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The thermodynamics of the phonon drag in ionic semiconductors
Georgescu,
Physica
L.
35
107-l 13
1967
THE THERMODYNAMICS
OF THE PHONON
DRAG IN
IONIC SEMICONDUCTORS by L. GEORGESCU Faculty
of Physics,
University
of Bucharest laboratory
Bucharest,
of molecular physics,
Rumania
Synopsis Acoustical
and optical
of the thermodynamic are considered into account
phonon principles
as charge carriers. the contribution
drag in ionic semiconductors of irreversible The expression
of acoustical
processes.
are studied on the basis The
of thermoelectric
and optical
phonon
electrons
and holes
coefficients,
taking
drag, is given.
In a metal or semiconductor, the normal vibrations of the lattice are replaced by a system of quasiparticles, phonons, a system generally considered as being in thermodynamic equilibrium. In the phenomena of transport, the displacement of the phononic system from the state of equilibrium, gives birth to the acousticall) 2) a) and opticala) phonon drag. In studying the phonon drag in the phenomena of transport, only the contribution of acoustical phonons was considered, both in the case of semiconductors5) and of metals6) 7). The thermoelectric phenomena were thoroughly studied from the thermodynamic point of view, starting with Callena). The thermoelectric coefficients are functions of the phenomenological coefficients; both the interaction mechanisms of the charge carriers and the type of semiconductor, degenerated or non-degenerated, are included in the phenomenological coefficients. Thus very general relations between the thermodynamic flows and forces are obtained. A certain charge carrier scattering mechanism, intended for the particular study of some special processes in solid bodies, can be demonstrated also phenomenologically. On this line, Prices) had worked out thermodynamically the acoustical phonon drag with semiconductors, the terms related to the phonon drag being expressed in terms of supplementary coefficients in the expression of the current density and of the flow of energy. Based on the thermodynamic principles of irreversible processes la) rr), the author tries to emphasize phenomenologically the possibility of acoustical and optical phonon drag existence, by explicitly considering two scatter-
107 -
108
L. GEORGESCU
ing mechanisms: the scattering of charge carriers by acoustical and optical phonons. The considered thermodynamical system is an ionic type semiconductor, having electrons in the conductive and holes in the valence bands. The concentration of charge carriers is assumed to be low in order to be able to write linear relations between the thermodynamical flows and forces. Generally, we may consider the ionic semiconductor with n types of charge carriers, and with an ek (k = 1, 2, . . ., n) electric charge per unit of mass to be subjected to a temperature gradient (VT # 0), to a gradient of chemical potential (0,~ f 0), and to the action of an electric field (E # 0). The magnetic field is zero (H = 0) ; chemical reactions within the system are excluded. The whole system is divided into two subsystems: the charge carrier subsystems (electrons and holes), and the phonon subsystem (acoustical and optical phonons). The local temperature of the two subsystems is considered to be the same (Tcarriers = TDhonoIls= T). The temperature gradient is the only thermodynamic force acting on the phononic system. The production entropy for the whole system (charge carriers and phonons) appears as 12) :
or an analogous
form: To = -Jb*VT
-
5
Jk*[vpk
- e&E]
k=l
were Jq is the total flow of heat (of the charge carriers plus phonons), Jh is the total flow of entropy (of the charge carriers plus phonons), Jk is the flow of material of the component K. Flow Jq and Jh are related:
J; = Jp -
I: ,LLk’Jk
(3)
k
being the chemical potential of the component k. The following phenomenological equations result from the expression the production entropy (1) :
pk
of
the thermodynamic forces appearing as Xi = -ezVp, - TV(pJT) (9’ is electric potential), X,, = -(VT/T) (independent from the index i), since we considered the temperature of the two subsystems as being equal). The
THERMODYNAMICS
phenomenological
OF THE PHONON
are
coefficients
DRAG
IN IONIC
second
109
SEMICONDUCTORS
degree
tensors
(Lkg = L$fl),
a, p = 1, 2, 3). In order to simplify the calculation, the semiconductors are considered to be cubically symmetric; thus only the components Lit;sl’ = Li:r2’ = L&3) of the tensor Lk B, are different from zero. The magnetic field of the system being zero, the Onsager reciprocity relations would read:
The total
current
and the total flow of heat are defined by the
density
relations :
I =
Jp = C Jqk.
c edk,
(7)
k
k
In the system under consideration, only the electrons and holes (ei f 0, e.2 # 0) participate to the total electrical current, the acoustical and optical phonons having no charge (es = e4 = 0). Similarly, the chemical potential of the phonons is zero (,~a = ,u4 = 0). By replacing in (4) the expression of thermodynamic forces, and considering relation (7), the total electrical current expression results as:
The coefficient of electrical conductivity ez(Li:’ + L&j,‘,‘)is not affected by the phonon drag, as it is determined exclusively by the charge carriers. In the stationary state with zero electrical current (I = 0) one finds from (8) for the coefficient of thermoelectric force a, the expression: 1 a=
x ek[L$’ -
the identity
(9)
ez(Lii) + L&i))
T (considering
Lii’,&!&]
i,k
__.
ag = x aioi and the fact that the electrons
and
holes form two independent subsystems; the Lrs type coefficients being null), From expression (9), the explicit form of the coefficient of thermoelectric force is : a=---*
1
Ly$ + Lg
eT
+ Lg) -
Lgu -
L’,1,’ + L&j -
Lg
+ Lg
+ L&’ + L&).&C + Lgp Ljl,’ + Lg
with the notations: electrons
ei = -e,
~1 = p
holes es = e, -_ruz = ,u -I- &G (EG width of forbidden
1 (10) (11)
band).
110
L.
GEORGESCU
From expression (lo), the effect of the electrons and of the phonon drag on the coefficient of thermoelectric force al and the effect of the holes and of the phonon drag on the coefficient of the thermoelectric force as, can be written separately : 1 cc1 -
or in a condensed
y
[
L’?’ L$2,’ + $kj eL$’
+ 2i!!i!
11
-
11
f
1
(12)
form: al=ae m2 =
where me, CY~are the coefficients charge carriers.
ah
+
ae,p
(14)
+
ah, p
(15)
of the thermoelectric
Coefficients CL~, p and Q,,p represent the coefficients force due to the acoustical and optical phonon drag:
force
due to the
of the thermoelectric
Relations (14) and (15) s h ow that the coefficient of the thermoelectric force is made up of the combined effect of the charge carriers and of the acoustical and optical phonons. In order to determine the form of the Peltier coefficient and of the coefficient of thermal conductivity, taking into consideration also the acoustical and optical phonon drag, it is necessary to determine the total flow of entropy. To this end, the expression of the flow of entropy, knowing the production entropy can be calculated directly. The relationship between the phenomenological coefficients in going over from the new forces to the old thermodynamic forces is as follows: q;’
= L$?
_q
zzzLz;’ + _&&Ji
Lit’ = Lp
+ (L$’ + LB;‘) #uk-
(18) LgugQ.
The same result is obtained if we calculate initially the total flow of heat in (5), the flow of entropy resulting from relation (3). Irrespective of the selected approach the following, generally known relation is obtained:
J& = ITI - AVT.
THERMODYNAMICS
OF THE PHONON
DRAG
IN IONIC
111
SEMICONDUCTORS
The contributions of phonon drag are also included coefficients 17 and A. Thus, the Peltier coefficient is:
in the thermoelectric
the Onsager relation Ta = --17 being fulfilled, if we compare relations (10) and (19). If the contribution of the carriers (both electrons and holes) to the coefficient of thermic conductivity, omitting the contribution of the phonon drag, is noted with A,, h, then:
1 WY and the total coefficient
of thermal
conductivity
will be
1=&,h+aL+~p
(21)
where AL is the coefficient of thermal conductivity due to the system of acoustical and optical phonons, while A, is the coefficient of thermal conductivity due to the acoustical and optical phonon drag. It can be expressed by: 2
Ap = 7’
Lg’ - Lg.’ Li1,’ + Li’,’ ‘[(Lg) + L’,2d)-
(Lpi + L~~~)l +
ccc% P In expression
(22) second degree drag factors
-
cLh, P).
(22)
have been neglected.
Conclzcsions. The contribution of acoustical and optical phonon drag in an ionic semiconductor has been emphasized based on the thermodynamic principles of irreversible processes. From a phenomenological point of view, the contribution of optical phonon drag is as effective as the contribution of acoustical phonon drag. The optical phonon drag is generally neglected in the literature as against the similar acoustical phonon drag. This could be acceptable at low temperatures, where the acoustical phonon drag is predominant, while the optical phonon drag is neglectable under such
112
L. GEORGESCU
circumstances. However, it would be quite wrong to conclude that the contribution of optical phonons to the phonon drag is negligible also at high temperatures. This is due to the fact, that there is a clear difference between acoustical and optical phonons. Acoustical phonons can be excited (that is moved out from the state of equilibrium) at low temperatures, under the influence of a strong interaction between carriers and acoustical phonons. Optical phonons can be excited at high temperatures, where they interact strongly with charge carriers, thereby contributing to the phonon drag 4). These general considerations substantiate the presence of terms characterising both the optical and the acoustical phonon drag in thermoelectrical coefficients. However, no conclusion can be reached phenomenologically with regard to the temperature range in which either of these phonon drags becomes predominant. The expression of the coefficient of thermoelectric force, due to the acoustical phonon drag (similar to relation (17)), can be derived on a variational basisis), for instance, and will be:
where m* is the effective mass of carriers, v the average velocity of sound, 76 carrier relaxion time, (1; phonon mean free path. For acoustical phonons, the dependence of mean free path A; upon temperature and upon the wave vector of phonons q is knowni4) : 4
-
Tn-5
.q-n
being a function of the symmetry of the crystal). Basically, the same reasoning could be made also for optical phonons ; however, the relationship between A, and temperature or wave vector in case of functional optical phonons, is not known. By analogy, a relationship similar to the one applying to acoustical phonons could be considered. In this case, the contribution of the optical phonon drag to the thermoelectric coefficients could be of the same order of magnitude as the contribution of the carriers, within a given temperature range4). As to the coefficient of thermal conductivity, we may assume on the basis of the same considerations, that the contribution of the optical phonon drag will be predominant at high temperatures and could be of the same order of magnitude as the carrier contribution, while negligible at low temperatures. A number of experimental results obtained in measurements of the coefficient of thermoelectric force at high temperaturesib) is) could be explained by considering the optical phonon drag. (PZ
Received
15-0-66
THERMODYNAMICS
OF THE
PHONON
DRAG
IN IONIC
113
SEMICONDUCTORS
REFERENCES Gurevich,
E. L., J. of Phys.
(Soviet
Union)
2 (1945)
1) 4
Herring,
3)
Appel,
4) 5)
Plivitu, C. N., Phys. Stat. Sol. 12 (1965) 265. Ter Haar, D. A. and Neaves, A., Phil. Msg. Suppl.
6)
Hanna,
7)
Ter
C., Phys.
96 (1954)
J., 2. Naturforsch.
12a
D. A. and Neaves,
H. B., Phys.
Rev.
410;
13a
(1958)
E. H., Proc. roy. A., Proc.
73 (1948)
P. I., Phil. Mag. 46 (1955)
477;
1U (1946) 67.
1163.
(1957)
I. I. and Sondheimer,
Haar,
8) Callen, 9) Price,
Rev.
386. 5 (1956)
Sot. A239
roy. Sot. A228
241. (1957) 247.
(1955)
568.
1349.
1252.
I., Etude thermodynamique des ph6nomenes irreversibles (Liege, 1947). 10) Prigogine, North Holland Publishing 11) de Groo t, S. R. and Masur, P., Non Equilibrium Thermodynamics, Company (Amsterdam, 1962). S. R., Mazur, P. and Tolhoek, H. A., Physica 19 (1953) 549. 12) de Groat, J. M., Electrons and Phonons, Oxford at the Clarendon Press (1960). 13) Ziman, 14)
Herring,
15)
Miller,
C., Phys.
16)
de Vroomen,
E., Komarek,
Rev.
95 (1954)
954.
K. and Cadoff,
A. R., van
Baarle,
I., J. appl. Phys. 32 (1961) 2457.
C. and Cuelenaere,
A. J., Physica
26 (1960)
19.
Physica
L.
35
107-l 13
1967
THE THERMODYNAMICS
OF THE PHONON
DRAG IN
IONIC SEMICONDUCTORS by L. GEORGESCU Faculty
of Physics,
University
of Bucharest laboratory
Bucharest,
of molecular physics,
Rumania
Synopsis Acoustical
and optical
of the thermodynamic are considered into account
phonon principles
as charge carriers. the contribution
drag in ionic semiconductors of irreversible The expression
of acoustical
processes.
are studied on the basis The
of thermoelectric
and optical
phonon
electrons
and holes
coefficients,
taking
drag, is given.
In a metal or semiconductor, the normal vibrations of the lattice are replaced by a system of quasiparticles, phonons, a system generally considered as being in thermodynamic equilibrium. In the phenomena of transport, the displacement of the phononic system from the state of equilibrium, gives birth to the acousticall) 2) a) and opticala) phonon drag. In studying the phonon drag in the phenomena of transport, only the contribution of acoustical phonons was considered, both in the case of semiconductors5) and of metals6) 7). The thermoelectric phenomena were thoroughly studied from the thermodynamic point of view, starting with Callena). The thermoelectric coefficients are functions of the phenomenological coefficients; both the interaction mechanisms of the charge carriers and the type of semiconductor, degenerated or non-degenerated, are included in the phenomenological coefficients. Thus very general relations between the thermodynamic flows and forces are obtained. A certain charge carrier scattering mechanism, intended for the particular study of some special processes in solid bodies, can be demonstrated also phenomenologically. On this line, Prices) had worked out thermodynamically the acoustical phonon drag with semiconductors, the terms related to the phonon drag being expressed in terms of supplementary coefficients in the expression of the current density and of the flow of energy. Based on the thermodynamic principles of irreversible processes la) rr), the author tries to emphasize phenomenologically the possibility of acoustical and optical phonon drag existence, by explicitly considering two scatter-
107 -
108
L. GEORGESCU
ing mechanisms: the scattering of charge carriers by acoustical and optical phonons. The considered thermodynamical system is an ionic type semiconductor, having electrons in the conductive and holes in the valence bands. The concentration of charge carriers is assumed to be low in order to be able to write linear relations between the thermodynamical flows and forces. Generally, we may consider the ionic semiconductor with n types of charge carriers, and with an ek (k = 1, 2, . . ., n) electric charge per unit of mass to be subjected to a temperature gradient (VT # 0), to a gradient of chemical potential (0,~ f 0), and to the action of an electric field (E # 0). The magnetic field is zero (H = 0) ; chemical reactions within the system are excluded. The whole system is divided into two subsystems: the charge carrier subsystems (electrons and holes), and the phonon subsystem (acoustical and optical phonons). The local temperature of the two subsystems is considered to be the same (Tcarriers = TDhonoIls= T). The temperature gradient is the only thermodynamic force acting on the phononic system. The production entropy for the whole system (charge carriers and phonons) appears as 12) :
or an analogous
form: To = -Jb*VT
-
5
Jk*[vpk
- e&E]
k=l
were Jq is the total flow of heat (of the charge carriers plus phonons), Jh is the total flow of entropy (of the charge carriers plus phonons), Jk is the flow of material of the component K. Flow Jq and Jh are related:
J; = Jp -
I: ,LLk’Jk
(3)
k
being the chemical potential of the component k. The following phenomenological equations result from the expression the production entropy (1) :
pk
of
the thermodynamic forces appearing as Xi = -ezVp, - TV(pJT) (9’ is electric potential), X,, = -(VT/T) (independent from the index i), since we considered the temperature of the two subsystems as being equal). The
THERMODYNAMICS
phenomenological
OF THE PHONON
are
coefficients
DRAG
IN IONIC
second
109
SEMICONDUCTORS
degree
tensors
(Lkg = L$fl),
a, p = 1, 2, 3). In order to simplify the calculation, the semiconductors are considered to be cubically symmetric; thus only the components Lit;sl’ = Li:r2’ = L&3) of the tensor Lk B, are different from zero. The magnetic field of the system being zero, the Onsager reciprocity relations would read:
The total
current
and the total flow of heat are defined by the
density
relations :
I =
Jp = C Jqk.
c edk,
(7)
k
k
In the system under consideration, only the electrons and holes (ei f 0, e.2 # 0) participate to the total electrical current, the acoustical and optical phonons having no charge (es = e4 = 0). Similarly, the chemical potential of the phonons is zero (,~a = ,u4 = 0). By replacing in (4) the expression of thermodynamic forces, and considering relation (7), the total electrical current expression results as:
The coefficient of electrical conductivity ez(Li:’ + L&j,‘,‘)is not affected by the phonon drag, as it is determined exclusively by the charge carriers. In the stationary state with zero electrical current (I = 0) one finds from (8) for the coefficient of thermoelectric force a, the expression: 1 a=
x ek[L$’ -
the identity
(9)
ez(Lii) + L&i))
T (considering
Lii’,&!&]
i,k
__.
ag = x aioi and the fact that the electrons
and
holes form two independent subsystems; the Lrs type coefficients being null), From expression (9), the explicit form of the coefficient of thermoelectric force is : a=---*
1
Ly$ + Lg
eT
+ Lg) -
Lgu -
L’,1,’ + L&j -
Lg
+ Lg
+ L&’ + L&).&C + Lgp Ljl,’ + Lg
with the notations: electrons
ei = -e,
~1 = p
holes es = e, -_ruz = ,u -I- &G (EG width of forbidden
1 (10) (11)
band).
110
L.
GEORGESCU
From expression (lo), the effect of the electrons and of the phonon drag on the coefficient of thermoelectric force al and the effect of the holes and of the phonon drag on the coefficient of the thermoelectric force as, can be written separately : 1 cc1 -
or in a condensed
y
[
L’?’ L$2,’ + $kj eL$’
+ 2i!!i!
11
-
11
f
1
(12)
form: al=ae m2 =
where me, CY~are the coefficients charge carriers.
ah
+
ae,p
(14)
+
ah, p
(15)
of the thermoelectric
Coefficients CL~, p and Q,,p represent the coefficients force due to the acoustical and optical phonon drag:
force
due to the
of the thermoelectric
Relations (14) and (15) s h ow that the coefficient of the thermoelectric force is made up of the combined effect of the charge carriers and of the acoustical and optical phonons. In order to determine the form of the Peltier coefficient and of the coefficient of thermal conductivity, taking into consideration also the acoustical and optical phonon drag, it is necessary to determine the total flow of entropy. To this end, the expression of the flow of entropy, knowing the production entropy can be calculated directly. The relationship between the phenomenological coefficients in going over from the new forces to the old thermodynamic forces is as follows: q;’
= L$?
_q
zzzLz;’ + _&&Ji
Lit’ = Lp
+ (L$’ + LB;‘) #uk-
(18) LgugQ.
The same result is obtained if we calculate initially the total flow of heat in (5), the flow of entropy resulting from relation (3). Irrespective of the selected approach the following, generally known relation is obtained:
J& = ITI - AVT.
THERMODYNAMICS
OF THE PHONON
DRAG
IN IONIC
111
SEMICONDUCTORS
The contributions of phonon drag are also included coefficients 17 and A. Thus, the Peltier coefficient is:
in the thermoelectric
the Onsager relation Ta = --17 being fulfilled, if we compare relations (10) and (19). If the contribution of the carriers (both electrons and holes) to the coefficient of thermic conductivity, omitting the contribution of the phonon drag, is noted with A,, h, then:
1 WY and the total coefficient
of thermal
conductivity
will be
1=&,h+aL+~p
(21)
where AL is the coefficient of thermal conductivity due to the system of acoustical and optical phonons, while A, is the coefficient of thermal conductivity due to the acoustical and optical phonon drag. It can be expressed by: 2
Ap = 7’
Lg’ - Lg.’ Li1,’ + Li’,’ ‘[(Lg) + L’,2d)-
(Lpi + L~~~)l +
ccc% P In expression
(22) second degree drag factors
-
cLh, P).
(22)
have been neglected.
Conclzcsions. The contribution of acoustical and optical phonon drag in an ionic semiconductor has been emphasized based on the thermodynamic principles of irreversible processes. From a phenomenological point of view, the contribution of optical phonon drag is as effective as the contribution of acoustical phonon drag. The optical phonon drag is generally neglected in the literature as against the similar acoustical phonon drag. This could be acceptable at low temperatures, where the acoustical phonon drag is predominant, while the optical phonon drag is neglectable under such
112
L. GEORGESCU
circumstances. However, it would be quite wrong to conclude that the contribution of optical phonons to the phonon drag is negligible also at high temperatures. This is due to the fact, that there is a clear difference between acoustical and optical phonons. Acoustical phonons can be excited (that is moved out from the state of equilibrium) at low temperatures, under the influence of a strong interaction between carriers and acoustical phonons. Optical phonons can be excited at high temperatures, where they interact strongly with charge carriers, thereby contributing to the phonon drag 4). These general considerations substantiate the presence of terms characterising both the optical and the acoustical phonon drag in thermoelectrical coefficients. However, no conclusion can be reached phenomenologically with regard to the temperature range in which either of these phonon drags becomes predominant. The expression of the coefficient of thermoelectric force, due to the acoustical phonon drag (similar to relation (17)), can be derived on a variational basisis), for instance, and will be:
where m* is the effective mass of carriers, v the average velocity of sound, 76 carrier relaxion time, (1; phonon mean free path. For acoustical phonons, the dependence of mean free path A; upon temperature and upon the wave vector of phonons q is knowni4) : 4
-
Tn-5
.q-n
being a function of the symmetry of the crystal). Basically, the same reasoning could be made also for optical phonons ; however, the relationship between A, and temperature or wave vector in case of functional optical phonons, is not known. By analogy, a relationship similar to the one applying to acoustical phonons could be considered. In this case, the contribution of the optical phonon drag to the thermoelectric coefficients could be of the same order of magnitude as the contribution of the carriers, within a given temperature range4). As to the coefficient of thermal conductivity, we may assume on the basis of the same considerations, that the contribution of the optical phonon drag will be predominant at high temperatures and could be of the same order of magnitude as the carrier contribution, while negligible at low temperatures. A number of experimental results obtained in measurements of the coefficient of thermoelectric force at high temperaturesib) is) could be explained by considering the optical phonon drag. (PZ
Received
15-0-66
THERMODYNAMICS
OF THE
PHONON
DRAG
IN IONIC
113
SEMICONDUCTORS
REFERENCES Gurevich,
E. L., J. of Phys.
(Soviet
Union)
2 (1945)
1) 4
Herring,
3)
Appel,
4) 5)
Plivitu, C. N., Phys. Stat. Sol. 12 (1965) 265. Ter Haar, D. A. and Neaves, A., Phil. Msg. Suppl.
6)
Hanna,
7)
Ter
C., Phys.
96 (1954)
J., 2. Naturforsch.
12a
D. A. and Neaves,
H. B., Phys.
Rev.
410;
13a
(1958)
E. H., Proc. roy. A., Proc.
73 (1948)
P. I., Phil. Mag. 46 (1955)
477;
1U (1946) 67.
1163.
(1957)
I. I. and Sondheimer,
Haar,
8) Callen, 9) Price,
Rev.
386. 5 (1956)
Sot. A239
roy. Sot. A228
241. (1957) 247.
(1955)
568.
1349.
1252.
I., Etude thermodynamique des ph6nomenes irreversibles (Liege, 1947). 10) Prigogine, North Holland Publishing 11) de Groo t, S. R. and Masur, P., Non Equilibrium Thermodynamics, Company (Amsterdam, 1962). S. R., Mazur, P. and Tolhoek, H. A., Physica 19 (1953) 549. 12) de Groat, J. M., Electrons and Phonons, Oxford at the Clarendon Press (1960). 13) Ziman, 14)
Herring,
15)
Miller,
C., Phys.
16)
de Vroomen,
E., Komarek,
Rev.
95 (1954)
954.
K. and Cadoff,
A. R., van
Baarle,
I., J. appl. Phys. 32 (1961) 2457.
C. and Cuelenaere,
A. J., Physica
26 (1960)
19.