einstein relation and other related parameters of randomly moving
TRANSCRIPT
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
158 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers
in materials with degenerated electron gas
( ) 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
American Journal of Modern Physics 2013; 2(3): 155-167 159
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
160 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers
in materials with degenerated electron gas
( ) ( ) ( )[ ] 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)
American Journal of Modern Physics 2013; 2(3): 155-167 161
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)
162 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers
in materials with degenerated electron gas
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)
American Journal of Modern Physics 2013; 2(3): 155-167 163
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
164 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers
in materials with degenerated electron gas
( ) 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
American Journal of Modern Physics 2013; 2(3): 155-167 165
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
166 Vilius Palenskis: Einstein relation and other related parameters of randomly moving charge carriers
in materials with degenerated electron gas
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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