Bachuk, V.V.

Freik, D.M.

Harun, L.T.

Lysiuk, Yu.V.

Yurchyshyn, I.K.

Electronic Sumy State University Institutional Repository

136                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              QUANTUM-SIZE EFFECTS IN NANOSTRUCTURES  ON THE BASE OF COMPOUNDS IV-VI  Dmytro M. Freik1*, Igor K. Yurchyshyn1, Vasyl V. Bachuk1,   Lidiya T. Harun1, Yuriy V. Lysiuk1  1 The Precarpathian National University, Shevchenko str., 57, 76018,  Ivano-Frankivsk, Ukraine.  ABSTRACT Thickness dependences of kinetic parameters of quantum wells (QWs) based on IV-VI compounds show non-monotonic oscillatory behavior that is associated with size quantization due to carrier movement restrictions in one direction. In work on basis of theoretical model of rectangular quantum well with infinitely high barriers we investigated the dependences of thermoelectric parameters on nanostructures thickness of compounds IV-VI (PbTe, PbSe, PbS, SnTe). It was shown, that in such structures it has place nonmonotonous, oscillation change of thermoelectric coefficients with well width. Thickness dependence of the Seebeck coefficient S, based on model of QW with infinitely high walls, are characterized by breaks with some peri-od. For quantum wells SnTe it was found good agreement between the values of theoret-ical and experimental oscillating amplitude in dependence S(d). Shown, that values ǻdexp of  oscillations  period  are  equal  to  QW  thickness  dmin, when the bottom of lowest subband coincides with the Fermi energy EF. This thickness dmin was  explained  as  the  minimum QW thickness,  when  quantum size  effects  are  the  main factor, that determines the nonmonotonous behaviour in thickness dependencies of thermoelectric parameters of the relevant structures. At smaller thicknesses important factor, affecting the change of kinetic parameters with thickness, is the phenomenon of percolation. It is shown that the percolation transition changes the value of the concen-tration of main type carrier in nanostructure, thereby changing the Fermi energies and hence the period of oscillations. Thus, the sudden change of concentration of the main type carrier results in a sharp change in oscillations period of thickness dependence of kinetic parameters.  Key words: nanostructures, quantum-size effects, thermoelectricity  INTRODUCTION In the process of creating highly efficient thermoelectric materials at re-cent times it has place an intensification of research in low-dimensional struc-tures [1,2] The behavior of thermoelectric parameters (electrical and thermal conduc-tivity k, Seebeck coefficient S) in bulk materials is usually described in terms                                                * e-mail: freik@pu.if.ua, tel: (+380342)596082 Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              137  of electronic and phonon properties that change greatly with reducing of sample size below the micrometer range, where quantum effects become more signifi-cant. Spatial restriction of acoustic phonons and the corresponding modification of group velocity leads to an increase in phonon relaxation rate, resulting in decrease lattice thermal conductivity kL. Also the nature of interactions between particles changes greatly.  Electronic restriction by reducing sample size leads to increased energy gap Eg, which in turn affects the transport properties. By quantization the con-duction band density of states changes abruptly with change of energy. De-pending on probability amplitude electrons can be found in any sub-bands (E1, E2, … En), with a certain probability hop from one band to another nearest band and occupy levels up to the Fermi energy. THEORETICAL MODEL  For quantum well with high walls, electrons are confined in the direction of oz, but in the x- and y-directions their movement is free.  In case of quantum well thermoelectric transport coefficients can be ob-tained from Boltzmann equations, which are written under the assumption that in steady state electronic distribution function remains constant and changes only by external forces and fields. Then system of electrons returns to equilib-rium through various relaxation processes with characteristic relaxation times. for quasi-two dimentional system it can be written [3]:  211FESeT eT* *.   (1) Transport coefficient Ƚ is defined by semi-classical approach, in which particles are confined in one-dimensional box. Thermal gradient and electric field are directed along ox-axis. Then:  1 (0)]*   ,                (2)  2 (1)]*  ,                                                   (3) where:   *( )* 21 02 1 ( )n FE Eys snxmT fE da m] WH HS Hfd w w¦ ³= . (4)  Here f - Fermi distribution function, İ=E-En, Ĳ - relaxation time in the case of scattering by acoustic phonons.  RESULTS AND DISCUSSION Basing on experimental data [4,5,6,7], which show nonmonotonous, oscil-latory character for dependences of Seebeck coefficient on the condensate thickness, it is natural to assume that such behavior is due to quantization of 138                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              carriers energy by restriction their movement in the potential well. Increasing well width on the value of half Fermi wavelength leads to new sub-band below the Fermi energy. At width when the new band is filled in the density of states there is a jump, which leads to oscillating behavior. Using the above theoretical model to describe behavior of Seebeck coeffi-cient in nanostructures PbS, PbSe, PbTe, SnTe at 300 K, quantum well width in the theoretical model was considered equal to the thickness of condensate in experimental dependences S (d). Calculations were carried out for different well widths at given values of Fermi energy, which is a function of concentra-tion and temperature. Fig. 1 shows d-dependences of Seebeck coefficient for the QW PbS, PbSe, PbTe (a) and SnTe (b) with experimentally determined oscillation period ǻdexp. It is seen that dependences of Seebeck coefficient S on well width are characterized by discontinuities with some period. Note that procedure of calculation the dependences S (d) for compounds IV-VI (fig. 1) takes into account different number of filled bands for various well widths at fixed Fermi energy.    ɚ ɛ Fig. 1 – Dependence of Seebeck coefficient on the width d of QW PbS, PbSe, PbTe (a) and SnTe (b) in the model infinitely deep potential well  In considered approach the experimental value of oscillation period is equal to minimum QW width dmin, from which quantum-size effects are the main factor that conditions nonmonotonous change of thickness dependencies TE-parameters in relevant structures. Evaluating value of Fermi energy on the basis of experimentally deter-mined values of carrier concentration it is possible to compare received oscilla-tion periods with its experimental values. Thus in contrast to PbS and PbTe QW for QW PbSe and SnTe it is obtained a good agreement with experiment: PbSe: ǻdtheor = 27 nm and ǻdexp§ 35 nm; SnTe: ǻdtheor = 3 nm and ǻdexp§ 2.9 nm [4,5,6,7]. Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              139  Assuming that percolation processes leads to the extrema points displace-ment of experimental dependence S (d) in comparison with theoretical, we can note a decrease their influence with increase SnTe thickness.  On the other hand, if percolation processes alter the main type carrier con-centration in nanostructure, thereby changing Fermi energy and hence oscilla-tion period, then abrupt change in carrier concentration would lead to abrupt changes in oscillation period of d-dependence of Seebeck coefficient. Since the oscillation period is associated with Fermi energy, it is possible to construct the dependence EF(d) (fig. 2). Fig. 2 shows that with changing well  CONCLUSIONS 1. Shown that dependences of Seebeck coefficient on width of quantum well  with  infinitely  high  walls  for  structures  based  on  IV-VI  compounds  are  characterized by discontinuities with some period.  2. Based on experimentally determined oscillation period ǻdexp of  the  thickness dependences of Seebeck coefficient Fermi energy values are deter-mined in QW structures of compounds IV-VI.  3. Shown that oscillation period value ǻdexp of quantum well is equal to the width of QW dmin, where the bottom of lowest sub-band coincides with Fermi energy EF. We have given to the width dmin the content of minimum QW width from which quantum-size effects are the main factor that determines non-monotonous change of thickness dependencies for thermoelectric parameters in corresponding nanostructures.  4. For QW SnTe it was found good agreement between theoretical and ex-perimental values of oscillation amplitude in dependence S(d).  5. Reducing the value of difference between the experimental and theoreti-cal extremum points with increasing SnTe QW width is explained by the reduc-ing of the percolation phenomena impact in such growth.  6. For SnTe nanostructures shown that dependences S(d) with different os-cillation periods are in good agreement with experiment at different intervals of change the condensate thickness. Thus the dependence of Fermi energy on the  Fig. 2 – Dependence of Fermi energy on SnTe well width, obtained by comparing the theoretical dependences S (d) for different oscillation periods with experi-mental 140                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              well width is received. Revealed a sharp change of Fermi energy in the range from 2 to 3 nm.   REFERENCES [1] The physics of low-dimensional semiconductors. An introduction (J.H. Davies), Cambridge university press, 1998. [2] Dresselhaus M.S., Ghen G., Rang M.I. et al., Adv. Mater., 2007, 19, P. 1043-1053. [3] CRC Handbook of Thermoelectrics, chapters 4-6, (Bhandari C.M. in ed. D.M. Rowe), CRC Press, 1995. [4] E.I. Rogacheva, O.N. Nashchekina, Y.O. Vekhov et al. Thin Solid Films, 2003, 423, P. 115-118. [5] E.I. Rogacheva, T.V. Tavrina, O.N. Nashchekina, S.N. Grigorov, K.A. Nasedkin, M.S. Dresselhaus, S.B. Cronin, 2002, 80(15), P. 2690-2692. [6] Rogacheva E.I, Nashchekina O.N, Grigorov S.N. et al. Institute of Physics Publish-ing. Nanotechnology, 2003, 14, P. 53–59. [7] E.I. Rogacheva, O.N. Nashchekina, A.V. Meriuts et al., Appl. Phys. Lett., 2005, 86, 063103.    

Quantum-size effects in nanostructures on the base of compounds IV-VI

SocioEconomic Challenges Journal (Sumy State University)

136                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I             
 
QUANTUM-SIZE EFFECTS IN NANOSTRUCTURES  
ON THE BASE OF COMPOUNDS IV-VI  
Dmytro M. Freik1*, Igor K. Yurchyshyn1, Vasyl V. Bachuk1,   
Lidiya T. Harun1, Yuriy V. Lysiuk1 
 
1 The Precarpathian National University, Shevchenko str., 57, 76018,  Ivano-Frankivsk, 
Ukraine. 
 
ABSTRACT 
Thickness dependences of kinetic parameters of quantum wells (QWs) based on 
IV-VI compounds show non-monotonic oscillatory behavior that is associated with size 
quantization due to carrier movement restrictions in one direction. 
In work on basis of theoretical model of rectangular quantum well with infinitely 
high barriers we investigated the dependences of thermoelectric parameters on 
nanostructures thickness of compounds IV-VI (PbTe, PbSe, PbS, SnTe). It was shown, 
that in such structures it has place nonmonotonous, oscillation change of thermoelectric 
coefficients with well width. Thickness dependence of the Seebeck coefficient S, based 
on model of QW with infinitely high walls, are characterized by breaks with some peri-
od. For quantum wells SnTe it was found good agreement between the values of theoret-
ical and experimental oscillating amplitude in dependence S(d). 
Shown, that values ?dexp of  oscillations  period  are  equal  to  QW  thickness  dmin, 
when the bottom of lowest subband coincides with the Fermi energy EF. This thickness 
dmin was  explained  as  the  minimum QW thickness,  when  quantum size  effects  are  the  
main factor, that determines the nonmonotonous behaviour in thickness dependencies of 
thermoelectric parameters of the relevant structures. At smaller thicknesses important 
factor, affecting the change of kinetic parameters with thickness, is the phenomenon of 
percolation. It is shown that the percolation transition changes the value of the concen-
tration of main type carrier in nanostructure, thereby changing the Fermi energies and 
hence the period of oscillations. Thus, the sudden change of concentration of the main 
type carrier results in a sharp change in oscillations period of thickness dependence of 
kinetic parameters. 
 
Key words: nanostructures, quantum-size effects, thermoelectricity 
 
INTRODUCTION 
In the process of creating highly efficient thermoelectric materials at re-
cent times it has place an intensification of research in low-dimensional struc-
tures [1,2] 
The behavior of thermoelectric parameters (electrical and thermal conduc-
tivity k, Seebeck coefficient S) in bulk materials is usually described in terms 
                                               
* e-mail: freik@pu.if.ua, tel: (+380342)596082 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              137 
 
of electronic and phonon properties that change greatly with reducing of sample 
size below the micrometer range, where quantum effects become more signifi-
cant. Spatial restriction of acoustic phonons and the corresponding modification 
of group velocity leads to an increase in phonon relaxation rate, resulting in 
decrease lattice thermal conductivity kL. Also the nature of interactions between 
particles changes greatly.  
Electronic restriction by reducing sample size leads to increased energy 
gap Eg, which in turn affects the transport properties. By quantization the con-
duction band density of states changes abruptly with change of energy. De-
pending on probability amplitude electrons can be found in any sub-bands (E1, 
E2, … En), with a certain probability hop from one band to another nearest band 
and occupy levels up to the Fermi energy. 
THEORETICAL MODEL  
For quantum well with high walls, electrons are confined in the direction 
of oz, but in the x- and y-directions their movement is free.  
In case of quantum well thermoelectric transport coefficients can be ob-
tained from Boltzmann equations, which are written under the assumption that 
in steady state electronic distribution function remains constant and changes 
only by external forces and fields. Then system of electrons returns to equilib-
rium through various relaxation processes with characteristic relaxation times. 
for quasi-two dimentional system it can be written [3]: 
 
2
1
1FES
eT eT
?? ?
?
.   (1) 
Transport coefficient ? is defined by semi-classical approach, in which 
particles are confined in one-dimensional box. Thermal gradient and electric 
field are directed along ox-axis. Then:  
1 (0)?? ? ? ,                (2) 
 2 (1)?? ? ,                                                   (3) 
where:  
 
*
( )
* 2
1 0
2 1 ( )
n FE E
ys s
nx
mT fE d
a m
? ?? ?? ?
??
?
??
?? ?? . (4) 
 
Here f - Fermi distribution function, ?=E-En, ? - relaxation time in the case 
of scattering by acoustic phonons.  
RESULTS AND DISCUSSION 
Basing on experimental data [4,5,6,7], which show nonmonotonous, oscil-
latory character for dependences of Seebeck coefficient on the condensate 
thickness, it is natural to assume that such behavior is due to quantization of 
138                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I             
 
carriers energy by restriction their movement in the potential well. Increasing 
well width on the value of half Fermi wavelength leads to new sub-band below 
the Fermi energy. At width when the new band is filled in the density of states 
there is a jump, which leads to oscillating behavior. 
Using the above theoretical model to describe behavior of Seebeck coeffi-
cient in nanostructures PbS, PbSe, PbTe, SnTe at 300 K, quantum well width in 
the theoretical model was considered equal to the thickness of condensate in 
experimental dependences S (d). Calculations were carried out for different 
well widths at given values of Fermi energy, which is a function of concentra-
tion and temperature. Fig. 1 shows d-dependences of Seebeck coefficient for 
the QW PbS, PbSe, PbTe (a) and SnTe (b) with experimentally determined 
oscillation period ?dexp. It is seen that dependences of Seebeck coefficient S on 
well width are characterized by discontinuities with some period. Note that 
procedure of calculation the dependences S (d) for compounds IV-VI (fig. 1) 
takes into account different number of filled bands for various well widths at 
fixed Fermi energy. 
 
  
? ? 
Fig. 1 – Dependence of Seebeck coefficient on the width d of QW PbS, PbSe, PbTe 
(a) and SnTe (b) in the model infinitely deep potential well 
 
In considered approach the experimental value of oscillation period is 
equal to minimum QW width dmin, from which quantum-size effects are the 
main factor that conditions nonmonotonous change of thickness dependencies 
TE-parameters in relevant structures. 
Evaluating value of Fermi energy on the basis of experimentally deter-
mined values of carrier concentration it is possible to compare received oscilla-
tion periods with its experimental values. Thus in contrast to PbS and PbTe 
QW for QW PbSe and SnTe it is obtained a good agreement with experiment: 
PbSe: ?dtheor = 27 nm and ?dexp?? 35 nm; SnTe: ?dtheor = 3 nm and 
?dexp?? 2.9 nm [4,5,6,7]. 
Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I              139 
 
Assuming that percolation processes leads to the extrema points displace-
ment of experimental dependence S (d) in comparison with theoretical, we can 
note a decrease their influence with increase SnTe thickness. 
 
On the other hand, if percolation processes alter the main type carrier con-
centration in nanostructure, thereby changing Fermi energy and hence oscilla-
tion period, then abrupt change in carrier concentration would lead to abrupt 
changes in oscillation period of d-dependence of Seebeck coefficient. Since the 
oscillation period is associated with Fermi energy, it is possible to construct the 
dependence EF(d) (fig. 2). Fig. 2 shows that with changing well  
CONCLUSIONS 
1. Shown that dependences of Seebeck coefficient on width of quantum 
well  with  infinitely  high  walls  for  structures  based  on  IV-VI  compounds  are  
characterized by discontinuities with some period.  
2. Based on experimentally determined oscillation period ?dexp of  the  
thickness dependences of Seebeck coefficient Fermi energy values are deter-
mined in QW structures of compounds IV-VI.  
3. Shown that oscillation period value ?dexp of quantum well is equal to the 
width of QW dmin, where the bottom of lowest sub-band coincides with Fermi 
energy EF. We have given to the width dmin the content of minimum QW width 
from which quantum-size effects are the main factor that determines non-
monotonous change of thickness dependencies for thermoelectric parameters in 
corresponding nanostructures.  
4. For QW SnTe it was found good agreement between theoretical and ex-
perimental values of oscillation amplitude in dependence S(d).  
5. Reducing the value of difference between the experimental and theoreti-
cal extremum points with increasing SnTe QW width is explained by the reduc-
ing of the percolation phenomena impact in such growth.  
6. For SnTe nanostructures shown that dependences S(d) with different os-
cillation periods are in good agreement with experiment at different intervals of 
change the condensate thickness. Thus the dependence of Fermi energy on the 
 
Fig. 2 – Dependence of Fermi energy on 
SnTe well width, obtained by comparing 
the theoretical dependences S (d) for 
different oscillation periods with experi-
mental 
140                  Nanomaterials: Applications and Properties (NAP-2011). Vol. 1, Part I             
 
well width is received. Revealed a sharp change of Fermi energy in the range 
from 2 to 3 nm. 
 
 
REFERENCES 
[1] The physics of low-dimensional semiconductors. An introduction (J.H. Davies), 
Cambridge university press, 1998. 
[2] Dresselhaus M.S., Ghen G., Rang M.I. et al., Adv. Mater., 2007, 19, P. 1043-1053. 
[3] CRC Handbook of Thermoelectrics, chapters 4-6, (Bhandari C.M. in ed. 
D.M. Rowe), CRC Press, 1995. 
[4] E.I. Rogacheva, O.N. Nashchekina, Y.O. Vekhov et al. Thin Solid Films, 2003, 423, 
P. 115-118. 
[5] E.I. Rogacheva, T.V. Tavrina, O.N. Nashchekina, S.N. Grigorov, K.A. Nasedkin, 
M.S. Dresselhaus, S.B. Cronin, 2002, 80(15), P. 2690-2692. 
[6] Rogacheva E.I, Nashchekina O.N, Grigorov S.N. et al. Institute of Physics Publish-
ing. Nanotechnology, 2003, 14, P. 53–59. 
[7] E.I. Rogacheva, O.N. Nashchekina, A.V. Meriuts et al., Appl. Phys. Lett., 2005, 86, 
063103.  
  


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Quantum-size effects in nanostructures on the base of compounds IV-VI

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