einstein relation and other related parameters of randomly moving

American Journal of Modern Physics 2013; 2(3): 155-167

Published online May 30, 2013 (http://www.sciencepublishinggroup.com/j/ajmp)

doi: 10.11648/j.ajmp.20130203.20

Einstein relation and other related parameters of randomly moving charge carriers in materials with degenerated electron gas

Vilius Palenskis

Dept. of Radiophysics, Physics Faculty, Vilnius University, Vilnius, Lithuania

Email address: [email protected]

To cite this article: Vilius Palenskis. Einstein Relation and Other Related Parameters of Randomly Moving Charge Carriers in Materials with Degenerated

Electron Gas. American Journal of Modern Physics. Vol. 2, No. 3, 2013, pp. 155-167. doi: 10.11648/j.ajmp.20130203.20

Abstract: In literature there are many cases when for interpretation of conductivity of metals, superconductor in the normal

state and semiconductors with degenerated electron gas the classical statistics is applied. It is well known that electrons obey

the Pauli principle and that electrons are described by the Fermi-Dirac statistics, i. e. in all kinetic processes only the electrons,

which energy is close to the Fermi energy level, can take part, and the other part of electrons, which energy is well below the

Fermi level can not change its energy, and they can not be scattered, and can not take part in kinetic processes. From this

position the classical statistics in principle can not be applicable for characterization of conductivity of metals and other

materials with degenerated electron gas. There we tried as simple as possible on the ground of Fermi distribution, and on

random motion of charge carriers, and on the well known experimental results to characterize the effective density of

randomly moving electrons, their diffusion coefficient, mobility, and velocity at the Fermi level, and many other

characteristics. There we present the general expressions for various kinetic parameters which are applicable for materials

both without and with degenerated electron gas. There also will be presented the above mentioned parameter values for

different metals. The general expressions, based on Fermi distribution of free electrons, are applied for calculation of kinetic

coefficients in donor-doped silicon at arbitrary degree of degeneracy of electron gas under equilibrium conditions. The

obtained results for the diffusion coefficient and drift mobility are discussed together with practical approximations

applicable for non-degenerate electron gas and materials with arbitrary degree of degeneracy. In particular, the drift mobility

of randomly moving electrons is found to depend on the degree of degeneracy. When the effective density is introduced, the

traditional Einstein relation between the diffusion coefficient and the drift mobility of randomly moving electrons is

conserved at any level of degeneracy. The main conclusions and formulae can be applicable for holes in acceptor-doped

silicon as well. A special attention was paid to the Hall effect and to its characteristics obtained from this effect.

Keywords: Conductivity, Thermal Noise, Diffusion Coefficient, Drift Mobility, Hall Mobility, Donor-Doped Silicon,

Einstein Relation

1. Introduction

In this report we attempt to call somebody’s attention to

problems of conductivity interpretation of metals, normal

superconductors and semiconductors with degenerated

electron gas. Every experimenter knows that conductivity of

homogeneous materials is defined by free randomly moving

charge carrier (electron) density and by their drift mobility.

They also know that electrons obey the Pauli principle and

that electrons are described by the Fermi-Dirac statistics.

The latter principles let to ones explain the experimental

results of the heat capacity of electrons in metals [1–8]. The

main conclusions of these investigations were that randomly

move only the small part of electrons, which energy is close

to the Fermi level, and that electrons, which energy is well

below the Fermi level, can not change their energy E

because the Fermi distribution function f(E)=1. It is very

strange that till now (during about 80 years) in many books

[1–8] for solid state physics in expression for conductivity of

metals ∗= mne /2 τσ (her e is the electrical charge of

electron, n is the total density of free electrons in the

conduction band, τ is the average time of relaxation of

electrons, ∗m is the effective mass of electron) and for

superconductors in the normal state [9–13] the total density

of free electrons is included. Thus, this expression of

156 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers

in materials with degenerated electron gas

conductivity is in principle wrong from the view of Fermi

statistics. If n is not the effective density of randomly

moving charge carriers, then the quantity ∗m/eτ is not

their drift mobility. On the other hand, it is well also known

that the thermal noise due to random moving of electrons is

completely described by the real part of conductance

(Nyquist formula) [14, 15]. Thus, electron heat capacity and

thermal noise of metals unambiguously show that in all

kinetic phenomena take place only that part of free electrons,

which energy is close to the Fermi level, because only these

electrons can change their energy under influence of

external fields. There the question arises: how from

measurement of the Hall effect of metals we can get the total

density of free electrons, if in the electric current only the

electrons that have energies close to the Fermi energy take

part? There are many other questions: how find the effective

density neff of randomly moving electrons, their diffusion

coefficient D and drift mobility µdrift, the Fermi energy EF,

velocity of electrons vF at the Fermi level? These problems

are very important for every researcher of solid state

materials, especially for superconductors in the normal state

and for semiconductors with very high doping level. In this

work, we try to present the answers to these questions and

other related problems.

Silicon is the main material for electronics during the past

fifty years. Its basic parameters have been intensively treated

theoretically and experimentally [16-39]. Diffusion

coefficient and drift mobility are necessary for

characterization of material quality and electronic transport

properties. In spite of the importance of these quantities for

device applications, such as bipolar transistors, the results on

heavily doped regions are quite limited, while accurate

values of electron mobility and diffusion coefficient are

essential for advanced engineering. The theoretic problems

for silicon and gallium arsenide are usually treated through

Monte Carlo simulation [17-19]. The ratio of the classical

value of the diffusion coefficient over the mobility for the

majority- and minority-carriers satisfies the Einstein relation

and equals e/kT where k is the Boltzmann constant, T is

the absolute temperature, and e is the elementary charge

[20-26]. Most published papers assume this relation to hold

only for non-degenerate materials. Therefore, at thermal

equilibrium, the diffusion coefficient D of charge carriers in

a degenerate semiconductor is often related to the drift

mobility µ through the modified relation [17, 18, 20-23]:

µFdd

1

E/n

n

eD = , (1)

where n is the total density of free electrons in the

conduction band and FE is the Fermi energy. Similar

relation is used for holes. Expression (1) can be rewritten

as:

µenE

nDe =

F

2

d

d, (2)

where σµ =en is the conductivity. As it will be shown

later the quantity Fdd E/n in left side of (2) is

proportional to the effective density of randomly moving

electrons, while the expression σµ =en contains the total

electron density n; this is wrong because the main

contribution comes from the randomly moving electrons,

located near the Fermi energy, while the contribution due to

the electrons located deep below the Fermi level is next to

zero because of limitations induced by the Pauli principle

[40]. The randomly moving electrons dominate not only in

electric conduction and electron diffusion, but also in

electron heat capacity, electron thermal noise, electron heat

conduction and other dissipative phenomena [40-42].

The electron Hall factor determined as the ratio of the

Hall mobility over the drift mobility for non-degenerate

semiconductors yields the values above unity [27, 28]. But

in some cases the Hall factor of the holes varies between

0,882 to 0,714 at 300 K over the acceptor density range 1814 10310 ⋅≤≤ AN cm

-3 [29]. The Hall factor tends to

unity when the degenerate case is approached since the

averaging of relaxation time τ over energy yields the value at

the Fermi energy. In spite of huge amount of investigations,

there are some unsolved problems of drift mobility and

diffusion coefficient, and Hall effect in silicon with high

level of degeneracy of electron gas.

2. Presentation of Characteristics of Randomly Moving Charge Carriers and Their Analysis

2.1. The Effective Density of Randomly Moving Electrons

It is well known that Fermi distribution function for

electrons reads as

( ) ( ) ,kT/E

Efη−+

=exp1

1 (3)

where E is the electron energy, η is the chemical potential,

k is the Boltzmann’s constant, and T is the absolute

temperature. This function means the probability that level

with energy E is occupied by electron. The total density of

free electrons n in conduction band is

( ) ( ) ,EEfEgn d0

∫∞

= (4)

where g(E) is the density of states in the conduction band.

From the Fermi-Dirac statistics directly follows that

effective density of electrons neff participating in the random

motion and, thus, in the conductivity depends not only on the

density of electrons g(E) and Fermi distribution function

f(E), but it depends also on the probability f1(E)=1-f(E) that

American Journal of Modern Physics 2013; 2(3): 155-167 157

any of such electron can leave the occupied energy level at a

given temperature [40, 41]:

( ) ( ) ( )[ ] EEfEfEgn d-10

eff ⋅= ∫∞

. (5)

Usually for calculation of different kinetic processes in

materials, instead of probability function

( ) ( ) ( ) ( )[ ]EfEfEfEf −⋅=⋅ 11 , ones use the derivative of

the Fermi distribution function ( ( ) εε ∂∂− /f ) (here

kT/E=ε ) [2-6], but [40, 41]

)](1)[()()(

EfEfE

EfkT

f −=∂

∂−=∂

∂−εε

(6)

0,0 0,1 0,2 0,3 0,4 0,5 0,6

0,0

5,0x1019

1,0x1020

1,5x1020

g(E)f(E)(1-f(E))

g(E)f(E)

g(E

), g

(E)f

(E),

g(E

)(1

-f(E

)),

eV

-1cm

-3

E, eV

g(E)

η=20

η=10η=0

a)

b)

Figure 1. Illustration of density of states g(E) and products g(E)f(E), and

g(E)f(E)[1-f(E)] (left axis) as functions of energy for (a) semiconductors with

parabolic energy band g(E)=CE1/2 for three Fermi energy values EF/kT: 0, 10

and 20 at T=300 K: the area under the curve g(E)f(E) represents the total

density of free electrons n (4) in conduction band, and the shaded area

represents the effective density of randomly moving electrons neff (5); (b) for

metals and normal state superconductors with composite density of states:

the light grey area represents the total density of electrons n (4) and the dark

grey area represents the effective density of randomly moving electrons neff

(5). Additionally there are represented the Fermi functions f(E) and (1-f(E))

dependency on energy (dashed curves, right scale).

The probability function( ) ( )[ ]EfEf −⋅ 1

is more

understandable in the case of the interpretation of the

effective density of randomly moving charge carriers (5).

Function (ε∂−∂ /f

) means that only such electrons can

take part in conduction process for which (0>∂−∂ ε/f

).

Illustration of expressions (3)–(5) is presented in Fig. 1.

From this figure it is clearly seen that the effective density of

randomly moving electrons neff (5) is many times smaller

than the total density of free electrons n.

Now let us to evaluate the expression (5). For

non-generate materials the probability ( ) ( ) 111 ≈−= EfEf ,

because ( ) 1<<Ef , and, therefore, all electrons in the

conduction band n participate in the random motion and in

the conduction process:

( ) ( ) EEfEgnn d0

eff ∫∞

== . (7)

This is the case where the classical statistics is applicable.

In the case of high degeneracy and considering that

expression ( ) ( )[ ]EfEf −⋅ 1 has a sharp maximum at the

energy E equal to the chemical potential η, (5) can be

written as:

( ) ( ) nkTEgkTgn <<≈= Feff η , (8)

where g(EF)=g(E) at E=EF.

The chemical potential η and the Fermi energy EF are

related by the following relation [5]:

,E

kTE

−=

2

FF

2

π

3

11η (9)

and the difference between these quantities is only about

0,01 % even at room temperature. So, in many cases for

calculation of different quantities ones use the Fermi

distribution function in the following form:

( ) ( )[ ] 1exp1

−−+= kT/EEEf F. (10)

The density of states at Fermi energy ( )FEg can be

determined from of the experimental results of the electron

heat capacity [5-7] (for silicon see [34]):

( ) eff

22

F

2

3

π

3

πnkTkEgTcV === γ , (11)

where



( ) 2F

2

3

πkEg=γ , (12)

and

( )22π

3

kEg F

γ= . (13)

Thus, for metals and other materials with high

degenerated electron gas

( )k

c

k

TkTEgn V

22Feffπ

3

π

3 === γ; (14)

i. e. neff is proportional to temperature. Experimental

results of conductivity and electron heat capacity and

calculated effective density neff of randomly moving

electrons from the (14) at T=295 K for different metals are

presented in Table 1. The ratio n/n ffe obviously shows

that neff amount in the total free electron density n is about a

few percents and smaller.

Comparison of the effective density of randomly moving

electrons neff with the total density of free electrons n in nSi

is presented for three temperatures in Fig. 2. Here

( ) ( ) ( )( )( ) ( )∫

∫∞

∞ −=

0

0eff

d

d1

EEfEg

EEfEfEg

n

n. (15)

The effective density of randomly moving electrons neff

almost equals the total density of free electrons, if the

shallow ionized donor density is smaller than 1018

cm-3

.

2.2. The Electric Conductivity and Diffusion Coefficient

The electric conductivity σ and the diffusion coefficient

D of charge carriers are related by the following expression

[42, 43]:

T

nDe

∂∂=η

σ 2. (16)

By simple calculation of the derivative n on parameter

(chemical potential) η we have:

( ) ( )[ ] E

kT/EEgDe d

exp11

0

2

ηησ

∂−+∂=

−∞

∫

( ) ( )( )[ ] .E

kT/E

kT/EEg

kT

Ded

exp1

exp2

0

2

−+−= ∫

∞

ηη

(17)

1014

1015

1016

1017

1018

1019

1020

1021

0,1

1n

eff

/n

n, cm-3

200 K

300 K

400 K

nSi

Figure 2. Ratio between the effective density of randomly moving electrons

neff and the total density of free electrons n as a function of total density of free electrons in nSi for three temperatures: T: 200 K, 300 K and 400 K

Table 1. Electrical conductivity and electron heat capacity parameters of different metals.

Element Number of valency electrons

Total density of valency electrons n, (1022 cm-3)

Electrical conductivity σσσσ (105 ΩΩΩΩ-1cm-1)

Electron heat capacity coefficient γγγγ (10-6 J/(K2cm3))

Density of states g(EF) (1022 eV-1 cm-3)

Effective density of free electrons neff (1022cm-3)

Ratio neff/n

Li 1 4,70 1,07 125 3,21 0,0814 0,0173

Na 1 2,65 2,11 58,4 1,49 0,0379 0,0143

K 1 1,40 1,39 45,9 1,17 0,0298 0,0213

Rb 1 1,15 0,80 43,2 1,10 0,0280 0,0244

Cs 1 0,91 0,50 45,7 1,17 0,0297 0,0326

Cu 1 8,47 5,88 98,0 2,51 0,0636 0,00751

Ag 1 5,86 6,21 62,9 1,61 0,0408 0,00697

Au 1 5,90 4,55 71,4 1,83 0,0464 0,00786

Be 2 24,7 3,08 34,9 0,89 0,0226 0,00092

Mg 2 8,61 2,33 93,0 2,38 0,0604 0,00701

Ca 2 4,61 2,78 111 2,85 0,0724 0,0157


Sr 2 3,55 0,47 108 2,76 0,0702 0,0198

Ba 2 3,15 0,26 71,7 1,83 0,0466 0,0148

Zn 2 13,2 1,69 69,8 1,78 0,0453 0,00343

Cd 2 9,27 1,38 52,9 1,35 0,0344 0,00371

Al 3 18,1 3,65 135 3,45 0,0877 0,00485

Ga 3 15,4 0,67 50,5 1,29 0,0328 0,00213

In 3 11,5 1,14 108 2,75 0,0699 0,00607

Tl 3 10,5 0,61 85,2 2,18 0,0553 0,00527

Sn 4 14,8 0,91 109 2,79 0,0710 0,00480

Pb 4 13,2 0,48 163 4,17 0,106 0,00802

Conductivity and electron heat capacity data are taken from books [5-7, 44]. Conductivity and neff are presented for T = 295˚ K.

Table 2. Kinetic parameters of various metals

Element Diffusion coefficient D (cm2/s)

Drift mobility, µµµµdrift

(cm2/Vs)

Electron velocity vF (107 cm/s)

Fermi energy EF (eV)

Ratio EF/kT

Wave vector kF (108 cm-1)

Free pass length lF=vFττττF (Å)

Li 20,9 820 4,9 0,68 26,9 0,42 128

Na 88,4 3470 10,1 2,90 114 0,87 263

K 74,1 2910 9,3 2,43 95,7 0,80 240

Rb 45,3 1780 7,2 1,49 58,5 0,63 188

Cs 26,7 1050 5,6 0,88 34,5 0,48 144

Cu 147 5770 13,0 4,81 189 1,12 338

Ag 241 9490 16,7 7,92 312 1,44 434

Au 156 6120 13,4 5,11 201 1,16 349

Be 216 8480 15,8 7,08 279 1,36 410

Mg 61,2 2410 8,4 2,01 79,1 0,73 219

Ca 61,0 2400 8,4 2,00 78,8 0,73 218

Sr 10,6 418 3,5 0,35 13,7 0,30 91

Ba 8,86 348 3,2 0,29 11,4 0,28 83

Zn 59,2 2330 8,3 1,94 76,5 0,71 215

Cd 63,7 2510 8,6 2,09 82,3 0,74 223

Al 66,1 2600 8,7 2,17 85,3 0,76 227

Ga 32,4 1270 6,1 1,06 41,9 0,53 159

In 25,9 1020 5,5 0,85 33,5 0,47 142

Tl 17,5 688 4,5 0,57 22,6 0,39 117

Sn 20,4 800 4,9 0,67 26,3 0,42 126

Pb 7,20 283 2,9 0,24 9,30 0,25 75

The data for calculation at T=295 K were used from Table 1.

On the other hand, the probability function

( ) ( )[ ]EfEf −1 can be presented as [40, 41]:

( ) ( )[ ] ( )( )[ ]2

exp1

exp1

kT/E

kT/EEfEf

µµ

−+−=− . (18)

Thus, now the conductivity (17), accounting (18), can be

presented in the following form:

( ) ( ) ( )[ ] ,nkT

DeEEfEfEg

kT

Deeff

2

0

2

d1 =−= ∫∞

σ (19)

where



( ) ( ) ( )[ ] EEfEfEgneff d10

−= ∫∞

(5a)

is the effective density of randomly moving charge carriers

(see (5)). The expression (19) unambiguously shows that

electric conductivity in all cases is determined by the

effective density of randomly moving charge carriers (5),

but not by the total free electron density in the conduction

band. The total free electron density for determination of

conductivity can be used only for materials with

nondegenerated electron gas, when the total density of free

electrons is smaller than 1018

cm-3

(Fig. 2).

2.3. Einstein Relation and Related Quantities

The conductivity (19) can be presented also as

eff

2

drifteff nkT

Deen == µσ , (20)

where µdrift is the drift mobility of randomly moving charge

carriers in isotropic materials with one type of charge

carriers (electrons or holes). From this general relation (20)

follows the Einstein’s relation

e

kTD =driftµ . (21)

This expression is true for both degenerated and

non-degenerated homogeneous materials with one type of

free randomly moving charge carriers (for electrons or for

holes). Equation (21) can be expressed as

T/D ϕµ =drift ; (22)

where e/kTT =ϕ is the thermal potential. In principle,

(21) and (22) show that the current components caused by

drift and diffusion of randomly moving charge carriers at

equilibrium compensate one another, and the resultant total

current is zero. It is true for all homogeneous materials

under equilibrium conditions. The presented relations (5),

(16) and (19) show that (1) and (2) are not applicable to

degenerate electron gas in silicon.

From (8) and (20) for highly degenerated materials we

obtain that diffusion coefficient of randomly moving

electrons

( )( )F2 Ege/D σ= . (23)

Thus, on the ground of experimental results of the

conductivity and the electron heat capacity we can evaluate

the diffusion coefficient D (from (23)) and drift mobility

driftµ (from (20) or (21)) for randomly moving charge

carriers in materials with highly degenerated electron gas.

The calculated results for different metals at T=295 K are

presented in Table 2.

The diffusion coefficient usually also is determined by the

average of square velocity of charge carriers and by their

average relaxation time (for metals free pass time):

.vD >><<= τ2

3

1 (24)

For highly degenerated electron gas

( ) ( ) ( )[ ]( ) ( ) ( )[ ] F

2F

0

0

2

3

1

d1

d1

3

1 ττ

vEEfEfEg

EEfEfvEgD =

−

−=

∫

∫∞

∞

, (25)

i. e. the diffusion coefficient is determined by the square

velocity and free pass time of randomly moving charge

carriers at the Fermi level. From (8), (20) and (25) ones

obtain

( ) F2FF

2

3

1 τσ vEge= . (26)

This expression is well known for metals and is obtained by

solving the Boltzmann kinetic equation [2, 3].

From (21) and (24) one obtains that drift mobility

kT

ve

3

2

drift

>><<= τµ . (27)

Including to this relation the conduction effective mass m* of

charge carriers the drift mobility can be presented in the

following form:

( )( )kT/

vm/

m

e *

* 23

21 2

drift

><⋅><= τµ (28)

or

( ) ** m

e

kT/

E

m

e ><=><⋅><= τατµ εdrift23

, (29)

where <E>=(1/2)m∗<v

2> is the average kinetic energy of

randomly moving charge carriers. Thus, we obtained the

fundamental relation for drift mobility of randomly moving

charge carriers in homogeneous materials with one type of

charge carriers (electrons or holes).The relation (29) is true

for non-generated and for degenerated materials, including

metals. The factor

( )kT/

E

23ε

><=α (30)


shows, how many times the average kinetic energy >< E

of randomly moving charge carriers is larger than ( )kT/ 23 .

For non-generated material 1ε =α and

*m

e ><= τµdrift , (31)

it is the classical case, but for highly degenerated

semiconductors, metals and superconductors in the normal

state

13

2 Fε >>=

kT

Eα , (32)

i. e. the drift mobility of random moving charge carriers in

metals at room temperature in many cases is about hundreds

times larger (see ratio EF/kT values in Table 2) than it

follows from the classical expression (31).

1014

1015

1016

1017

1018

1019

1020

1021

-0,6

-0,4

-0,2

0,0

0,2

0,4

EF,

<E

>, e

V

n, cm-3

200 K

300 K

400 K

nSi

EF

<E> 200 K 300 K 400 K

~n2/3

Figure. 3. Fermi energy EF and average kinetic energy <E> of randomly

moving electrons as a function of total free electron density in nSi for three

temperatures T: 200 K, 300 K and 400 K.

From equation (29) follows that the drift velocity driftv of

electrons in the electric field E can be presented as

EEvkT)/(

vm)/(

m

e *

* 23

212

drifttdrif

><⋅><== τµ . (33)

A very important behavior of electrons in materials follows

from this expression: the drift velocity of electrons in the

electric field E is proportional to the square velocity of

randomly moving electrons. This result can also be obtained

by solution of the Boltzmann kinetic equation.

Fig. 3 illustrates the dependence of the Fermi energy EF

and the average kinetic energy <E> of randomly moving

electron on the total electron density n for n-type Si at three

temperatures: 200 K, 300 K and 400 K. The average kinetic

energy <E> of randomly moving electrons equals

( ) ( ) ( )( )( ) ( ) ( )( )∫

∫∞

∞

−

−>=<

0

0

d1

d1

EEfEfEg

EEfEfEEgE . (34)

The average kinetic energy of randomly moving electrons

in nSi equals <E>=(3/2)kT for the total electron densities n

smaller than 1018

cm-3

, but for 2010>n cm

-3 <E>≈EF and

does not depend on temperature.

From (20) and (29) follow the general expression for

conductivity

( )kT/

E

m

eenen

*t

23effdrifeff

><⋅><== τµσ . (35)

For non-generated material ( 1=εα and nne =ff ):

*m

een

><= τσ , (36)

as in classical case, but for highly degenerated

semiconductors and metals ( ( )kTEgn Feff = ):

( ) ( ) F2FF

2Feff

3

1

23ττσ vEge

kT/

E

m

een

*=⋅><= . (37)

This expression is the same as (26). The correctness of

presented diffusion coefficient and drift mobility values in

Table 2 also follows from the electronic heat conductivity

for randomly moving electrons in metals [3, 5, 6]

kDnTvlvcV eff

2

F2F

33

1

3

1 πγτκ === , (38)

where l is the free pass length of randomly moving electrons;

cV is the electron heat capacity. From (37) and (38) we obtain

the well known Wiedemann-Franz law [2, 3]

2

3

2

=e

k

T

πσκ . (39)

So, the diffusion coefficient calculated from (38) gives the

same results as from (23).

In behalf that Einstein relation is valid for metals shows

this fact that Einstein relation for metals can be obtained

from the Wiedemann-Franz law:

2

3

2

3

2

3

2

driftdrifteff

eff

===e

k

T

D

e

k

Ten

kDn

T

πµ

πµ

πσκ

(40)

and now

e

k

T

D =driftµ . (41)



2.4. Hall Mobility and Its Comparison with the Drift

Mobility

Now compare the obtained drift mobility results with

that from the Hall effect measurements. On the ground of

classical statistics for highly degenerated materials the Hall

mobility usually is expressed as [45]

** m

er

m

e

B

FH

HH

ττµ =⋅><==xΕ

Ε, (42)

where HE is the Hall electric field strength in y direction

perpendicular to the current flow direction; xE is the

applied external electric field strength in x direction; B is

the magnetic flux density in z direction; 22

H ><>=< ττ /r is the Hall factor, which depends on

charge carrier scattering mechanism, but for high

degenerated charge carriers rH=1.

The derivation of (42) is done on the ground of classical

free electron distribution, but such approximation for

electrons is valid only for isotropic materials with

non-degenerated electron gas. Now there a short

description of Hall effect considering that electrons obey

the Fermi distribution will be presented. In this case

accounting (33), the current density in x direction is

driftx vEx effx en−== σj

( ) xEkT/

E

m

eeneff

23

><><= ∗τ

, (43)

but not (as it is in many books of Physics of Solid States):

xx xxm

τeenenσ EvE ∗

><=== driftj . (44)

The expressions (43) and (44) coincide only when the total

free electron density is smaller than 1018

cm-3

(see Fig. 4).

From (43) follows that

effdrifx

en

xt

jv −= . (45)

The drift velocity also can be expressed as

><= ∗ τm

Fv eff

drift , (46)

where effF is the effective force acting to the mobile

electrons in material in applied external electric field E.

From (30), (33) and (46), when ones operate with average

quantities of current density and drift velocity, follows such

very important relation:

Eεαττ ∗∗><>=<=

m

e

m

Fv eff

drift (47)

and

EF eεα=eff . (48)

Here must be pointed, if ones use the relation EF e= ,

instead of the drift velocity (45) he must use the electron

velocity ∗= m/kv xx ℏ (where kx is the wave vector of

electron), and for current density jx he has to solve the

Boltzmann kinetic equation.

The use of the effective force (48) in many cases

facilitates the solution of the problem by using drift velocity.

Thus, at equilibrium condition in perpendicular magnetic

field (here B is magnetic flux density):

Bee H driftx vE =− εα (49)

and

Bne

xH

effεαj

E −= . (50)

On the other hand,

BR xHH jE = , (51)

From (50) and (51) follows that

effH

1

neR

εα−= . (52)

The expression (51) can be presented as

BRBR xHxHH EjE σ== , (53)

and

σHx

H RB

=E

E. (54)

This is completely right expression in all cases. From (35),

(52) and (54) follows that

Hm

eR

Bµτσ =><−== ∗H

x

H

E

E. (55)


If one uses the classical expressions for the total free

electron density (7), drift mobility (31), electrical

conductivity (36), and current density (44), which are not

applicable for Fermi particles in the case of highly

degenerate materials, he obtains that

enR

1H −= , (56)

but for the Hall mobility he obtains the same result as (55). It

is due to that two errors compensate one another: how many

times the drift mobility is decreased, the same times the

density of randomly moving electrons is increased (compare

(52) and (29)).

In general case in (55) there must be included the Hall

factor rH due to different mechanisms of charge carrier

scattering:

HH rm

eRr

B ∗><−== τσH

x

H

E

E, (57)

1013

1014

1015

1016

1017

1018

1019

1020

1021

0,1

1

10

~n-2/3

<E

>/(

3/2

)kT

, n

eff

/n

n, cm-3

T = 300 K

neff

/n

<E>/(3/2)kT

~n2/3

Figure 4. Comparison of the average kinetic energy <E> and the effective

density neff of randomly moving electrons as a function of total free electron

density n for parabolic energy band, i. e. when g(E)~E1/2, at T= 300 K.

In the case of materials with degenerated electron gas rH=1:

HF

Hx

H µτσ =−== ∗m

eR

BE

E, (58)

The correct expressions are (52) and (55), they are valid

for degenerated and non-degenerated homogeneous

materials with one type of free charge carriers (electrons or

holes).

From (58) it is seen that Hall mobility for highly

degenerated materials is determined by average relaxation

time of electrons at Fermi level. According to investigations

in the work [46], for all isotropic metals and highly

degenerated semiconductors the charge carrier relaxation

time at Fermi level at room temperature is general and equal

( )kT/F ℏ=τ ; (59)

where π2/h=ℏ is the Plank’s constant. Expression (59)

also explains the experimental results of silicon that Hall

mobility of electrons and holes did not depend on the doping

concentration, when the doping concentration exceeds 1019

cm-3

, because at these doping concentrations we have a

material with high level of degeneracy of charge carriers. At

T=295 K ≈Fτ 2,6 10-14

s, and the Hall mobility

( )∗≈ m/m0H 46µ cm2/Vs.

Now from (25) and (59) we can find electron velocity at

the Fermi level vF and the Fermi energy EF considering that

effective mass m* of electron is equal to free electron mass

m0, i. e. 220 /vmE FF = . The calculation results for different

metals are presented in Table 2. In this table for different

metals there also are presented the free pass length of

electrons FFF vl τ= , the obtained values are close to

experimental data [47].

Usually for evaluation of the Fermi energy it is

considered that Fermi surface is spherical, and density of

states is [5, 15]:

( ) ( )( ) EmmEg ∗∗= 223πℏ (60)

and

( )( ) 3222 32/

F nmE π∗= ℏ (61)

But for many metals and superconductors in normal state

the density of states has composite view (Fig. 1(b)), and

there is no definite relation between the total density of

charge carriers n and the Fermi energy EF [48].

From comparison of (29) and (58) follows that for high

degenerated materials the drift mobility (29), can tens or

hundred times be larger than the Hall mobility (58).

In Fig 4. there is presented the comparison between the

average kinetic energy <E> and the effective density neff of

randomly moving electron as a function of total free

electron density for parabolic energy band, when the

density of states ( ) E~Eg . From this figure it is seen

that



( ) 123

eff =⋅><n

n

kT/

E (62)

or

( ) eff23 n

n

kT/

E =><=εα (63)

From Fig. 4, and from (59) and (61) it is seen that

characteristics obtained from classical description of Hall

effect coincide with that obtained on the base of the Fermi

statistics only when the total free charge density is smaller

than 1018

cm-3

. The charge carrier density obtained from the

Hall effect measurements only in particular cases is equal

to the total free charge density for metals with spherical

Fermi surfaces, but even in these cases the factor εα

characterizes not the charge density, but the drift mobility.

For multivalent metals and superconductors in the normal

state with composite density of states (Fig 1b) the relation

(60) and (61) are not valid.

As shown in (8), the randomly moving charge carrier

density effn in high degenerated materials is not constant,

but is proportional to the temperature, and the relaxation

time close to room temperature T/~ 1Fτ . Thus, in this

temperature range the drift mobility µdrift~1/T2. These

investigations show that ones can be very careful in the

interpretation of the Hall effect and conductivity

measurement results. For example, there are many strange

interpretations of charge carrier density and drift mobility

of superconductors in the normal state from Hall effect and

conductivity temperature measurements [9-13].

2.5. Experimental Possibilities of Determination

Conductivity Components in Materials with Highly

Degenerated Electron Gas

Demonstration of experimental results is presented on the

base of silicon. From (23) the diffusion coefficient of

randomly moving charge carriers in highly degenerated

materials is expressed as

( )FEgeD 2σ= . (64)

1014

1015

1016

1017

1018

1019

1020

1021

100

1000

µdrift

=1/eρneff

µH cl

µdrift

µdrift calc

µ, c

m2/V

s

n, cm-3

µH cl

nSi

T=300 K

Figure 5. Hal mobility µH (from [35,36]) and drift mobility µdrift (65) as

functions of the total density of free electrons in nSi at T=300 K. The results

of resistivity ρ measurement have been taken from [36-39]. The drift

mobility µdrift calc is calculated by (29).

1014

1015

1016

1017

1018

1019

1020

1021

10

100

1000

D, cm

2/s

;

µ drift, cm

2/V

s

n, cm-3

D

µdrift

nSi

T=300 K ~n2/3

Figure 6. Comparison of diffusion coefficient and the drift mobility of

randomly moving electrons on total density of free electrons in nSi at

T=300 K.

Thus, from the experimental results of the conductivity and

electron heat capacity it is possible to evaluate the electron

diffusion coefficient in highly degenerated materials. The

drift mobility and diffusion coefficient of randomly moving

electrons at any level of degeneracy is determined as:

( ) ( )effeffdrift 1 ne/en/ ρσµ == . (65)

eff2 ne

kTD

ρ= (66)

where neff can be obtained from (8) or from (52). In Fig. 5

there the Hall mobility and drift mobility of electrons and

their dependencies on the total density of free electrons at

temperature T=300 K in silicon are compared. Comparison

of drift mobility and diffusion coefficient dependencies on

the total density of free electrons is presented in Fig. 6.

Though these dependencies are complex, but their ratio

always is the same. The relaxation time dependence on the

total density of free electrons in silicon is shown in Fig.7


1014

1015

1016

1017

1018

1019

1020

1021

10-14

10-13

τ, s

n, cm-3

τF

nSi

T=300K

Figure 7. Relaxation time dependence on total density of free electrons in

nSi at T=300 K. s1062 14−⋅≈≈ .kT/F ℏτ . This dependence can be

described by ( )1713105111051 ⋅+⋅+= −

./n/.Fττ s, here n is in

cm-3.

2.6. Kubo Formula, Thermal Noise and Einstein Relation

Einstein relation (21) also follows from Nyquist formula

for the thermal noise. The spectral density of the current

fluctuations Si0 at low frequencies ( ( )τπ21 /f << , here τ

is the relaxation time of charge carriers) can be presented as

[14, 49, 50]

L

ADne

L

AkT

RkTSi eff

20 44

14 === σ , (67)

where R is the resistance of the sample of the test material;

A and L are respectively the cross-section and the length of

the sample. This relation is applicable for all homogeneous

materials at equilibrium. The both equalities of (67) can be

obtained also from the general Kubo formula for

conductivity [2, 15, 40]:

( ) ( ) 110d

1 ττσ >⋅+<= ∫∞

tjtjkT

xxx . (68)

i.e. the conductivity is defined by the autocorrelation

function on time ( ) ( ) ( ) >⋅+=< tjtjk xxjx 11 ττ of the

current density jx(t) fluctuations on time t due to random

motion of charge carriers. For stationary processes the

autocorrelation function depends only on time difference 1τ .

Relation (68) can be presented in the form of Nyquist

formula:

( ) ( ) 110d444 ττσ ∫

∞== jxx kL/AR/kTL/AkT . (69)

The right side of this expression according to

Wiener-Khintchine theorem is the spectral density of the

current fluctuations, because at low frequencies

( ( )k/f πτ21<< 1cos ≈kτω (here kτ is the

autocorrelation time).

In (68) the current density fluctuations can be replaced by

the independent and randomly moving charge carrier

velocity fluctuations:

( ) ( ) 110 1d2 ff ττσ >⋅+∫ <

= ∞

=∑ tl

vtvkT/e en

l lx

( )1

d0 1eff2 ττ∫

∞

=

vkkT/ne . (70)

Considering Wiener-Khintchine theorem at low frequencies

( ( )k/f πτ21<< the relation (70) can be presented as:

( )( ) 0eff2 4 vx SkT/ne ⋅=σ , (71)

where 0vS is the spectral density of charge carrier velocity

fluctuations.

The particle random (Brownian) motion (the variance of

the displacement) during time t can be described by

Einstein’s relation:

Dttx 2)(2 >=< , (72)

where D is the diffusion coefficient of the particle. If the

charge carrier random velocity is v(t), then the displacement

∫= t't'tvtx

0)d()( (73)

and

∫ ∫ ><>=< t t''t't''tv'tvtx

0 0

2 dd)()()( . (74)

Now one can transform the variables: 'tt =0 and

't''t −=1τ :

( ) ∫ ∫−

−>=< t tt

t vkttx0 110

2 0

0)d(d ττ , (75)

where ( )1τvk is the autocorrelation function of the charge

carrier velocity fluctuations. If the observation time is much

longer than the autocorrelation time kτ , i.e. kt τ>>0 and

ktt τ>>− 0 , then

∫∫∫∞∞

∞− =>=<00 0

2 )d(2)d(d)( ττττ vv

tktkttx . (76)

Considering the Wiener-Khintchine theorem the spectral

density of charge carrier velocity fluctuations can be

expressed as:

∫∞=0

dπ2)cos(4)( τττ fkfS vv . (77)

At low frequencies ( )1π2 <<kfτ :

∫∞≈00 d)(4 ττvv kS . (78)

From (72), (76) and (78) follows that



DSv 40 = , (79)

and from (71) and (79):

( ) drifteffeff2 µσ enkT/Dnex == . (80)

and

e/kT/D =driftµ .

This derivation once more confirms that Einstein relation is

correct for all homogeneous materials with nondegenerated

and degenerated electron gas. The presented derivation

unambiguously show that, under equilibrium conditions,

the Einstein relation for one type of charge carriers in

homogeneous materials is always fulfilled, if the Nyquist

formula (67) is valid. This conclusion is independent

neither of non-parabolicity of conduction band nor of the

mechanisms of electron scattering.

3. Conclusions

This work tries to attract attention to interpretation of

transport properties of metals and degenerate

semiconductors and colossal errors caused when classical

statistics is applied for estimation of transport parameters for

such materials. On the ground of detail analysis of thermal

fluctuations in homogeneous materials with different level

of degeneracy of electron gas it is presented the general

expression for drift mobility of charge carriers in materials

with one type of charge carriers (electrons or holes). The

presented expression for drift mobility is valid in all cases

for materials with different level of degeneracy of electron

gas: for non-generated semiconductors, degenerated

semiconductors, metals and superconductors in normal state.

There are explained the relation between Hall mobility and

drift mobility of charge carriers at different level of

degeneracy of electron gas. It is shown that the Einstein

relation between the diffusion coefficient and the drift

mobility of charge carriers is valid even in the case of high

degeneracy of electron gas.

The general expressions for effective density of randomly

moving electrons, their diffusion coefficient, drift mobility,

and conductivity are derived from the Fermi distribution in

relaxation time approximation and illustrated for electrons in

silicon. The calculated values of effective density of

randomly moving electrons, their diffusion coefficient and

drift mobility for silicon at any degree of degeneracy there

are presented for first time. The well known results on

resistivity and Hall effect measurement are discussed, the

comparison between the Hall mobility and the drift mobility

of randomly moving electrons is made and their specific

behavior at high level of degeneracy is explained. The

presented expressions are valid for holes as well as.

It is unambiguously shown that under equilibrium

conditions the Einstein relation is always valid for one type

of charge carriers in homogeneous materials, if the Nyquist

formula is valid. This relation does not depend neither on the

non-parabolicity of conduction band nor on mechanism of

scattering of charge carriers: the expressions for the Nyquist

formula and the Einstein relation remain the same.

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einstein relation and other related parameters of randomly moving

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