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

AlcGal-cAs 0.468c2 + 1.042c +1.439 0.29c + 1.87

GaAsl-cPc 0.21c2 + 1.142c + 1.439 0.211c2 + 0.144c + 1.87

IncGal-cAs 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

AlcGa1-cAs 4.07 – 0.896 (Egd – Eg2)

GaAs1-cPc 4.07 - 0.85 (Egd – Eg2)

IncGa1-cAs 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 III-V 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 III-V 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 built-in field that operates on electrons is proportional to the slope of the conduction band edge Ec as

The effective built-in 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

Rgbulknono CD 11

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

Shockley-Read-Hall 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 quasi-Fermi 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 space-charge 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 space-charge 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 steady-state

(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

Voltage-Current 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 space-charge bulk regions and large space-charge junction region of small dimension

Using eqns. (1.14) the continuity equation in steady-state can be expressed as

(1.38a)

(1.38b)

with Ln as the diffusion length of electrons, and Gn and Gp as the generation rates.

Steady-state 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 p-region,

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 space-charge 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)

Capacitance-Voltage Characteristics

The forthcoming calculations employ Poisson’s equation

xdx

d

s

2

2

(1.42)

From the above equation for a p-n junction diode one gets

dxx

wxwqNV

nw

o n

nndnn

1

(1.43)

where Vnn denotes the potential drop across the n-region.

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 p-region. 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 Cut-off Frequency of a Transistor

The cut-off 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 cut-off frequency is expressed as

34*2

2143.2

vw

D

n

pc (1.64)

From eqn. (1.64) it reveals that the alpha cut-off 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 p-n heterojunction

(i)

(ii)

Non-equilibrium energy band diagram for an abrupt p-n heterojunction.

Schematic representation of the emission-recombination model for A p-n heterojunction.

Schematic representation of the tunneling model based on an energy band diagram of anderson.

Energy band diagram for an abrupt p-n heterojucntion under forward bias. The tunneling-recombination process is schematically indicated for both

single-step and multi-step process.