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VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 17-23<br /> <br /> Calculation of the Ettingshausen Coefficient<br /> in a Rectangular Quantum Wire with an Infinite Potential<br /> in the Presence of an Electromagnetic Wave<br /> (the Electron - Optical Phonon Interaction )<br /> Cao Thi Vi Ba, Tran Hai Hung*, Doan Minh Quang, Nguyen Quang Bau<br /> Faculty of Physics, VNU University of Science, 334 Nguyen Trai, Hanoi, Vietnam<br /> Received 11 October 2017<br /> Revised 24 October 2017; Accepted 25 October 2017<br /> <br /> Abstract: The Ettingshausen coefficient (EC) in a Rectangular quantum wire with an infinite<br /> potential (RQWIP)in the presence of an Electromagnetic wave (EMW) is calculated by using a<br /> quantum kinetic equation for electrons. Considering the case of the electron - optical phonon<br /> interaction, we have found the expressions of the kinetic tensors  ik , ik , ik , ik . From the kinetic<br /> tensors, we have also obtained the analytical expression of the EC in the RQWIP in the presence<br /> of EMW as function of the frequency and the intensity of the EMW, of the temperature of system,<br /> of the magnetic field and of the characteristic parameters of RQWIP. The theoretical results for<br /> the EC are numerically evaluated, plotted and discussed for a specific RQWIP GaAs/GaAsAL. We<br /> also compared received EC with those for normal bulk semiconductors and quamtum wells to<br /> show the difference. The Ettingshausen effect in a RQWIP in the presence of an EMW is<br /> newly developed.<br /> Keywords: Ettingshausen effect, Quantum kinetic equation, RQWIP, Electron - phonon<br /> interaction, kinetic tensor.<br /> <br /> 1. Introduction<br /> Nowadays, the theoretical study of kinetic effects in low-dimensional systems is increasingly<br /> interested, especially on the electrical, magnetic and optical properties of the low-dimensional systems such<br /> as: the absorption of electromagnetic waves, the acoustomagnetoelectric effect, the Hall effect, ... These<br /> results show us that there are some significant differences from the bulk semiconductor that the previous<br /> researches studied [1-12]. Among those, the Ettingshausen effect has just been researched in bulk<br /> semiconductors [13] and only been studied on the theoretical basis in 2-D systems [14]. Furthermore, no<br /> research has been done on the Ettinghausen effect in 1-D systems such as quantum wires so far. In this<br /> paper, the calculation of Ettingshausen coefficient in the Rectangular quantum wire with an infinite<br /> <br /> _______<br /> <br /> <br /> Corresponding author. Tel.: 84-903293995.<br /> Email: haihung307@gmail.com<br /> https//doi.org/ 10.25073/2588-1124/vnumap.4236<br /> <br /> 17<br /> <br /> C.T.V. Ba et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 17-23<br /> <br /> 18<br /> <br /> potential in the presence of magnetic field, electric field under the influence of electromagnetic wave is<br /> done by using the quantum kinetic equation method that brings the high accuracy and the high efficiency.<br /> Comparing the results obtained in this case with in the case of the bulk semiconductors and quantum<br /> wires, we see some differences. To demonstrate this, we estimate numerical values for a GaAs/GaAsAl<br /> quantum wire.<br /> 2. Calculation of the Ettingshausen coefficient in a Rectangular quantum wire with an infinite<br /> potential in the presence of an electromagnetic wave<br /> In a model, we consider a wire with rectangular cross section (Lx  Ly) and the length Lz. The effective<br /> mass of electron is denoted as m. The RQWIP is subjected to a crossed dc electric field E1  ( 0,0,E1 ) and<br /> magnetic field B  ( B,0,0 ) in the presence of a strong EMW characterized by electric field<br /> E( t )  E0 sin(  t ) (with E0 and  are the amplitude and the frequency of LR, respectively). Under these<br /> condition, the wave function and energy spectrum of confined electron can be written as:<br /> <br />   ,k ( x, y,z) <br /> <br /> 1 i kz<br /> e<br /> Lz<br /> <br /> 0  x  Lx<br /> 2<br /> n x<br /> 2<br /> l y<br /> <br /> sin(<br /> )<br /> sin(<br /> ) when <br /> Lx<br /> Lx<br /> Ly<br /> Ly<br /> <br /> 0  y  Ly<br /> <br /> (1)<br /> <br /> and   ,k ( x, y,z)  0 if else.<br /> k z2  2 2  n2 l 2 <br /> 1<br /> 1  eE1 <br />  ( k ) <br /> <br />  2  2   c ( N  ) <br /> <br /> <br /> 2m<br /> 2m  Lx Ly <br /> 2<br /> 2m  c <br /> 2<br /> <br /> 2<br /> <br /> (2)<br /> <br /> eB<br /> is the cyclotron frequenciesn;  and ‟ are the<br /> m<br /> quantum numbers (n,l) and (n,l‟) of electron; N, N‟ are the Landau level (N=0,1,2,…). These expressions<br /> differ from the equivalent expressions in bulk semiconductors [14] and quantum wells [13].<br /> The Hamiltonnian of the electron - optical phonon interaction system in the above RQWIP can be<br /> written as:<br /> e<br /> H     ( k  A( t ) )a,k a ,k   q bq bq <br /> c<br /> q<br />  ,k<br /> (3)<br /> 2<br /> 2 <br />   Cq I  , ' ( q ) a ,k  q a ',k ( bq  bq )  ( q )a,k  q a ',k<br /> where kz is the electron wave momentum; c <br /> <br />  , ',k ,q<br /> <br /> q<br /> <br /> <br />  ,k<br /> <br /> Where a<br /> <br /> <br /> q<br /> <br /> and a ,k ( b and bq ) are the creation and the annihilation operators of electron (optical<br /> <br /> phonon); k is the electron wave momentum; q is the phonon wave vector; q are optical phonon<br />  1<br /> 1 <br />   (here V is<br /> <br />    0 <br /> is magnetic permeability of high frequency dielectric,  0 is magnetic<br /> <br /> frequency; Cq the electron – optical phonon interaction constant: | Cq |2 =<br /> the unit normalization volume,  <br /> <br /> e2o<br /> 2 0 q 2V<br /> <br /> permeability of static dielectric; I , ' ( q ) is the electron form factor, which is determinned by [8], different<br /> from that in cylindrical quantum wire;   q  is the potential undirected:<br /> <br /> C.T.V. Ba et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 17-23<br /> <br />   q    2 i  ( eE  c [ q,h ])<br /> 3<br /> <br /> <br />  q <br /> q<br /> <br /> 19<br /> <br /> (4)<br /> <br /> ( h is unit vector in the direction of magnetic field).<br /> Through some computation steps, the quantum kinetic equation takes the form:<br /> e<br /> <br />  m kn<br />  ,k<br /> <br /> ,k<br /> <br /> δ(ε  ε ,k )<br /> <br /> τ<br /> <br /> <br />  n ,k<br /> e<br /> kF<br /> <br /> m  ,k  k<br /> <br /> <br /> e<br />  c  h , kn ,k δ(ε  ε ,k<br /> m<br />   ,k<br /> <br /> <br /> 2 e<br />  δ(ε  ε ,k ) <br />  Cq<br /> m  , ' ,q ,k<br /> <br /> <br /> <br /> 2<br />   n ',q  k  n ,k   1 <br /> 2 2<br /> <br /> <br /> <br /> <br /> 2<br /> <br /> <br /> ) <br /> <br /> 2<br /> <br /> I  , ' ( q ) N q k <br /> <br /> <br /> 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />    ',k  q    ,k  o   <br /> <br /> o<br />  ',k  q<br />  ,k<br /> 4 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> 2<br /> 2 <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> n<br /> <br /> n<br /> 1<br /> <br />     ',k  q    ,k  o <br /> o<br />    ',k  q<br />  ',k  q<br />  ,k<br />  ,k  <br /> 4 2<br /> 2 2 <br /> <br /> <br /> <br /> <br /> <br /> 2<br /> 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />    ',k  q    ,k  o  <br /> o<br />  ',k  q<br />  .k<br /> 4 2<br /> 4 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> (5)<br /> <br /> <br /> <br /> <br /> <br />   ( ε  ε<br /> <br />  ,k<br /> <br /> <br /> <br /> )<br /> <br /> Equation (5) we put:<br /> R(  )  <br />  ,k<br /> <br /> Q(  )  <br /> <br /> S(  ) <br /> <br /> e<br /> kn δ(ε  ε ,k )<br /> m  ,k<br /> <br />  n ,k<br /> e<br /> kF<br /> <br /> m  ,k  k<br /> <br /> (6)<br /> <br /> <br />   F<br />  T;<br />  δ(ε  ε ,k ) ; F  e.E1 <br /> T<br /> <br /> <br /> (7)<br /> <br /> 2<br /> 2 e<br /> 2<br /> C(q) I  , ' ( q ) N q k <br /> <br /> m  , ' ,q ,k<br /> <br /> <br /> 2 <br /> 2<br />   n ',q  k  n ,k   1 <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />    ',k  q    ,k  o   <br /> <br /> o<br />  ',k  q<br />  ,k<br /> 2 2 <br /> 4 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> 2<br /> 2 <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> n<br /> <br /> n<br /> 1<br /> <br />     ',k  q    ,k  o <br /> o<br />    ',k  q<br />  ',k  q<br />  ,k<br />  ,k  <br /> 4 2<br /> 2 2 <br /> <br /> <br /> <br /> <br /> <br /> 2<br /> 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />    ',k  q    ,k  o  <br /> o<br />  ',k  q<br />  .k<br /> 4 2<br /> 4 2<br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> (8)<br /> <br /> <br /> <br /> <br /> <br />   ( ε  ε<br /> <br />  ,k<br /> <br /> <br /> <br /> ).<br /> <br /> We obtain the following equations:<br /> R(  ) <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br /> <br />   (  ) Q(  )  S(  ),h h .<br /> 2 2<br /> c<br /> <br /> <br /> <br /> <br /> <br /> ( )<br /> Q(  )  S(  )  c  (  ) h ,Q(  )  h ,S(  ) <br /> 1  c2 2 (  )<br /> <br /> (9)<br /> <br /> C.T.V. Ba et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 17-23<br /> <br /> 20<br /> <br /> After some approximate developing and computation steps, we obtain the expression of Ettinghausen<br /> coefficient as follows:<br />  xx xy   xy xx<br /> 1<br /> (10)<br /> P<br /> H  xx   T  xx   xx  T  K L <br /> xx<br />  xx<br /> <br /> <br /> <br /> <br /> <br /> <br /> Here:<br /> ea<br /> eb<br /> 2<br /> ea<br /> eb<br /> 2<br />  ( 1  c2 2 )<br /> ;  xy <br /> .c  .c .<br /> 2<br /> 2<br /> 2 2<br /> 2<br /> 2<br /> 1  c <br /> m<br /> 1  c <br /> m<br /> 1   2 2 <br /> 1   2 2 <br /> <br /> (11)<br /> <br /> eb<br /> 2<br /> .( 1  c2 2 ).<br /> 2<br /> mT<br /> 1   2 2 <br /> <br /> (12)<br /> <br />  xx <br /> <br /> b<br /> 2<br /> b<br /> 2<br /> .( 1  c2 2 ).<br /> ;<br /> <br /> <br /> .<br /> <br /> <br /> .<br /> xy<br /> c<br /> 2<br /> 2<br /> m<br /> m<br /> 1   2 2 <br /> 1   2 2 <br /> <br /> (13)<br /> <br /> T <br /> <br /> b<br /> <br /> .( 1  c2 2 ).<br /> 2<br /> mT<br /> 1   2 2 <br /> <br /> (14)<br /> <br />  xx <br /> <br /> c<br /> <br />  xx <br /> <br /> c<br /> <br /> c<br /> <br /> c<br /> <br /> xx<br /> <br /> 2<br /> <br /> c<br /> <br /> 2<br /> <br /> c<br /> <br /> 1/ 2<br /> <br /> e Lx  2m <br /> a<br /> <br /> 2 <br /> 4m    <br /> <br /> b<br /> <br /> 2 eNo<br /> m<br /> <br /> 2<br /> <br /> 1  eE1   2 2  n2 l 2 <br /> 1  <br />  <br /> <br /> <br /> exp  β  ε F <br />  2  2   c  N    <br /> <br />  <br /> 2m  c <br /> 2m  Lx Ly <br /> 2  <br /> <br /> <br /> <br />  <br /> <br /> (A A<br /> <br />  <br /> 1<br /> <br /> 2<br /> <br /> , '<br /> <br />  1<br /> 1 <br />  A3  A4  A5  A6  A7  A8 ) <br />   I .I  , ' e B<br />   o <br /> <br /> (16)<br /> <br /> 2<br /> <br />  <br /> 2<br />  2 2  n2 l 2 <br /> 1  1  eE   <br /> <br /> <br /> I  exp   F <br />  2  2   c  N   <br />    , I , '   I , ' ( q ) dq<br />  <br /> 2m  Lx Ly <br /> 2  2m  c   <br /> <br /> <br /> <br />  <br /> <br />  Lx kBTe2   B2<br /> A1 <br /> e<br /> 8 2 3<br /> <br /> 11<br /> <br /> A2  <br /> A3 <br /> <br />     B211<br /> <br /> B<br /> e<br />  ( 2B11m )1/ 2 K 1 (  11 )2 <br /> <br /> 2 <br /> 2<br />  2 m<br /> <br />  Lx kBTe4 Eo2 B11    B2<br /> e<br /> 16m2 (  / 8m )3/ 2  4<br /> <br /> 11<br /> <br /> <br /> 1 <br />  <br /> <br /> B11 <br /> <br /> <br />  Lx kBTe4 Eo2 B13    B2 <br />  Lx kBTe4 Eo2 B14    B2 <br /> 1 <br /> 1 <br /> e<br /> <br /> <br /> ,<br /> A<br /> <br /> e<br /> <br /> <br />  <br /> <br /> 4<br /> 2<br /> 3/ 2<br /> 4<br /> 2<br /> 3/ 2<br /> 4<br /> 16m (  / 8m ) <br /> B13 <br /> 16m (  / 8m ) <br /> B14 <br /> <br /> <br /> 13<br /> <br />  Lx kBTe2   B4<br /> A5 <br /> e<br /> 8 2 3<br /> <br /> 15<br /> <br /> A6  <br /> <br /> 14<br /> <br />     B215<br /> <br /> B<br /> e<br />  ( 2B15 m )1/ 2 K 1 (  15 )2 <br /> <br /> 2 <br /> 2<br />  2 m<br /> <br />  Lx kBTe4 Eo2 B15    B2<br /> e<br /> 16m2 (  / 8m )3 / 2  4<br /> <br /> 15<br /> <br /> <br /> 1 <br />  <br /> <br /> B15 <br /> <br /> <br /> (15)<br /> <br /> (17)<br /> <br /> C.T.V. Ba et al. / VNU Journal of Science: Mathematics – Physics, Vol. 33, No. 4 (2017) 17-23<br /> <br /> A7 <br /> <br />  Lx kBTe4 Eo2 B17    B2 <br /> 1<br /> e<br />  <br /> 16m2 (  / 8m )3 / 2  4<br /> B<br /> 17<br /> <br /> <br /> B11 <br /> <br /> 17<br /> <br />  n' 2  n2 l' 2  l 2<br /> <br /> <br /> 2m  L2x<br /> L2y<br /> <br /> 2<br /> <br /> 2<br /> <br /> <br />  Lx kBTe4 Eo2 B18    B218<br /> ,<br /> A<br /> <br /> e<br />  8<br /> 2m2 (  / 8m )3 / 2  4<br /> <br /> <br /> 21<br /> <br /> <br /> 1 <br />  <br /> <br /> B18 <br /> <br /> <br /> <br />   c  N'  N   o ,<br /> <br /> <br /> B13  B11   , B14  B11   , B15  B11  2o , B17  B15   , B18  B15  <br /> <br /> Here   1 / ( kBT ) ; hx  0,hy  0,hz  1; K L ,  ,T ,k B , 0 ,  ,  F :is the lattice heat conductivity, the<br /> momentum laxation time, the temperature, the Boltzmann constant, the static dielecttric constant, the high<br /> frequency dielectric constant, and the Fermi level, respectively. The expressions of the kinetic tensors<br />  ik , ik , ik , ik (11-14) and of the EC (10) as well as functions of the frequency and the intensity of the<br /> EMW, of the temperature of system, of the magnetic field and of the characteristic parameters of RQWIP<br /> are different from those in bulk semiconductors and quamtum wells. It is newly developed in the quantum<br /> theory of Ettinghausen effect.<br /> 3. Numerical results<br /> We will survey, plot and discuss the expressions for the case of a specific GaAs/GaAsAl quantum well.<br /> The parameters used in the calculations are as follows:<br /> <br />   10.9,   12.9, 0  36.25meV ,   5320 kg.m 3 ,  3.10 13 s 1 ,<br />  F  50meV ,  10 12 s,Lx  8.10 9 m,Ly  7.10 9 m,m  0,067.m0 ( m<br /> <br /> 0<br /> <br /> is the mass of a free electron )<br /> <br /> In Fig. 1, we show the dependence of the EC on the laser frequency. From the figure, we see that the<br /> EC in RQWIP decreased is nonliner with the frequency, however, the EC in the quantum wells increased<br /> with the frequency [14]. This also demonstrates its difference in bulk semiconductors [13].<br /> In Fig. 2, we show the dependence of the EC on laser amplitute. We found that the EC in RQWIP<br /> decreased is nonliner with laser amplitude. This is similar in the case of quantum wells, however, the EC in<br /> the quantum wire has decreased much faster than in quantum wells and in bulk semiconductors [13,14].<br /> <br /> Fig 1. The dependence of EC on laser<br /> frequency.<br /> <br /> Fig 2. The dependence of EC on<br /> laser amplitute.<br /> <br />