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PELTIER EFFECT IN SEMICONDUCTORS
In recent years, devices based on the Peltier effect, which is
the basis for solid-state thermoelectric cooling, have
evolved rapidly to meet the fast-growing industry of elec-
tronics. The main point arises from the fact that the heat
extraction or absorption occurs at the contact between two
different conducting media when a direct current (DC)
electric current ﬂows through this contact. A comprehen-
sive study of the mechanisms of heating and cooling origi-
nated by an electrical current in semiconductor devices is
reported. The thermoelectric cooling in n-n,p-p, and p-n
junction contacts, as well as inhomogeneous bulk semi-
conductors, are analyzed. Both degenerate and nondegen-
erate electron and hole gases are considered. The role of
recombination and nonequilibrium charge carriers in the
contact cooling (heating) effect is discussed. Along with the
above, special attention is paid to several aspects of non-
equilibrium thermodynamics of thermoelectric phe-
nomena involved in Peltier effect in semiconductors that
demand a careful examination. The formulation of an
adequate self-consistent theoretical model describing the
Peltier effect is also presented.
The Peltier effect was discovered in 1834 in a contact of
two metals, and it was described as the heat extraction or
absorption that occurs at the contact metals when a DC
electric current ﬂows through it (1). As long as the
attempts to repeat the experiments to conﬁrm this new
effect failed to give positive results, it was not recognized
by physicists until 1838 when E. H. Lentz managed to
verify visually the Peltier effect (2).
The ﬁrst theoretical descriptions of thermoelectric
effects were based on the thermodynamic ideas
applied to the simplest models of thermoelectric circuits.
R. Klausius and W. Thomson initiated the development of
a theory almost simultaneously. The ﬁrst result of these
investigations was the paper by W. Thomson (3). This
paper marked the beginning of the development of
the phenomenological theory of thermoelectricity. In
1931, L. Onsager used the thermodynamics of irreversible
processes to describe thermoelectric effects, but his results
were not different from those obtained by W. Thomson (4).
In the middle of the twentieth century, A. F. Ioffe and his
colleagues led the revival of interest in thermoelectricity (5).
One of the ﬁrst materials they studied was PbTe, and they
quickly understood that the most prospective materials for
Peltier effect were semiconductors.
The Peltier effect can be used to create a refrigerator
(see Figure 1) that is compact and has no circulating ﬂuid
or moving parts; such refrigerators are useful in applica-
tions where their advantages outweigh the disadvantage of
their very low efﬁciency (6–14).
Cooling devices play a crucial role in everyday life. We
use them to preserve food and drugs, among many other
purposes. Often, persons cannot travel freely because they
use a medicine that must be kept in a refrigerator. It is easy
to understand that a 10-cubic-foot refrigerator or an icebox
are not acceptable solutions to allow the mobility of persons
with certain chronic diseases. Compact refrigerators with
eased low power supply requirements (low-power DC
instead of wall alternating current [AC] or lead batteries
used in cars) would be very useful. They would be used in
many different applications, such as in cryogenic skin
surgery, high-availability electronic systems, and defense.
In addition, those refrigerators should have a long life and
Thermoelectric cooling may advantageously replace the
current cooling methods based on chloroﬂuorocarbons
(CFCs) that are suspected as responsible for the break-
down of the ozone layer. Thermoelectric cooling has some
distinct beneﬁts, as follows:
1. Switch from heating to cooling by inverting the
polarity of the applied bias.
2. Vibration and acoustic noise free.
3. Modulation of the cooling power through the supply
4. Maintenance free.
5. Free of moving parts.
6. Functioning freely and independently of the
Several interesting results have been reported on the
progress in materials science with applications in thermo-
electrics: Skutterudites, clathrates compounds, half-Heus-
ler alloys, compounds of layered bismuth telluride
structures, and low-dimensional structures (such as quan-
tum dots and superlattices) have been investigated as a
means to increase the thermoelectric ﬁgure of merit.
The traditional theory of the Peltier effect is presented
in several books (5–9, 14) and reviews (10, 11, 15, 16), but
many important aspects of the Peltier effect are far from
being covered satisfactorily in common books. In this
article, we will present the current status of the theory
of Peltier effect.
TRADITIONAL THEORY OF PELTIER EFFECT
From the earliest papers on thermoelectrics right up to the
latest publications, this effect has been deﬁned as an
absorption of heat (in addition to the Joule heat) of the
junction of two conductors or its cooling through which a
DC electric current ﬂows (see, for example, References 5, 6,
14, and 16–18). The absorption of this heat or its evolution
depends on the direction of the electric current, and per
unit time, it is equal to
where P1;2are the Peltier coefﬁcients of the conducting
materials at each side of the junction and Jis the total
For a nondegenerate system of electrons (subscript n)
and holes (subscript p) (14), the Peltier coefﬁcient is
J. Webster (ed.), Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright #2014 John Wiley & Sons, Inc.
3GW8206 03/26/2014 0:6:45 Page 2
where Tis the temperature in energy units; m
chemical potentials of electrons and holes that are meas-
ured from the bottom of the conduction band and the top of
the valence band, respectively (m
, where e
the band gap); eis the hole charge; and g
exponents in the momentum relaxation times (19):
where eis the carrier energy. The constant quantities t0
and gn;pfor different relaxation mechanisms can be found
in Reference 19.
For degenerate electron and hole gases
To understand the physical meaning of equations 2 and 4,
it is convenient to rewrite these expressions as follows:
n;pis the mean kinetic energy of the carriers in the
0eðÞare the Fermi-Dirac distribution functions.
Notice that, similar to the case of the chemical potentials,
the mean energies ehi
n;pare measured from the bottom of
the conduction band and from the top of the valence band,
respectively. Often, the chemical potentials mn;pare inter-
preted as the potential energies of electrons and holes (14).
For nondegenerate statistics ( m
while for the degenerate case ( m
In some papers (see, for example, References 15 and 17), it
is emphasized that the Peltier effect manifests itself in the
thermally uniform systems, i.e., when the temperature
gradient is absent.
Thus, the authors of Reference 17, when describing the
Peltier effect, wrote the following:
If an electric current is driven in a bimetallic
circuit maintained at uniform temperature,
then heat is produced at one junction and
absorbed at the other (Figure 2). This is because
an isothermal electric current in a metal is accom-
panied by a thermal current q¼Pj, where jis the
electric current density. Because the electric cur-
rent is constant in the closed circuit and the
Peltier coefﬁcients differ from metal to metal,
the thermal currents in the two metals are not
equal, and the difference must be produced at one
junction and supplied to the other to keep the
Once the discussion of the physical principles of the
Peltier effect is concluded, we will brieﬂy revise its
Figure 1. Schematic of a thermoelectric couple.
jq = ∏j
Figure 2. Schematic circuit of Peltier effect for constant T.
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applications. Actually there are many of them, for several
reasons. Thermoelectric devices use DC electric current,
and changing its direction switches the device from the
cooling mode to the heating one or vice versa. Precise
temperature control <0:1C
can be achieved with
Peltier coolers. Having such advantages, thermoelectric
coolers ﬁnd a lot of applications in areas such as consumer
products (recreational vehicle refrigerators, automobile
seat coolers, portable picnic coolers, motorcycle helmet
refrigerators, and residential water coolers/puriﬁers), lab-
oratory and scientiﬁc equipment (coolers of infrared detec-
tors, laser diode coolers, integrated circuit coolers, heat
density measuring), industrial equipment (personal com-
puter [PC] microprocessors, ﬁne temperature control),
medical instruments (portable and ﬁxed pharmaceutical
refrigerators, blood analysis, insulin coolers), restaurants,
and restaurant industry products (hotel room refrigera-
tors, noiseless air conditioners, ice makers). This list is not
complete and can go on and on.
At the same time, it is necessary to point out some
disadvantages of thermoelectric coolers. First, they have
a low efﬁciency compared with conventional refrigerators.
Current thermoelectric devices operate at about 10% of
Carnot efﬁciency, whereas the efﬁciency of a compressor-
based refrigerator is about 30%. Thus, the use of thermo-
electric coolers is restricted to applications where their
unique advantages outweigh their low efﬁciency. Although
some large-scale applications have been considered,
Peltier coolers are generally used in applications where
small size is needed and the cooling demands are not too
great, such as cooling electronic components (see, for the
sake of example, References 20 and 21).
Summing up, one could say that the traditional
explanation of the Peltier effect presupposes the following
1. An isothermal state of the structure (bimetallic cir-
cuit) exists, through which an electric current ﬂows;
consequently, there is total absence of thermal ﬂuxes
that would be proportional to a temperature gradi-
ent. The only thermal ﬂuxes are those accompanying
the electric current.
2. If a thermal interaction between this structure and
the external heat reservoirs is necessary to maintain
the previously described isothermal state (a Peltier
heat source and a Peltier heat sink), then it becomes
necessary to take into account the external sources of
An important question arises regarding these consider-
ations: What do authors imply by the concepts of thermo-
electric “cooling” or “heating” in the traditional theories?
The presence of heat ﬂuxes from the external reservoirs
into the structure and vice versa from the structure to the
external reservoirs does not describe heating or cooling
processes under an isothermal condition for the structure.
Another question is as follows: Can the Peltier effect
occur in the thermoelectric circuit without a thermal
interaction with the surroundings at the junctions; i.e.,
can the Peltier effect occur as a physical phenomenon in a
structure adiabatically insulated?
The answer given by the cited works to the second
question is negative. But, it can be readily shown that
the Peltier effect in an adiabatic structure may take place
beyond the traditional interpretation of the Peltier effect.
PHYSICS OF PELTIER EFFECT IN n-nAND p-pSTRUCTURES
The origin of the Peltier effect is not associated with the
external Peltier heat sources or the Peltier heat sinks. This
effect can also exist in an adiabatically insulated thermo-
electric structure. It is necessary to understand the con-
cepts of “cooling” or “heating” of the junctions as the
decrease or increase in the junction’s temperature com-
pared with the equilibrium one in absence of electric
current. Of course, this statement contradicts the hypoth-
esis isothermal conditions, from where a natural question
immediately arises: Why is a nonuniform temperature
proﬁle established across the structure in the absence of
external heat sources when an electric current ﬂows per-
pendicularly to the two media interface?
Let us consider the simplest structure composed of two
uniform semiconductors through which the electric cur-
rent ﬂows in the direction 0x (Figure 3). Assume that the
electric contacts x¼d1and x¼d2are kept at the equi-
librium temperature T0, that the lateral surfaces are
insulated adiabatically, and that the structure’s cross-
sectional area is equal to the unit measurement every-
where. For simplicity, we shall assume initially that a heat
resistance of the junction located at x¼0 can be neglected
(this assumption will be removed later on); this assump-
tion implies temperature continuity at the interface in
where T1;2ðxÞare the temperature distributions in each
region of the structure.
Usually, the thermal ﬂux Pjis named the Peltier heat
ﬂux (16). For the theory stated, it is more convenient to
name this heat ﬂux a “drift heat ﬂux” because it is associ-
ated with the charge drift transport in the external electric
ﬁeld, and accordingly, for holes the “drift heat ﬂux” and
current have the same direction while they are opposite for
electrons. An important feature of the drift heat ﬂux is that
Figure 3. Schema of the simplest Peltier device consisting of two
Peltier Effect in Semiconductors 3
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it is associated with particle drift in a single direction; this
is in strong contrast with the movement in two opposite
directions in the thermal diffusion ﬂux that will be ana-
According to the Le Ch^
atelier-Braun principle in the
irreversible thermodynamics (24), “some internal ﬂuxes
appear in the system in the stationary state when an
external inﬂuence affects this system, and these internal
ﬂuxes weaken the results of this inﬂuence.” Applying this
principle to our problem, one can say that the discontinuity
in the drift ﬂuxes at the junction x¼0 that appears because
of the different Peltier coefﬁcients (P16¼ P2) must lead to
other thermal ﬂuxes tending to reduce this discontinuity
These thermal ﬂuxes can only be thermal diffusion
ﬂuxes (qdif ¼kdT=dx,kbeing the thermal conductivity)
because the other drift heat ﬂuxes are absent. The
“diffusion heat ﬂux” nature greatly differs from the one
of the drift heat ﬂux. To understand the origin of the
diffusion heat ﬂux, let us consider a material with a
nonhomogeneous temperature proﬁle across it; under
these conditions, the equilibrium in the microscopic ﬂuxes
and balance will be removed. Inside the material, two
ﬂuxes of quasiparticles will take place: one of particles
moving from the hotter area into the cooler one and a
second ﬂux of particles in opposite direction. The total
particle ﬂux balance is null in the absence of electrical
current or when particles are not charged (phonons). But
because the energy of carriers in the hotter area is larger
than the ones in the cooler one, the total heat ﬂux is
nonzero and is directed from the hot side toward the
cold side. This ﬂux is the diffusion heat ﬂux.
Thus, a temperature heterogeneity arises inevitably in
the structure in the presence of an electric current.
Because of this temperature distribution, the temperature
of the junction can be lower (thermoelectric cooling) or
higher (thermoelectric heating) than the equilibrium one.
Because we are not interested in carrying calculations
in real structures, let us keep considering the simple
structure of Figure 3, assuming that it is composed of
two n-type semiconductors and using the same subscript
convention. Both drift heat ﬂuxes q1;2
dr ¼P1;2jﬂow in this
structure in the direction opposite to the electric current
direction, as it is pointed in Figures 4 through 6, because
P1;2<0 in the n-type materials (14).
Let us suppose that P2
jjfor deﬁniteness. In this
case, the drift thermal ﬂux increases when coming from
side 2 into side 1 (Figure 4). According to the Le Ch^
Braun principle, a thermal diffusion ﬂux q1
(k1being the thermal conductivity of the left side) should
arise in the ﬁrst side, tending to compensate this increase
of drift heat ﬂux. It is clear that the direction of this ﬂux
should be opposite to the direction of the drift ﬂux
dr ¼P1j. We name these arising thermal diffusion ﬂuxes
the “induced thermal diffusion ﬂuxes.”
In turn, this means that the temperature of the junction
from the side of the ﬁrst sample decreases in comparison
with the temperature on the surface x¼d1. The condi-
tion of the temperature continuity (eq. 9) requires a reduc-
tion in temperature in the junction from the side of the
second side too, as compared with the temperature on the
surface x¼d2. As a result, an induced thermal diffusion
dif ¼k2dT2=dx arises in the second sample, where
k2is the thermal conductivity of the right side. Let us
notice that this thermal ﬂux direction coincides now with
the direction of a drift heat ﬂux q2
dr ¼P2jin this sample
(Figure 4), tending to decrease the discontinuity in drift
The corresponding temperature distributions in the
structure are qualitatively represented in Figure 5. It is
obvious that for P2
and the chosen electric current
direction, the junction temperature T(x¼0) decreases as
compared with the equilibrium temperature T0. The low-
ering of the temperature caused by the emergence of the
induced thermal diffusion ﬂuxes is the essence of the
Peltier thermoelectric cooling.
If the condition P2
takes place and the electric
current has the same direction, then the drift heat ﬂux
decreases when passing through the junction. In this case,
the induced thermal diffusion ﬂux arises in the ﬁrst sample
again, but its direction changes to the opposite compared
with the previous case (Figure 6). Now this ﬂux tends to
= – κ1
= ∏1 j
(2) = ∏2 j
(2) = – κ2
x = 0
Figure 4. Heat ﬂuxes in an n-nstructure assuming P2
k1;2is thermal conductivity in each region.
Figure 5. Temperature distribution in an n-nstructure assuming
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increase the drift thermal ﬂux q1
dr ¼P1j, which has
decreased compared with q2
Further reasoning is similar to that used in the analysis
of the previous situation, and it leads to the temperature
distributions in the structure presented in Figure 7. In this
case the junction temperature increases in comparison
with the equilibrium temperature T0, and this situation
corresponds to the Peltier thermoelectric heating.
Thus, we have to understand the Peltier effect as the
lowering or raising of the junction temperature (depending
on the current direction) as a result of the appearance of
the induced thermal diffusion ﬂuxes in the structure, but
not as an evolution or absorption of the additional heat on
To demonstrate an additional physical meaning (25) of
the Peltier effect, a metal-n-type semiconductor contact is
normally used (see Figure 8). It is easy to verify that the
surplus of the electron energy in the semiconductor in
comparison with the metal is De¼e
The obtained result is clear for this type of contact. At
the same time, these contacts are not really used in appli-
cations. The consideration of other contacts does not have
the same obvious understanding, and a priori the following
What does the potential energy for the degenerate gas
of the charge carriers mean?
What is the physical meaning of the energy change in
a metal-p-semiconductor contact and in a p-n
Why does the approach stated above ignore the work
of the built-in electric ﬁeld within the contact?
It is easy to show that the value e
min any type of
contacts is different on the left and on the right from
the contact plane. It means that single electron tran-
sitions cannot determine the work of the refrigerator.
Therefore, the description of the contact refrigeration
based on the consideration of the separate electron
transitions is not valid. One needs to take into
account the statistics of these transitions. How can
one reﬂect this fact in the preceding scheme?
In the usual scheme of n-nor p-pjunctions, the contact
heating or cooling is explained by the electron (hole) tran-
sitions between ehi–levels (see Figure 9). The latter are
counted from the common Fermi level.
Some additional assumptions will be introduced. Let us
consider ﬁrst the n-ncontact between nondegenerate semi-
conductors (Figure 10). If the momentum scattering in
both semiconductors is the same, then (see eq. 2):
(1) = – κ1 qdif
(2) = – κ2
(1) = ∏1 jddr
(2) = ∏2 j
x = 0
Figure 6. Heat ﬂuxes in an n-nstructure assuming P2
Figure 7. Temperature distribution in an n-nstructure assuming
Figure 8. Heating or cooling of a metal-n-type semiconductor
Figure 9. Energy band diagram of n-nor p-pcontacts in equili-
Peltier Effect in Semiconductors 5
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where xiand x0
i(I¼1, 2) are work functions and electron
afﬁnities of materials 1 and 2, wc¼x2x1
contact voltage, and Dec¼x0
2is the work of the
valence force. As is known, this is a result of the length
of the forbidden region that is position dependent. It means
that the contact cooling or heating is connected with the
work of the built-in electric ﬁeld W
and the work of the
valence forces W
. The total work is W¼QP¼WcþWn,
where Wc¼Jwcand Wn¼JDec=e. Here, the valence force
is caused by the abrupt change of the conduction band.
Similar reasoning is valid for the p-pjunction.
If we have a contact between degenerate and nonde-
generate semiconductors, i.e., the n
then the Peltier effect does not depend on the contact
properties, and it is determined only by the parameters
of the nondegenerate semiconductor (see Figure 8).
In the case of a contact between two degenerate n-orp-type
semiconductors (or two metals), the Peltier effect depends
on the contact properties again (see Figure 11 and eq. 4),
and the work of the built-in electric ﬁeld jointly with the
work of the valence forces occurs.
Comparing equations 10 and 12, we see that the Peltier
heat in the contact between degenerate materials is essen-
tially less than that in the contact between the nondegen-
Let us also note that the work of the built-in electric
ﬁeld and the valence forces will give rise to the qualita-
tively different temperature of the contacts in the cases of
nondegenerate and degenerated branches of the circuit. In
where T(E) is the nonequilibrium temperature, Eis the
built-in electric ﬁeld, and T
is the room temperature. In
The quantitative calculation of the thermoelectric cooling
or thermoelectric heating requires a solution to the heat
transfer equation with the corresponding boundary condi-
tions. The next section is devoted to this calculation.
ENERGY AND HEAT BALANCE EQUATIONS FOR
Let us now remove the assumption about the absence of the
thermal resistance at the junction x¼0. Instead, we shall
describe this junction by the coefﬁcient of the surface
thermal conductivity (26–29), which can take arbitrary
values. Later, we shall show that the junction is described
not only by the surface thermal conductivity but also by
some other surface parameters.
The searched temperature distribution in the structure
can be obtained from the energy balance equation. In a
steady state, it is given by (16)
where w¼qþcjis the energy ﬂux of charge carriers
(electrons or holes), q¼krTþPjis a total heat ﬂux of
these carriers, c¼wm=eis the electrochemical poten-
tial, wis the electric potential, mis the chemical potential,
and eis the electron charge.
Now equation 15 can be rewritten in the form
We have made the assumption that nonequilibrium carri-
ers are absent in the structure (30), i.e., rj¼0. Let us note
that it is necessary to take into account the nonequilibrium
carriers in pnstructures (17, 18).
Figure 10. Energy band diagram of the contact of two nonde-
generate semiconductors. e
is the vacuum level, and x
the work functions and electron afﬁnities, respectively.
Figure 11. Energy band diagram of the contact between two
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Using the equation for the electric current (31)
to eliminate the electrochemical potential c(sis electric
conductivity and ais the Seebeck coefﬁcient), we arrive at
the heat balance equation
It is easy to see, from the shape of this equation, that there
are only two sources of heat: one of them is the Joule heat,
and the remaining source of heat can be named the Thom-
son heat because of its proportionality to the product of j
Because the temperature gradient has appeared from
the Peltier effect, we can say that the Thomson effect
occurs as a consequence of the Peltier effect in the ther-
moelectric cooling (heating) phenomenon.
Let us note that according to our deﬁnition following
from the heat balance equation (eq. 18), the Thomson
coefﬁcient is equal to the Seebeck coefﬁcient a. From
this deﬁnition, it follows that the Thomson heat occurs
even in the case when the Seebeck coefﬁcient does not
depend on temperature. Moreover, it is possible to claim
that thermoelectrics contains only one fundamental
parameter, which is the Seebeck coefﬁcient.
Thus, only one thermoelectric relationship was named
the second thermoelectric relationship, P¼aT(15, 16). As
it was shown in Reference 16, it follows from the Onsager’s
principle of kinetic coefﬁcients symmetry. The ﬁrst ther-
moelectric relationship actually determines the Thomson
coefﬁcient t. As it follows from equation 18, it can be
reduced to the trivial equality t¼a.
The Thomson coefﬁcient was determined by the
equation t¼@P=@Tain the previous publications
(5–9, 14, 18, 31, 41). The authors of these works formed
the term by combining j@P=@TrTfrom divq with the term
ajrTpresented on the right-hand side of equation 18. This
is incorrect because ajrTis the source of heat while
j@P=@TrTis just a part of the thermal ﬂux change.
This is trivially wrong from the formal point of view. We
cannot state that the left-hand side of equation 18
is still the complete divergence of the thermal ﬂux vector
q¼krTþPjwhen the term j@P=@TrTis moved to the
right-hand side of equation 18 or that it is the divergence of
another vector in this case.
Now, the sense of the Thomson heat is very transparent.
It is the charge carriers heating or cooling in the thermo-
electric ﬁeld Ete ¼arT. In a certain sense, it can be named
the Joule effect associated with the thermoelectric ﬁeld.
The Joule heating takes place in the electric ﬁeld. It is
important to note that the Joule effect leads to heating in
any case, whereas the Thomson effect may lead to either
heating or cooling depending on the relative directions of
the electric current and the temperature gradient. Subse-
quent development of these ideas can be found in Refer-
Equation 18 demonstrates that the Peltier source of
heat is absent everywhere in the structure volume, as it
was pointed out previously in this article. For this reason,
the statement about the Peltier’s heat absorption or
generation loses its physical sense in the bulk of the
structure (the analysis of this situation at the junction is
Let us now obtain the boundary conditions for equation
18 (22, 23). Because equation 18 is an equation of the
second order in relation to unknown temperatures T1ðxÞ
and T2ðxÞfor two-layer samples, it is necessary to obtain
four boundary conditions. Two of them correspond to the
assumptions that were made before
The next boundary condition describes the total heat ﬂux
continuity at the boundary of the two media
q1x¼0ðÞ¼q2x¼0ðÞðÞin the most simple case of the
absence of heat sources and sinks on the interface x¼0.
Last, the fourth boundary condition determines the heat
ﬂux through the interface x¼0 with the surface thermal
The right-hand side of equation 22 determines the total
thermal ﬂux ﬂowing into the interface of the two media.
The expression on the left-hand side of the equation deter-
mines the same thermal ﬂux in the surface layer x¼0 (for
simpliﬁcation, we neglect the surface Peltier coefﬁcient
(33) in the given work, supposing it negligibly small in
comparison with the bulk Peltier coefﬁcient).
TEMPERATURE DISTRIBUTION IN n-nAND p-p
To obtain the required temperature distributions we will
use the heat ﬂux continuity equation (eq. 18) in the linear
approximation of j, which for a homogeneous medium in
one dimensional case reduces to
Together with the boundary conditions (19–22), equation
23 results in the following temperature distributions:
The second term in the square brackets of equation 24
determines the contribution of the Peltier effect to the
temperature distribution in the structure with a ﬁnite h.
This term deﬁnes the Peltier thermoelectric heating or
cooling considered in the introduction. As it has been
shown in Reference 25, the value ðP1P2Þjdetermines
the change in the kinetic energy ﬂux on the junction. This
Peltier Effect in Semiconductors 7
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change is completely dependent on the electric potential
barrier parameters on the junction, such as the work
functions of the contacting semiconductors and their elec-
Let us note, as it has not been previously mentioned,
that the Peltier effect depends strongly on the junction
surface thermal conductivity. The contribution of this
effect to the total effect of thermoelectric cooling increases
with an increase in surface thermal conductivity and it
slackens with a decrease in surface thermal conductivity.
The term in the square brackets of the numerator of
equation 24 determines a new cooling or heating effect that
cannot be reduced to the Peltier effect. First, it is deter-
mined not by the difference between the Peltier coefﬁcients
but by each Peltier coefﬁcient separately. This implies that
this effect can be observed in a structure composed of a
single material and radically distinguishes it from the
Peltier effect. Besides, this effect is not associated with
the electric potential barrier on the interface x¼0, being
determined exclusively by the junction surface thermal
The heat balance equation (eq. 18) is in the general case
a nonlinear equation of the density of the electric current j.
This follows from the nonlinear terms describing the Joule
and the Thomson heat. In the case of the Thomson heat, it
is necessary to take into account that the temperature
gradient in the problem considered is not created by any
external sources. It arises only as a result of an electric
current ﬂow. Thus, both the Joule and the Thomson heat
sources are nonlinear terms in j. As it was shown in
Reference 32, the magnitudes of both heat sources are of
the same order. The Thomson heat even exceeds the Joule
heat in good thermoelectric materials, i.e., those materials
that exhibit a ﬁgure-of-merit ZT, where Zis Z¼a2s=k(5),
exceeding the unity (more details about ZT will be given in
the section titled, “Applications”). Besides, both the
coefﬁcient of thermal conductivity and the Seebeck
coefﬁcient depend on temperature, which is also a function
of the electric current.
At the same time, it is easy to observe that the non-
equilibrium temperature distributions occur even in the
linear approximation with respect to the electric current.
So, thermoelectric cooling or heating are linear effects in
an electric current and manifested themselves “in the pure
state” only at small values of current when the Joule and
the Thomson heat do not play a noticeable role. For this
reason, it is convenient to consider the problem of thermo-
electric cooling just in the linear approximation to the
electric current j.
Let us notice that all the equations obtained above
remain true for the structures composed of semiconductors
with the p-type conductivity. It is only necessary to con-
sider that P1;2>0 in this case.
The Peltier effect occurs at the junction of semiconduc-
tors with electron and hole conductivities too. However,
the thermoelectric processes in this case depend essen-
tially on the recombination rates in areas close to the
junction (see References 25, 30, 34–36). In the case of an
inﬁnitely strong recombination, all the equations above are
correct. The essential difference between this case and the
previous ones is that now the material with the n-type
conductivity is characterized by the Peltier coefﬁcient
Pn<0, whereas the material with the p-type conductivity
is characterized by the Peltier coefﬁcient Pp>0. In the
last case, both areas close to the junction will be heated or
cooled (depending on the direction of the electric current)
simultaneously and the effect of thermoelectric cooling
(heating) is more intense.
In fact, let us assume for simplicity that the structure
represented in Figure 3 is composed of p- and n- materials
identical in length and with equal values of the coefﬁcients
of thermal conductivity d1¼d2¼d;k1¼k2¼k; that the
recombination rate on p-njunction is inﬁnitely high; and
that the Peltier coefﬁcients of both materials are equal in
absolute magnitude (P1¼P2).
In this case, the Peltier effects are
. The last equation shows that the
Peltier effects are commensurable in the structure
COOLING IN INHOMOGENEOUS SEMICONDUCTORS
(BULK PELTIER EFFECT)
Under certain conditions charge carriers in inhomogeneous
semiconductorscan be cooled by a static electricﬁeld instead
of being heated, and consequently the carrier average
energy of drops below the lattice temperature T0(37).
This effect can be explained as follows. Let assume that
the impurity concentration Nin a semiconductor depends on
a coordinate (x, for example). If the Debye radius is the
smallest parameter having the dimension of length in the
problem, the carrier concentration at each point will coin-
cide with the impurity concentration(26, 27), thus establish-
ing an electron concentration gradient (dN/dx).
It is well known that an associated electrostatic internal
ﬁeld is then directed along the gradient. An external
electric ﬁeld applied to the sample will induce a current
in the same direction as this ﬁeld (31). If the current
direction is opposite to the direction of the concentration
gradient and the external ﬁeld is weaker than the ﬁeld
associated to the gradient dN/dx, then the carriers will
move in a (total) ﬁeld that is oriented against their motion.
Then, the electric ﬁeld will obviously remove energy from
the carriers, which are thus “cooled.” The energy removed
from the carriers must be liberated outside the sample, to
the current source, for example.
The described effect is somewhat analogous to the
Peltier effect, although the former is of spatial character.
Let us as assume an isotropic quadratic electron dis-
persion law and that collisions between carriers and scat-
tering centers are quasielastic (with the momentum-
transfer average free carrier path lple(38)).
The concentration of impurities, and therefore that of
the carriers, depends on xand it is given, for simplicity, by
N¼N0 exp x=lðÞ. The magnitude of N
is such that the
frequency nee of interelectronic collisions greatly exceeds
8 Peltier Effect in Semiconductors
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the frequency neof collisions between electrons and scat-
tering centers accompanied by energy transfer.
Under the previous assumptions, our problem may be
considered one dimensional (i.e., all quantities will depend
only on the xcoordinate).
The electron temperature as a function of the coordi-
nates can be determined by the equation of heat balance,
which can be obtained from the following kinetic equation
(see References 26 and 27, for example):
dq=dx ¼jE NðxÞneðTeT0Þ(26)
q¼N=N0keTe1þgeE Te=NdN=dx ð2þgÞdTe=dx½
is the heat ﬂux in electron subsystem and Eis the electric
ﬁeld strength in the sample (Ehas only an xcomponent).
is the current and Teis the electron temperature. The
are, respectively, the electronic heat and the electric con-
ductivity at Te¼T0.n¼n0ðT0Þðe=T0Þgis the frequency of
collisions with momentum transfer (26, 27) between
electrons and scattering centers. ne¼n0eðT0ÞðTe=T0Þr1
(26, 27) is the frequency of collisions with energy transfer
(26, 27) between electrons and scattering centers. rand g
are numerical coefﬁcients with magnitudes close to one (for
which explicit expressions are given in Reference 19),
whose values depend on the scattering mechanism with
respect to the carrier energy and momentum, respectively.
Because under stationary conditions the current
density jdoes not depend on the coordinate, it is
convenient to express the ﬁeld Ein terms of j.Using
equation 28, we obtain Eand substituting it and qinto
equation 27, we can ﬁnally obtain the expression for the
This equation must be supplemented with the boundary
conditions for temperature in the planes x¼0 and x¼L,
where Lis length of sample (see eq. 21).
In the current range 0 <j<j
, where jc¼2le=lð5=2þ
leT0is the average temperature Te
TeðxÞdx,ifle<l(37). As the ratio
le=lincreases, the minimum of Te
as a function of jis
shifted to higher values of j;Te
min can be of the order of
unity when l
Note that the one-temperature approximation consists
of assuming that the temperatures of all quasiparticles
(electrons, holes, and phonons) in each spatial point
coincide. Until now, this approximation has been widely
used for research of the thermoelectric cooling phenome-
non. However, the criteria for applicability of such an
approximation are usually not discussed. Nevertheless,
there are many phenomena in which the temperatures
of current carriers and phonons are different from each
other (as it was shown in this section).
As it is well known (22), the Peltier effect, which under-
lies thermoelectric cooling, is a contact phenomenon. It
occurs in a subsystem of electrical-charged quasiparticles
(electrons and holes). That is, the electron and/or the hole
cooling takes place near an interface. Then, the lattice is
also cooled by means of energy interactions between elec-
trons (holes) with phonons near the interface. As a result of
this two-step process, different temperatures are set in the
subsystems of current carriers and phonons.
Thus, it is obvious that the multitemperature approxi-
mation, when each subsystem of quasiparticles is charac-
terized by its own temperature, is more appropriate for the
investigation of the thermoelectric cooling phenomenon
PELTIER EFFECT IN p-nJUNCTIONS
Conventionally, a p-nstructure is used to make a thermo-
electric refrigerator (5, 6, 9, 41) because of the thermo-
electric drift ﬂuxes directed (at a corresponding direction of
a current from the n-region to p-region) from the interface
toward the edge in both layers of the p-nstructure that
strengthens the cooling phenomenon (22). Traditionally,
the studies of the Peltier effect do not consider the non-
equilibrium charge carriers, so that only majority charge
carriers and their electric current are taken into account in
the expressions for heat ﬂuxes in n- and p-regions, even
though the current of minority charge carriers near the p-n
junction has the same order of magnitude than the current
of majority charge carriers (25, 42) (see Figure 12).
In this way, the thermal generation and extraction of
minority charge carriers must take place near the interface
to allow the ﬂow of electric current (42). As a consequence,
nonequilibrium charge carriers will settle (25, 35, 36).
Let us examine the p-njunction (Figure 13). It is easy to
see From Figure 13 that
Figure 12. Distribution of currents in p-njunctions.
Peltier Effect in Semiconductors 9
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The generation or recombination at the contact causes
the Peltier heating or cooling in this case.
At the same time, there are a lot of situations where this
rate is weak enough. What has happened with the Peltier
effect in this case? The situation is similar to the electric
current or thermoelectric current in bipolar semiconduc-
tors. The Fermi quasilevels will set up causing the
redistribution of carriers concentrations and, accordingly,
modifying the electric ﬁeld proﬁle. As a result, the nature
of the Peltier effect can drastically change.
It follows from Figure 14 that under the chosen current
direction cooling of the junction switches to heating in
absence of recombination.
The equivalent electric circuits are presented in
Figures 15 and 16 for the two limiting cases, very strong
and weak recombination, respectively.
The thermoelectric behavior of p-njunction for Peltier
effect was investigated for the ﬁrst time in Reference 43.
Experimentally, this phenomenon was discovered in 1974
by Ponomarenko and Stafeev (44). The investigation of
the thermoelectric effect was undertaken in the following
years in a number of studies (45–53). Unfortunately in
all these articles, several assertions were incorrect and
led to erroneous results. Some of the mistakes were
already pointed out and discussed in Ref. 54; this short
paper is free from fundamental errors, but it presents
neither new equations nor a quantitative analysis of the
thermoelectric phenomena. Common mistakes in previ-
ous studies, additional to those discussed in Ref. 54, can
be found in the modeling of the recombination used, in
the energy balance equation, in the boundary conditions,
and so on (35, 36). The analysis of the thermoelectric
cooling that is given in References 43–54 is only approxi-
mate because it ignores several factors as we will discuss
Figure 13. Energy band diagram of a p-njunction.
Figure 14. Energy band diagram a p-njunction without
Figure 15. Strong recombination: Carrier movement (left) and schematic of the equivalent circuit (right).
Figure 16. Weak recombination: Carrier movement (left) and schematic of the equivalent circuit (right).
10 Peltier Effect in Semiconductors
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HEAT BALANCE EQUATION IN BIPOLAR
The energy balance equation has never been sufﬁciently
investigated in bipolar semiconductors; nevertheless, even
if in the linear approximation with respect to the gradient
of the temperature (i.e., assuming a small mismatch of the
temperatures in both regions) a signiﬁcant population of
nonequilibrium charge carriers will always set up and,
consequently, generation-recombination processes must
be taken also into account (55).
Let us note that in this case (as it will be shown next),
the recombination has a dual role. On the one hand, it is
the source or the absorbent (sink) of heat, and on the other
hand, it bears a signiﬁcant inﬂuence on the drift heat
ﬂuxes. Consequently, the expressions for the sources of
the total heat ﬂux and the diffusion heat ﬂux are distinctly
The energy ﬂux density in a unipolar semiconductor is
(see eqs. 15 and 16):
Let us recall that, under the approximation of quasineu-
trality, nonequilibrium charge carriers in a homogeneous
unipolar semiconductor will not exist in the linear approxi-
mation with respect to the perturbation (55). Consequently,
the variation of the electrochemical potential dc reduces to
the variation of the electrical potential dw (56) because
cn;p¼wð1=eÞmn;p, where wis the electrical potential, e
is the elementarypositive charge, and mnðmpÞis the chemical
potential of electrons (holes). Thisexplains why, as it follows
directly from equation 30, the variation of the energy ﬂux dw
is equal to the variation of the heat ﬂux dq, dw¼dq under
steady-state conditions (divj ¼0).
The heat ﬂux density qis equal to the sum of the density
of the drift heat ﬂux qdr ¼Pjand the density of the
diffusion one qdiff ¼krT. Let us remark that formally
heat ﬂux is a vector (q), in here we will drop the vector
notation as, for the sake of simplicity, we are considering a
In a bipolar semiconductor (57, 58), the drift heat ﬂux is
divided in the drift heat ﬂuxes of electrons and holes,
qdr;n¼Pnjnand qdr;p¼Ppjp, whereas the diffusion heat
ﬂux is equal to the sum of the diffusion heat ﬂuxes of
electrons (holes) qdiff ;n¼knrT(qdiff;p¼kprT) and the
diffusion heat ﬂux of phonons, qdiff ;ph ¼kphrT.
qdiff ¼qdiff;nþqdiff ;ph ¼ knþkph
qdiff ¼qdiff;pþqdiff ;ph ¼kpþkph
From equation 31, it follows that the energy ﬂux in a
bipolar semiconductor is
In this expression of the energy ﬂux in a bipolar semi-
conductor is the energy ﬂux of electrons, holes, and pho-
nons. Let us note that cnand cpare independent Fermi
quasilevels of electrons and holes (in equilibrium
cncp¼eg=e, where egis the energy gap) (14).
The expression for the energy ﬂux win a bipolar semi-
conductor can be written in a reduced form as follows:
where q¼qdr þqdiff (34)
If the transport energy processes take place in a bipolar
semiconductor, then the rate of change of the energy
density uin the semiconductor is as follows:
The right-hand side of equation 35 corresponds to an
energy change caused by energy transport. If the processes
that take place in the semiconductor are stationary, then
the energy of each region does not change with time and
equation 35 reduces to the following:
divw ¼0 (36)
Now a question arises: “What value plays the role of the
source of the heat ﬂux?” To answer to this question, the
expression for divq needs to be found. By substitution of
the energy ﬂux, w, from equation 33 into equation 36, we
obtain the following expression (let us recover the vector
nature of several magnitudes to avoid misunderstand-
Taking into account the carrier continuity equations (55),
in which the rates of the total recombination are the same
and under total absence of external generation (see next
where Ris the recombination rate. Using equation 38,
equation 37 transforms into the following:
Using the expressions for the electrical current (34)
where snðspÞis the electrical conductivity of electrons
(holes), and Pn;p¼Tan;p. The gradients of the electrochem-
ical potentials can be written as follows:
Substituting the expression in equation 41 into equation
39, the divergence of the heat ﬂux adopts the following
Peltier Effect in Semiconductors 11
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From equation 42, we can see that recombination creates
the source of the heat ﬂux ðmnþmpÞR. Because mnþmpis
negative in nondegenerate semiconductors, if the
recombination rate is positive, then this source of heat
ﬂux is negative ðmnþmpÞR<0, while in the case of ther-
mal generation (R<0), it is positive ðmnþmpÞR>0. These
paradoxical phenomena will be explained subsequently.
Because under a linear approximation mnþmp¼eg, the
source of heat ﬂux caused by recombination transforms
into egRand we obtain the following:
To ﬁnd the sources of the diffusion heat ﬂux, the diver-
gence of the diffusion heat ﬂux divqdiff must be obtained.
Substituting the heat ﬂux qfrom equation 34 into equation
42 and taking into account equation 38, we obtain in the
linear approximation the following:
Taking into account, that (58)
It is worth noting that the source of the diffusion heat ﬂux
caused by the recombination is positive, ðgnþgpþ
5ÞT0R>0 because gnþgp<5 (19). This means that
recombination causes the increment of the temperature
(heating) in agreement with common sense.
It should be borne in mind that the absolute value of the
Peltier coefﬁcient, equation 45, of minority carriers can
largely exceed the one of majority carriers. Because the
Fermi quasi-levels depend on the concentration of majority
and minority charge carries, the Peltier coefﬁcients will
depend on the coordinate in the space-charge layer near
the p-njunction (rn
D, where rn;p
Dis the Debye
radius in the n- and p-regions as indicated by the super-
script) even in linear approximation with respect to the
current because of the spatial dependence of the equili-
brium concentrations near the p-njunction (59).
The expression for the diffusion heat ﬂuxes qn;p
diff is as
diff ¼ kn;p
p, and kn;p
ph are, respectively, the electron,
hole, and phonon heat conductivities in the n- and the
p-regions as indicated by the superscript.
Because kn;pkph in nondegenerate semiconductors,
equation 46 reduces to
Taking into account the considerations presented previ-
ously, the heat balance in equation 43 can be rewritten as
Because the current densities can be calculated as
(see eq. 40) follows:
the calculation of mn;mp, and wis required.
The macroscopic description of the transport of non-
equilibrium charge carriers is performed using the conti-
nuity equations for the electron and hole current densities
(see eq. 38) and the Poisson equation (30)
where ris the space charge, eis the permittivity, and Ris
the recombination rate in n- and p-regions.
RECOMBINATION IN SEMICONDUCTORS
To make use of the system of equations 50–52, we should
specify the dependences of partial currents, the space-
charge density, and the recombination rates on the non-
Unfortunately, many publications devoted to these
problems contain obvious errors; these fallacies are caused
by the recombination itself that is often described by the
incorrect expressions as follows:
Here n¼n0þdnand p¼p0þdpare the electron and
hole concentrations, where dnand dpare the concentra-
tions of nonequilibrium charge carriers, and tnand tpare
the lifetimes of nonequilibrium charge carriers, which are
key parameters of the semiconductor under consideration
(60, 61). However, because the condition for the total
current continuity div j¼0 should be satisﬁed under static
conditions, an additional condition dn=tn¼dp=tparises;
the latter condition does not follow from any physical
concepts and is difﬁcult to interpret. In certain cases
(see, for example, monograph in Reference 62), this condi-
tion is used to reduce the number of variables, which is
completely wrong. Sometimes, this condition is also con-
sidered as an expression relating to the lifetimes of charge
carriers (60). The latter approach is not physically mean-
ingful and, in addition, is hardly constructive because the
carrier lifetimes no longer constitute the semiconductor
parameters; rather, the lifetimes are functions of the non-
equilibrium carrier concentrations that, in turn, should be
determined, which leads to considerable mathematical
Another approach based on the assumption that R
, where dpand t
are the concentration and
12 Peltier Effect in Semiconductors
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lifetime, respectively, of the nonequilibrium minority car-
riers, is widely accepted (63). In this case, the condition for
the total-current continuity is identically satisﬁed; how-
ever, another basic contradiction takes place. This contra-
diction becomes especially evident if we consider a majority
carrier injection. Physically, it is obvious that the injected
nonequilibrium majority carriers recombine. At the same
time, formally, the recombination rate is equal to zero
because there are no nonequilibrium charge carriers
The problem is appreciably complicated if the tempera-
ture distribution in the sample is nonuniform (i.e., the
temperature is coordinate dependent). As a result, it
becomes unclear from which value the nonequilibrium-
carrier concentration is measured (i.e., what is meant by
the quantities n
). A method to overcome this
difﬁculty has been suggested recently (34); however, this
method is rather artiﬁcial and is applicable only in the case
of a known (ﬁxed) temperature ﬁeld.
It is relatively straightforward to obtain the following
expressions for recombination processes (64) by rigorous
consideration of transitions between valence and conduc-
where xis the recombination factor and niis the carrier
concentration of the intrinsic semiconductor. By lineariza-
tion of these expressions for the case of weak deviation
from the condition of thermodynamic equilibrium (the
changes in carrier concentrations as a result of current
are small in comparison with their equilibrium values), we
have (30) (TðxÞ¼T0þdTðxÞand dTðxÞT0):
where (65, 66) t¼½xðn0þp0Þ1,l¼2ni
Let us note that this baseless idea is widespread;
namely, the presence of only interband recombination is
a sufﬁcient condition for dn¼dpto be fulﬁlled (42, 61, 62).
However, there is no proof of this conclusion; moreover, the
case of injection obviously contradicts it.
Thus, the presence of a temperature gradient results in
the appearance of an additional term in expressions for
recombination rates. This term takes into account the
change in the rate of thermal generation (which, as it is
well known, is proportional to the squared concentration of
the intrinsic semiconductor at a given temperature).
The situation becomes even more complicated if
recombination resulting through the impurity centers
(traps) is taken into account. Within the framework of
the Shockley–Read–Hall model and with the assumption
that the carriers of impurity centers are characterized by
the temperature T(r), the recombination can be given by
the following equations (see References 61, 63, 64, and 67):
is the impurity concentration; xnand xpare,
respectively, the electron and hole capture coefﬁcients
etis the impurity energy level; nnðTÞ¼1=4ð2mnT=
ph2Þ3=2; npðTÞ¼1=4ð2mpT=ph2Þ3=2; and mnand mpare
the electron and hole effective masses. The concentrations
n1and p1are the parameters characterizing the impurity
level and, physically, represent the electron or hole con-
centrations when the Fermi level in the semiconductor
coincide with the impurity level. nnand npare the densities
of state at the bottom of the conduction band and top of the
Unless otherwise indicated, and with no loss of gener-
ality, we shall refer to a semiconductor that contains
impurities with a single energy level able to capture elec-
trons. It follows from equation 56 that one more unknown
value nt(ntis the concentration of electrons captured by
the impurities) arises when the recombination takes place
through the impurity centers.
Subtracting the second equation 53 from the ﬁrst, we
obtain (30, 55) the following:
The charge conservation in steady state can be written as
divj ¼divðjnþjpÞ¼0 (58)
From equations 57 and 58, we obtain the following rela-
As it follows directly from equations 59 and 56, the
deviation of the concentration of the electrons trapped
in the impurity level dntfrom the equilibrium value n0
depends on the deviations of the electron and hole
concentrations from their equilibrium values through
The recombination rates R
are actually deﬁned
as the difference between the rates of capture of electrons
and holes and their thermal generation. These two mech-
anisms cannot create nonequilibrium carriers without an
external source of excitation. From equations 59 and 56, we
obtain the following expression for dnt:
To obtain equation 60, we showed that n1and p1can be
presented in the following form: n1¼n10 þdn1and
p1¼p10 þdp1. Here, dn1¼n10=T0ð3=2þet=T0ÞdTand
By substitution of equation 60 into equation 56, we
obtain equation 55, where t1¼xnxpNtn0þp0
Peltier Effect in Semiconductors 13
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We can rewrite the Poisson equation (eq. 52) for nonequi-
librium variations in the following form:
where the electrical potential w¼w0þdw, bulk electrical
charge r¼r0þdr, and w0and r0are the equilibrium
electrical potential and bulk charge.
For intrinsic semiconductors dr ¼eðdpdnÞ, and for
semiconductors of n-orp-type, dr ¼eðdpdndntÞ.In
the last case, it is necessary to use equation 60.
Quasineutrality (QN) is a basic concept in semiconductor
device analysis and is used widely in the literature on
transport phenomena. Its physical meaning has been exten-
sively reviewed (see, for instance, Reference 68). It can be
readily seen from Poisson’s equation that dr=r0ðrd=LÞ2
(rdis the Debye length and Lis the characteristic length of
the change of the chemical potential). If ðrd=LÞ21, then
dr !0. Under these conditions, the differential Poisson
equation becomes the following algebraic equation:
which does not need boundary conditions. Additionally,
this algebraic equation establishes a relationship between
the excess of both kinds of carrier and renders the Poisson
equation redundant in the system of equations 50–52.
Despite the widespread use of the QN hypothesis, it is in
general mistakenly reduced to a simpliﬁed situation in
which both excesses of majority and minority carriers
are the same: dndp(69, 70). This assumption, which is
valid in some situations, can also lead to important errors in
other cases (55). From the equation dr ’0, it follows natu-
rally that quasineutrality does not necessarily imply
Therefore, quasineutrality can be identiﬁed with the
expression dn¼dponly in intrinsic semiconductors or in
doped ones if the variation in the trapped charge is weak.
From Reference 55, it follows that if rd!0, then w0ðxÞ
tends to a constant value and is null elsewhere. Therefore,
w0ðxÞcan be asymptotically approximated in the p-njunc-
tion by a step function with a discontinuity at the semi-
conductor interfaces. In electrical terms, this means that
in quasineutrality, if the Debye length is very small (i.e.,
when considering highly doped semiconductors), then the
space charge spread over a length comparable with the
Debye length in the semiconductor can be replaced by a
surface space charge (boundary layer function (71)) that
supports the discontinuity of the electric potential.
Because the Poisson equation cannot provide w0, removing
it from the system of equations 50–52 will not lead to any
loss of physical information. Furthermore, in the limiting
case of rd!0 we are in QN, and the Poisson equation does
not need to be solved; instead, the value of w0can be
obtained by taking advantage of the fact that the electro-
chemical potential is constant across the structure in
equilibrium (see Reference 72).
It should be stressed that beyond the fact that the
quasineutrality concept is useful when dealing with trans-
port phenomena, QN is important in semiconductor
devices because it is the basis of the operation of most
semiconductor devices. In bipolar semiconductors, the
movement of charge packets requires a quasineutral
nature of the packet (73, 74).
EQUATIONS FOR COOLING AND HEATING IN p-n
In the linear approximation with respect to the electric
current, the heat balance equation is given by (see eq. 43)
Equation 64 represents the total heat ﬂux conservation
law that can be formulated as follows: Any change in the
drift component of the heat ﬂux is accompanied by a
change in the thermal diffusion component of the same
heat ﬂux, and the heat bulk sources and sinks are given by
the recombination processes.
The expression for q
in bipolar semiconductors is
As introduced previously, the Peltier coefﬁcients in non-
degenerated semiconductors are
The expression for the diffusion heat ﬂuxes qn;p
diff is as
follows (see eq. 47):
ph is the phonon heat conductivity in the n- and the
p-regions as indicated by the superscript.
Taking into account the considerations presented pre-
viously, the heat balance equation can be rewritten as
Because the current densities can be calculated as (see
the calculation of mn,mp, and wis required. It means
that we need equations 50–52. Under quasineutral approx-
imation, instead of equation 52 we have equation 62.
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Because by deﬁnition (see Reference 72), n¼nn exp
½mn=T;p¼np exp½mp=T, where mn;p¼m0n;pþdmn;p, and
The system of equations 50, 51, 68, and 63 deﬁnes
the mathematical framework of thermoelectric cooling,
and it must be complemented with the appropriate bound-
ary conditions that describe the electric currents, the
heat ﬂux, and the electric potential through the interfaces
(75–77). Let us assume that in the y- and z-direction the p-n
junction is adiabatically isolated. Then, the boundary con-
ditions in the remaining direction (i.e., the p-njunction
interface is orthogonal to the x-axis, and assuming that the
interface is located at x¼0, the nregion located between
and x¼0, the pregion between x¼0 and x¼l
Assuming that an ideal metal–semiconductor contact is
placed at x¼l
, we can write the following boundary
conditions for the excess of temperature and carrier densi-
ties (hereafter a superscript nor pin a magnitude refers to
the nor pregion, respectively):
These boundary conditions are justiﬁed because of the high
value of the thermal conductivity of metals and the intense
recombination at the metal–semiconductor interface. Sim-
ilar boundary conditions may be written as follows at the
metal–semiconductor interface at x¼l
where Vis the applied voltage. These boundary conditions
assume that the semiconductor is at equilibrium in x¼l
and in x¼l
; in other situations, the electric potential (i.e.,
eqs. 72 and 75) cannot be rigorously deﬁned (78). At the p-n
junction interface, we can introduce the following addi-
tional boundary conditions (75, 76):
In the preceding equations, the superscript n(p) in mag-
nitudes refers to the n(p) region and subscript (nor p)
refers to the carriers (electrons or holes); additionally,
subscript 0 in a magnitude denotes equilibrium. Because
the total current density is constant across the junction, it
is enough to formulate boundary conditions for one of the
two current densities. These boundary conditions are
obtained assuming, respectively, continuity of the electro-
chemical potential at the interface, that both heat and
electrical conductivities are very large at the junction,
and the absence of surface recombination. In fact, because
the p-ninterface is inside the depletion region, this latter
assumption is not realistic, and boundary conditions with
ﬁnite conductivities need to be used (75). In this work, we
use equations 76–79 for the sake of simplicity.
SIMPLIFICATION OF THE MODEL OF PELTIER EFFECT
IN TWO LIMITING CASES
In this section, we will analyze the thermoelectric cooling
in a p-njunction in the two limiting cases: strong and weak
Let us now consider that volume recombination is weak. In
this case, the conditions ln;p
Dare the dif-
fusion lengths in n-pregions) are fulﬁlled, which means
that the weak recombination is correct for thin ﬁlm p-n
structures. Formally R¼0 when t!1. Under this condi-
tion, the right-hand side of equation 43 becomes trivially
null and along with equations 65 and 67 transforms into
Equations 50 and 51 transform into
From equation 81, it follows that jn;pare not spatially
dependent and jn
p¼j0, where j0is the whole
current through the p-nstructure. From the boundary
conditions for currents (75, 76), it follows that
It is not difﬁcult to understand that the concentrations
of the nonequilibrium carriers (dnand dp) are maxima in
It may seem that the calculation of thermoelectric cool-
ing does not require the use of equation 81 in the absence of
recombination because there are no other unknown func-
tions in equation 80. So, it seems that thermoelectric
cooling does not depend on the nonequilibrium carrier
concentrations. However, the boundary conditions to equa-
tion 80 must be formulated for heat ﬂuxes (eqs. 65 and 67).
The drift heat ﬂuxes depend on the current of majority and
minority carriers (eq. 65). The latter essentially depends
on the distribution of nonequilibrium carrier concentration
because of the terms ðrmn;pÞ=e. Therefore, there are no
reasons to assume a priori that jn
Peltier Effect in Semiconductors 15
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The problem is reduced to the calculation of the currents
in the electrical circuit composed of two circuits connected
in parallel (see Figure 16). One of them is composed of two
n-type semiconductors connected in series with the con-
centrations nnand np, whereas the other is composed of
two p-type semiconductors connected in series with the
concentrations pnand pp. At a given direction of the
current (from n-top-region), heating instead of cooling
takes place at weak recombination.
With respect to what was said previously, let us notice
that the classic theory for current-voltage characteristic
through the p-njunction (6, 14) obeys the following
where the saturation current (j
) varies in direct proportion
to the capture coefﬁcients. It means the current j
the p-njunction is equal to zero when the recombination is
absent under any voltage.
The above means that the model (eq. 82) is not correct
for weak recombination. At the same time, equation 81
(together with eqs. 80 and 63) will give the correct expres-
sion for the current density j0(at least under weak bias
A main result is that the temperature deviation from
equilibrium at the junction may be obtained as follows:
The expression of His
From equation 83, it follows that a positive current will
generate heat instead of cooling at the junction, which is in
strong contrast with the conventional results.
Let us assume that the recombination is very strong. The
physical meaning is that ln;pln:p
D!0. From a
mathematical point of view, we have t!0 in equation 55.
Because the recombination rate (R) cannot be inﬁnite at
t!0, from equation 55 follows that
At the same time, the magnitude Ris ﬁnite but not deﬁned.
Adding equation 38 we have
It is important to emphasize that the nonequilibrium
charge carrier concentrations (dnand dp) are not equal
to zero in the considered approximation. Therefore, there
is no reason to state that jn
Therefore, because ln;p
D!0, Rdiffers from zero only at
the interface. The volume equation (eq. 43) transforms
again into equation 80.
Once again, just like in the weak recombination case,
the right-hand side of equation 43 also becomes zero but for
different physical reasons. But as in the case of weak
recombination, the heat ﬂux depends on the nonequili-
brium carrier concentrations. The latter are deﬁned by
equations 85, 86, and 80 with the corresponding boundary
conditions. In the previous case, it was noted that equation
82 is not correct when the recombination is weak enough.
Also, it is not difﬁcult to understand that equation 82 is not
correct in the case of strong recombination. It follows from
the expression for j
that js!1when t!0 at any applied
voltage V. The last statement is not correct from a physical
point of view.
The method described allows the calculation of the
current-voltage characteristic of the p-njunction in the
case of strong recombination in the linear regime with
respect to the applied voltage V.
The temperature deviation at the junction has been
analytically obtained as follows:
This expression clearly differs from the commonly used
expression as follows (6, 9, 14, 18):
The differences are not only in magnitude but also in sign.
Contrary to equation 88, indicating that the positive values
only predict a decrease in temperature with j
tion 87 predicts that a p-njunction under the same bias
conditions (positive values of j
) may be heated or cooled
depending on the values of the Peltier coefﬁcients and
electrical conductivities in the p-njunction. Moreover,
equation 87 clearly shows the paramount importance of
the nonequilibrium carriers at both sides of the junction
(neglected in equation 88) because they control the sign of
Finally, let us emphasize that only when the following
two criteria are met simultaneously:
Equation 87 reduces to equation 88.
16 Peltier Effect in Semiconductors
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Research and development of bulk homogeneous materials
for thermoelectric applications began to increase starting
in the 1950s and resulted in commercial solid-state power
generation and cooling systems (5). With the introduction
of materials and concepts based on nanostructuring, the
ﬁeld has witnessed truly dramatic growth over the past 15
years. Nanostructuring of semiconductors in the form of
quantum wells and superlattices started in the 1960s and
early 1970s for electrical and optical devices. In contrast,
the primary beneﬁt to date of nanostructuring for thermo-
electric materials has been the impact on thermal propert-
ies rather than electronic (79).
The performance of thermoelectric materials is quanti-
ﬁed by the previously introduced unitless ﬁgure of merit
ZT, which is deﬁned as Z¼a2s=k. The theory of thermo-
electric efﬁciency was outlined by Altenkirch (80, 81). It
was based on a simple procedure of taking into account
Joule losses and losses caused by thermal conductivity.
The resultant expression was given in the form proposed
by Ioffe (5). The process of energy conversion was studied
by Harman and Honig (82), Anatychuk (9), Nolas et al. (6),
and Tauc (14).
In the 1950s, it was discovered that alloys of Bi
a value of ZT 1 near room temperature, and they have
played a dominant role in the ﬁeld of thermoelectrics
through today. Although each magnitude in the expression
of ZT can individually be changed by several orders of
magnitude, the interdependence and coupling between
these properties have made it extremely difﬁcult to
increase ZT above the unity despite ﬁve decades of active
research. The thermoelectrics community is widely target-
ing ZT >3 to make these solid-state systems competitive
with traditional mechanical energy conversion systems
In 1993, Hicks and Dresselhaus (13, 83) pioneered the
concept that quantum conﬁnement of electrons and holes
in low-dimensional materials could dramatically increase
ZT above the unity by independently changing the
magnitudes in the ﬁgure of merit (a2s). Although this
ignited intense research into nanostructured thermo-
electric materials, there are still many debates about the
exact role that low dimensionality and nanostructures
could play in improving thermoelectrics. Table 1 summa-
rizes major ZT milestones achieved over the past decades
It is important to note that some of the ZT values in
Table 1 have not yet been independently veriﬁed or repro-
duced, and corroboration of these key breakthroughs
would be helpful to the thermoelectrics community.
The ﬁeld of thermoelectrics presents an important chal-
lenge to synthetic chemists, physicists, and materials sci-
entists. The discovery of new and promising materials
requires a combination of theoretical guidance, keen chem-
ical intuition, synthetic chemistry expertise, materials
processing, and good measurement skills. This powerful
combination can be effectively achieved by reaching across
scientiﬁc disciplines (99).
A common theme among many emerging thermoelectric
materials is the concept of nanostructuring to improve
the thermoelectric performance. In these systems, the
enhanced thermoelectric performance is attributable to a
strong decrease in lattice thermal conductivity rather than
an increase in the electrical power factor (a2s). Thus, in
certain cases, nanodots clearly play a signiﬁcant role in
reducing lattice thermal conductivity, probably by effec-
tively scattering phonons that otherwise would have rela-
tively long mean free paths. In many of these cases, it has
been clearly demonstrated that the reduction in thermal
conductivity far exceeds any concomitant reduction in the
power factor caused by electronic carrier scattering, thus
resulting in enhanced ZT values (99). Thin-ﬁlm super-
lattices and nanowires make up a signiﬁcant part of cur-
rent thermoelectric research.
In review (100), strategies to improve the thermo-
electric ﬁgure of merit, new discussions on device physics
and applications, and assessments of challenges on
these topics were presented. Understanding of phonon
transport in bulk materials has advanced signiﬁcantly
Table 1. Thermoelectric Figure-of-Merit ZT as a Function of Temperature and Year Illustrating Important Milestones
Material ZT T (K) Year Reference
(supperlattice) 2.4 300 2001 84
PbSeTe (nano dot superlattice) 1.6 300 2002 85
2.2 800 2004 86
0.77 300 1958 87
Te/PbTe (multi quantum well) 1.2 300 1996 88
0.8 225 2000 89
Si NW (nanowire) 1.0 200 2008 90
Si NW (nanowire) 0.6 300 2008 91
BiSbTe 1.4 373 2008 92
1.7 700 2006 93
1.48 705 2009 94
PbTe (Ta-doped) 1.5 773 2008 95
1.35 900 2006 96
SiGe 1.3 1173 2008 97
SiGe NW 0.46 450 2012 98
Although there have been several demonstrations of ZT >1 in the past decade, no material has yet achieved the target goal of ZT 3. The
material systems that have achieved ZT >1 have all been based on some form of nanostructuring.
Peltier Effect in Semiconductors 17
3GW8206 03/26/2014 0:6:49 Page 18
as the ﬁrst-principles calculations are applied to thermo-
electric materials, and new experimental tools are being
developed. As a result, some new strategies have been
developed to improve electron transport in thermoelectric
A thermoelectric device consists of heavily doped semi-
conductor legs that are connected electrically in series and
thermally in parallel. A fundamental understanding of
heat and charge carrier transport inside the thermoelectric
legs will lead to new strategies to design and fabricate
high-efﬁciency thermoelectric materials. The device efﬁ-
ciency depends not only on materials but also on an opti-
mum choice of the legs’ size, conﬁguration, and contacts.
Finding new applications for thermoelectric devices in
places that they are superior to other technologies is
another challenge for the thermoelectric community.
Table 2 summarizes the main advantages and disad-
vantages of thermoelectric coolers.
As a motivating conclusion of this section, it is impor-
tant to examine more closely the meaning of ZT and its
limits as a ﬁgure-of-merit in thermoelectric devices. ZT is a
good parameter to characterize thermoelectric materials.
When a thermoelectric device consists of just one homoge-
neous material (as it is often the case in thermoelectric
generators), ZT will be truly useful to characterize the
efﬁciency of the device. Nevertheless, in thermoelectric
cooling (Peltier effect), it is necessary to have a contact
between two materials. In this situation, nonequilibrium
electrons and holes will set up. Which parameter that could
characterize the efﬁciency of cooling system as a whole was
not known until today.
The Peltier effect is the heat extraction or absorption that
occurs at the contact between two different conducting
media when a DC electric current ﬂows through this
contact. Devices based on the Peltier effect, which is the
basis for solid-state thermoelectric cooling, have evolved
rapidly to meet the pace of the ever-growing industry of
electronics. In recent years, new solid-state cooling solu-
tions based on new materials and devices (such as gra-
phene, Si and SiGe nanowires, etc.) have been proposed. To
drive the quest for new high-efﬁciency cooling devices, a
suitable knowledge of the theory behind thermoelectric
cooling is of paramount importance.
Thermoelectric phenomena, such as the Peltier effect,
are at the crossroads of several ﬁelds of physics and
engineering such as solid-state electronics, nonequili-
brium thermodynamics, transducer devices, and so on.
A comprehensive study of the mechanisms of heating and
cooling originated by an electrical current in semi-
conductor devices has been presented in this article.
Thermoelectric cooling in n-n,p-p,andp-njunction con-
tacts and inhomogeneous bulk semiconductors was ana-
lyzed. Both degenerate and nondegenerate electron and
hole gases were considered. The eminent roles of
recombination and nonequilibrium charge carriers in
heating/cooling of solid-state devices, usually ignored
in the Peltier effect’s literature, has been discussed.
Along with the above, special attention has been paid
to several aspects of nonequilibrium thermodynamics of
the thermoelectric phenomena involved in Peltier effect
in semiconductors that demanded a careful examination.
In particular, some inaccuracies in the traditional theory
widely extended in the literature were discussed, and a
recently formulated self-consistent theoretical model,
better describing the Peltier effect, was presented.
Finally, a glimpse of applications was given with special
emphasis on the evolution of experimental values of the
ﬁgure of merit ZT in the past decades. A reﬂection was
made about the lack of ﬁgure of merit for thermoelectric
devices involving two or more materials (such as those
based on the Peltier effect).
Yu. G. Gurevich thanks CONACYT-Mexico for ﬁnancial
support. J. E. Velazquez-Perez wants to thank Spanish
Ministerio de Econom
ıa y Competitividad (MINECO) and
FEDER for ﬁnancial support under grant TEC2012-32777.
We would like also thank Associate Editor Ms. Cassandra
Strikland for her valuable help and assistance in the
processing of this manuscript.
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J. E. VELAZQUEZ-PEREZ
Universidad de Salamanca
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