2 + 1.142c + 1.439 0.211c2 + 0.144c + 1.87
IncGalcAs 0.33c2 – 1.41c + 1.439 0.27c + 1.87
where ab and bb are experimental values and Eg2 is the bandgap of material 2. Based on the available measurements the variation of bandgap of ALGaAs, GaAsP and InGaAs with composition may be written as
(1.2)
(ii) Electron affinity
22 ggdacc EEK
AlcGa1cAs 4.07 – 0.896 (Egd – Eg2)
GaAs1cPc 4.07  0.85 (Egd – Eg2)
IncGa1cAs 4.07 – 0.769 (Egd – Eg2)
where c2 is the electron affinity for one binary semiconductor and Ka is a proportionality constant associated with the alloy being considered. The electron affinity for a few GaAs based semiconductors is given below.
(1.3)
(iii) Effective masses
1*2
1*1
1* 1
ppp mcmcm for holes
1*2
1*1
1* 1
cdcdcd mcmcm for electrons (direct valley)
1*2
1*1* 1
cicilci mcmcm for electrons (indirect valley)
where m1*, m2* and m* are the effective masses in material 1, material 2 and the alloy respectively. The suffices p, cd and ci are used for brevity to indicate holes, electrons in direct valley and electrons in indirect valley respectively
(1.4a)
(1.4b)
(1.4c)
(iv) Mobility
*2
*2
2*p
p
p
pp
p
pp m
m
m
q
11121* lhm mK
The hole mobility of an alloy may be expressed as
where < p > and < p2 > are relaxation times for holes in the alloy and in material 2 respectively.
In the IIIV semiconductors it is assumed that mobility is controlled primarily by a polar optical phonon scattering process of the form
where Km is a proportionality constant for all materials
(1.5a)
(1.5b)
23
*
*2
11
12
12
2,
p
p
lh
lhpp m
mNfcN
23
*
*2,
cd
cdd m
mKcN
23
*
*2,
ci
cii m
mKcN
Addition of above two eqns. gives an approximation to the mobility
An empirical function fp2(N) where N is the total impurity concentration is included to account for the dependence of mobility on doping level.
m*p2, εh2 and εl2 are known parameters of material 2, and mp, εh and εl are given as functions of compositions.
where with fn2(N) as an empirical function relating electron mobility to doping for material 2.
(1.6)
(1.7a) (1.7b)
11112
122
lhlhn NfK
diddn RRcN 1,
123
*
*
exp1
kT
EE
m
mR gigd
cd
cid
An effective mobility can be approximated as
where Rd, the fraction of electrons in the direct valley, is given by
with Egi as the indirect energy gap for IIIV alloys.
(1.8)
dx
EEd
qE cF
effn
1
Fields Due to Bandgrading
dx
EEd
qE VF
eff p
1
dx
dE
qEE g
effeff pn
1
The effective builtin field that operates on electrons is proportional to the slope of the conduction band edge Ec as
The effective builtin field that operates on holes is proportional to the slope of the valence band edge EV as
The difference in the fields may be represented as
where Eg is the bandgap.
(1.9a)
(1.9b)
(1.10)
Interface Recombination
21
22
21
22
ll
lli
d aa
aaKD
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Rgbulkpopo CD 11
The density of dangling bonds at an abrupt junction is given by
where al1 and al2 are the lattice constants of the two materials and Ki = 4 for [100] interface, for [110] interface and Ki = 4/3 for [111] interface
CR is the capture coefficient
ShockleyReadHall model
(1.12)
(1.11)
(1.13)
Carrier Transport in Heterojunction
Carrier densities at equilibrium:
cpnn FEnqdx
dnqDJ
vppp FEpqdx
dpqDJ
current equations can be written as
where (Jn, n, Dn, n ) and (Jp, p, Dp, p ) are the current densities, carrier densities, diffusion constants and mobilities for electrons and holes respectively. Fc and Fv are the fields due to variation of conduction and valence band edge and the effective density of states in two bands.
(1.14a)
(1.14b)
Fc and Fv may be described as gradients of two potentials Vn and Vp which are defined as
io
co
cn n
N
q
kT
qV ln
io
vo
gcp n
N
q
kT
q
EV ln
where Nc and Nv are the effective densities of states for conduction and valence bands respectively, ψo is some reference potential, and nio is the intrinsic carrier concentration of some reference material.
Fc and Fv can be written as
dx
dVF n
c dx
dVF p
v
(1.15)
(1.16)
(1.17a) (1.17b)
dx
Vd
kT
q
dx
dn
nn
1
nio
VkT
q
n
n
ln
putting E = dψ/dx one gets
From the above equation
Similarly for holes
pio
VkT
q
n
p
ln
pnio
VVkT
q
n
np
2
ln
(1.18)
(1.19)
(1.20)
(1.21)
Addition of equns. (1.19) and (1.20) results
io
v
io
cgpn n
N
q
kT
n
N
q
kT
q
EVV lnln
kTENNnp gvc exp
nnio VkT
qnn exp
ppio VkT
qnp exp
where n and p are the quasiFermi potentials.
Combining eqns. (1.15) and (1.16)
Using eqns. (1.21) and (1.22)
(1.22)
Equation (1.23) is valid at equilibrium condition. When there is a finite current, n and p can be expressed as follows:
(1.23)
(1.24a)
(1.24b)
Density of Excess Carriers
Narrow Base Diode
dx
dn
nq
kTFE c
1
01
p
pvc qD
J
dx
dn
nq
kTFF
kT
qp
dx
dp
(1.25)
On substitution of eqn. (1.25), eqn. (1.14b) becomes
(1.26)
It is convenient to write
T
Gvc V
V
dx
dFF
kT
q(1.27)
io
v
io
cGG n
N
q
kT
n
N
q
kTVV lnln
q
kTVT where and
01
p
p
T
G
qD
J
V
V
dx
d
dx
dn
np
dx
dp
Using eqn. (1.27), eqn. (1.26) becomes
(1.28)
Equation (1.28) differs from the usual equation for excess hole density in the base region of a graded diode in that the eqn. (1.28) has an extra term p d(V′G/VT)/dx. For spacecharge neutrality, (p + Nd) is equal to n
TGdp
pTGd VVN
qD
JVVNp
dx
d expexp
For the boundary condition p = 0 at x = wn,
Assuming a low excess carrier density eqn. above may be expressed as
01
p
p
T
Gd
d qD
J
V
V
dx
d
dx
dN
Np
dx
dp(1.29a)
Equation (1.29a) may be rewritten as
(1.29b)
TGdp
TG
w
x
dp
VVNDq
dxVVNJ
xp
n
exp
exp
(1.30)
Equation (1.30) gives the distribution of the excess hole concentration in terms of bandgrading and doping density.
dp
w
x
dp
NDq
dxNJ
xp
n
1exp2
Tad
iob VV
N
np
For a graded doped diode the excess hole concentration in terms of an arbitrary impurity distribution is given by
The hole concentration at the boundary of the spacecharge region, pb may be expressed in terms of the applied voltage Va. Using Boltzmann’s distribution as
(1.31)
(1.32)
nw
o
TGd
TGTaiopp
dxVVN
VVVVnDqJ
exp
exp1exp2
The hole current is then given as
(1.33)
dxVVN
D
dxVVN
DVVVVnqJ
TG
w
o
a
n
TG
w
o
d
pTaTGio
pn
expexp
1expexp2
dxN
D
dxN
DVVnqJ
pn w
o
a
nw
o
d
pTai 1exp2
The total current density J is thus given as
whereas for junction with nonuniform doping the total current density is given as
(1.34)
Comparing eqns. (1.30) and (1.31), and eqns. (1.34) and (1.35) it can be concluded that behaviour of graded heterojunction device is similar to that of graded doped homojunction device if one puts Nd(x) = Nd exp (V′G / VT).
(1.35)
Finite Recombination
dx
dJqp
dt
dpq p
p
01
22
2
pT
Gd
d L
p
dt
dp
V
V
dx
d
dx
dN
Np
dx
d
dx
pd
(1.36)
In the general case one has to solve the continuity equation with appropriate boundary conditions to find the minority carrier density and hence the minority carrier current.
For the case of graded material the continuity equation for the holes takes the following form
(1.37a)
Substitution of eqn. (1.36) in eqn. (1.29a) gives
0ln22
2
pL
pN
dx
dp
dx
d
dx
pd
xaxbCxbCp 11211 expexpexp
where , C1 and C2 are constants to be obtained from the given boundary conditions.
212211
pLab
This gives in steadystate
(1.37b)
where N = Nd exp(V′G / VT) and Lp is the diffusion length.
Knowing Nd and the nature of bandgrading, eqn. (1.37b) may be solved to obtain the value of p. Assuming d(ln N)/dx to be a constant of magnitude 2a1, the solution of eqn. (1.37b) is
VoltageCurrent Characteristics of a Diode
02
2
n
nn
nn
nn G
n
dx
dnE
dx
dEn
dx
dD
02
2
p
pp
pp
pp G
p
dx
dpE
dx
dEp
dx
dD
ccTcn NdxdNVqdxdxEE
vvTgcp NdxdNVqdxdEqdxdxEE
where
and
Initially the diode is divided into two parts: small spacecharge bulk regions and large spacecharge junction region of small dimension
Using eqns. (1.14) the continuity equation in steadystate can be expressed as
(1.38a)
(1.38b)
with Ln as the diffusion length of electrons, and Gn and Gp as the generation rates.
Steadystate carrier density in the bulk:
01
22
2
nn
n
T L
n
dx
dnE
dx
Edn
Vdx
nd
01
22
2
pn
p
T L
p
dx
dpE
dx
Edp
Vdx
pd
Rearranging eqns. (1.38) one obtains
(1.39a)
(1.39b)
where and and with no and po as the electron and hole mobilities of some reference material.
nonnn EE poppp EE
In eqns. (1.39) E′n and E′p in the third term are replaced by their average values, and dE′n/dx and dE′p/dx in the second term are assumed to be constant for each section.
0
0
2221
1211
pp J
p
AA
AA
xJ
xp
xba
ab
V
ExaA
T2
2
22211 exp
sinhcosh
where
xbaDq
xaA
p2
2
212 exp
sinh
xbV
Eba
a
xaDqA
Tp 2
2
222
2
221 exp
sinh
xba
a
V
EbxaA
T2
2
22222 exp
sinhcosh
For low field E the solutions for p(x) and Jp(x) are expressed as
(1.40a)
With 2b2 = Eo/VT and 212222
pLba
Eo is the average field, and p(0) and Jp(0) are the hole density and current density respectively at the initial point of the section.
pn
p
n lJ
ln
BB
BB
xJ
xn
2221
1211
where lp is the length of the pregion,
xba
xab
V
ExaB
T
3
3
33311 exp
sinhcosh
xbaDq
xaB
n
33
312 exp
sinh
xbbV
Ea
a
xaDqB
Tn
33
23
3
321 exp
sinh
xba
xab
V
ExaB
T
3
3
33322 exp
sinhcosh
(1.40b)
with x′ = lp – x, b3 =  Eo/VT and 212233
nLba
Eo is the average field, and Jn(lp) and n(lp) are the electron current density and electron density at the extreme end of the section.
Low recombination spacecharge region:
dxVDq
JVp T
w
o p
pT
n
expexp
dxVDq
JVn T
w
o n
nT
p
expexp
For a junction region it is assumed that the recombination is small except at the interface. Solving for p and n, one gets
(1.41a)
(1.41b)
CapacitanceVoltage Characteristics
The forthcoming calculations employ Poisson’s equation
xdx
d
s
2
2
(1.42)
From the above equation for a pn junction diode one gets
dxx
wxwqNV
nw
o n
nndnn
1
(1.43)
where Vnn denotes the potential drop across the nregion.
xhx nn exp0 (1.44)
Differentiating eqn. (4.43) one gets
h
whqN
dw
dV n
n
d
n
nn
1exp
0(1.45)
The total charge Q can be represented as follows
pand wNwNqQ (1.46)
From eqns. (1.45) and (1.46) the capacitance for the reverse biased junction is expressed as
p
nn
a
dnnT
xw
N
N
h
whC
1exp
02 (1.47)
The total voltage can be expressed as
ppnn VVV (1.48)
where Vpp is the potential drop across the pregion. Substituting eqn. (1.43) in eqn. (1.48) and using
p
pn
a
d
n
n
n
ndwx
N
N
wh
wh
h
wqNV
2
11exp1
0(1.49)
For convenience two variables Xn and Yn are defined as follows
,
0
d
nn qN
VX
T
nn C
Y02
(1.50)
On substitution of eqns. (1.47) and (1.49), equns. (1.50) result in
p
nn
a
d
n
nnn
wx
N
N
wh
wh
hwX
2
11exp1
p
nn
a
d
n
nnn
wx
N
N
wh
whwY
1exp
(1.51a)
(1.51b)
Similarly for the second case for which εn(x) increases away from the junction i.e. the bandgap is decreasing away from the junction Xn and Yn can be obtained as
n
a
d
n
nnn w
N
N
wh
wh
hwX
2
exp11
1
p
n
a
d
n
nnn
x
N
N
wh
whwY
exp1
(1.52a)
(1.52b)
Alpha Cutoff Frequency of a Transistor
The cutoff frequency has been arbitrarily defined at 3 db or at the frequency where the intrinsic carrier transfer
707.0* (1.53)
The impurity distribution is assumed to be exponential and is expressed as follows
odd xxNN exp0 (1.54)
Let of eqn. (1.27) be a linear function of x which can be represented as
xwVmVV nToGG 0 (1.55)
where moVT/wn is the slope. For convenience let v* be defined as
oo
p mx
wv * (1.56)
From equns. (1.53) to (1.56)
nT
Gd
d w
v
V
V
dx
d
dx
dN
N
*1
(1.57)
which may be combined with eqn. (1.14) to give
npp w
vp
dx
dpqDJ
*
(1.58)
The hole concentration at the emitter and is, therefore,
*
*exp1
v
v
qD
wJp
p
npe (1.59)
Using eqn. (1.58) the continuity equation can be expressed as
02
*
2
2
ppp L
pp
D
j
dx
dp
w
v
dx
pd (1.60)
Solution of eqn. (1.60) with appropriate boundary conditions yields
21*2*
21*2*
*
2sinh
21sinh
2exp
jv
jvwx
w
xvpxp
n
po
(1.61)
where
pp
n j
D
w
2*
(1.62)
The intrinsic current transfer ratio *( = ic / ie) is finally obtained as
21***
*
0*
* 2cosh2exp
jvv
pw
v
dx
dpqD
dx
dpqD
onxo
p
wxp
p
121*2*
21*2** 2sinh22
jvjvv (1.63)
Neglecting recombination (1/p << ) the alpha cutoff frequency is expressed as
34*2
2143.2
vw
D
n
pc (1.64)
From eqn. (1.64) it reveals that the alpha cutoff frequency increases as the slope of bandgrading is increased and the effective base width is decreased.
Equilibrium energy band diagrams before and after the formation of an abrupt pn heterojunction
(i)
(ii)
Nonequilibrium energy band diagram for an abrupt pn heterojunction.
Schematic representation of the emissionrecombination model for A pn heterojunction.
Schematic representation of the tunneling model based on an energy band diagram of anderson.
Energy band diagram for an abrupt pn heterojucntion under forward bias. The tunnelingrecombination process is schematically indicated for both
singlestep and multistep process.