1. Introduction

Most interpretations of optical measurements on

compound semiconductors such as GaSb require physical models and associated input parameters that

describe how carrier densities vary with dopant concentrations and measured Fermi energies. In this paper, we

report on a method that gives closed form analytic

expressions for the carrier densities in the conduction

sub-bands for GaSb at room temperature. The method

is based on an iterative and self-consistent solution of

the charge neutrality equation with full Fermi-Dirac

statistics for the carriers at finite temperature and on the

use of statistical analyses to give analytic expressions

that represent the calculated data sets.

The method reported here is related to earlier work

on n-type GaAs presented in reference [1]. Reference

[1] gives the results predicted by an effective two-band

model, one equivalent conduction band and one equivalent valence band at Γ, that includes the densities of

states modifications due to high concentrations of

dopants and due to many-body effects associated with

carrier-carrier interactions. The method given below for

GaSb is a four-band model. But, because of computational limitations, it does not include the densities of

states modifications due to high concentrations of

dopants and due to many-body effects.

2. Theory

The electron n and hole concentrations h in units of

cm–3 at thermal equilibrium are given, respectively, by

(1)

where f0(E) = {1 + exp[(E – EF)/kBT]}–1 is the FermiDirac distribution function, EF is the Fermi energy in

eV, ρc(E) and ρv(E) are, respectively, the electron denVolume 108, Number 3, May-June 2003

Journal of Research of the National Institute of Standards and Technology

193

[J. Res. Natl. Inst. Stand. Technol. 108, 193-197 (2003)]

Dependence of Electron Density on Fermi

Energy in N-Type Gallium Antimonide 

Volume 108 Number 3 May-June 2003

Herbert S. Bennett and Howard

Hung

National Institute of Standards

and Technology,

Gaithersburg, MD 20899-8120

USA

herbert.bennett@nist.gov

howard.hung@nist.gov

The majority electron density as a function

of the Fermi energy is calculated in zinc

blende, n-type GaSb for donor densities

between 1016 cm–3 and 1019 cm–3. These

calculations solve the charge neutrality

equation self-consistently for a four-band

model (three conduction sub-bands at Γ, L,

and X and one equivalent valence band at

Γ ) of GaSb. Our calculations assume parabolic densities of states and thus do not

treat the density-of-states modifications

due to high concentrations of dopants,

many body effects, and non-parabolicity of

the bands. Even with these assumptions,

the results are important for interpreting

optical measurements such as Raman

measurements that are proposed as a nondestructive method for wafer acceptance

tests.

Key words: band structure; dopants; electron density; Fermi energy; gallium antinomide; Raman measurements.

Accepted: April 11, 2003

Available online: http://www.nist.gov/jres

0c 0 v n fE EE h fE EE ( ) ( )d and [1 ( )] ( )d , ρ ρ

+∞ +∞

−∞ −∞

= =− ∫ ∫
sity of states for the conduction band and the hole density of states for the valence band, kB is the Boltzmann

constant, and T is the temperature in kelvins. The calculations incorporate the Thomas-Fermi expression for

the screening radius,

(2)

and the charge neutrality condition

NI = n – h, (3)

to compute self-consistently the Fermi energy EF and

the screening radius rs for given values of the ionized

dopant concentration NI and temperature T. The static

dielectric constant is ε and the permittivity of free space

is ε0. The ionized dopant concentration is positive for ntype material (donor ions) and negative for p-type

material (acceptor ions). The results reported here are

for uncompensated n-type material. The results for the

screening radius rs are not reported here because they

are not needed to extract carrier concentrations from

Raman measurements.

In this paper, we use the four-band model that has

three conduction sub-bands centered at the Γ, L, and X

symmetry points in the Brillouin zone and one equivalent valence band centered at the Γ symmetry point. We

do not include the detailed nonparabolicity of the GaSb

energy bands at Γ. Unlike GaAs, the GaSb conduction

Γ, L, and X sub-band masses and energy spacings are

such that for donor densities of technological interest,

the conduction sub-band at L is the one that is most

populated. The non-parabolicity of the conduction Γ

sub-band in GaAs is discussed in Ref. [2]. If we were

to use the Kane three level k · p model [2], which does

not include the conduction sub-bands at L and X, we

would be able to include the non-parabolicity of the

conduction Γ sub-band. However, because the conduction Γ sub-band band in GaSb is not the dominant band

for determining the Fermi energy, its non-parabolicity

correction may not have a significant effect on the

results given below and may lie within the uncertainties

associated with the band masses quoted in the literature

for GaSb.

The heavy hole mass mhh and light hole mass mlh for

the two degenerate sub-bands at the top of the valence

band are combined to give an effective mass

mvΓ = (mhh

3/2 + mlh3/2)

2/3, (4)

for the valence topmost sub-band. The values of these

parameters are given in Table 1.

The zero of energy is at the bottom of the conduction

Γ sub-band. The bottoms of the conduction L and X

sub-bands are, respectively, at EcL and EcX. The top of

the degenerate valence Γ sub-band is at –EG, where EG

is the bandgap of GaSb. The split-off valence sub-band

at Γ due to spin-orbit coupling and the non-parabolicity factor of the conduction Γ sub-band are neglected.

The probabilities for typical carriers in equilibrium to

occupy appreciably these states are low. This means

that the Fermi energies should be sufficiently above the

valence sub-band maximum at Γ. Placing exact limits

on the Fermi energies for which the four-band model is

valid would be tenuous, because knowledge of how the

various sub-bands move relative to one another due to

the dopant concentrations considered here and due to

many body effects is not adequate.

The general expression [3] for the temperature

dependence of conduction sub-band minima relative to

the top of the valence band at Γ is

Ei = Ei0 – [AiT2/(T + Bi)] (5)

in units of eV, where i = Γ, L, or X. The values for the

coefficients Ei0, Ai, and Bi are listed in Table 2.

The general expression for the parabolic densities of

states for electrons and holes per band extrema and per

spin direction is given by

(6)

where Ne is the number of equivalent ellipsoids in the

first Brillouin zone, the volume of the unit cell is

V = aL

3

, aL is the lattice constant, m* is one of the effective masses listed in Table 1 for the appropriate band

extrema, and m0 is the free electron mass. Because eight

permutations of the wave vector in the (111) direction

exist, there are eight L sub-band ellipsoids with centers

located near the boundary of the first Brillouin zone.

Also, because six permutations of the wave vector in

the (100) direction exist, there are six X sub-band ellipsoids with centers located near the boundary of the first

Brillouin zone. Since about half of each ellipsoid is in

the neighboring zone, the number of equivalent subbands NcL for the L sub-band is four and the number of

equivalent sub-bands NcX for the X sub-band is three.

In terms of a four-band model for room temperature

n-type GaSb, the total density of states ρc(E) for the

majority carrier electrons in n-type GaSb then becomes

Volume 108, Number 3, May-June 2003

Journal of Research of the National Institute of Standards and Technology

194

2

2 0

s cv

0

4 e d()[ ( ) ( )]d , d

f E r E EE

E

π

ρ ρ

εε

+∞

−∞

=− − ∫

e

3 2 3/2

0

4 () , (8 )( / 2 * )N VE E

m m

π

ρ

π = =
ρc(E) = ρcΓ(E) + ρcL(E) + ρcX(E), (7)

where ρcΓ(E), ρcL(E), and ρcX(E) are the sub-band densities of states for the conduction Γ, L, and X sub-bands

with effective masses of mcΓ, mcL, and mcX, respectively.

The density of states for the minority carrier holes is

ρv(E) = ρvΓ(E) (8)

with an effective mass of mvΓ.

3. Results

Tables 1 and 2 contain the input parameters for the

calculations of the Fermi energy as a function of the

dopant donor density. We solve self-consistently, by

means of an iterative procedure, Eq. (3) with Eqs. (6),

(7) and (8). The independent variable is the temperature

T. The Fermi energy is varied for a given temperature

until Eq. (3) is satisfied. Figure 1 presents the calculated data graphically for 28 values of donor densities

between 1016 cm–3 and 1019 cm–3. Figure 2 gives the

electron densities in the conduction sub-bands at Γ and

L and the total electron density as functions of the

Fermi energy. Figure 2 does not show the electron density in the conduction sub-band at X, because it is less

than 10–3 times the total electron density. Because

mcΓ << mcL and EcL is much closer to EcΓ than it is to EcX,

the electron density in the conduction L sub-band

exceeds the electron density in the conduction Γ subband at room temperature. The solid curve in Fig. 2 is

the same curve as given in Fig. 1. Figure 2 shows that

the majority of electrons is in the conduction L subband and that the density of electrons in the L sub-band

approaches the total density of electrons as the donor

density approaches 1019 cm–3. Hence, even though

GaSb is intrinsically a direct semiconductor, the results

from Fig. 2 suggest that electrons for n-type GaSb in

Volume 108, Number 3, May-June 2003

Journal of Research of the National Institute of Standards and Technology

195

Table 1. Input parameters for intrinsic zinc blende GaSb at 300 K. The energies of the extrema of the conduction and valence sub-bands are referenced to the bottom of the conduction sub-band at the Γ symmetry point in the Brillouin zone of the reciprocal lattice space. The mass of the

free electron is m0. These data are from Ref. [3]

Parameter Symbol Value Units

Lattice constant aL 6.09593 × 10–8 cm

Dielectric constant in vacuum ε 8.854 × 10–12 F/m

Static dielectric constant ε0 15.7

Bandgap EG = |EvΓ| 0.726 eV

Bottom of the conduction L sub-band EcL 0.084 eV

Bottom of the conduction X sub-band EvX 0.31 eV

Top of the degenerate valence Γ sub-band –EvΓ –0.726 eV

Spin-orbit splitting Eso 0.80 eV

Top of the split-off (spin-orbit splitting) valence Γ sub-band –EsoΓ = – EvΓ – Eso –1.526 eV

Effective mass of conduction Γ sub-band mcΓ 0.041 m0

Transverse L sub-band mass mtL 0.11 m0

Longitudinal L sub-band mass mlL 0.95 m0

Effective mass of conduction L sub-band mcL = (mlL mtL

2

)

1/3 0.226 m0

Transverse X sub-band mass mtX 0.22 m0

Longitudinal X sub-band mass mlX 1.51 m0

Effective mass of conduction X sub-band mcX = (mlX mtX

2

)

1/3 0.418 m0

Light hole mass of degenerate valence Γ sub-band mlh 0.05 m0

Heavy hole mass of degenerate valence Γ sub-band mhh 0.4 m0

Effective mass of degenerate valence Γ sub-band mvΓ 0.41 m0

Splitoff band mass of the valence sub-band at Γ mso 0.14 m0

Number of equivalent conduction L sub-bands NcL 4

Number of equivalent conduction X sub-bands NcX 3

Table 2. Coefficients for the temperature dependence of the conduction band extrema that are used in Eq. (5). These data are from Ref.

[3]

Parameter Symbol Value Units

Γ sub-band EΓ0 0.813 eV

Γ sub-band AΓ 3.78 × 10–4 eV/K

Γ sub-band BΓ 94. K

L sub-band EL0 0.902 eV

L sub-band AL 3.97 × 10–4 eV/K

L sub-band BL 94. K

X sub-band EX0 1.142 eV

X sub-band AX 4.75 × 10–4 eV/K

X sub-band BX 94. K
the vicinity of the Fermi surface will behave as though

they have many characteristics of electrons in an indirect semiconductor.

For illustrative purposes, we give here only the

results for fitting the logarithm to the base 10 of the

total electron density n in units of cm–3 to a polynomial

in EF, namely,

(9)

The analytic fits for the electron densities in the Γ, L,

and X sub-bands are available by sending an email to

herbert.bennett@nist.gov. During the fitting analyses,

we rely substantially on graphics and keep the number

of fitting parameters to a minimum, subject to the constraint that the residual standard deviation Sres is acceptably small, i.e., Sres ≤ 0.01. The standard deviation is a

measure of the “average” error in a fitted model and

thereby is a metric for assessing the quality of the fit. A

smaller Sres indicates a better fit. The residual standard

deviation for a model Yf = f (Z) is

(10)

where Yj are the calculated data values, the are the

predicted values from the fitted model, N is the total

number of data points (here N = 28), and P is the total

number of parameters to be fitted in the model. We use

the NIST-developed DATAPLOT [4] software for both

the exploratory graphics and for the statistical analyses.

Table 3 gives the four fitting parameters for the cubic

l = 3 polynomial fit to the log10(n) as shown in Eq. (9)

and the associated residual standard deviation Sres =

0.0066. In general, the values of Sres decrease monotonically with increasing number l of terms in the polynomial given in Eq. (9). But, care must be taken to avoid

fitting noise in data sets. The general guideline for

Volume 108, Number 3, May-June 2003

Journal of Research of the National Institute of Standards and Technology

196

Fig. 1. The calculated Fermi energy for n-type GaSb at 300 K as a

function of the donor density. The Fermi energy is relative to the

majority conduction band edge at the Γ symmetry point in the first

Brillouin zone.

Fig. 2. The calculated electron densities in the conduction sub-bands

at Γ and L and the total electron density as functions of the Fermi

energy. The Fermi energy is relative to the majority conduction band

edge at the Γ symmetry point in the first Brillouin zone.

3 2l

10 0 1 F 2 F l F log ( cm ) ... .... n a aE a E aE − = + + ++

2

res

1

[ ( ) /( )],

N f

j j

j

S Y Y NP ==− − ∑

f Yj

Table 3. The four fitting parameters for a cubic polynomial fit of the theoretical calculations for the total

electron density in n-type, zinc blende GaSb at 300 K as a function of the Fermi energy relative to the bottom of the conduction Γ sub-band. The ratio is the estimated value divided by the estimated standard deviation. The residual standard deviation is Sres = 0.0066

Fitting parameter Estimated value Estimated standard Units Ratio

deviation

a0 17.7504 0.1774 × 10–3 1.001 × 105

a1 15.6775 0.5416 × 10–2 eV–1 2.895 × 103

a2 –11.4745 0.4723 × 10–1 eV–2 –2.43 × 102

a3 –41.3848 0.8535 eV–3 –4.849 × 101
many data sets is that when the absolute value of the

ratio R of the estimated parameter value divided by its

estimated standard deviation is less than about 2, then

the rate of decrease in Sres with increasing l tends to

decrease. For the data given in Fig. 1, when l = 2 or

P = 3, Sres = 0.0644; when l = 3 or P = 4, Sres = 0.0066;

and when l = 4 or P = 5, Sres = 0.0063. Because the

change in values of Sres between l = 3 and l = 4 is not

significant, we use the fitting parameters for the cubic

l = 3 case in this paper. Also, the ratio R for the parameter a4 when l = 4 is –1.917, and such a value for R

means that proceeding with higher l values probably is

not warranted. A figure that compares the calculated

total electron density as a function of the Fermi energy

with the fitting results from Eq. (9) for a cubic polynomial is not given because the two curves lie on top of

one another within the line widths of each curve. Fits to

the calculated electron densities for each of the conduction sub-bands are available from the author upon

request. Also, since the screening radii for the carriers

from Eq. (2) are not needed when interpreting the proposed measurements considered here, the corresponding screening radii are not presented in this paper.

4. Conclusions

The results from Sec. 3 are consistent with the findings of experimental work reported in the literature

such as Refs. [5] and [6]. Interpreting experiments for

GaSb requires at least a three-band model and under

some conditions may require a four-band model. And

finally, even though GaSb is intrinsically a direct semiconductor, our results show that electrons for n-type

GaSb in the vicinity of the Fermi surface will have

some characteristics that are similar to those for electrons in an indirect semiconductor.

Acknowledgments

The authors thank Jeremiah Lowney, formerly of

NIST, for many helpful discussions and assistance during the course of this work. We acknowledge many discussions with James Maslar and Wilbur Hurst concerning physical models for interpreting Raman measurements. We thank Alan Heckert and James Filliben for

help with the statistical analyses of data sets and for

guidance in using DATAPLOT. We benefited substantially from having access to the NIST Centralized

Computing Facility and from the support given by

many NIST Information Technology Laboratory (ITL)

members such as Mary-Lou Blessing, William George,

Denis Lehane, William Mitchell, and James Sims.

5. References

[1] H. S. Bennett, J. Appl. Phys. 83, 3102 (1998).

[2] J. S. Blakemore, J. Appl. Phys. 53, R123 (1982).

[3] A. Ya. Vul, in Handbook Series on Semiconductor Parameters,

Volume I, M. Levinshtein, S. Rumyantsev, and M. Shur, eds.,

World Scientific Publishing, Singapore (1996) pp. 125-146.

[4] J. J. Filleben and A. N. Heckert, the DATAPLOT software for

graphics and detailed statistical analyses runs on both UNIX and

WINTEL platforms. It has both command-line versions and

graphical user interface (GUI) versions. It is available by downloading from http://www.itl.nist.gov/div898/software/dataplot/.

In addition, the NIST-SEMATECH Engineering Statistics

Handbook at http://www.itl.nist.gov/div898/handbook/ is based

in part on DATAPLOT. This latter WWW site has tutorials that

explain in some detail the statistical analysis methods used in this

paper.

[5] A. Baraldi, F. Colonna, C. Ghezzi, R. Magnanini, A. Parisini, L

Tarricone, A. Bosacchi, and S. Franchi, Semicond. Sci. Technol.

11, 1656 (1996).

[6] V. W. L. Chin, Solid-State Electronics 38, 59 (1995).

About the authors: Herbert S. Bennett is a NIST

Fellow in the Semiconductor Electronics Division of

the NIST Electronics and Electrical Engineering

Laboratory. His interests include industrial consensus

building for technology roadmaps on semiconductors

and on optoelectronics packaging, device physics for

improved understanding of electronic, optoelectronic,

and magnetic materials, and assessing the quality of

digital video images.

Howard Hung is a computational scientist in the

Scientific Applications and Visualization Group,

Mathematical and Computational Sciences Division, of

the NIST Information Technology Laboratory. The

National Institute of Standards and Technology is an

agency of the Technology Administration, U.S.

Department of Commerce.

Volume 108, Number 3, May-June 2003

Journal of Research of the National Institute of Standards and Technology

197