NBS SPECIAL PUBLICATION 400-47

The Theoretical and

Experimental Study of the

Temperature and Dopant Density

Dependence of Hole Mobility,

Effective Mass, and

Resistivity in Boron-Doped Silicon
NATtONAL BUREAU OF STANDARDS

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Semiconductor Measurement Technology:

The Theoretical and Experimental Study of the

Temperature and Dopant Density Dependence of Hole Mobility,

Effective Mass, and Resistivity in Boron-Doped Silicon

Sheng S. Li

Department of Electrical Engineering

University of Florida

Gainesville, Florida 32611

This activity was supported by The Defense Advanced Research Projects Agency

and The National Science Foundation

U.S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Jr., Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued November 1979
, Library of Congress Cataloging in Publication Data

Li. Sheng S 1938- The theoretical and experimental study of the temperature and

dopant density dependence of hole mobihty, effective mass, and

resistivity in boron-doped silicon. (Semiconductor measurement technology) (NBS special publi-

. cation ; 400-47) "This activity was supported by the Defense Advanced Research Projects Agency and the National Science Foundation."

Bibliography: p. Includes index. Supt. of Docs, no.: CI 3. 1 0:400-47

I. Holes (electron deficiencies). 2. Semiconductor doping. 3. Silicon—defects. 1. United States. Defense Advanced Research

Projects Agency. 11. United States. National Science Foundation.

III. Title. IV. Series. V. Series: United States. National Bureau of Standards. Special publication ; 400-47. QC 1 00. U 57 no. 400-47 [QC6 1 1 .6. H6] 602'. I s [537.6"22] 78-24 1 85

National Bureau of Standards Special Publication 400-47

Nat. Bur. Stand. (U.S.), Spec. Publ. 400-47. 50 puge.s (Nov. 1979)

CODbN: XNBSAV

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1979

For sale by the Superintendent of Documents. U.S. Government Printing Office, Washington, D.C. 20402

Stock No. 003-003-02140-7 Price $2.25

(Add 25 percent additional for other than U.S. mailing).

S^ U.S. GOVERNMENT PRINTING OFFICE: 1979 O—311-046 (310)
TABLE OF CONTENTS

Page

1. Introduction ]_

2. Theoretical Approach 2

A. Valence Bands and Hole Effective Masses 2

B. Mobility Formulations and Scattering Mechanisms 3

i. Lattice Scattering Mobility 4

ii. Ionized Impurity Scattering Mobility 6

iii. Neutral Impurity Scattering Mobility 7

iv. Effect of Hole-Hole Scattering 8

V. Hole Mobility vs. Dopant Density and Temperature 8

vi . Resistivity as. Dopant Density and Temperature 9

3. Results and Discussions n

4. Conclusions 22

5. References 23

Appendix 2 5

LIST OF FIGURES

Page

1. Hole mobility VS. hole density for boron-doped silicon at 300 K 12

2. Theoretical calculations of the ratio of ionized and total boron density vs. boron density with temperature as a parameter 13

3. The calculated hole mobility VS. hole density for boron-doped silicon with

temperature as a parameter 15

4. The calculated hole mobility vs. temperature for boron-doped silicon with

dopant density as a parameter lo

5. Resistivity vs. dopant density for boron-doped silicon at 300 K 17

6. Resistivity vs. temperature for ei.;/it boron-doped silicon slices 18

7. Theoretical calculations of resistivity vs. temperature for boron-doped

silicon with dopant density as a parameter 19

8. Theoretical calculations of resistivity vs. dopant density for boron-doped

silicon with temperature as a parameter . . 20

9. The mobility reduction factor, Yj^-^^ . as a function of dopant density for borondoped silicon, for 100 < T -' 400 K, showing the effect of h-h scattering on ionized impurity scattering mobility as is calculated from eq (21) 21

il . Simplified valence band structure of silicon based on Kane's [A-12| calculations

and measured properties of the valence band (after Barber [A-9] ) , . 28

iii
Page

A2 . Temperature dependence of the density-of-states effective masses m , * + . . Dl

"^D2 ' ^^'^ ^D3 ^'^^ individual bands, and the combined density-of-states effective mass m* of holes in silicon 35

D

* * A3. Temperature dependence of the conductivity effective masses m , m , and

the individual bands, and the combined conductivity effective mass m*

of holes in silicon

0 Boltzmann's constant (= 1.38 x 10"^ 3)

m Free electron mass (= 9.1 ^ 10

0 m*^ effective mass in I;eavy hole band (in units of m^)

m*2 Conductivity effective mass in light-hole band (in units of m^)

m*^ Conductivity effective mass in spin-orbit-split (SO) band (in units of m^)

m*, Density-of-states effective mass in heavy-hole band (in units of m ) Dl o

m*., Density-of-states effective mass in light-hole band (in units of m ) D2 0 m*^ Density-of-states effective mass in SO band (in units of m ) D3 0 m* Total density-of-states effective mass of hole (in units of m^

)

Total boron density (cm~^)

N~ Ionized boron density (cm~^)

N Neutral boron density (cm~^) N

N Effective density of valence band states (cm~^)

36

A4 . Temperature dependence of the Hall effective masses , i^j^^' ^^'^

^ti3

^'^ individual bands, and the combined Hall effective mass m* of holes m silicon . 37 H A5 . The acceptor density dependence of the combined conductivity effective

mass of holes in silicon as a function of temperature 38

A6. The acceptor density dependence of the combined Hall effective mass of

holes in silicon as a function of temperature 39

LIST OF SYMBOLS*

E Energy of holes (eV)

E^ Acceptor energy level (eV)

E Fermi energy level (eV) F

E^ Energy band gap (eV)

E^ Binding energy of neutral acceptors (eV)

E^ Valence band edge (eV)

h Planck's constant (= 6.625 x 10"3'+) , -fi = h/2iT

Unless stated otherwise, all units are in MKS

i V
p Hole density (cm"^)

p' Effective screening hole density (cm~^)

q Electronic charge {= 1.6 x lO"-*-^)

T Absolute temperature

e Reduced energy of carrier (= E/1. T)

0 Free space permittivity {= 8.854 x 10"-'^^)

e Permittivity of silicon (= 4tt x 11.7 x e ) s 0

9 y Lattice scattering mobility (cm /V-s) L Ionized impurity scattering mobility (cm^/V-s)

o Neutral impurity scattering mobility (cm /V-s)

y Combined lattice and ionized impurity scattering mobility (cm^/V-s)

LX

V Total hole conductivity mobility (cm /V-s)

P

p Resistivity (H-cm)

T Relaxation time

V
PREFACE

This study was carried out at the University of Florida as a part of the Semiconductor

Technology Program in the Electron Devices Division at the National Bureau of Standards.

The Semiconductor Technology Program serves to focus NBS research to enhance the performance, interchangeability , and reliability of discrete semiconductor devices and integrated

circuits through improvements in measurement technology for use in specifying materials and

devices in national and international commerce and for use by industry in controlling device

fabrication processes. This work was supported by the Defense Advanced Research Projects

Agency, ARPA Order 2397, Program Code 7Y10, through the National Bureau of Standards Semiconductor Technology Program.

ACKNOWLEDGMENTS

The author would like to express his special thanks to Dr. W. M. Bullis for his critical review of the manuscript and for his valuable comments. He would also like to acknowledge stimulating discussions and assistance on the part of Dr. M. G. Buehler and W. R. Thurber during the course of this study. The planar four-probe test structures used in the

present resistivity measurements were fabricated by Y. M. Liu of NBS. R. L. Mattis and

W. R. Thurber provided the information of dopant density in these test samples. Thanks

are also due L. C. Linares for effective mass calculations and for checking other theoretical calculations described in this work. The author gratefully acknowledges the critical

review and comments of Dr. A. H. Kahn of NBS. Typing and proofreading of the final draft

was done by Jo Halapatz, Jane Walters, and Elaine Cohen.

vi
Semiconductor Measurement Technology

THE THEORETICAL AND EXPERIMENTAL STUDY OF

THE TEMPERATURE AND DOPANT DENSITY

DEPENDENCE OF HOLE MOBILITY, EFFECTIVE MASS,

AND RESISTIVITY IN BORON-DOPED SILICON

By

SHENG S. LI Department of Electrical Engineering

University of Florida

Gainesville, FL 32611

ABSTRACT

Theoretical expressions for computing resistivity and conductivity mobility

of holes as functions of dopant density and temperature have been derived for

boron-doped silicon. The model is applicable for dopant densities from 10^" to

3 X 10^^ cm~^ and temperatures between 100 and 400 K.

Using a three-band [i.e., heavy-hole, light-hole, and spin-orbit split (SO)

bands] model, the hole mobility was calculated by properly combining the contributions from scattering by lattice phonons, ionized impurities, and neutral impurities. In addition, the effects of hole-hole (h-h) scattering and nonparabolicity of the light-hole band were taken into account in the mobility formulation .

To verify our theoretical calculations, resistivity measurements on nine

boron-doped silicon slices with dopant densities from 4.5 x 10-^ to 3.2 x 10-^^ cm were performed for 100 < T < 400 K, using a planar four-probe square-array

test structure. Agreement between our calculated and measured resistivity values

was within 6 percent over the entire range of dopant density and temperature

studied here. Excellent agreement (within ±5 percent) between our calculated hole

mobility values and those of Thurber et at. was obtained for N, < 10-*^^ cm ^ for

boron-doped silicon, while discrepancies between our calculated values and those

of Wagner were found for boron densities greater than lO-*-^ cm~^. These discrepancies are attributed to Wagner's neglect of the effect of deionization of boron

impurities at higher dopant densities (i.e., he assumed hole density equal to the

total boron density)

.

Finally, formulations for the density-of- states effective mass, conductivity

effective mass, and Hall effective mass are described, and the results are applied

to the calculations of hole masses in boron-doped silicon for 10-''^

<_ N^ <_ 10^®

cm~3 and 50 < T £ 500 K.

Key Words: Boron-doped silicon; dopant density; effective mass; hole mobility; ionized impurity scattering mobility; lattice scattering mobility; neutral

impurity scattering mobility; p-type silicon; resistivity; scattering mechanisms.

1 . INTRODUCTION

Although a considerable amount of work on mobility and resistivity has been published

[1-11] , because of the complexity of the valence band structure and of the various scatter

ing mechanisms involved, only limited amount of work has been reported on mobility calcula

tions in p-type silicon. For example, Ottaviani et al . [1] have developed a microscopic

theoretical model applicable only to computation of ohmic mobility in high purity p-type

silicon in which nonpolar optical and acoustical phonon scattering is dominant. In their

theoretical analysis, they have shown the necessity of considering band nonparabol icity in

the mobility formulation, and have pointed out that the band warping effect is secondary i
importance to that of nonparabolicity . The lattice mobilities for holes in p-type silicon

are well established for temperatures between 50 and 400 K [1-6] . Braggins [4] has analyzed

carrier concentration and hole mobility in boron-doped silicon from Hall effect and resistivity measurements. His calculations, which are based on a three-band model and the re- laxation time approximation, yield impurity densities which appear to agree well with Hall

effect data for boron densities less than 5 x 10^^ cm~^. Large discrepancies exist among

the published hole mobility data [7-10] for dopant densities greater than 10-^^ cm~^; no adequate theoretical model is available for mobility and resistivity calculations in p-type

silicon that is valid over the full range of dopant densities and temperatures studied here.

The purpose of this paper is to present theoretical and experimental investigations of

the dopant density and temperature dependence of hole mobility and resistivity in borondoped silicon for 10^^ 1 1 3 x 10^^ cm-^ and 100 £ T ^ 400 K.

To overcome difficulties in effective mass calculations and mobility formulation, the

valence band structure of silicon was approximated by a three-band model consisting of a spherical, parabolic heavy-hole band; a spherical, nonparabolic light-hole band; and a spherical, parabolic split-off band [12]. Together with this model, the relaxation time

approximation was used to develop theoretical expressions for mobility of holes in p-type

silicon. Scattering mechanisms due to lattice (i.e., acoustical and optical phonons)

,

ionized impurity, and neutral impurity scattering are considered over the entire range of

dopant densities and temperatures studied. In addition, the effects of hole-hole (h-h)

scattering on the lattice and ionized impurity scattering mobilities are also taken into

account. In view of the complex nature of heavy doping effects and the uncertainties in

accountino for hole density and impurity density at high dopant densities, the present

analysis has been restricted to boron densities up to 3 x 10^^ cm ^ , a dopant density range

in which the use of Boltzmann statistics is valid.

In section 2 the theoretical approach is described, presenting the results of effective mass calculations and mobility and resistivity formulations. The appropriate scattering mechanisms for holes are discussed. In section 3 the theoretical and experimental re- sults are compared and discussed. Section 4 summarizes the main conclusions derived from

this work. The Appendix is a paper by L, C, Linares and S. S. Li on the temperature dependence of effective masses in p-type silicon.

2 . THEORETICAL APPROACH

Presented in this section are the results of calculations of hole effective mass

and the mobility and resistivity formulations for p-type silicon. Theoretical expressions

for hole mobility are derived by considering contributions from lattice, ionized impurity,

and neutral impurity scattering. The effects of h-h scattering and light-hole band non- parabolicity are also taken into account in the mobility formulation.

A . Valence Band and Hole Effective Masses ' •

' ' ' >

The heavy-hole and light-hole bands of silicon have an extremum at k = 0 and are degenerate there. The constant energy surfaces for this case are warped spheres. The derivation

2
of an expression for a valence band effective mass which incorporates the nonparabolic

nature of the individual bands begins with a simplified model of the warped structure based

on the work of Kane [13] and Barber [12]. In this model, the heavy-hole band is characterized by holes with an energy-independent but direction-dependent effective mass, and the

light-hole band is characterized by holes with an effective mass which is both energy- and

direction-dependent. The split-off (SO) band is separated from the heavy- and light-hole

bands at k = 0 by an energy A = 0.044 eV, and is characterized by an isotropic, energydependent effective mass. Thus, the heavy-hole band is treated as parabolic for the range

of temperatures considered in this work, while the light-hole band is parabolic only for

very low values of k. At higher values of k, the light-hole band has an energy-dependent

curvature which takes on the same characteristics as the heavy-hole band for energies

greater than 0.02 eV.

Based on the above band model, effective masses and density-of-states effective m.asses have been calculated for holes in each of the three valence bands. Details of the calculations are provided in the Appendix; the results are summarized in table 1.

Table 1. Density-of-States and Conductivity Effective Masses of Holes in p-Type Silicon for

N = 10^5 cm~3 and 100 < T < 400 K.

A - —

T(K) 100 150 200 250 300 350 400

"^Dl 0 5503 0 5547 0 5592 0 5638 0 5685 0 5733 0 5782

0 2481 0 3087 0 3527 0 3865 0 4118 0 4332 0 4496

%3 0 238 0 240 0 242 0 244 0 246 0 248 0 250

m* 0 6570 0 7023 0 7340 0 7624 0 7865 0 8081 0 8265

TM^1 0 4271 0 4322 0 4375 0 4429 0 4484 0 4541 0 4599

^^2 0 3284 0 4 3 37 0 4886 0 5172 0 5324 0 538 3 0. 5380

0 2399 0 2427 0 2475 0 2512 0 2532 0 2549 0 2597

m*

C

0 3592 0 4181 0 4497 0 4673 0 4785 0 4865 0 4904

Note: Definitions of density -of -states and conductivity effective masses of holes i'- ecc'i band are given in the Appendix.

B . Mobility Formulations and Scattering Mechanisms

The calculation of hole mobility in p-type silicon can be carried out by first computing

the hole mobilities in the heavy-hole, the light-hole, and the SO band separately; the

overall mobility of holes is then calculated as a weighted average of these one-band mobilities over the individual hole densities in each valence band. The scattering mechanisms

concidered in this work include (a) intraband optical and acoustical phonon scattering,

(b) intraband ionized and neutral impurity scattering, (c) interband acoustical phonon

scattering, and (d) the h-h scattering. In addition, the effect of nonparabolicity of the

light-hole band is also taken into account in the mobility calculations. Following is a

3
discussion of the various scattering mechanisms cited above and a presentation of the appropriate mobility formulation for each scattering process in p-type silicon.

i. Lattic e Scattering Mobility

The published hole conductivity mobility data for lightly doped p-type silicon exhibit

a T ^^-^^ temperature dependence for 150 < T < 400 K [1-6], Since acoustical phonon scattering only leads to theoretical mobility following a T ' ^ law, additional scattering due

to nonpolar optical phonons , interband acoustical phonons, and the effects of hole-hole

scattering and nonparabolicity of the valence band should be included to explain this discrepancy [2] , The nonparabolic band structure in the light-hole band and the temperature

dependence of the energy band gap lead to a dependence of hole effective masses on temperature in each band; this is shown in table 1.

The relaxation times for scattering by acoustical phonons include both the possibility

of interband as well as intraband transitions; interband scattering has been considered

only between the light- and heavy-hole bands. The treatment of the acoustical phonons has

been based on the theory of Bir, Normantas, and Pikus [14] where the relaxation times can be

expressed in terms of a single constant, , which controls the overall magnitude of the

scattering. Both transverse and longitudinal phonons participate in the scattering.

Braggins [4] has applied their theory to evaluate the lattice mobility for p-type silicon.

His results for the acoustical phonon relaxation times may be written as

]. 0 .959 * „ 3/2 ] /? ' (1)

1 1.61 * 3/2 3/2 1/2 (2) m ^ T e • ' T ^ T D2 ac2

* 5/2

0.224 %1 *3/2 3/2 1/2 (3)

T T

* Dl

acl2 m

m_, T e

D.:

and

T T 0 ac3 0 2^/V/Vwhere ^ 1 _ °

T A

' O IT h P u

s s

0.224 * 3/? ^3/2 :./2 m_ T e , (4)

T 1 '2 o

1 1/2 3/2 (5)

(e - A/k T) ^ T

(6)

Equations (1) to (4) represent the acoustical phonon scattering times for the heavy- and

light-hole bands taking into account the interband and intraband transitions, while eq (5)

is the expression for the intraband acoustical phonon scattering in the split-off band.

4
The relaxation time for scattering by the nonpolar optical phonons is given by [14]

-1 3/2

:i . Dl ]/9l 1^'' 1/2

WGT I + 1^ - Q/r^ " ^o (

^ ®^'^) (7)

where 9 is the Debye temperature, n = [exp (G/T) o

-1

1] is the phonon distribution function, and W is a constant which determines the relative coupling strength of the holes to the optical phonon mode compared to the acoustical phonon mode. In eg (7) the band index,

i = 1, 2, or 3 , denotes the heavy-hole, light-hole, or the SO ba.nd, respectively. This

notation will be used for other scattering processes as well throughout the text. The

lattice mobility in each of the three valence bands can be calculated from

q<T . > (8) ^Li

wnere

and

Li

Li

4 3/2 -e ^

# . e

' e de

-1 -1

T . + T . aci oi (9)

Here t . can be evaluated from eqs (1) through (6) and t , fror.i er, (7) . aci oi ' The total lattice mobility, based on a three-band model, can be expressed by

3/2

exp -A/k T^ o

'

-A/k t"! - (10)

0

where y. , , P,.-,. and y are lattice mobilities calculated from^eq (3) for the heavy-hole,

light-hole, and the SO band, respectively. Values of u^^ calculated from eqs (1) through

(10) for boron-doped silicon are listed in table 2 for 100 <_ < 400 K. Table 3 lists the

physical constants used in the present calculations.

Table 2. Hole Lattice Mobility for Boron-Doped Silicon Calculated

From ea (0) , Using the Parameters Listed in Table 3.

T(K) 100 150 200 250 300 350 400

cm- 6020 2287 1204 706 465 336 261

^L^vTs'

5
Table 3. Material Parameters Used in Calculating Lattice

Mobility in Boron-Doped Silicon.

Optical phonon temperature, 6, (K) [1] 73S .0

Acoustical deformation potential constant, 8 . 718

E^{eV) [1]

W: Coupling constant between optical and

acoustical phonon mode* 0 .4

Material density, (g/cm^) [1] 2 . 329

Velocity of acoustical wave, (cm/s) [1] 9 .04 X 105

T

0 5 .8 X 10- 10

W = — : D = 8.3025 x 10° eV/cm is the optical deformation

^ ... 2k2 02 ° :., . . 0 1

' potential constant. '

. :~^~r:

ii. Ionized Impurity Scattering Mobility '

In pure silicon at room temperature, the main contribution to carrier mobility comes

from lattice scattering. However, as dopant density increases or temperature decreases,

the role of ionized impurity scattering becomes more important. Theories for ionized

impurity scattering have been developed previously by Brooks [15] and Herring [16] and by

Conwell and Weisskopf [17] . The Brooks-Herring (B-H) formula has been widely used in

mobility calculations for several semiconductors including silicon. The B-H formula can be

written as

2^/2 m=.2 (koT)3/2

p_ = : 10 2,

^3/2 TT

where i = 1, 2, or 3 is the band index and

TT" - q3 m*. N G(b. ) ^ Ci I i' (11)

b. G(b.) = ;.n(b.H-l) - ,

^

_ (12)

1

24Tfm*. (k^T)2

b. = : X 10-6 (1,)

q2h2p' :

and ^ p' - p + N-(l - N~/N ), for N = 0 ita)

The total ionized impurity scattering mobility is calculated from

6
^11 ^ ^12

3/2

+ M I3|

D3'

Dl/

exp -A/k T

' o

-1

3 /2

1 + D2

Dl,

exp fl/ k T

o

(15)

where p^^, and li^^ are calculated from eq (11) for the heavy-hole, light-hole, and the

SO band, respectively.

Note that eq (11) does not include such effects as the variation of dielectric constant around the impurity due to a position-dependent screening which may become important

for the heavily doped case. Since analysis is restricted to < 3 x 10"° cm , neglecting

this variation should not introduce an appreciable- error in the mobility calculations.

iii . Neutral Impurity Scattering Mobility

Although neutral impurity scattering may not be as important as other scattering

mechanisms discussed above in the range of dopant density and temperature considered here,

it becomes important at low temperatures or at high dopant densities. For completeness,

this scattering mechanism is included in the mobility formulation. The mobility formula

derived by Sclar [18] is employed here for neutral impurity scattering. Sclar used his

calculation of the scattering from a three-dimensional square well to estimate the

influence of a weakly bound state on the scattering. From h: s derivation one obtains

1/2 , 1/2

where

V . = 0.82 p . Ni El

2

3 \E

: T

o N

o

= 1.135 X 10-1 9 (m„ ) (g. )2 Do OS

(16)

(17)

and

q^ m

Di

Ei

5 N E h- N s

X 10-2 (18)

Equation (18) is the mobility expression derived by Erginsoy [19] for neutral impurity

.scattering, based on the partial wave expansion approximation.

The total neutral impurity scattering mobility can be computed from

^NI ^ ^N2

3/2

^ ^'N3 1

3/2

• exp

exp I -A/k T

o

1 -1

-A/k T

0 (19)

7
where V-^-^t ^-^2' ^^'^ ^n3

^'^'^ calculated from eqs (16) through (18) .

iv . Effect of Hole-Hole Scattering

The mobility formulas given in eqs (1) through (19) neglect the effect of h-h scattering. The importance of including electron-electron (e-e) scattering in n-type silicon has

been discussed in detail previously [20] . In this section the effect of h-h scattering in

p-type silicon is discussed. The e-e and h-h scattering events, while not directly producing a change in total momentum, provide a mechanism for the redistribution of momentum

gained from the electric field. The maximum effect of h-h scattering can be readily calculated (by classical treatment) if the distribution function is Maxwellian and centered about

s the drift velocity. Assuming x = aE and by analogy of e-e scattering, it can be shown that

the ratio of hole mobilities with and without h-h scattering is given by

O <T><T >

For acoustical phonon scattering with s = - ^, eq (20) predicts a value for r, , =

2 2 hh

9tt/32 - 0.88, while for ionized impurity scattering s = — and y = 37r/32 - 0.29.

It is noted that given by eq (20) shows no dependence on hole density and temperature. A more rigorous treatment to account for the effect of carrier-carrier scattering

on the ionized impurity scattering mobility has been developed by Luong and Shaw (L&S) [21] , using a one-particle-like approximation from the Hartree-Fock theory. Applying their

(L&S) theory to the case of the effect of h-h scattering on ionized impurity scattering,

the B-H formula [i.e., eq (11)] is reduced by a factor which can be expressed in closed

form:

y^^r = (\/P') tl - exp(-pVN~)] , (21)

where ig the ionized acceptor density, and p' is the screening hole density given by

eq (14)

In contrast to the classical formula [i. 2., eq (20)], eq (21) predicts a dependence

of ^^1^1 °" hole density and, hence, temperature. Values of Yj^j^j ni^Y vary between 0.432 and

0.632, depending on the degree of ionization of the shallow acceptor impurities.

A comparison of the values, y, , , calculated from eqs (20) and (21) indicates that the

hhl

classical formula overestimates the h-h scattering effect. Furthermore, by correcting

lattice and ionized impurity scattering mobilities separately, the effect of h-h scattering

may also be overestimated. To overcome these difficulties, a semi-empirical formula is

derived based on eq (20) to correct the effect of h-h scattering on the lattice scattering

mobility, and eq (21) has been retained for correcting the effect of h-h scattering on ionized impurity scattering mobility. This will be discussed later.

V . Hole Mobility vs. Dopant Density and Temperature

Following is a discussion of the mobility calculations for p-type silicon for dopant

densities froui 10^^ t^, x 10-^° cm and temperatures between 100 and 400 K. The combined
hole mobility due to both lattice and ionized impxjrity scattering contributions is calculated according to the mixed scattering formula [22]

•^LI "

"^lI-"- x2[Ci(X)cosX + sinX(Si(X) - ^)

1 1 , (22)

where

v2 ^ 6p^/iJj • - (23)

Equation (22) is obtained by averaging the reciprocal sum of t"-^ and t"^ over the Maxwellian

L L distribution. Here, Ci (X) and Si(X) are the cosine and sine integrals of X, respectively;

and are given, respectively, by eqs (10) and (15) when the effect of h-h scattering

is neglected.

The effect of h-h scattering can be included by incorporating the mobility reduction

factor, Yj^j^' given in eq (20) for lattice scattering mobility and in eq (21) for ionized

impurity scattering mobility, into eq (23) to correct the values of p and y • L I Equation (22) allows the calculations of y for 10^^ < N < 3 x 10^^ cm"^ and

LI — A — 100 < T < 400 K. When neutral impurity scattering is included, the total mobility may be

computed from the expression

V = + , (24) p LI N

where y^^ and y^ are given by eqs (22) and (19) , respectively.

The use of eq (24) for mobility calculations may be considered as a good appro.ximation

since neutral impurity scattering, being energy-independent, does not affect any contributions from y LIand y . Calculations of y indicated that the contribution from neutral

N impurity scattering to total hole mobility is less than 10 percent over the entire range

of dopant densities and temperatures studied here. As a result, the error introduced by

this approximation is expected to be negligibly small.

The procedure for incorporating the effect of h-h scattering on the lattice scattering

mobility for p-type silicon is similar to that used for n-type silicon [21]. For example,

for acoustical phonon scattering, it is assumed that Yj^j^ decreases linearly with increasing

1 ^ _ Q 97T dopant density from y, , = 1 at N = 10^~^ cm ^ to a maximum value of y, , = - O.BO for

hh A hh 32

> 3 X 10^^ cm~3, as predicted by eq (20) [i.e., y, , = 1-0 - 0.008 N /(2 x 10^^) is used A — ^ J ' ' hh A

' in the calculation] . This provides the best fit to the experimental data for the range of

dopant densities and temperatures studied. This approximation may also offset the possible

overestimation of the h-h scattering effect by the classical formula discussed earlier. For

ionized impurity scattering, y, , is calculated from eq (21) which is applicable for the

hhl

entire dopant density and temperature range considered here. The mobility reduction factor

for optical phonon scattering is evaluated numerically using eq (20) . For neutral impurity

scattering, no correction is needed since is independent of energy.

vi. Resistivity vn. Dopant Density and Temperature ' The resistivity for p-type silicon is calculated from

9
1 (25)

.: : .• iJ. - w > . p = ,

where u is the hole mobility given by eq (24), and the hole density for the nondegenerate

P case is E -E

~

P = Pi + P2 + P3 = \ exp j^-^- ' (26) L ° 3/2

where N =2

* 2TTm k T

D o ^ ' h'

is the effective density of valence band states, and p- , pj , and

Pg are hole densities in the heavy-hole, light-hole, and the SO bands, respectively.

In order to calculate hole mobility and resistivity from eqs (24) and (25) , it is

necessary to know the exact ionized and neutral impurity densities so that individual

scattering contributions and hole density can be evaluated. This is done as follows:

The ionized boron densities for boron-doped silicon are computed by solving the charge

balance equation for the Fermi energy by the iteration procedure. Since the minority carrier density is negligible, the charge balance equation for uncompensated p-type silicon is

* simply

P - n" , -

.'

.

, (27)

where [23] , N

^- ^ A (28) - • ^ 1 + [4 + 2 exp(- -~^)]exp [ {E-E^) /k^T]

. -

. ' o

is the ionized acceptor density, and A = 0.044 eV. Equation (28) is used to include the

contribution from the heavy- and light-hole bands and the SO band.

Experimental evidence indicates that the acceptor ionization energy E^ is not a constant, but decreases with increasing dopant density [24,25]. A study by Penin et at. [25]

of heavily doped silicon over the temperature range from 4 to 300 K shows no evidence of an ionization energy for shallow impurities such as phosphorus or boron at impurity densities

1 ° - ^ greater than 3 x lo cm . The dependence of ionization energy on dopant density for

boron-doped silicon can be expressed by [22 J

E = U.0438 - 3.037 x IQ-S N^^^ . (29) A A

With the help of eqs (26) through (29) , the hole density and fraction of ionized

boron density can be calculated as functions of dopant density and temperature.

* - + A more general form of the charge balance equation is - N = p - n, which should be

used if n and are not negligible compared to p and N^, respectively.

10
3. RESULTS AND DISCUSSIONS

Using the theoretical expressions derived in section 2, the hole mobility and resistivity have been calculated as a function of dopant density and temperature in boron-doped

silicon for 10^^ £ N^ £ 3 x 10^^ cm"^ and 100 T ^ 400 K. The results are compared with

experimental data. Figure 1 shows the hole mobility plotted as a function of hole density

at 300 K for boron-doped silicon. Curve 1 is the theoretical calculation using eqs (1) through (24) . Open circles are hole mobility data published by Thurber et al. [26] and

solid triangles are given by Horn [27] . These results are in excellent agreement with our calculated values. Curves 2 and 3 are reproduced from Wagner [9] and Caughey and Thomas

[28] , respectively. Note that the calculated mobility values are within 5 percent of the

values reported by Wagner for N^_^10^'' cm~^. However, for N^ l_ 10^^ cm"^, the calculated

values are substantially higher than those of Wagner [9] (e.g., the deviation is 2.2 percent

1 •? q 1 ft _ at 10 cm" , and increases to 22.6 percent at 10 cm ). To explain this discrepancy,

note that VJagner [9] obtained an empirical mobility expression based on measurements of

boron-implanted silicon. The amount of boron retained in the silicon was determined by integration of data obtained from incremental sheet resistance and Hall effect measurements on layers which were sectioned by anodic oxidation. A mobility value as a function of boron

density was derived which made the integrated dose equal to the implanted dose. This was done with the assumption that all of the boron was electrically active and ionized. This

assumption is valid only at low dopant density or at high temperature where full ionization

of boron atoms prevails. For example, the theoretical calculations indicate that at 300 K,

nT/N^ = 0.933 at N = 10^'' cm" ^ , and N~/N = 0.756 at N = 10^^ cm"^- Failure to correct A A A A A A this ionization effect at higher dopant density or lower temperature in the mobility calculations results in a large error in mobility values derived from resistivity measurements.

Curve 3 was reproduced using the Caughey and Thomas [28] empirical formula fitted to Irvin's mobility curve [29] . This mobility curve is based on data from Hall effect measurements, which give hole density within the uncertainty of the Hall scattering factor

and on data from chemical techniques which give total dopant density. The lower mobility

values shown in curve 3 as compared to Wagner's may be due in part to the fact that in the

region 10^^ < < 10^^ cm" ^ Sze and Irvin's [8] mobility curves include data on galliumdoped silicon, which is expected to have a lower mobility value for the same dopant density

due to the deeper ionization energy of gallium as compared to boron.

From the above analysis, it is clear that values of carrier mobility calculated from

measurements of resistivity and dopant density are correct only if the carrier density is

equal to the dopant density. This is not always the case. Figure 2 shows the ratio of

ionized and total boron impurity density as a function of boron density for 100 ^ T £ 400 K.

It is clearly shown in this figure that the deionization of boron impurities is significant

for 10-^ <N <3xlO^^ cm~^ and at low temperatures. In this range the carrier density

A —

is equal to the ionized boron density.

The mobility formulas presented in section 2 were used to compute the hole mobility as

a function of dopant density and temperature for dopant densities between 10^^ and 3 x 10^^

11
T

1 1 r I I T—

CO

esi

'4

m

ci

— C\J rO

cr

UJ

CD(T

IDX

o

o

X

00

E

o

O Ll — CO

LjJ

o

UJ

"2d

_ o

o

o

o

o

o

o

rO O

CJ O

O

roO

O —

(s-A/^ujo) AiniaoiAi 3nOH

CNM

0)>

(T< • U 3

4-1 •H O X

+J 0) OJ 4-1 o C

00 m y e

!^ w

P OJ 0 6

0

w c H Eh H

(-H T3 Ti C 0) 0) U) > cn M

P 0) u U IS) 3 13 o U) o e o 0

ro 1-4 >i MH 4J

4-1 H

tn •H

0) 0 n 13 UH 01 4-1 t-H > C

•H (0

to P ft u o

T3 13

OJ Ti

ft c 01 0 to TD

1

4-1 c

-

'—

0 0 4-1

5-4 0 rH

X! iHQ) p U C

0 CP 01

<B >i

4-1

4-1 e •H H 0 W

W i-4 c

c n-t 0)

0) 13

T}

0) 0)

0) o iH p 0

0

0

CP

ft c

to 01 •H

!-i

>i m en

4-1 H w

•H n3 fH (N •H >.

X! 01 X! 0 >

g 13 p CU

0) o 4-1 4-1 0 0

in rH c ft 0

•r4 0) rH 4-1 U

a rti 01 iHp ro o

CP rH 13 H ra C

ti u

12
20

10

14 J5 .16 .17 18

10" 10" 10' 10

TOTAL BORON DENSITY (cm"^)

10

19

Figure 2. Theoretical calculations of the ratio of ionized and total boron density vs. boron density with temperature as a parameter.

13
cm and temperatures between 100 and 400 K. The results are displayed in figures 3 and 4,

respectively.

Due to lack of available measured mobility data, comparisons of theory and experiment

over the range of dopant density and temperature reported here were unobtainable. In order

to verify the adequacy of the theoretical model developed in this work, resistivity measure- ments were performed on nine boron-doped silicon slices with dopant densities ranging from

4.5 X lo'^'^ to 6 X lO-*-^ cm~^ for temperatures between 100 and 400 K, using a planar fourprobe square-array test structure [30] as discussed in a previous paper [20]. The results

are compared with the theoretical calculations.

The resistivity for boron-doped silicon as calculated from eqs (24) through (29) is

shown in figures 5 through 8. Figure 5 shows the plot of resistivity as a function of

dopant density of p-type silicon at T = 300 K. The solid line (curve 1) is the theoretical

calculation; the solid dots are the experimental data for boron-doped silicon. Note that

each experimental data point represents the mean value of resistivity measured on five to seven selected test cells with a maximum variation of resistivity less than 5 percent.

The net dopant density in these specimens was determined from measurements of capacitance as

a function of voltage on a junction diode test structure near the four-probe square array

resistor. Details of this method can be found elsewhere [3r)] . The dotted curve (curve 3) was reproduced from Irvin's resistivity curve using the Caughey- Thomas formula [28], and

the dashed curve (curve 2) is VJagner's for boron-doped silicon [9]. The theoretical calculations agree with Wagner's resistivity data within 6 percent for ^ 3 x 10 cm . Irvin's resistivity values are higher than both Wagner's values and the values calculated

and measured here for N > 10^^ cm~^.

A Figure 6 shows resistivity as a function of temperature from 100 to 400 K for the

eight boron-doped silicon slices with boron density ranging from 4.5 x 10-^^ to 3.2 x 10^^

cm~^. The results show that the agreement between the calculated and measured values is

within ±6 percent over the entire range of dopant density and temperature studied. Figure

7 shows the calculated resistivity as a function of dopant density for temperatures between

100 and 400 K at 50-K intervals. Figure 8 displays the calculated resisti^'it" as a function of temperature for dopant density from lO-*-^ to 5 x 10^^ cm~^ in more regular steps.

To illustrate the effect of h-h scattering on the ionized impurity scattering mobility,

the mobility reduction factor, v , /is calculated from eq (21). The results, displayed hhl

in figure 9, for temperatures between 100 and 400 K, show that h-h scattering increases with

increasing dopant density and decreasing temperature, where the deioniza'cion effect becomes more prominent .

. ,

Finally, it is worth noting that both mobility and resistivity depend more strongly

on temperature for the lightly doped case where lattice scattering is dominant, and become

less dependent on temperature for dopant density greater than lO-*-^ cm~^ where neutral impurity scattering prevails. This trend can be seen in figure 4 (mobility) or figure 8 (resistivity) . The resistivity and mobility calculations for boron-doped silicon also

14
15
100 150 200 250 300 350 400

TEMPERATURE (K)

Figure 4. The calculated hole mobility vs. temperature for boron-doped silicon

with dopant density as a parameter.

16
10'

E

o

>

^ 10'

a:

10"

10

-3

I

I

I

I I Mil 1

I I I I I I I Mll| 1 I I I II

T= 300 K

• THIS WORK

IRVIN [29]

WAGNER [9]

' 'III ml I I I Ml III I I I I I ml I I I 1 .1 .^14 .J5 ,J6 ,J7 ,J8 ,J9

10 10 10 10 10 10

TOTAL BORON DENSITY (cm"^)

Figure 5. Resistivity vs. dopant density for boron-doped silicon at 300 K. Solid

line is our theoretical calculation, broken line is from Wagner [9] and dotted line

from Irvin [29]

.

17
TEMPERATURE (K)

Figure 6. Resistivity vs. temperature for eight boron-doped silicon slices.

Solid lines are the theoretical calculations and solid dots are the experimental

data taken from measurements on the four-probe square-array test structures fabricated on these slices.

18
TEMPERATURE (K)

Figure 7. Theoretical calculations of resistivity vs. temperature for borondoped silicon with dopant density as a parameter.

19
TOTAL BORON DENSITY (cm"^)

Figure 8. Theoretical calculations of resistivity vs. dopant density for

doped silicon with temperature as a parameter.

20
0.64

BORON DENSITY (cm"^)

Figure 9. The mobility reduction factor, "Yj^j^-j-- as a function of dopant density

for boron-doped silicon, for 100 < T < 400 K, showing the effect of h-h scattering on ionized impurity scattering mobility as is calculated from eq (21)

.

21
reveal that contributions from the SO band are negligible for temperatures below 200 K. Using a two-band model in the mobility calculations should not introduce any significant

error over the range of dopant densities and temperatures reported here.

4. CONCLUSIONS

Using a three-band model, theoretical expressions have been derived to compute hole

mobility and resistivity as functions of dopant density and temperature for boron-doped

silicon. In addition to considering contributions from scattering by lattice phonons,

ionized impurities, and neutral impurities, the model also takes into account the effect

of h-h scattering on both lattice and ionized impurity scattering mobilities. The non- parabolicity of the valence band structure is included in the effective mass calculations.

Resistivity analysis for nine boron-doped silicon samples shows that, for dopant

densities less than 3 x IQ^S cm~'^, agreement between the calculated and measured values of

resistivity is within ±6 percent for 100 ^ T £ 400 K. A comparison between the calculated

mobility values and those of Wagner at 300 K shows that agreement is within ±5 percent for N < 1.4 X 10-^' cm~^, but that differences as large as 27 percent occur at N = 2.5 x 10^^

cm~^. This di:-- repancy can be eliminated if the effect of deionization of boron impurity were included in Wagner's calculation. From this study, it was found that the theoretical

expressions derived in this work are adequate for mobility and resistivity calculations in

boron-doped silicon for 10^^ ^ £ 3 x 10^^ cm~^ and 100 £ T £ 400 K. It is important to point out that incorporation of both h-h scattering as well as deionization effects in the

mobility and resistivity calculations plays a major role in bringing good agreement between

theory and experiment in the range of dopant densities from 5 x 10^^ <^ 3 x 10^^ cm~^

for which previous theories failed.

The mobility and resistivity calculations further reveal that contributions from the SO

band are negligible for T < 200 K. Using a two-band model (i.e., neglecting the SO band

contribution) in the mobility calculations should not introduce any noticeable error (i.e.,

less than 5 percent) over the range of dopant densities and temperatures studied.

Finally, a rigorous formulation for correcting the effect of h-h scattering on lattice

and ionized impurity scattering together is needed to further improve our mobility formulations. The extension of the present work to higher dopant densities and lower temperatures

requires consideration of additional factors not included in this work.

22
5 . REFERENCES

1. Ottaviani, G. , Reggiani, L. , Canali, C, Nava, F. , and Alberigi-Quaranta , A., Hole

Drift Velocity in Silicon, Phys. Rev. B 12, 3318-3329 (1975)

.

2. Costato, M. , and Reggiani, L. , Ohmic Mobility of Holes in Silicon, Nuovo Cimento 68,

64-74 (1970).

3. Morin, F. J., and Malta, J. '°

. , Electrical Properties of Silicon Containing Arsenic and

Boron, Phys. Rev. 96, 28-35 (1954)

.

4. Braggins, T. T. , Analysis of Carrier Concentration and Mobility in Boron-Doped Silicon,

Ph. D. dissertation. University Microfilms International, Ann Arbor, Michigan (1975).

5. Prince, M. , Drift Mobility Measurements in N- and P-Type Silicon, Phys. Rev. 93 , 1204-1206 (1954)

.

6. Ludwig, G. W. , and Watters , R. L. , Drift and Conductivity Mobility in Silicon, Phys.

Rev. 101, 1699-1701 (1956)

.

7. Sclar, N. , Resistivity and Deep Impurity Levels in Silicon at 300 K, IEEE Trans. Electron Devices ED-24, 709-712 (1977).

8. Sze, S. M. , and Irvin, J. C, Resistivity, Mobility, and Impurity Levels in GaAs , Ge

,

and Si at 300 K, Solid-State Electronics 1^, 599-602 (1968)

.

9. Wagner, S., Diffusion of Boron from Shallow Ion Implants in Silicon, J. Electro diem.

Soo. 119 , 1570-1576 (1972)

.

10. Tsao, K. Y., and Sah, C. T. , Temperature Dependence of Resistivity and Hole Conductivity Mobility in P-Type Silicon, Solid-State Electronics 19_, 949-953 (1976).

11. Brown, D. M. , and Bray, R. , Analysis of Lattice and Ionized Impurity Scattering in

p-type Germanium, Phys. Rev. 121 , 1593-1602 (1962)

.

12. Barber, H. D. , Effective Mass and Intrinsic Concentration in Silicon, Solid-State

Electronics _10, 1039-1051 (1967)

.

13. Kane, E. O. , Energy Band Structure in p-type Germanium and Silicon, J. Phys. Chem.

Solids 1, 82-99 (1956)

.

14. Bir, G. H. , Normantas, E., and Pikus, G. E., Galva;iomagnetic Effects in Semiconductors

With Degenerate Zones, Soviet Phys. Solid-State ^, 867-877 (1962)

.

15. Brooks, H., Theory of the Electrical Properties of Germanium and Silicon, Advances in

Electronics and. Electron Physics, L. Marton, Ed., vol. 7, pp. 85-182 (Academic Press,

New York, 1955)

.

16. Herring C, , Transport Properties of a Many-Valley Semiconductor, Bell Syst. Tech. J. 34 , 237-290 (1955)

.

17. Conwell, E. , and Weisskopf, V. F. , Ionized Impurity Scattering in Semiconductors,

Phys. Rev. 388-390 (1950) .

18. Sclar, N., Neutral Impurity "cattering in Semiconductors, Phys. Rev. 104 , 1559-1561

(1956)

.

19. Erginsoy, C. , Neutral Impurity Scattering in Semiconductors, Phys. Rev. 79, 1013-j;::l4

(1950)

.

23
20. Li, S. S., and Thurber, W. R. , The Dopant Density and Temperature Dependence of Hole

Density and Resistivity in n-type Silicon, Solid-State EZectvontos 20, 609-616 (1977)

See also S. S. Li, SemiconductoT Measurement Technology: The Dopant Density and Temperature Dependence of Electron Mobility and Resistivity in N-Type Silicon, NBS Special

Publication 400-33 (1977)

21. Luong, M. , and Shaw, A. W., Quantum Transport Theory of Impurity Scattering-Limited

Mobility in N-Type Semiconductors Including Electron-Electron Scattering, Phys. ReV

B 4, 2436-2441 (1971)

22. Debye, P. P., and Conwell, E. M. , Electrical Properties of N-Type Germanium, Phys. Rev.

93_, 693-706 (1954) .

23. Li, S. S., The Dopant Density and Temperature Dependence of Hole Mobility and Resistivity in Boron-Doped Silicon, Solid-State Electronics 21, 1109-1117 (1978).

24. Pearson, G. L. , and Bardeen, J., Electrical Properties of Pure Silicon and Silicon

Alloys Containing Boron and Phosphorus, Phys. Rev. 865-883 (1949).

25. Penin, N. A., Zhurkin, B. G. , and Volkov, B. A., Influence of the Concentration of

Donors and Acceptors on the Electrical Conductivity of Heavily Doped n-Type Silicon,

Sov. Phys. Solid-state 1_, 2580-2584 (1966). .

26. Thurber, W. R. , Mattis, R. L. , and Liu, Y. M. , Resistivity-Dopant Density Relationship

for Sil icon. Semiconductor Characterization Techniques, Proceedings Volume 78-3,

P. A. Barnes and G. A. Rozgonyi, Eds., pp. 81-92 (ECS, Princeton, New Jersey, 1978).

27. Horn, F. H., Densitometric and Electrical Investigation of Boron in Silicon, Phys.

Rev. 97, 1521-1525 (1955)

28. Caughey, D. M. , and Thomas, R. E., Carrier Mobilities in Silicon Empirically Related

to Doping and Field, Proc. IEEE _55, 2192-2193 (1967) .

29. Irvin, J. C, Resistivity of Bulk Silicon and of Diffused Layers in Silicon, Bell

System Tech. J. 41, 387-410 (1962) .

30. Buehler, M. G., Semiconductor Measurement Technology: Microelectronic Test Pattern

NBS-3 for Evaluating the Resistivity-Dopant Density Relationship of Silicon, NBS

Special Publication 400-22 (1976); Buehler, M. G. , and Thurber, W. R. , A Planar Four

Probe Test Structure for Measuring Bulk Resistivity, IEEE Trans. Electron Devices

ED-2 3, 968-974 (1976).

24
APPENDIX

TEMPERATURE DEPENDENCE OF THE COMBINED DENSITY OF STATES,

CONDUCTIVITY, AND HALL EFFECTIVE MASS OF HOLES IN P-TYPE SILICON*

by

L. C. Linares and S. S. Li

Department of Electrical Engineering

University of Florida

Gainesville, FL 32611

ABSTRACT

A derivation involving the use of the Boltzmann transport theory is applied

* to obtain expressions for the density-of -states effective mass, m , the conductivity effective mass, m^ , and the Hall effective mass, m^, of holes in p-type

silicon. The valence band model as determined by Kane consists of a parabolic

heavy-hole band, a non-parabolic light-hole band, ana the split-off (SO) band.

V-tlues of effective mass calculated from this model reveal the temperature and

dopant density dependence of the effective mass due to the anisotropic, non- parabolic shape of the bands.

Calculations are carried out for dopant densities from lO-*^ to lO-'-'^ cm_~^ and

for temperatures from 50 to 400 K for boron-doped silicon. Conductivity effective mass increases from a value of 0.268 m at 50 K to a value of 0.490 m at

0 0 400 K, while Hall effective mass increases from 0.182 m at 50 K to 0.526 m

... 0 0 at 400 K. For temperatures above 100 K, m shows little change with dopant

density, but m varies by as much as 35 percent over the range of dopant densiH ties considered.

1 . INTRODUCTION

The interpretation of transport properties in silicon and the mc '.eling of silicon

junction devices depend on an accurate knowledge of values of effective mass. The complex

valence band structure of silicon leads to difficulties in the study of transport properties

of holes in this material. Thus, the development of a model incorporating the non-parabolic

and anisotropic nature of the band into a single parameter, the combined hole effective

mass, would greatly simplify the study of mobility, resistivity, and the Hall effect in

silicon. This is a reasonable procedure for including the band non-parabolicity in calculations of relaxation time, and has been applied effectively by Radcliffe fA-i: to study

acoustic phonon scattering, and by Barrie ;a-2j to study optical phonon and impurity scattering in non-parabolic bands [A-3] . It is the purpose of this paper to derive such a theoretical model for hole effective mass calculations in silicon.

* This research was supported in part by the Advanced Research Project Agency Order No. 2397

through the National Bureau of Standards' Semiconductor Technology Program Contract No. 7-35741 and in part by the National Science Foundation Grant No. ENG 72-81828.

25
Lax and Mavroides [A-4] have derived expressions for density-of-states effective masses

* * m^^ and m^^ the heavy-hole band and the light-hole band, respectively, which lead to the

generally accepted and quoted value, m„ = 0.591 m . This value, however, can only be con-

* D 0 * sidered applicable at 4.2 K, where m , = 0.537 m and m ^ = 0.153 m . From the published

Dl o D2 o experimental data it is noted that both electron and hole effective masses are dependent on temperature and dopant density [A-5, A-6]. Below 50 K, hole effective mass remains constant

as indicated in high frequency magnetoconductivity experiments [A-7]. However, at higher

temperatures and for higher acceptor impurity densities two mechanisms are responsible for

the temperature dependence of the effective mass: the thermal expansion of the lattice and

the explicit effect of temperature. The effect of the thermal expansion can be estimated

from the stress dependence of the effective mass [A-8], and has been shown to be negligible

[A-5, A-9] . The explicit temperature effect, however, is of great importance. It consists

of the temperature variation of the Fermi distribution function in a non-parabolic band,

the temperature-dependent distribution function of the split-off band, and the temperature

variation of the curvature at the band extremum due to the interaction between lioles and

lattice vibrations.

Following the work of Lax and Mavroides [A-4] , but using Fermi-Dirac statistics and a simplified model of the valence band structure for silicon, Barber [A-9] obtained an expression for the density-of-states effective mass, m^,^, which is temperature and hole density

dependent. Barber, however, did not apply the non-parabolic model of the valence band to the study of conductivity or Hall effective mass in p-type silicon.

* * Costato and Reggiani [A-10] also developed expressions for m^^ and m^ , the band conductivity effective mass, which show a variation with temperature, but they neglected the

effects of the split-off band and the temperature variation of the band curvature.

In this paper the expressions for density-of-states effective mass, conductivity

effective mass, and Hall effective mass of holes are derived based on the following

* definitions. The density-of-states effective mass, m , enters in tlie normalization of the

distribution function; the conductivity effective mass, m^, is the mass of a mobile charge

carrier under the influence of an external electric field; and the Hall effective mass,

* m , is the mass of a mobile charge carrier under the application of external electric and

Hmagnetic fields. (The reason for these particular definitions of effective masses is that

the primary application of this work is to generate improved theoretical calculations of

Hall mobility, resistivity, and conductivity mobility [A-11].) These expressions were used

to calculate hole effective masses in p-type silicon over a wide range of temperature and

dopant density. Since the crystal structure of silicon has cubic symmetry, the ohmic

mobility and the low-field Hall coefficient are isotropic. An angular average of the

effective masses may be performed taking into account separately the warping of the individ-

* * * ual bands so that expressions for m , m , and m of isotropic form can be derived. Values ^ D C H

^ calculated from these expressions differ from one another because of the warping and non- parabolicity , and consequently effective mass in each band depends on temperature and

dopant density in its own way.

26
The valence band structure of silicon is presented in section 2, In section 3 expres-

* * * sions for m^, m^., and are derived. In section 4 numerical values of effective mass are analyzed, and in section 5 the validity of the model is examined.

2 . THE VALENCE BAND STRUCTURE OF SILICON

The theoretical calculations by Kane [A-12] have established some basic features of

the valence band of silicon. It consists of heavy-hole and light-hole bands, degenerate at

k = 0, and a third band displaced down in energy at k = 0 by spin orbit coupling.

The heavy-hole band is characterized by holes with an energy-independent, but

direction-dependent effective mass. The light-hole band is characterized by holes with an energy- and direction-dependent effective mass. These two bands can be described by the

E vs. k relationship [A-13]

1/2N

E(k) = --^ {Ak2 ±1 B2k'+ + C2(k2k2 + k2k2 + k2k2)

X y X z y z (A-1)

where A, B, and C are the experimentally determined inverse mass band parameters,

? ? ? ^'^ k = (k^ + k^ + k^) , and the upper sign is associated with the holes in the light-hole

band, while the lower sign is associated with the holes in the heavy-hole band. Values of

A, B, and C are obtained by cyclotron resonance measurements at 4 K [A-5, A-14]

.

Although warped, the bands are parabolic for small values of k. 'However, for larger

values of k, the bands become non-parabolic, and along the <100> and <111> directions the

heavy-and light-hole bands are parallel over most of the Brillouin zone. This situation,

however, is not strictly valid for general directions [A-6] . The assumption of overall

parallelism while questionable in III-V compounds, is reasonable in the case of Ge and Si

[A-15, A-16] . The split-off band is separated at k = 0 by an energy A = 0.044 eV [A-l?]

,

and is characterized by an effective mass which is independent of energy and direction. If

the anisotrophy is small, the square root in eq (A-l) may be expanded [A-4] and tne energy

surfaces may be expressed by

E = E, - ^—

(A + B' ) j (6,(})) , {A-2; V ^m

0 where B' = (b2 + 0^/6)^^^, v ^- T c2/2B ' (A±B ' ) , 6 and ^ are the spherical coordinates, E^

is the top of the valence band, and

j(e,4.) = 1 + yY|sin'+e (cos'+cf) + sin'*'.})) + cos'*e - 2/3 | . (A-2a)

Follov7ing the work of Barber lA-9j , we have used the simplified model of the band

structure illustrated in figure Al . In this model the heavy-hole band is considered para-

* bolic and thus the mass m^ is a constant, equal to its value at 4.2 K. For energies within

0.02 eV the light-hole band is considered parabolic with a constant slope corresponding to

* the value of m^ at 4.2 K. For higher energies the light-hole band is assumed to take on approximately the same slope as that of the heavy-hole band, but remains separated from the

heavy—hole band by A/3 eV [A— 12] . The exti'apolation of these two constants creates the kink

in the light-hole band at 0.02 eV. Because of the change in slope, the light-hole band has

27

an energy-varying effective mass and in general can only be described in terms of partial

Fermi-Dirac integrals [A-9] . Although the split-off band is parabolically disturbed, the

apparent effective mass at the top of the valence band is a function of temperature due to the energy displacement at k = 0. Theoretical and experimental studies [A-18, A-19] have

* shown that at high temperatures the heavy-hole band is not parabolic and thus m^ is not

energy- and temperature-independent. However, within the range of temperatures considered

here, the assumption of parabolicity for the heavy-hole band based on Kane's model [A-12] is

reasonable. Other studies [A-20, A-21] support the validity of this model for the valence

band of silicon.

3 . EFFECTIVE MASS FORMULATION

In the case of spherically symmetric energy surfaces, all of the carriers respond in

the same way to a given set of applied forces. The effective mass is a scalar and thus has

the same value for the nail effect, conductivity, and density-of-states masses. For non- spherical energy surfaces, however, this is not the case. The mixed response of carriers

to a set of applied forces is reflected in differences between the different kinds of effective masses. ship

The density-of-states effective mass, m , , is defined from the relation-

-1

k T m . o Dl

Fi/2(n) , (A-3a)

where n V F k T

o

E is the top of the valence band and E^ is the Fermi-level, k is the

V F o

Boltzman constant, Y\/ e'-^^ d£

1+exp ( e-n)

E -E

V

k T

o and i = 1 or 2 refers to the heavyhole and light-hole bands, respectively. For the split-off band, the density-of-states ef- fective mass, m ^, is defined by

D3 ^

ni k T D3 o i/2

Fl/.(n2) , (A-3b)

where ri' n - A/k T.

0 The electric current density in the presence of electric and magnetic fields can be

expressed by [^'--4^

{A-4)

a I. -I- a E.H„ + a E.H„H + .

where E., H„, H are the electrical and maanf^tic field components and the a's represent

j £ m single-energy-surface conductivity coefficients. The first coefficient in eq (A-^!-) is the

zero-magnetic field electrical conductivity, and the second coefficient is associated

with the non-directional Hall effect. We use a to define the conductivity effective mass, m . , by the relationship

0^. = P.

':2<x > i

Ci

(A-5)

29
and a to define the Hall mobility effective mass by means of [A-22] H

a„. = p. Hi 1

q^<T?>

1

(m . ) 2 Hi

(A-6)-

Equations (A-5) and (A-6) apply both to the parabolic and non-parabolic case;:. Both

effective mass and scattering relaxation time are energy dependent and are being averaged

ever energy and direction, but we have separated them in order to solve for effective mass

in terms of relaxation time. By defining P. as in (A-3) , we model the bands as parabolic

* * * ^ with m ., m and m , containing information relative to the true, non-parabolic shape of

Di Ci Hi ^ ^

' the bands. To obtain m . [A-4, A-91, the number of holes in each band is computed from

Di

P, =-^3 /f (k) d3k , (A-7)

'/here f is the Fermi-Dirac distribution function. The conductivity in the direction of an applied electric field may be expressed by [7,-22]

a =

c 3 2 / ^^ak'

iTT -11

9f (E) P d^k •(A-8)

while the conductivity corresponding to the Hall effect is 'given by [A-23;

f

T Sf (E) p

3E

d3k

9k^

{A-9)

Since eqs {A-7) through {A-9) do not use the effective mass, they are valid both for parabolic and non-parabolic band structures. '

. " ;

In order to derive expressions for the effective masses, each warped, non-parabolic

band is expressed by eqs (A-7), (A-8), and (A-9). Corresponding parabolic heavy-hole,

light-hole, and split-off bands are represented by eqs (A-3) , (A-5) , and (A-6) . Thus, by

substituting eq (A~2) into (A-7) , (A-8) , and (A-9) and equating these to eqs (A-3) , (A-5)

,

* * * and (A-6), we can solve for m , m and m^, . . These are discussed separately as follows: UX t^X til (a) The heavy-hole band

In this band, the effective masses are given by .

Dl (A-B-)

[f (- .) ] 2/3

(A-10)

3^^1/2 (n) ^ (-Y) (A-11) CI (a-B-) (n) f, (-Y)

^ 3 /2 1

30
and m

0 Hi (A-B")

'^V^^'^^ f(-Y)

2^,, (n) f (->) L ^3/2 2

n 1/2

(A-12)

where y is defined in [A--2] by

f(Y) = (1 + O.OSy + O.OieSSy^ + 0.000908y^ + ...) ,

f (y) = (1 + 0.01667Y + 0.041369y2 + 0.00090679y^ + 0.00091959y^ + ...)

f2(Y) = (1 - O.Oir.G-'Y + 0.017556y- - 0.0069857y^ + 0.0012610y'* + •••),

and

(A-12a)

*v'2-''' =/ T- (A-12b)

J [1 + exp(e-n)

I

0

Since the heavy-hole band was assumed parabolic, the integrals containing t in eqs

(A-7) through {A-9) are identical and can be canceled out. The anisotropic nature of the

band is represented by the terms in v, A, and B'. The temperature dependence is introduced

by the use of the Fermi-Dirac distribution function, and is only relevant at high dopant

densities

.

(b) The light-hole band

In the light-hole band, as modeled by figure Al , the effective masses of holes are obtained in terms of partial Fermi-Dirac integrals [A-9]

.

Thus

,

3/2 [ ^'''^o* 1/2

F]/2(n) I fA+B')3'2 0 l+exp(c-n) (A-B')-/^

1/2 /e de I

l+exp(e, -n^ C/k T

o

(A-13)

f (+Y)

3m^<T,> {A+B')3/2 o/ l+exp(£-n) (A-B ') ^/-^ "cA^T l+exp{£i-ni)

de ^ f (-Y) y

m

C2 2 O'k^T , oo 3/2

f (+Y) /' T2e'^' 2exp(£-ri) de f, (-y) /* T2e-, exp ( e ; -n i :^(+Y) /* Toe'*' exp(£-ri)de f ^ (-Y) f ) de

1/2 J 2 \/2 1 J

(A+B') 0 [l+exp(e-n)] (A-B') C/k T f 1+exp ( £

, -n i )

]

o

(A-14)

and

31
H2

3<t2>

2^

f (+Y)

(A+B

?/k T

0 f (-Y)

(A-B') / 1+exp { e

. -n^

1/2

(A+B') f (+y) /Ot,/k T

T

.:: V

[

exp ( e-n) de

(A-B '

)

i--^ T^e ^^exp (e T -Pn ) de

f (-Y

2 [1+exp ( ^-n)

7

C/k,

;.l + exp (e j-rii

)

(A-15)

where e - A/3k T, ? = ? - b/l,, r.. = r\ - //3k T, and A and E, are defined m figure Al,

0 - o In this case, because eqs (A-7) through (A-9) were expressed in terms of nartial FermiDirac integrals and eqs (A-3) , (A-5) , and (A-6) were expressed in terms of complete FermiDirac integrals, the dependence on t does not cancel out. Thus, the non-parabolicity of the

light-hole band introduces a dependence on the scattering relaxation time.

(c) The split-off band

Although the split-off band is parabolic, the apparent effective mass in this band will

also exhibit a temperature dependence due to the energy displacement at k = 0. The energy

of a hole in the third band is given by

2v2

E = E ^^k

V 2m (A-16)

where A is the split-off energy (0.044 eV) and A is one of the inverse mass band parameters. Substituting eq (A-16) into eqs (A-7) , (A-8) , and (A-9) , and then equating to eqs

(A-3) , (A-5) , and (A-6) for the split-off band, we obtain

03

(A-17)

and

C3

3 < I" >F ( n ) 3 1/2^ '2^ / [1+exp ( e -T\ ) ] ^ (A-18)

H3

3''T >F (n ) 3 1/2 ^ '2

1 cr

iJ r

^^2"' "Gxpi, e,,-n ,)de.

[1+exp (e2""i2'

-1/2

(A-19)

where k T.

0 The combined hole density-of-states effective mass can be determined by assuming that

the total number of holes in the valence band is equal to the sum of the holes in the

individual bands:

Pi + P, + P3 ; (A-20)

thus ,

32
3/2

(m , ) Dl

3/2

+ (m ^) D3

3/2

'1/2 (n2)/Fi/2 (n; 2/:

(A-21)

In the nondegeneracy limit, (n << 1), ^1 /?_ ^ ^2 ) /^^i/? (n) = exp{-A/k T). This combined

effective mass is defined as the mass corresponding to the density of states of an effective

single equivalent parabolic valence band. This concept is useful in calculations where the

effective density of states at different temperatures can be calculated from one m^. The

explicit temperature variation of the band curvature is included by assuming that the

density of states near the band edges varies in a similar manner as the temperature dependence of the energy gap [9] energy gap at 0 K.

Thus, (m ) ^/'^ is proportional to E /E where E is the

D GO G GO

To evaluate the total band equivalent conductivity and the Hall effective masses, we assume that, in valence band conduction, the total number of holes in motion is equal to the sum of the holes moving on tlie separate energy surfaces, and that tnese holes can be modeled

as moving on a single spherical energy surface. Thus, the ohrnic and the Hall conductivities

in the equivalent valence band are given by

C CI C2 C3

and

"'h = ^Hl ^H2 + ^H3

(A-22)

(A-23)

respectively

.

Substituting eqs (A-5) and (A-6) into (A-22) and (A-23) , it follows that

m =

H

(A-24)

(A-25)

Equations (A-24) and (A-25) are valid for the nondegenerate case.

4 . RESULTS

Equations (A-21) , (A-24) , and (A-25) have been evaluated numerically as a function of

temperature and acceptor doping density for p-type silicon. Values of the band parameters.

4. 27, = 0.63 and C i.93, were determined at 4.2 K by Hensel and Feher [A-6]

and Balslev and Lewaetz IA-14]. The scattering relaxation time, t, was calculated by including lattice, ionized impurity, neutral impurity, and hole-hole scattering mechanisms

[A-11]. In the case of acoustic phonon scattering, both interband and intraband transitions were considered. In order to simplify the calculations and maintain tractability

,

anisotropies in the relaxation time were ignored. A rigorous analysis of the cO;.di:cLivities

33
for nonisotropic scattering would be extremely difficult to carry out because no relaxation

time is expected to exist in the usual sense [A-24]

* Figure A2 shows the dependence of m^^ with temperature in the range from 100 to 400 K. The slight temperature dependence due to the explicit temperature variation of the curvature at the edge of the band results in an effective mass increase of about 5 percent

* in each band at 400 K. This can be seen in the slope of nij-j-j^ - The temperature dependence

due to non-parabolicity is very apparent in the shape of the m^^ curve.

The temperature dependence of the conductivity effective mass and the Hall effective

mass is shown in figures A3 and A4 , respectively, for dopant density equal to 10 ^'^ cm~^.

One consequence of the non-parabolicity of the light-hole band is an increase in the com- bined conductivity effective mass as temperature increases from 50 to 400 K. This happens

because with increasing thermal energy k T, more holes reside in the less parabolic regions

of the light-hole band. The results plotted in figure A3 show an increase in m of about

* ^ 83 .percent in this temperature range. The temperature dependence of m can be attributed

H mainly to the non-parabolicity of the light-hole band. In the temperature range from 50 to

* * 400 K, m^ increases from 0.182 to 0.526 m^ . The slight temperature dependence of m^,^ and m is due to the explicit temperature effect and results in increases of 7.7 percent and

H-i- * * 3.76 percent in m and m , respectively. A larger temperature variation occurs in the

CI HI case of the split-off band because of the additional effects of the energy displacement

at k = 0. , _^ * * Figures A5 and A6 show the variation of m and m. with dopant density and temperature.

C Pi For T > 100 K, m^ varies less than 10 percent in the dopant density range from 10 to

lO-*-^ cm~^. Since the influence of non-parabolicity is reduced in degenerate material [A-19] , it follows as shown in figures A5 and A6 that the variation of effective mass with temperature is much stronger at low dopant densities. At lower temperatures there is a much

greater change in effective mass due to variations in scattering relaxation time with percentage of ionized impurities.

5. DISCUSSION

The idea of temperature-dependent mass is supported by a number of experimental data.

Cardona et at, [A-5] found an increase of about 12 percent in optical effective mass between

90 and 300 K in heavily doped p-type silicon. Cyclotron-resonance studies conducted by

Hensel and Feher [A--6] show that when carrier heating populates deeper regions of the lighthole band, the non-parabolic nature of this band at higher values of k results in an in- crease in the effective mass of holes.

The model used here in the calculation of hole density-of-states effective mass is

* * identical to that of Barber [A-9] , and consequently our results for m and m are in excel- ^ J Di D lent agreement with those of Barber [A-9]. We have extended Barber's work to the calcula-

* * * tions m and m, in p-type silicon. The increase of m by 83 percent at 400 K shown in

C H C figure A3 is much larger than that reported by Costato and Reggiani (15 percent) [A-10]

Their calculation was done over a similar range of temperatures, and their value at 50 K, 34
0.90

0.80

0.70

0.60

0.50

0.40

0.30

0.20

TTT 0.10

- fTIo^

"

mg|

n]D2^-——

171^3

100 200 300 400

TEMPERATURE (K)

Figure A2. Temperature dependence of the density-of-states effective masses ^^-^i '"^2

and m_, in the individual bands, and the combined density-of-states effective mass m of^^oles in silicon.

D A rected for the temperature variation of curvature at the edge of the band.

N = 10 11+ Note that the effective masses have been cor35

10a

Xi

rH

ro -aH

>

•H

73G

•H

0)

4-1 C

•H

ro

•X E

e

E

u

i-H or-\ LU

* X

5 II cr tn

0)

(/) to AT

njE

01>

licon ::

rH

4-1

•-HW LU

U

0)

U-) C

•H CL

M-l

0) W

OJ ^ (—

1

0£

LJ

h0

X.

<DUC

0) 'ac

0)

OJ TS

OJ

!h

-P

ro

0)a

E

0) H

(°uj/>) ssviAi 3AI103JJ3 nnvH

OJ X

•H C

37
toI wwmE>

•HPo

0)

0)o

0)oc

0) TIC

01

0) X)>.

-p •Hwc

(U t)uo

4-)

0)p

+J

a (c 0) u V 0) O ft m g

0) 4-1

E-i M-io • c

in o

f^C -r-t 4-) 0) U °UU/^^IJU

38
roIEo oe>

•Po

(U

M-( LM

(1)

0)c

•rH £1eou

ttj

4-) ino

Q)UG

<U T>C

0)

TD

>,

-P•HWC 0)

0) ^( T! 3

-M

M « O U

4-1 (U ft a

Q)

g

O 0) O 4J

<1) oO

•H • 4-) ID U

flj

C

(D M-J 3 m

•rH

W

h rd

39
nij^ - 0.33 m , is somewhat higher than our calculated value. The discrepancies between our

* results and those of Costato and Reggiani are due mainly to the correction of m^^ for the

explicit temperature dependence of the energy gap, the inclusion of the SO band, and the

consideration of unequal relaxation times in the three bands. Note that our calculations

of effective masses were achieved through more rigorous mathem.atical derivations, while

those of Costato and Reggiani followed a more empirical curve-fitting type of procedure.

The experimental values of density-of-states effective masses of holes in p-type sili- con have been published by several authors fA-5, A-6] , but very little data can be found

for the conductivity and the Hall effective masses, laaking it difficult to properly assess

the value of bur calculations. Magneto-kerr effect measurements conducted by Hauge [A-25]

* indicate that m could increase by as much as 48 percent in the range of temperatures from

0 to 300 K. This is smaller than our calculated percentage increase in m^, but it is impossible to compare our calculations with Hauge ' s experimental results, because our effective mass definition was chosen to be mainly applicable to the study of the Hall and con- ductivity mobility in the low field limit, and this may not apply to the measurements of

Hauge [A-25]

From the results of this work, it can be seen that the approximation of a constant ef

fective mass seems to be inadequate to describe transport properties of holes in silicon

above 50 K. There is a substantial increase in the effective mass of holes from 50 to 400

due to the non-parabol ic ity of the light-hole band, and a smaller, though not negligible,

contribution due to the explicit temperature dependence and the effects of the split-off

band. The validity of this model for the calculation of density-of-states effective mass

has been well established [A-9] . Barber [A-9] has shown that when the temperaturedependent effective masses are substituted into the theoretical expression for intrinsic

carrier density in silicon, the agreement with reported measurements of is within the

limits of experimental error. Application of this model to theoretical calculation of

mobility and resistivity in /j-type silicon ['.-11] has provided excellent agreement between

theoretical and experimental values (resistivity within ±6 percent) over a temperature

range from 100 to 400 K and dopant density range from 10-^^ to 3 -< 10^^ cm~^. This calcula

tion is limited to applications in ohmic mobility and low field Hall effect.

40
REFERENCES

A-1. Radcliffe, J. M. , On Lattice Scattering in Homopolar Semiconductors, Proc. Phys. Soc. London B 68, 675-680 (1955).

A-2 . Barrie, R. , Electronic Conduction in Solids with Spherically Symmetric Band Structure, Proa. Phys. Soc. London B 69, 553-561 (1956).

A-3 . Replacement of the energy-independent effective mass of the parabolic band by the

energy-dependent effective mass of the non-parabolic band is not the only correction

which should be made in calculating the relaxation time. The scattering probability

must also be modified by using the proper carrier wave functions for a non-parabolic

band. [Zawadski, W. , and Szymanska, W. , Phys. Status Solidi B 45, 415 (1971)].

A-4. Lax, B., and Mavroides, J. G. , Statistics of Gal vanomagnetic Effects in Germanium and

Silicon with Warped Energy Surfaces, Phys. Rev. 101 , 1650-1657 (1955).

A-5. Cardona , M. , Paul, W. , and Brooks, H., The Temperature Dependence of Polarizability

of the Free Carriers in Germanium and Silicon, Helv . Phys. Acta 33, 329-346 (1960).

A-6. Hansel, J. C. , and Feher, G., Cyclotron Resonance Experiments in Uniaxially Stressed

Silicon: Valence Band Inverse Mass Parameters and Deformation Potentials, Phys. Rev.

129, 1041-1062 (1963)

.

A-7. Amstrong, D. B., and Champlin, K. S., J. Appl. Phys. 37.' (1966).

A-8. Nathan, J. I., Paul, W., and Brooks, H., Am. Phys. Soc. Bull. 3_, 14 (1958).

A-9. Barber, H. D., Effective Mass and Intrinsic Concentration in Silicon, Solid State

Electronics 10_, 1039-1051 (1967).

A-10. Costato, M. , and Reggiani, E., Temperature-Dependence of the Combined Effective Mass

of Holes in Silicon, Letteve al Nuovo Cimento 3, 239-245 (1970).

A-11. Li, S. S., The Dopant Density and Temperature Dependence of Resistivity and Hole

Mo",:.ility in Boron-Doped Silicon, Solid State Electronics 2]^, 1109-1118 (1978).

A-12. Kane, E. O. , Energy Band Structure in p-Type Germanium and Silicon, J. Phys. Chern. Solids 1 , 82-99 (1956)

.

A-13- Dresselhaus, G., Kip, A. F., and Kittel, C, Spin-Orbit Interaction and the Effective

Masses of Holes in Germanium, Phys. Rev. 95, 568 (1954).

A-14. Balslev, I., and Lewaetz, P., On the Interpretation of the Observed Hole Mass Shift

with Uniaxial Stress in Silicon, Phys. Letters 19, 6 (1965).

A-15. Wiley, J. D., Mobility of Holes in III-V Compounds, Semiconductors and Semimetals,

R. K. Willardson and A. C. Beer, Eds., Vol. 10, 91-194 (Academic Press, New York,

1975)

.

A-16. Cardona, M. , and Pollak, F. H., Energy-Band Structure of Germanium and Silicon: The

k.p Method, Ph^s. Rev. 142, 530-543 (1966).

A-17. Swerdling, S., Button, K. J., Lax, B., and Roth, L. M. , Internal Impurity Levels in Semiconductors: Experiments in p-Type Silicon, Phys. Rev. Letters ^, 173 (1960).

A-18.. Gagliani, G., and Reggiani, L. , Nonparabolicity and Intrinsic Carrier Concentration

in Si and Ge , Nuovo Cimento 30, 207-216 (1975).

A-19. Miyazawa, H., Suzuki, K. , and Maeda , H., Evidence for the <100> Swelling Constant

Energy Surface for Heavy Holes in Silicon, Phys. Rev. 131, 2442-2444 (1963;.

41
A-20. Balslev, I., Influence of Uniaxial Stress on the Indirect Absorption Edge in Silicon

and Germanium, Phys. Rev. 143 , 636-647 (1966).

A-21. Adler, E., and Erlbach, E. , Polarization Dependence of the Indirect Piezoabsorption

Coefficient in Ge and Si, Phys. Rev. Letters 16_, 87 (1966).

A-22. Seeger, K. , Semiconductor Physios (Springer-Verlag , New York, 1973).

A-23. Bube, R. H., Electronic Properties of Crystalline Solids (Academic Press, New York,

1974).

A-24. Long, D. , Galvanomagnetic Effects in p-Type Silicon, Phys. Rev. 107 , 672-677 (1957).

A-25. Hauge , P., Magneto-kerr Effect Study in P-Type Silicon, Ph.D., Thesis, The University

of Minnesota, Minneapolis, Minnesota (1967).
NBS-114A iREv. B-781

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. SP-400-47

2, Gov't' Accession No. ?. ftacifjtettt's Accession No. s^^M

4. TITLE AND SUBTITLE

Semiconductor Measurement Technology

The Theoretical and Experimental Study of the Temperature

and Dopant Density Dependence of Hole Mobility, Effective

Mass, and Resistivity in Boron-Doped Silicon

5. Publication Date November 1979

6. Performing Organization Code

7. AUTHOR(S)

Sheng S. Li

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

University of Florida

Electrical Engineering Dept.

Gainesville, FL 32611

10. Project/Task/Work IJnit No.

11. Contract/Grant No. 7-35741 - ARPA

ENG76-81828 - NSF

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS rsfreef. City. State, ziP)

NBS - Washington, DC 20234

ARPA - Arlington, VA 22209

National Science Foundation - Washington, DC 20550

13. Type of Report & Period Covered Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

This work was funded by the Defeijse Advanced Research Projects Agency under

ARPA Order 2397, Program Code 7Y±0 .

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of moat significant information. If document includes a signi ficant bibliography or

literature survey, mention it here.) Theoretical expressions for computing resistivity and conductivity mobility of holes

as functions of dopant density and temperature have been derived for boron-doped

silicon. The model is applicable for dopant densities from 10^3 to 3 X 10^^ cm ^ and

temperatures between 100 and 400 K. Using a three-band [i.e., heavy-hole, light-hole, and spin-orbit split (SO) bands]

model, the hole mobility was calculated by properly combining the contributions from

scattering by lattice phonons, ionized impurities, and neutral impurities. In addition, the effects of hole-hole (h-h) scattering and nonparabolicity of the light-hole

band were taken into account in the mobility formulation.

To verify our theoretical calculations, resistivity measurements on nine boron-doped

silicon slices with dopant densities from 4.5 x lO^'^ to 3.2 x lO-'-^ cm~^ were performed for 100 ^ T <^ 400 K, using a planar four-probe square-array test structure.

Agreement between our calculated and measured resistivity values was within 6 percent

over the entire range of dopant density and temperature studied here. Excellent

agreement (within ±5 percent) between our calculated hole mobility values and those

of Thurber et at. was obtained for N. < 10-^^ cm These discrepancies are attrib- ^ A —

uted to Wagner's neglect of the effect of deionization of boron impurities at

(over)

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper nami

separated by semicolons) Boron-doped silicon; dopant density; effective mass; hole mobility; ionized

impurity scattering mobility; lattice scattering mobility; neutral impurity

scattering mobility; p-type silicon; resistivity; scattering mechanisms.

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16. ABSTRACT (continued)

higher dopant densities (i.e., he assumed hole density equal to the total boron

density)

Finally, formulations for the dens ity-of-states effective mass, conductivity effective mass, and Hall effective mass are described, and the results are applied to the

calculations of hole masses in boron-doped silicon for 10-^^ < N < 10^^ cm

^ and 50 < T < 500 K. - A -

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