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PELTIER EFFECT IN SEMICONDUCTORS

INTRODUCTION

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

be maintenance-free.

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

bias.

4. Maintenance free.

5. Free of moving parts.

6. Functioning freely and independently of the

position.

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

QP¼ðP1P2ÞJ(1)

where P1;2are the Peltier coefﬁcients of the conducting

materials at each side of the junction and Jis the total

electric current.

For a nondegenerate system of electrons (subscript n)

and holes (subscript p) (14), the Peltier coefﬁcient is

Pn;p¼1=eðgn;pþ5=2ÞTmn;p

hi

(2)

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

n,p

are the

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

p

¼e

g

m

n

, where e

g

is

the band gap); eis the hole charge; and g

n,p

are the

exponents in the momentum relaxation times (19):

tn;peðÞ¼t0

n;p

e

T

gn;p(3)

where eis the carrier energy. The constant quantities t0

n;p

and gn;pfor different relaxation mechanisms can be found

in Reference 19.

For degenerate electron and hole gases

Pn;p¼p2

3egn;pþ3

2

T2

mn;p

(4)

To understand the physical meaning of equations 2 and 4,

it is convenient to rewrite these expressions as follows:

Pn;p¼1

ee

hi

n;pmn;p

hi (5)

where e

hi

n;pis the mean kinetic energy of the carriers in the

ﬂux,

ehi

n;p¼R

1

0

tn;peðÞe5=2df n;p

0eðÞ

de

de

R

1

0

tn;peðÞe3=2df n;p

0eðÞ

de

de

(6)

where fn;p

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

jj

T,m<0)

e

hi¼gþ5

2

T(7)

while for the degenerate case ( m

jj

T,m>0)

e

hi¼mþp2T2

3mgþ3

2

(8)

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

temperature uniform.

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.

T0T0

T0T0

j

j

j

jq = ∏j

j

q

jq

Figure 2. Schematic circuit of Peltier effect for constant T.

2 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:45 Page 3

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

ðÞ

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

(22, 23):

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

heat.

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

x¼0.

T1ðx¼0Þ¼T2ðx¼0Þ(9)

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

different media.

Peltier Effect in Semiconductors 3

3GW8206 03/26/2014 0:6:45 Page 4

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-

lyzed below.

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

(22).

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

jj<P1

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^

atelier-

Braun principle, a thermal diffusion ﬂux q1

dif ¼k1dT1=dx

(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

q1

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

ﬂux q2

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

heat ﬂuxes.

The corresponding temperature distributions in the

structure are qualitatively represented in Figure 5. It is

obvious that for P2

jj

<P1

jj

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

jj

>P1

jj

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

qdif

ddr

dT1

dx

(1)

(1)

= – κ1

= ∏1 j

j

ddr

(2) = ∏2 j

qdif

dT2

dx

(2) = – κ2

x = 0

Figure 4. Heat ﬂuxes in an n-nstructure assuming P2

jj

<P1

jj

,

k1;2is thermal conductivity in each region.

T(x)

T1(x)T2(x)

0

T0

d2

–d1x

Figure 5. Temperature distribution in an n-nstructure assuming

P2

jj

<P1

jj

.

4 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:45 Page 5

increase the drift thermal ﬂux q1

dr ¼P1j, which has

decreased compared with q2

dr ¼P2j.

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

the junction.

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

hi

mn¼gþ5=2ðÞT

mnand P1P2¼De=e.

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

questions arise:

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

junction?

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

hi

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):

P1P2¼m1m2

e¼x0

1x1x0

2x2

e¼wcþDec

e(10)

qdif

dT1

dx

(1) = – κ1 qdif

dT2

dx

(2) = – κ2

ddr

(1) = ∏1 jddr

(2) = ∏2 j

x = 0

j

Figure 6. Heat ﬂuxes in an n-nstructure assuming P2

jj>P1

jj.

T(x)

0

T2(x)

T1(x)

T0

–d1d2

x

Figure 7. Temperature distribution in an n-nstructure assuming

P2

jj>P1

jj.

Figure 8. Heating or cooling of a metal-n-type semiconductor

contact.

Figure 9. Energy band diagram of n-nor p-pcontacts in equili-

brium (j¼0).

Peltier Effect in Semiconductors 5

3GW8206 03/26/2014 0:6:46 Page 6

where xiand x0

i(I¼1, 2) are work functions and electron

afﬁnities of materials 1 and 2, wc¼x2x1

ðÞ=eis the

contact voltage, and Dec¼x0

1x0

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

c

and the work of the

valence forces W

n

. 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

þ

-nor p

þ

-pjunction,

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).

P1P2¼mn

e(11)

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.

Now,

P1P2¼p2

3egþ3

2

T2

m1m2

m1m2

ðÞ(12)

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-

erate semiconductors.

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

the former

Wc/3

2TEðÞT0

ðÞ (13)

where T(E) is the nonequilibrium temperature, Eis the

built-in electric ﬁeld, and T

0

is the room temperature. In

the latter,

Wc/p2

4

T2EðÞT2

0

m(14)

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

UNIPOLAR SEMICONDUCTORS

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)

rw¼0 (15)

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

rqþjrc¼0 (16)

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

0

is the vacuum level, and x

i

and x

i0

are

the work functions and electron afﬁnities, respectively.

Figure 11. Energy band diagram of the contact between two

metals.

6 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:46 Page 7

Using the equation for the electric current (31)

j¼sðrcþarTÞ(17)

to eliminate the electrochemical potential c(sis electric

conductivity and ais the Seebeck coefﬁcient), we arrive at

the heat balance equation

rq¼j2=sþajrT(18)

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

and rT(32).

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-

ence 32.

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

presented next).

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

T1ðx¼d1Þ¼T0(19)

T2ðx¼d2Þ¼T0(20)

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.

k1rTðx¼0ÞP1j¼k2rT2ðx¼0ÞP2j(21)

Last, the fourth boundary condition determines the heat

ﬂux through the interface x¼0 with the surface thermal

conductivity h(33).

hT1ðx¼0ÞT2ðx¼0Þ

½

¼k1rTðx¼0ÞþP1j(22)

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

STRUCTURES

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

d2T=dx2¼0 (23)

Together with the boundary conditions (19–22), equation

23 results in the following temperature distributions:

T1;2¼T0(1P1;2hd2;1

k2;1

P1P2

ðÞ

jd

1;2x

T0k1;21þhd1

k1þd2

k2

hi

);d1x0

0xd2

:ð24Þ

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

3GW8206 03/26/2014 0:6:46 Page 8

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-

tron afﬁnities.

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

properties.

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

Tðx¼0Þ¼T01P

jj

d

T0kj

(25)

where P

jj

¼P1

jj

¼P2

jj

. The last equation shows that the

Peltier effects are commensurable in the structure

considered.

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

0

is such that the

frequency nee of interelectronic collisions greatly exceeds

8 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:46 Page 9

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)

where E¼dw=dx,

q¼N=N0keTe1þgeE Te=NdN=dx ð2þgÞdTe=dx½

(27)

is the heat ﬂux in electron subsystem and Eis the electric

ﬁeld strength in the sample (Ehas only an xcomponent).

j¼N=N0sTegETe=e1=NdN=dx ð1þgÞ1=edTe=dx½

(28)

is the current and Teis the electron temperature. The

parameters

ke¼4Gð7=2þgÞN0

3ﬃﬃﬃﬃ

p

pmn0Tg

0

;s¼4Gð5=2þgÞN0e2

3ﬃﬃﬃﬃ

p

pmn0Tg

0

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

electron temperature.

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

c

, where jc¼2le=lð5=2þ

gÞexpðL=lÞ1

expð2L=lÞ1

ek

leT0is the average temperature Te

hi

<T0

where Te

hi

¼1=LR

L

0

TeðxÞdx,ifle<l(37). As the ratio

le=lincreases, the minimum of Te

hi

as a function of jis

shifted to higher values of j;Te

hi

min can be of the order of

unity when l

e

/l1 (37).

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

(39, 40).

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

P1P2¼mnþmp

e(29)

Figure 12. Distribution of currents in p-njunctions.

Peltier Effect in Semiconductors 9

3GW8206 03/26/2014 0:6:46 Page 10

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

next.

Figure 13. Energy band diagram of a p-njunction.

Recombination.

Figure 14. Energy band diagram a p-njunction without

recombination.

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

3GW8206 03/26/2014 0:6:47 Page 11

HEAT BALANCE EQUATION IN BIPOLAR

SEMICONDUCTORS

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

different.

The energy ﬂux density in a unipolar semiconductor is

(see eqs. 15 and 16):

w¼qþcj(30)

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

one-dimensional case.

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

rT

qdiff ¼qdiff;pþqdiff ;ph ¼kpþkph

rT(31)

From equation 31, it follows that the energy ﬂux in a

bipolar semiconductor is

w¼qdr;nþqdiff;nþcnjnþqdr;pþqdiff ;pþcpjpþqdiff;ph

(32)

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:

w¼qþcnjnþcpjp(33)

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:

@u

@t¼divw (35)

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-

ings):

divq¼rcn

ðÞjncnrjn

ðÞrcp

jpcprjp

(37)

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

section)

rjn¼eR;rjp¼eR (38)

where Ris the recombination rate. Using equation 38,

equation 37 transforms into the following:

divq¼mnþmp

Rrcn

ðÞjnr cp

jp(39)

Using the expressions for the electrical current (34)

jn;p¼sn;prcn;p

þan;prT

(40)

where snðspÞis the electrical conductivity of electrons

(holes), and Pn;p¼Tan;p. The gradients of the electrochem-

ical potentials can be written as follows:

rcn;p

¼ 1

sn;p

jn;pan;prT(41)

Substituting the expression in equation 41 into equation

39, the divergence of the heat ﬂux adopts the following

form:

divq ¼mnþmp

Rþ1

sn

j2

nþ1

sp

j2

pþanjnrTþapjprT

(42)

Peltier Effect in Semiconductors 11

3GW8206 03/26/2014 0:6:48 Page 12

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:

divq ¼egR(43)

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:

divqdiff ¼ðgnþgpþ5ÞT0R(44)

Taking into account, that (58)

Pn;p¼1/eðÞgn;pþ5/2

Tmn;p

tn;peðÞ¼t0n;pe/TðÞ

gn;p

(45)

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<x<rp

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

follows:

qn;p

diff ¼ kn;p

nþkn;p

pþkn;p

ph

rT(46)

where kn;p

n;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

qn;p

diff ¼kn;p

ph rT(47)

Taking into account the considerations presented previ-

ously, the heat balance in equation 43 can be rewritten as

follows:

kphDTþPnrjnþPprjpþjnrPnþjprPp¼egR

(48)

Because the current densities can be calculated as

(see eq. 40) follows:

jn¼snrwrmn=eþanrTðÞ

jp¼sprwþrmp=eþaprT

(49)

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)

rjn¼eR (50)

rjp¼eR (51)

Dw ¼4pr

e

(52)

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-

equilibrium-carrier concentration.

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:

divjn¼Rn;divjp¼Rp(53)

with Rn¼dn=tn;Rp¼dp=tp.

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

difﬁculties.

Another approach based on the assumption that R

n

¼

R

p

¼dp/t

p

, where dpand t

p

are the concentration and

12 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:48 Page 13

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

(dp¼0).

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

0

and p

0

). 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-

tion bands:

Rn¼Rp¼R¼xðnp n2

iÞ(54)

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):

R¼1=tp0

n0þp0

dnþn0

n0þp0

dpldT

(55)

where (65, 66) t¼½xðn0þp0Þ1,l¼2ni

n0þp0dni=dT ¼

n0p0

n0þp0

1

T0ð3þeg=T0Þ.

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):

Rn¼xn½nðNtntÞn1nt

Rp¼xp½pntp1ðNtntÞ (56)

where N

t

is the impurity concentration; xnand xpare,

respectively, the electron and hole capture coefﬁcients

n1¼nnðTÞexp½et=T;p1¼npðTÞexp½ðetegÞ=T, where

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

valence band.

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:

divðjnþjpÞ¼eðRnRpÞ(57)

The charge conservation in steady state can be written as

follows (31):

divj ¼divðjnþjpÞ¼0 (58)

From equations 57 and 58, we obtain the following rela-

tionship (30):

Rn¼Rp(59)

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

t

depends on the deviations of the electron and hole

concentrations from their equilibrium values through

R

n

and R

p

.

The recombination rates R

n

and R

p

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:

dnt¼n2

t0

Ntðxnnoþxpp10Þxn

p0

p10

dnxpdp

xnn10 et=T0þ3=2ðÞxpp0

eget

T0þ3=2

dT=T0

(60)

To obtain equation 60, we showed that n1and p1can be

presented in the following form: n1¼n10 þdn1and

p1¼p10 þdp1. Here, dn1¼n10=T0ð3=2þet=T0ÞdTand

dp1¼p10=T03=2eteg

T0

dT.

By substitution of equation 60 into equation 56, we

obtain equation 55, where t1¼xnxpNtn0þp0

xnðn0þn10Þþxpðp0þp10 Þ.

Peltier Effect in Semiconductors 13

3GW8206 03/26/2014 0:6:48 Page 14

QUASINEUTRAL APPROXIMATION

We can rewrite the Poisson equation (eq. 52) for nonequi-

librium variations in the following form:

d2dw=dx2¼4pdr=e(61)

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:

dr 0(62)

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

dndp, but

dp¼AdnBðn0þp0ÞdT=T0(63)

where

A¼xnðNtn0þn2

t0p0=p10ÞþxpNtp10

xnNtn0þxpðNtp10 þn2

t0Þ;

B¼

xnn10ðet=T0þ3=2Þxpp0

eget

T0þ3=2

xnNtn0þxpðNtp10 þn2

t0Þ

n2

t0

n0þp0

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

JUNCTIONS

In the linear approximation with respect to the electric

current, the heat balance equation is given by (see eq. 43)

the following:

rq¼egR(64)

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

dr

in bipolar semiconductors is

qdr ¼PnjnþPpjp(65)

As introduced previously, the Peltier coefﬁcients in non-

degenerated semiconductors are

Pn;p¼1

egn;pþ52

=Tmn;ph (66)

The expression for the diffusion heat ﬂuxes qn;p

diff is as

follows (see eq. 47):

qn;p

diff ¼kn;p

ph rT(67)

where kn;p

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

follows:

kphDTþPnrjnþPprjpþjnrPnþjprPp¼egR

(68)

Because the current densities can be calculated as (see

eq. 49),

jn¼snrwrmn=eþanrTðÞ

jp¼sprwþrmp=eþaprT

(69)

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.

14 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:48 Page 15

Because by deﬁnition (see Reference 72), n¼nn exp

½mn=T;p¼np exp½mp=T, where mn;p¼m0n;pþdmn;p, and

dn¼n0=T0½dmnþð3=2m0n=T0ÞdT,dp¼p0=T0½dmpþ

ð3=2m0p=T0ÞdT.

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

x¼l

n

and x¼0, the pregion between x¼0 and x¼l

p

) are

given below.

Assuming that an ideal metal–semiconductor contact is

placed at x¼l

n

, 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):

dTnln

ðÞ¼0 (70)

dnln

ðÞ¼0 (71)

dwðlnÞ¼0 (72)

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

p

:

dTplp

¼0 (73)

dpl

p

¼0 (74)

dwðlpÞ¼V(75)

where Vis the applied voltage. These boundary conditions

assume that the semiconductor is at equilibrium in x¼l

n

and in x¼l

p

; 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):

dwn0ðÞ

dmn

n0ðÞ

e1

e

@mn

n0

@TdTn0ðÞ

¼dwp0ðÞ

dmp

n0ðÞ

e1

e

@mp

n0

@TdTp0ðÞ ð76Þ

dwn0ðÞþ

dmn

p0ðÞ

e1

e

@mn

n0

@TdTn0ðÞ

¼dwp0ðÞþ

dmp

p0ðÞ

e1

e

@mp

n0

@TdTp0ðÞ

qn0ðÞ¼qp0ðÞ (77)

jn

n0ðÞ¼jp

n0ðÞ (78)

dTn0ðÞ¼dTp0ðÞ (79)

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

recombination.

Weak Recombination

Let us now consider that volume recombination is weak. In

this case, the conditions ln;p

Dln;prn;p

D(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

DT¼0 (80)

Equations 50 and 51 transform into

rjn;p¼0 (81)

From equation 81, it follows that jn;pare not spatially

dependent and jn

nþjn

p¼jp

nþjp

p¼j0, where j0is the whole

current through the p-nstructure. From the boundary

conditions for currents (75, 76), it follows that

jn

nþjp

n¼jn

0,jn

pþjp

p¼jp

0(jn

0þjp

0¼j0).

It is not difﬁcult to understand that the concentrations

of the nonequilibrium carriers (dnand dp) are maxima in

this case.

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

njn

pand jp

pjp

n.

Peltier Effect in Semiconductors 15

3GW8206 03/26/2014 0:6:49 Page 16

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

expression:

j0¼jsexp eV

T

1

(82)

where the saturation current (j

s

) varies in direct proportion

to the capture coefﬁcients. It means the current j

0

through

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

voltage).

A main result is that the temperature deviation from

equilibrium at the junction may be obtained as follows:

dTnð0Þ/j0lnPp

nþPn

p

2

Pp

n

2lp

sp

pþPn

p

2ln

sn

n

þj0lnHPp

n

2lp

sp

nþln

sn

p

!

ln

sn

pþlp

sp

p

!"#

þj0lnHPn

p

2lp

sp

nþln

sn

p

!

lp

sp

nþlp

sn

n

"#

þj0lnHPn

pþPp

n

2lnlp

sp

nsn

p

"#

(83)

The expression of His

H¼T0

lnlp

kn

phlpþkp

phln

(84)

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.

Strong Recombination

Let us assume that the recombination is very strong. The

physical meaning is that ln;pln:p

D!0rn;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

p0

n0þp0

dnþn0

n0þp0

dp¼ldT(85)

At the same time, the magnitude Ris ﬁnite but not deﬁned.

Adding equation 38 we have

r jn

nþjn

p

¼0;r jp

nþjp

p

¼0 (86)

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

njn

pand jp

pjp

n.

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

s

that js!1when t!0 at any applied

voltage V. The last statement is not correct from a physical

point of view.

Pn

n

Pp

nsp

n

sp

p

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:

dTnð0Þ/j0lnsn

nsp

pPn

n

Pp

p

þsn

psp

pPn

psn

nsp

nPp

n

hi

(87)

This expression clearly differs from the commonly used

expression as follows (6, 9, 14, 18):

dTnð0Þ/j0sn

nsp

pPn

nPp

p

ln(88)

The differences are not only in magnitude but also in sign.

Contrary to equation 88, indicating that the positive values

of j

0

only predict a decrease in temperature with j

0

, equa-

tion 87 predicts that a p-njunction under the same bias

conditions (positive values of j

0

) 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

dTnð0Þ.

Finally, let us emphasize that only when the following

two criteria are met simultaneously:

Pn

n

Pp

nsp

n

sp

p

;Pp

p

Pn

psn

p

sn

n

(89)

Equation 87 reduces to equation 88.

16 Peltier Effect in Semiconductors

3GW8206 03/26/2014 0:6:49 Page 17

APPLICATIONS

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

2

Te

3

had

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

(79).

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

(79).

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

Bi

2

Te

3

/Sb

2

Te

3

(supperlattice) 2.4 300 2001 84

PbSeTe (nano dot superlattice) 1.6 300 2002 85

AgPb

m

SbTe

mþ2

2.2 800 2004 86

Bi

2

Te

3

0.77 300 1958 87

Pb

1x

Eu

x

Te/PbTe (multi quantum well) 1.2 300 1996 88

CsBi

4

Te

6

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

Na

1x

Sb

m

Sb

y

Te

mþ2

1.7 700 2006 93

In

4

Se

3d

1.48 705 2009 94

PbTe (Ta-doped) 1.5 773 2008 95

Ba

8

Ga

16

Ge

30

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

materials.

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.

SUMMARY

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).

ACKNOWLEDGMENTS

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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YU. G. GUREVICH

CINVESTAV-IPN

J. E. VELAZQUEZ-PEREZ

Universidad de Salamanca

Peltier Effect in Semiconductors 21