Sensors and Materials, Vol. 8, No. 6 (1996) 389-408
MYUTokyo
S &M0253

Review of the Thermoelectric Efficiency of Bulk
and Thin-Film Materials
Friedemann Volklein
Fachhochschule Wiesbaden
Fachbereich Physikalische Technik
Am Brtickweg26, D-65428 Riisselsheim, Germany(Received March 29, 1995; accepted March 18, 1996)
Key words: thermoelectric efficiency, thermoelectric sensors and generators, thermoelectric prop­
erties of bulk and thin-film materials

The thermoelectric prope11ies of bulk and thin-film materials are reviewed. We
demonstrate that the thermoelectric figure of merit z is the crucial material parameter which
determines the efficiency of thermoelectric generators, coolers or sensors. Theoretical
approaches for the optimization of thermoelectric materials are discussed. The most usefulbulk and thin-film materials for the room, medium, and high temperature ranges and someexperimental data are presented.

1.

Introduction

Today a means must be found to apply the full power of modem materials science to theimportant problem of efficient and environmentally friendly energy conversion. Such
conversion can be performed by thermoelectric devices. Their applications range fromaerospace power generation to coolers and sensors. Currently, the advance of micro­
electromechanical systems (MEMS) is linked with the development of new them1oelectric
microcomponents. Developments in materials science present a variety of new experimental
techniques with the potential to crack the z · T barrier and to improve the parameters ofthermoelectric bulk and thin-film devices.

389

Sensors and Materials, Vol. 8, No. 6 (1996)

390
2.

Definition of the Thermoelectric Efficiency z

The thermoelectric efficiency (or thermoelectric figure of merit) z is the crucial
material parameter for the optimization of devices for thermoelectric energy conversion(generators, coolers) and for thermoelectric sensors. Figure 1 shows a typical thermoelec­
tric power generator. <1l Its coefficient of performance <P (power supplied to the load/heatabsorbed at hot junction) is determined by the current flow I, the load resistance RL, theSeebeck coefficients CXa, <Xp, the thermal conductance G(A.n, \) and the electrical resistanceR(pn, pp) of the thermocouple and the temperatures TH, Tc of the hot and cold junctions.
Here, An, \ are the thermal conductivities and Pn, pp are the electrical resistivities of theindividual thermoelectric legs.

(1)

The maximum efficiency is <1l<Pmax =

TH - TcTH�l + Z · T - l
�1 + Z · T +Tc/ TH(2)

where T =(TH+ Tc)/2 represents the average temperature. <Pmax is the efficien9y of an ideal
reversible engine (first term) reduced by the irreversible losses of heat conduction andJoule heating in the thermocouple (second term). The second term is determined by only

Heat

-

--

R [,

Fig. 1. Schematic of typical thermoelectric power generator. Refrigeration is achieved by replacing
the resistor with a current source, and reversing the current flow.< 1 lSensors and Materials, Vol. 8, No. 6 (1996)

391

one crucial material parameter, the thermoelectric efficiency of the thermocouple Z, where(3)

For the purpose of comparing materials it is convenient to define the figure of merit for anindividual material asa2

az. <Jz = p·l =;.,-,(4)

where <J is the electrical conductivity.
For thermoelectric coolers the coefficient of performance 17 is defined as the heatremoved by the cold junction devided by power input, and it depends on the parameters andon the material properties which we also found to be significant for thermoelectric
generators:

1] =

(a0

-

2
ap) ·I·Tc - __!_/ · R(pn,Pp)- G(ln, Ap)· (TH - Tc)l[l· R(Pn,Pp) + (� - ap )· (TH - Tc)]

(5)

The maximum efficiency is

1Jmax

Tc
"11 + Z · T -T H/ T c-Vl+Z·T+l
T H -T c(6)

and the maximum temperature difference which can be achieved by the thermoelement is
1
2(TH-Tc)max =-·Z·Tc.
2(7)

Figure 2 demonstrates the importance of Z for the maximum temperature difference. Cll
T8 is chosen as a parameter since this is usually held fixed in cooling applications.
Figure 3 represents the model of an ideal thermoelectric sensor and the corresponding
heat balance.c2, l The sensor consists of a sensitive area F (with radiation conductance Gp)
coupled via two thermoelectric legs (thermal conductance GL) to a heatsink of constantambient temperature T0• Without thermal input the average temperature of Fis T0, but thetemperature fluctuates around this value. This fluctuation causes the "temperature noise,"which ultimately limits the minimum detectable signal. When heating power N (which can
3392

Sensors and Materials, Vol. 8, No. 6 (1996)
200
TH = soo· K

180160 --140400 ° K

s 120

�c.,

I

�"'
...._,

80

20

0.001

0.002

z (K-1)

0.003

Fig. 2. Maximum temperature drop (TH - Tc)max as a function of thermoelectric efficiency Z fordifferent hot junction temperatures TH (adapted from ref. (1)).

arise from radiation, electrical heating or other sources) is absorbed by F, the temperature
increases to T1 • The heating power N is dissipated by conduction via GL and by radiation
via GF. The corresponding heat fluxes are NT and NF, respectively. For small temperature
differences with (T1 -T0)/T0 << 1, the heat fluxes NT and Np can be expressed in terms of the
conductances GL and GF. Assuming zero convection heat losses (operation in highvacuum) and negligible radiation losses of the thermoelectric legs in comparison to GL we
obtain the heat balance equation(8)

Sensors and Materials, Vol. 8, No. 6 (1996)

393

Absorbing area F
conductance GF

Fig. 3.

Model of an ideal thermoelectric sensor.

and hence the steady state temperature increase of the sensor(9)
Here RT denotes the thermal resistance. GL is determined by the dimensions (length l, crosssections Am Ap) and the thermal conductivities of the legs, GL = Mn / l + ApA.p I l, and GF by
the Stefan-Boltzmann law, GF = 4 e <J, TJ F, where e denotes the emissivity of F and <J, theStefan-Boltzmann constant. The steady state sensitivity (responsivity) Sis the ratio of thesignal voltage U = ( <Xri - CXp) • (T1 - T0) to the incoming heating power N:(10)
The noise equivalent power (NEP) relates the sensitivity S to the noise voltage UN. For
our ideal sensor we assume only thermal noise (Nyquist noise) arising from the resistanceR = l Pn lA n + l pp !A P. Hence,(11)
Here, 11/ denotes the amplified noise bandwidth and k the Boltzmann constant. The
detectivity D = 1 / NEP is usually referred to the sensitive area F and 11/= 1 Hz, resulting inSensors and Materials, Vol. 8, No. 6 (1996)394
the specific detectivity:(12)
The specific detectivity of our ideal sensor with a black sensitive area (e = 1) reaches the
maximum, when the heat flux NT through the thermocouple legs equals the heat flux NF
radiated from the receiver area and the heat fluxes through the legs are equal. In this casethe maximum detectivity is<2, l
3(13)

We compare eq. (13) with the general limit of the detectivity D* iim = 1 /4�<J, · k · Tg forthermal sensors associated with the fluctuations of the background radiation<4• l and find thespecific detectivity of the optimized thermoelectric sensor
5

D* max = D*lim -,Jz · To /2.

(14)

Nowadays, thermoelectric materials with Z · T0"' 1 are available. Therefore, the specificdetectivity D*max = (1 /2) · D*lim = 1 · 10 10 cm (Hz) 1'2 w -1 seems to be the ultimate limit of
the ideal thermoelectric sensor.

3.

Aspects of Optimization of the Thermoelectric Figure of Merit

The Seebeck coefficient a, the electrical conductivity <J and the thermal conductivity ;\,are the most impo1tant transport properties characterizing thermoelectric materials. It is ofinterest to compare the figure of merit z for different classes of materials.<6-9l
For metals the Seebeck coefficient becomes<10Ja=<J]

n2k2T Jln
[
3q
JE

(15)
E=Ee

which for monovalent metals at room temperature can be approximated<11l by a"' rr:- k2 TI
qEF. Here, E denotes the electron energy measured relative to the Fermi energy EF andqis
the elementary charge. The thermal conductivity for metals is ;t = Ac+ .Ai,"' Ac where ;\,L is
the lattice contribution and Ac is the electronic contribution to the thermal conductivity.
From the Wiedemann-Franz law Ac I <J = rr:- k2 TI 3 q2 = L · T with the Lorentz number L =2.4 · 10-s V2 / K2 we obtain z = a2 /LT"' 3 · 10-6 K- 1 at EF = 4.4 eV and 300 K. Thus metals
are not the most desirable materials for thermoelectric devices with high efficiency.
The Seebeck coefficient for semiconductors and insulators can have much higher

395

Sensors and Materials, Vol. 8, No. 6 (1996)

values than that for metals, particularly when the Fermi level lies deep within the forbidden
gap, i.e. for large Ee -EF or EF -Ev, where Ee, Ev are the energies associated with theconduction and valence band edges, respectively. The limiting value of a is determined
mainly by the value of the energy gap. As the Fermi level approaches midgap, contribu­
tions from charge carriers of both bands are subtractive and lower a. For this ambipolar or
intrinsic conduction, the Seebeck coefficient is given by a= ( £XnO"n + £Xi,O-p) / ( O"n + ap), wherean, av denote the partial conductivities of the corresponding bands. High Seebeckcoefficients are likely to be found in materials with large energy gap, such as insulators,
where large Ee -EF or EF -Ev leads to a""' 1 mV /Kor more. Electrical conductivities forinsulators, however, are very low and lead to much smaller values of a2 athan that of mostsemiconductors, as illustratedC11l by Fig. 4.
From inspection of Fig. 4 we learn that there is some optimum doping level or charge
carrier concentration n0p, where z achieves a maximum. Therefore, in order to determine
this optimum level we must relate the Seebeck coefficient and the electrical conductivity to�t).... :....�
....u

�

C:
Q)

z

a.

·u ::::,.;::

cr-�....

0

-

u

0

u
C'CI
� u
u

Q)

E

-�
.0 ....
u

0

Q)

Cl)
Cl)

....
Q)

�
w

C)

I

Insulators 1�....
'>....
u
::::,

"C
C:

0

u
C'CI

E
....

II

N

"C
C:

Q)

"' i::$t)"opt

Semiconductorsu..

MetalsI- - - - 7 --------AL

Q)

.c:

I-

Charge carrier concentration
Fig. 4.

Thermoelectric properties of metals, semiconductors and insulators.c11J

Sensors and Materials, Vol. 8, No. 6 (1996)396
the charge carrier concentration. We obtain
CT= qµ n,(16)a= ±(kl q) · [A+ ln(NJ n) + X],(17)

where n andµ denote the carrier concentration and mobility, respectively, N; denotes theeffective density of states in the conduction or valence band and A depends on the carrier
scattering mechanism and typically ranges between 1 and 4. The positive sign holds forholes, the negative for electrons. For some semiconductors such as silicon in a certain
temperature range ( 10- 500 K) the phonon drag effect contributes to the Seebeck coeffi­
cient. The phonon drag contribution X ranges from 0 for highly doped silicon to 5 for low­
doped silicon at 300 K. For polysilicon films the phonon drag effect can be neglected.<12J
The condition ;\,, << At, holds for some practical thermoelectric materials, such as silicon
and the SiGe solid solutions. Under this condition and neglecting phonon drag, we
maximize a2afrom d(a 2cr)/ Jn = 0. This leads to O'.op, = ±2 · (klq) = ±172µV /K, whichis independent of temperature and scattering mechanism, and with eq. ( 17) yields
nopt =

N; · exp(A- 2).

(18)

Assuming Ni = 2 · 10 19 cm-3 and acoustic mode lattice scattering (A = 2) the optimum
dopant level is n0v, = 2 · 1019 cm- 3, and for ionized impurity scattering (A= 4) we obtain n0p, =1.48 · 1020 cm- 3• Unfortunately, these optimum carrier concentrations are too high for theoriginal assumption of nondegenerate statistics to be valid. Therefore we must invoke themore general expressions

a=

±(�)[
q

(r + 2)Fr+i ( s)(r+l) Fr (s)

S+x]

(19)

(20)
involving the Fermi integrals=
Fr ( s) = foxrexp (x - s) dx,(21)

where r characterizes the scattering mechanism (with r = 0 for acoustic mode lattice
scattering and r = 2 for ionized impurity scattering) and s represents the reduced Fermi
levels for electrons Sn = (EF- Ee) I kT and for holes Sp = (Ev - EF) /kT. If we take into
consideration the charge carrier and lattice thermal conductivities

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Sensors and Materials, Vol. 8, No. 6 (1996)

A, = Ae + A,L = L(s{ ¾JO". T +

(22)

A,L

with the Lorentz number L = L( 0 · (k/ q)2 and neglect phonon drag effect, the dimensionlessfigure of merit z · T can be expressed by<13Js]2(r + 2) F,+1(s)[( r + 1) F, ( s) _z.T=L(s) + (r!//3 · F,(s))(23)

and is related to fundamental material parameters by jJ oc (m*) 312 • µI ).,L· Figure 5 showsz · T calculated according to eq. (23) as a function of the reduced Fermi level and of theparameter./3. The figure of merit increases with increasing./3, which suggests that the ideal
thermoelectric material should satisfy the somewhat conflicting requirements of higheffective mass m*, high mobilityµ, and low lattice thermal conductivity.
In conclusion, the best thermoelectric material can be achieved by an optimal doping ofsemiconductor alloys. The alloying of two isostructural semiconductors (such as Bi3Ter
Bi2Se3 solid solutions or SiGe alloys) introduces strong point defect scattering, which
substantially lowers the lattice thermal conductivity AL.
5.0 r---,-- --,-----,---,----,r----,-------,-----,
I

4.0
•

\

\

Optimum values
\

J.O

?..0

1.0

-4

-J

-2

-I

0

4

�

Fig. 5. Dimensionless figure of merit z · T as a function of the reduced Fermi level t; for various
values of the material parameter./3 oc (m*)312 µI ).,L (adapted from ref. (13)).

398
4.

Sensors and Materials, Vol. 8, No. 6 (1996)

Overview of Thermoelectric Bulk Materials

Figure 6 gives an overview of figures of merit for the most common thermoelectricmaterials at various operation temperatures. We can distinguish three groups of "conven­tional semiconductor materials". This term is intended to imply that the transport proper­
ties in these materials are well described by the classical semiconductor theory:i) (Bi1- xSbx)z(Te1_ ySey h solid solutions for the room (and low) temperature range04. t 5>ii) PbTe binary compounds for the medium temperature rangeiii) Silicon-germanium (SiGe) alloys for high-temperature applications<16 -18>
The best z values around room temperature up to z = 3.3 · 10- 3 K-1 are achieved with
doped solid solutions of group V-VI compounds, since the formation of solid solutions
considerably reduces the lattice thermal conductivity.09> As an example, Fig. 7 shows thethermal conductivity and figure of merit for a p-type Bi0_5Sb 1 .5Te3 monocrystalline sample.
It should be pointed out that the V-VI solid solutions usually show anisotropic thermoelec­
hic properties with the best values perpendicular to the crystallographic c-axis.
Silicon-germanium alloys doped with phosphorus (for n-type materials) or boron (forp-type materials) are cmTently the most suitable materials for the high temperature range.
Efforts to improve these materials are partly motivated by the ease with which SiGe-based

--·r
3.0
2.6

t()
I

0

�
·-

\

'

', - Bi2 Te3 - 75Sb 2 Te3
:-..

2.2

E

{p-T.YPE)'� B i2 Te3 - 25Bi2 Se 3
' -....

4'N-PbTe

1.4

'..,

0
QJ

(n-TYPE)

V ', ,.

1. 8

L

Q)

'

1.0

3N-PbTe'z•J.:_½ ����

2P-PbTe

SiGe(n-TYPE)
lp-TYP I
�

CJ)

i.i::

0.6
0.2
0

100

200

300 400 500 600

700

800 900 1000°

Temperature ( C)

Fig. 6. Figure of merit z of selected thermoelectric materials as a function of temperature.<6l Dashedline: z · T= l.

399

Sensors and Materials, Vol. 8, No. 6 (1996)

z(J_)

3p
3P

3,0
4

l 2,

7 18 r----r---P--t-· �--1--------1 1,8 '<
I
_<
-s-I

C)

f2

V

� 12
I

"-.J

0,6
b)
0'------L---.J------L--__Jo

0

100

200

300

Temperature T (K)

4 00

T

K

Fig. 7. Thermal conductivity A and figure of merit z for p-Bi0_5Sb1 .sT� monocrystals perpendicular(.L) and parallel (II) to the crystallographic c-axis as a function of temperatureY 9>

materials advances can be translated into systems technology advances. Recently, figureof merit values which are approximately 30% higher than those for standard SiGe havebeen achieved in n-type SiGe alloys containing small amounts of GaP.<20lExperimental data of the type shown in Fig. 6 appear to indicate that z · T = l represents
a limiting value. However, there is no theoretical basis for this conclusion. A number ofinvestigators have attempted to place a theoretical upper bound on the value of z · Tusing
standard semiconductor theory.<8- -26l In fact, they found no theoretical limit. Therefore,
two possibilities should be discussed:i) z · Tl is possible and will be demonstrated experimentallyii) z · T = 1 represents a limit due to some as yet unknown principle.
21

»

400
5.

Sensors and Materials, Vol. 8, No. 6 (1996)

Thermoelectric Transport Properties of Thin Films

5.1 Classical thin-film transport theory

The thermoelectric properties depend on the carrier transport mechanisms. Therefore,
they are modified in thin films, where the carriers are scattered by film imperfections orsurface and grain boundaries. The well-known Fuchs-SondheimerC27•2 l and Mayadas­
Shatzkes<29l models have been extended to thermoelectric transport.<30J According to thesemodels, the Seebeck coefficient of polycrystalline metal films a(d,G) can be described by
8(24)
where d and G denote the film thickness and average grain size, respectively. Here,f is afunction of d and G, which relates the film thickness and grain size to the mean free path ofthe carriers. A rough approximation off is given by1[j(d' G)r "'l +3(1- p,)y(E)3R,y(E)+2(1- R,,)G
8d

(25)

where y(E) denotes the mean free path and p,, R, denote the scattering parameters forsurface and grain boundary scattering, respectively. In the framework of these models, theelectrical resistivity p(d,G) can be described byp (d, G) = Pbulk · [f(d, G)r1.

(26)

For the thermal conductivity A(d,G), the validity of the Wiedemann-Franz law has beenverified theoretically: <31l
A(d,G) · p(d,G) = L · T.

(27)

For polycrystalline semiconductor films an extended Mayadas-Shatzkes mode1<32J leads toa(d , G ) =±(!5.-)[(r + 2). F,+1(0 · g1(r, , ) _ s]
d G

q

(r+l)

F,(s)·g 0 (r, d , G )<J'= abulk · g o (r,d, G),\ = L(r,d, G) · T · a

AL = ALbulk ·gL(d,G),

(28)

(29)

(30 )

401

Sensors and Materials, Vol. 8, No. 6 (1996)

where the functions gm(r,d,G) describe the effects of thickness and grain size on a, er and,\. The function gL(d,G) is deduced using a model of the phonon scattering on grainboundaries and film surfaces.c33l In many cases the ratio g1 (r,d,G) I go(r,d,G) can beapproximated by unity. Then, the partial Seebeck coefficients of the conduction or valenceband equal that of the bulk material.c32l However, a distinct decrease of bulk a is observedin polycrystalline films due to suppression of phonon drag. For bipolar conduction, a canbe reduced by a shift of the partial conductivities cr0 crP. The influence of the various thin­film scattering mechanisms on a, er and A. leads to a very complex variation of thethermoelectric figure of merit with film structure and temperature.
,

5.2 Heterostructures

The modem semiconductor fabrication techniques enable the preparation of materials
with properties never previously achievable. Recently there have been theoretical andexperimental studies of thermoelectric effects in a high-mobility two-dimensional electrongas(34l such as can be prepared in heterostructures at low temperatures. Mensah andKangah<35l indicate that some enhancement of the Seebeck effect is expected due to thetwo-dimensional character of the confined carrier gas. Balmush et al. <36l and Dashevsky etaz.c37J have extended the conventional description of p-n-junctions to conditions withtemperature gradients and including nonlinear effects such as the Benedicks emf. Moizhes
and Nemchinsky<38l have investigated the influence of energy barriers, which are placed in
the current path. In this scheme a and z are improved in the direction perpendicular to theplane of the energy barriers. Hicks and Dresselhaus<39l have calculated z in one- and two­
dimensional quantum well structures. They estimate that two-dimensional quantum wellstructures of Bi2Terbased materials may have z values up to 13 times greater than those ofbulk materials. Other beneficial effects are also possible with heterostructures, such asincreased phonon scattering and carrier enhancement. Phonon scatte1ing due to acoustic
mismatch between layers and creation of localized modes at the interfaces are expected tocontribute to reduction in phonon heat transport. As a result, flexible preparation methods
such as molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition areavailable for the fabrication of layered structures and superlattices. However, experimental
attempts to apply these techniques to engineering the thermal and thermoelectric propertiesof thin films are still in their infancy. Theoretical investigations lead us to expect that afurther increase of z by thin-film techniques can be achieved in the near future.

6.

Thermoelectric Thin-Film Materials

It should be pointed out that the utility of thermoelectric thin-film materials depends not
only on the thermoelectric parameters, but also on technological aspects such as: reproducible
film preparation, long-term stability, simple and low-cost deposition, photolithographic
patterning, adherence to substrates and compatibility with standard IC technologies. The mostuseful thermoelectric films for room-temperature applications are (Bi,.1-xSbxMTe 1_ ySey)3
solid solutions. They can be prepared by thermal evaporation, flash evaporation, sputter­ing or MBE.<40-42l

402

Sensors and Materials, Vol. 8, No. 6 (1996)

Figures 8 and 9 show the figure of merit z and the lattice thermal conductivity of Bi 1_ xSbx
films as a function of the antimony concentration. <43) A band gap with the maximum band
width at an antimony concentration of 13 at.% is induced by alloying. Therefore, the a andz values increase in comparison with those of pure bismuth<32l and antimonyC44l films, which
are determined by the overlap of the conduction and valence bands. Furthermore, thealloying reduces the lattice thermal conductivity by mass fluctuation scattering of phonons
as demonstrated in Fig. 9.
The most efficient films at room temperature are p-Bi0_5 SbuTe3 films, which are preparedby flash evaporation.<41) After deposition the thermoelectric properties must be improved byannealing as demonstrated in Fig. 10. After several hours a value z = 2.9 · 10-3 K-1 can beachieved, which corresponds to the best bulk values.
Lead telluride (PbTe) films<45l for the medium temperature range are normally n-type
films. They can be prepared by thermal or flash evaporation. As a result of post-deposition
annealing, which is necessary for an improvement of the efficiency, a z value of 0.6 · 10- 3 K - 1
can be achieved at room temperature.
Metal silicides, such as CrSi films,<46l are useful materials for high-temperatureapplications. They can be deposited by magnetron sputtering. Their properties arestrongly dependent on film structure and, consequently, on post-deposition annealing
treatment, as shown in Fig. 11.

7
�

--t
I

0
..-.__,N

5
3
T= 200K
T=300K
T= 80K

1
0

10

20

30

40

50

Sb-conc.[At %]Fig. 8.

Figure of merit z of Bi1_xSbx films as a function of antimony concentration.c43l

403

Sensors and Materials, Vol. 8, No. 6 (1996)

15
13

AL (d=330nm)
10

20

30

40

50

Sb-conc.[At%]Fig. 9. Lattice thermal conductivity of Bi 1_xSb, films AL (thickness d = 330 nm) and of the bulk }.,Lb
vs. antimony concentration.<43J

7.

Silicon and Polysilicon Films

In view of the figure of merit, silicon and polysilicon films are not the most usefulmaterials, mainly because of their large thermal conductivity. Their z values are one ormore orders of magnitude smaller than those of the best materials. However, Si and poly­
Si are interesting for applications because of their well-established technology. Thermo­
electric microsensors and actuators based on silicon are becoming increasingly important
as components of MEMS.<2A7J The on-chip integration of such thermoelectric sensors andactuators with modem signal processing circuitries requires the use of silicon as a low-cost
thermoelectric material, which is compatible with standard IC technologies. Therefore,
recent investigations have been focused on the process-specific thermal and thermoelectric404

Sensors and Materials, Vol. 8, No. 6 (1996)

Q' 25

r-- -----

w
0
,-

N

-

w

�

'U

20

0

ex

15

0
,-

�

10 L

w

5

-

E

cr

0
,-

'i6

0

?-

-

-

0

2

.

z
3

5

4

Annealing time t (h)Fig. 10. Seebeck coefficient a, electrical conductivity a and figure of merit z of p-Bi0_5Sb 1 .5Te3
films as a function of annealing time for annealing in an argon atmosphere at T = 513 K. <41>properties of Si and poly-Si fihns.<12. 8. )The electrical conductivity, Seebeck coefficient and thermal conductivity of bulk Si
can be described by eqs. (16), (19), and (22). These equations demonstrate that thevariation of the carrier concentration by doping enables variation of a and CJ over a wide
range, whereas 1 is dominated by the lattice component AL and is therefore less influenced
by doping. Figure 12 shows a of bulk silicon as a function of the electrical resistivity. The
Seebeck coefficient can be related to the resistivity by a= 2.6 (kl q) ln (p I p0 with p0 = 5 · 10-0
Qm (see Fig. 12).
The electrical conductivity of poly-Si films can be described by the trapping states andgrain boundary scattering model. <50l Investigations of the Seebeckeffect in poly-Si films <12ldemonstrate that a can be interpreted using eq. (19) with a negligible phonon drag effect.
Figure 13 shows a of poly-Si films, doped with boron and phosphorus, as a function oftemperature. The curves are calculated according to eq. (19) with X = 0. The thermal
conductivity<49l of poly-Si films is about 5 times smaller than that of the bulk because ofphonon grain boundary scattering. However, we learn from Fig. 14 that A of poly-Si films
is also determined by the lattice component AL·
4 49

)

405

Sensors and Materials, Vol. 8, No. 6 (1996)

10.00
9.50

0

T a ,.,,.,,°C

0

9.00

------

8.50

C

7.50

y::

7.00

Eu

N

.,____,,"'"---------

>

8.00

650

0

•

900
950

CrSi 2

0

8S0

6.00
5.50

b

4.50

d

4.00

N

z

6.50

::J._
,.....,,
I.D

0

'v

5.00

3.50
3.00

2.50

2.00

200

300

400

500

600

700

T,K

Temperature T (K)
Fig. 11. Thermoelectric power factor a2 • u of annealed Cr022Si0_78 films in comparison with apolycrystalline CrSi2 film.C46l

8.

Conclusions

It appears that, although the experimental results to date indicate that z · T � l represents
an upper limit, there is no theoretical basis for this conclusion. Modem thin-film techniques
enable the preparation of layered structures on an atomic length scale with a variety ofnovel properties not possible with bulk materials. These techniquies may allow realizationof effects beneficial to thermoelectricity such as thermal conductivity reduction andmobility enhancement. Thermoelectric thin-film sensors and actuators based on silicon,
polysilicon and other materials will play an important role for MEMS, even if thethermoelectric figure of merit is not significantly increased.

406

Sensors and Materials, Vol. 8, No. 6 (1996)

2000

-

I,, ...

1500

1)'

"'• t

a., 1000

u
a.,
a.,

i,,

"'

.,, ,,, ·�

·z;

u

�

.,, "'

500

'

�

'

.

.,,�

/,
i,,

"'

0
0.00001

0.0001

0.01

0.001

0. 1

Resistivity (nm}Fig. 12. Seebeck coefficient of silicon as a function of the electrical resistivity at 300 K. Symbols,
experimental results; dashed line, a= 2.6 (kl q) In (p/ p0) with p0 = 5 · IQ-6 Qm.(48l
600

Pl

550
500

PII

450

-

Bl

Q' 400
350

BIi

� 300
250
200
150

50

100

150

200

250

300

350

400

450

T (K)
Fig. 13. Absolute value of the Seebeck coefficient a of boron- (BI and BII) and phosphorus- (PIand PII) doped polysilicon films and curves calculated from eq. (19) as a function of temperature.C12l

407

Sensors and Materials, Vol. 8, No. 6 (1996)
40

;-..,

30
J7. _J"l� � �-- "e-

- -

d�

Q'

�

-

-

20

AL

J

<--<

.,,

« 10
«'
0

- 50

100

150

200

250

300

350

400

T (K)

Fig. 14. Thermal conductivity A, electron contribution A,,, and lattice thermal conductivity ;\,L ofpolysilicon films as a function of temperature. c49J

References

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