The thermoelectric parameters have been investigated depending on the thickness of the layer of nanostructures
PbTe doped by Bi. Based on the theoretical model of quantum well (QW) with infinitely high walls, it is
demonstrated that this model explains nonmonotonous behaviour of the Seebeck coefficient S and electric conductivity
σ with the change of the well width.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3487

Freik, D.M.

Mezhylovska, L.Yo.

Yavorsky, Ya.S.

Yurchyshyn, I.K.

SocioEconomic Challenges Journal (Sumy State University)

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE NANOMATERIALS: APPLICATIONS AND PROPERTIES 
Vol. 1 No 2, 02NFC14(2pp) (2012) 
 
 
2304-1862/2012/1(2)02NFC14(2) 02NFC14-1  2012 Sumy State University 
Thermoelectricity of Nanostructures Based on Compounds of Lead Telluride 
 
D.M. Freik*, I.K. Yurchyshyn†, L.Yo. Mezhylovska‡, Ya.S. Yavorsky 
 
Physics and Chemistry Institute at PreCarpathian Vasyl Stefanyk National University  
57, Shevchenko Str., Ivano-Frankivsk, 76018, Ukraine 
 
(Received 11 June 2012; published online 25 August 2012) 
 
The thermoelectric parameters have been investigated depending on the thickness of the layer of nanostruc-
tures PbTe doped by Bi. Based on the theoretical model of quantum well (QW) with infinitely high walls, it is 
demonstrated that this model explains nonmonotonous behaviour of the Seebeck coefficient S and electric con-
ductivity σ with the change of the well width.  
 
Keywords: Nanostructures, Thermoelectricity, Electrical properties. 
 
  PACS numbers: 75.50.Lw, 73.63.Hs, 73.61.Ey 
 
 
                                                          
* freik@pu.if.ua 
† i.yurchyshyn@ukr.net 
‡ fcss@pu.if.ua 
1. INTRODUCTION  
 
Recently, in the process of creating highly efficient 
thermoelectric materials, the research in the low-
dimensional structures has been intensified [1]. The re-
ducing of the material dimension creates the conditions 
for observing the phenomenon of quantum-size effect, 
which increases the density of states near the Fermi 
energy. This allows to sufficiently save high electrical 
conductivity  at the relatively low Fermi energy EF, 
where there are high values of the Seebeck coefficient S. 
The behavior of thermoelectric parameters (the elec-
trical σ and thermal conductivity k, the Seebeck coeffi-
cient S) in bulk materials is usually described in terms of 
electronic and phonon properties that change greatly 
with the reducing of the sample size below  100 nm, 
where the quantum effects become more significant. The 
spatial restriction of the acoustic phonons and the ap-
propriate modification of the group velocity leads to the 
increase in phonon relaxation rate and results in the 
decrease of lattice thermal conductivity kL [2]. Besides, 
the nature of interactions between the charge carriers 
changes greatly. The theoretical calculations associated 
with the estimation of these effects on the transport coef-
ficients are usually performed within the framework of 
size quantum limit, which is based on the simplification 
that electrons occupy only the lowest sub-bands. In such 
an approach, consideration of the multiple sub-band 
structure of quantum well can lead to nonmonotonous 
changes of various kinetic parameters such as mobility 
, the Seebeck S and the Hall coefficients RH [3].  
The aim of this study is to verify the expediency of 
using the model of infinitely deep rectangular QW for 
compound PbTe: Bi, grown on substrates glass-ceramic.  
 
2. THEORETICAL MODEL AND RESULTS OF 
CALCULATIONS 
 
For the quantum well with high walls, the electrons 
are confined in the direction of OZ, but in the X- and Y-
directions their movement is autonomous. The electronic 
wave function and energy eigenvalues in terms of para-
bolic energy bands are determined by equations [3]:  
 1 2(2 ) exp( )sin( )x yik x jk y n z d  (1) 
 
2 2 2 2
2
* 2 *2 2z p
k
E n
m d m
, (2) 
where 2 2 2x yk k k , mz
* – effective mass of electron along 
confinement direction; * * *p x ym m m , mx
*, my* – effective 
masses of electron along the x and y axes,  – total volume 
of the layer, d – well width, n – quantum number that 
obtains values of natural numbers.  
At this condition it is possible to write down the ex-
pressions for the Seebeck coefficient and electric conduc-
tivity as:  
 
 1 2
3
B F
B
k E A A
S
e k T A
, (3) 
 
 
*
2
0 32 *
21
2
xB
y
mk T
e A
d m
, (4) 
 
where: 
 
 1
1 0
n FE E
n
n
f
A dx
x
, (5) 
 
 '2
1 0
n FE E
n
n
f
A E x dx
x
, (6) 
 
 3
1 0
n FE E
n
n
f
A x dx
x
. (7) 
 
The Fermi distribution function has a known form:  
 
 
1
1n
n x
f
e
, (8) 
 
where Bx k T  – reduced carrier energy, 
'
n nE . Here F BE k T  and 
'
n n BE E k T . 
It is possible to estimate the relaxation time in (4) 
using the n-type carriers mobility  in a bulk sample [4]: 
 D.M. FREIK, I.K. YURCHYSHYN, L.YO. MEZHYLOVSKA, YA.S. YAVORSKY PROC. NAP 1, 02NFC14 (2012) 
 
 
02NFC14-2 
 
 
0 /e m , (9) 
 
The Fermi energy value can be expressed through the 
well width (d) and the concentration of carrier in the con-
duction band [5]  
 
 
2
0 0
1
0
( 1)(2 1)
6 *
el
F
n n n
E d
m n
, (10) 
 
where 1 – the first quantized level, when n  1; 
n0  [(EF/ 1)1/2] –  the effective mass of carriers, nel –
 electronic concentration. 
If the Fermi lever coincide with the bottom of the band 
n0, then EF(d0)  1n02. At such widths (EF(d0)/ 1)1/2 is an 
integer. Putting this value in (10) for dn0 we get:  
 
 
0
1/3
0 0
0 0 1 2
0
( 1)(2 1)
1
6
n
n n
d d n
n
, (11) 
 
where d0  ( /2nel)1/3. A number n0 at this width is a whole 
part from the result of decision of equation (11) relatively 
to n0 when dn0  d.  
The influence of doping level by Bi on PbTe can be es-
timated through the proper change of effective mass in 
(10) using an empiric formula, what have been gotten as a 
result of approximation of experimental dependence of 
effective mass on an electronic concentration [6]: 
 
 
1
*
36
0
0,111 10 el
m
n
m
, (7) 
 
де m0 –  the mass of electron, nеl – an electronic concentra-
tion in units of cm-3. 
The account of the Fermi energy d-dependence in for-
mulas (3)-(8), and also the effective mass z-component d-
dependence in expressions (5)-(8) allow to get the proper 
dependences of Seebeck coefficient S and electric conduc-
tivity σ on the width of quantum well PbTe:Bi. Amount of 
levels below the Fermi energy is determined by d-
dependences of effective mass and EF, and also by well 
width d. Calculation procedure took into account both the 
change of Fermi energy and the change of amount of lev-
els below it at the change of well width. Also, at the calcu-
lation of conductivity it was assumed on a formula (4), 
that mx*  my*. 
Using the described higher theoretical model for de-
scription of the behaviour of thermoelectric coefficients in 
the nanostructures PbTe:Bi at 300 K the quantum well 
width in a theoretical model was considered to be equal to 
the condensate thickness in experimental dependences of 
the proper parameters. The resulting dependences of 
thermoelectric coefficients on the well width PbTe:Bi are 
characterized by an unmonotonous oscillation behaviour 
(Fig. 1). Thus the calculating values are well enough con-
formed to proper experimental dependences. 
 
 
 
Fig. 1 – Theoretical dependences of the Seebeck coefficient (a), 
electric conductivity (b) and thermoelectric power coefficient S2σ 
on the width of PbTe:Bi quantum well in a model of infinitely 
deep potential well at T  300 K 
 
3. CONCLUSIONS 
 
1. The analysis of experimental dependences for ther-
moelectric coefficients on the thickness of nanostruc-
utures PbTe:Bi on a class-ceramic at 300 K have 
been done. It is discovered that they are character-
ized by an unmonotonous oscillation behaviour. 
2. In the model of quantum well with infinitely high 
walls the dependences of Fermi energy and effective 
mass on the well width are calculated. On their basis 
it was investigated the character of oscillations peri-
od change for the density of states with well width 
growth. 
3. On the basis of correlations of Fermi energy and ef-
fective mass with the well width there were built d-
dependences of Seebeck coefficient, electric conduc-
tivity and thermoelectric power coefficient. It was 
shown that the theoretical d-dependences conform to 
the experimental. 
Work is executed at financing within the framework of 
the state budget themes №0111U001766 (2010-2012) of 
Ministry of education and science, youth and sport of 
Ukraine, №0110U007674 (2010-2012) SFFR of Ukraine 
and №0110U006281 (2010-2013) NAS of Ukraine. 
 
 
 
REFERENCES 
 
1. J.H. Davies: The physics of low-dimensional semiconductors. 
An introduction, (Cambridge university press: 1998). 
2. D.M. Rowe CRC Handbook of Thermoelectrics (CRC Press: 
1995). 
3. J.R. Drabble, H.J. Coldsmid, Thermal Conduction in Semi-
conductors (London: Pergaman Press: 1961). 
4. A.I. Anselm Introduction to the semiconductors theory (M: 
Science: 1978). 
5. B.M. Askerov Electronic transport phenomena in semiconduc-
tors (M.: Science.: 1985). 
6. I.N. Dubrovskaya, Yu.I Ravich, PSS 8 No5, 1455 (1966).
 


Thermoelectricity of Nanostructures Based on Compounds of Lead Telluride

Electronic Sumy State University Institutional Repository

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE NANOMATERIALS: APPLICATIONS AND PROPERTIES Vol. 1 No 2, 02NFC14(2pp) (2012)   2304-1862/2012/1(2)02NFC14(2) 02NFC14-1  2012 Sumy State University Thermoelectricity of Nanostructures Based on Compounds of Lead Telluride  D.M. Freik*, I.K. Yurchyshyn†, L.Yo. Mezhylovska‡, Ya.S. Yavorsky  Physics and Chemistry Institute at PreCarpathian Vasyl Stefanyk National University  57, Shevchenko Str., Ivano-Frankivsk, 76018, Ukraine  (Received 11 June 2012; published online 25 August 2012)  The thermoelectric parameters have been investigated depending on the thickness of the layer of nanostruc-tures PbTe doped by Bi. Based on the theoretical model of quantum well (QW) with infinitely high walls, it is demonstrated that this model explains nonmonotonous behaviour of the Seebeck coefficient S and electric con-ductivity σ with the change of the well width.   Keywords: Nanostructures, Thermoelectricity, Electrical properties.    PACS numbers: 75.50.Lw, 73.63.Hs, 73.61.Ey                                                             * freik@pu.if.ua † i.yurchyshyn@ukr.net ‡ fcss@pu.if.ua 1. INTRODUCTION   Recently, in the process of creating highly efficient thermoelectric materials, the research in the low-dimensional structures has been intensified [1]. The re-ducing of the material dimension creates the conditions for observing the phenomenon of quantum-size effect, which increases the density of states near the Fermi energy. This allows to sufficiently save high electrical conductivity  at the relatively low Fermi energy EF, where there are high values of the Seebeck coefficient S. The behavior of thermoelectric parameters (the elec-trical σ and thermal conductivity k, the Seebeck coeffi-cient S) in bulk materials is usually described in terms of electronic and phonon properties that change greatly with the reducing of the sample size below  100 nm, where the quantum effects become more significant. The spatial restriction of the acoustic phonons and the ap-propriate modification of the group velocity leads to the increase in phonon relaxation rate and results in the decrease of lattice thermal conductivity kL [2]. Besides, the nature of interactions between the charge carriers changes greatly. The theoretical calculations associated with the estimation of these effects on the transport coef-ficients are usually performed within the framework of size quantum limit, which is based on the simplification that electrons occupy only the lowest sub-bands. In such an approach, consideration of the multiple sub-band structure of quantum well can lead to nonmonotonous changes of various kinetic parameters such as mobility , the Seebeck S and the Hall coefficients RH [3].  The aim of this study is to verify the expediency of using the model of infinitely deep rectangular QW for compound PbTe: Bi, grown on substrates glass-ceramic.   2. THEORETICAL MODEL AND RESULTS OF CALCULATIONS  For the quantum well with high walls, the electrons are confined in the direction of OZ, but in the X- and Y-directions their movement is autonomous. The electronic wave function and energy eigenvalues in terms of para-bolic energy bands are determined by equations [3]:   1 2(2 ) exp( )sin( )x yik x jk y n z d  (1)  2 2 2 22* 2 *2 2z pkE nm d m, (2) where 2 2 2x yk k k , mz* – effective mass of electron along confinement direction; * * *p x ym m m , mx*, my* – effective masses of electron along the x and y axes,  – total volume of the layer, d – well width, n – quantum number that obtains values of natural numbers.  At this condition it is possible to write down the ex-pressions for the Seebeck coefficient and electric conduc-tivity as:    1 23B FBk E A ASe k T A, (3)   *20 32 *212xBymk Te Ad m, (4)  where:   11 0n FE EnnfA dxx, (5)   '21 0n FE EnnfA E x dxx, (6)   31 0n FE EnnfA x dxx. (7)  The Fermi distribution function has a known form:    11nn xfe, (8)  where Bx k T  – reduced carrier energy, 'n nE . Here F BE k T  and 'n n BE E k T . It is possible to estimate the relaxation time in (4) using the n-type carriers mobility  in a bulk sample [4]:  D.M. FREIK, I.K. YURCHYSHYN, L.YO. MEZHYLOVSKA, YA.S. YAVORSKY PROC. NAP 1, 02NFC14 (2012)   02NFC14-2   0 /e m , (9)  The Fermi energy value can be expressed through the well width (d) and the concentration of carrier in the con-duction band [5]    20 010( 1)(2 1)6 *elFn n nE dm n, (10)  where 1 – the first quantized level, when n  1; n0  [(EF/ 1)1/2] –  the effective mass of carriers, nel – electronic concentration. If the Fermi lever coincide with the bottom of the band n0, then EF(d0)  1n02. At such widths (EF(d0)/ 1)1/2 is an integer. Putting this value in (10) for dn0 we get:    01/30 00 0 1 20( 1)(2 1)16nn nd d nn, (11)  where d0  ( /2nel)1/3. A number n0 at this width is a whole part from the result of decision of equation (11) relatively to n0 when dn0  d.  The influence of doping level by Bi on PbTe can be es-timated through the proper change of effective mass in (10) using an empiric formula, what have been gotten as a result of approximation of experimental dependence of effective mass on an electronic concentration [6]:   1*3600,111 10 elmnm, (7)  де m0 –  the mass of electron, nеl – an electronic concentra-tion in units of cm-3. The account of the Fermi energy d-dependence in for-mulas (3)-(8), and also the effective mass z-component d-dependence in expressions (5)-(8) allow to get the proper dependences of Seebeck coefficient S and electric conduc-tivity σ on the width of quantum well PbTe:Bi. Amount of levels below the Fermi energy is determined by d-dependences of effective mass and EF, and also by well width d. Calculation procedure took into account both the change of Fermi energy and the change of amount of lev-els below it at the change of well width. Also, at the calcu-lation of conductivity it was assumed on a formula (4), that mx*  my*. Using the described higher theoretical model for de-scription of the behaviour of thermoelectric coefficients in the nanostructures PbTe:Bi at 300 K the quantum well width in a theoretical model was considered to be equal to the condensate thickness in experimental dependences of the proper parameters. The resulting dependences of thermoelectric coefficients on the well width PbTe:Bi are characterized by an unmonotonous oscillation behaviour (Fig. 1). Thus the calculating values are well enough con-formed to proper experimental dependences.    Fig. 1 – Theoretical dependences of the Seebeck coefficient (a), electric conductivity (b) and thermoelectric power coefficient S2σ on the width of PbTe:Bi quantum well in a model of infinitely deep potential well at T  300 K  3. CONCLUSIONS  1. The analysis of experimental dependences for ther-moelectric coefficients on the thickness of nanostruc-utures PbTe:Bi on a class-ceramic at 300 K have been done. It is discovered that they are character-ized by an unmonotonous oscillation behaviour. 2. In the model of quantum well with infinitely high walls the dependences of Fermi energy and effective mass on the well width are calculated. On their basis it was investigated the character of oscillations peri-od change for the density of states with well width growth. 3. On the basis of correlations of Fermi energy and ef-fective mass with the well width there were built d-dependences of Seebeck coefficient, electric conduc-tivity and thermoelectric power coefficient. It was shown that the theoretical d-dependences conform to the experimental. Work is executed at financing within the framework of the state budget themes №0111U001766 (2010-2012) of Ministry of education and science, youth and sport of Ukraine, №0110U007674 (2010-2012) SFFR of Ukraine and №0110U006281 (2010-2013) NAS of Ukraine.    REFERENCES  1. J.H. Davies: The physics of low-dimensional semiconductors. An introduction, (Cambridge university press: 1998). 2. D.M. Rowe CRC Handbook of Thermoelectrics (CRC Press: 1995). 3. J.R. Drabble, H.J. Coldsmid, Thermal Conduction in Semi-conductors (London: Pergaman Press: 1961). 4. A.I. Anselm Introduction to the semiconductors theory (M: Science: 1978). 5. B.M. Askerov Electronic transport phenomena in semiconduc-tors (M.: Science.: 1985). 6. I.N. Dubrovskaya, Yu.I Ravich, PSS 8 No5, 1455 (1966). 

https://essuir.sumdu.edu.ua/bitstream/123456789/34878/1/princon_2012_1_2_16.pdf

Thermoelectricity of Nanostructures Based on Compounds of Lead Telluride

Abstract

Similar works

Full text

Available Versions

SocioEconomic Challenges Journal (Sumy State University)

Electronic Sumy State University Institutional Repository