Specific thermoemf and Hall-effect in crystals with monopolar conductivity

Specific thermoemf and Hall-effect in crystals with monopolar conductivity

The Seebeck coefficient α and Hall constant RH are calculated for monopolar crystal as based of quantum kinetic equation. It is shown that α and RH in the case of simple isotropic band do not depend on relaxation characteristics in contrast to the result obtained for the same crystals by calculat...

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Опубліковано в: :Semiconductor Physics Quantum Electronics & Optoelectronics
Дата:2009
Автор: Boiko, I.I.
Формат: Стаття
Мова:English
Опубліковано: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2009
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/118609
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Цитувати:Specific thermoemf and Hall-effect in crystals with monopolar conductivity / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 1. — С. 47-52. — Бібліогр.: 6 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
id nasplib_isofts_kiev_ua-123456789-118609
record_format dspace
spelling Boiko, I.I.
2017-05-30T17:19:57Z
2017-05-30T17:19:57Z
2009
Specific thermoemf and Hall-effect in crystals with monopolar conductivity / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 1. — С. 47-52. — Бібліогр.: 6 назв. — англ.
1560-8034
PACS 71.20.Nr, 71.55. Eq, 72.20.My
https://nasplib.isofts.kiev.ua/handle/123456789/118609
The Seebeck coefficient α and Hall constant RH are calculated for monopolar crystal as based of quantum kinetic equation. It is shown that α and RH in the case of simple isotropic band do not depend on relaxation characteristics in contrast to the result obtained for the same crystals by calculations founded on relaxation time approximation.
en
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
Semiconductor Physics Quantum Electronics & Optoelectronics
Specific thermoemf and Hall-effect in crystals with monopolar conductivity
Article
published earlier
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
title Specific thermoemf and Hall-effect in crystals with monopolar conductivity
spellingShingle Specific thermoemf and Hall-effect in crystals with monopolar conductivity
Boiko, I.I.
title_short Specific thermoemf and Hall-effect in crystals with monopolar conductivity
title_full Specific thermoemf and Hall-effect in crystals with monopolar conductivity
title_fullStr Specific thermoemf and Hall-effect in crystals with monopolar conductivity
title_full_unstemmed Specific thermoemf and Hall-effect in crystals with monopolar conductivity
title_sort specific thermoemf and hall-effect in crystals with monopolar conductivity
author Boiko, I.I.
author_facet Boiko, I.I.
publishDate 2009
language English
container_title Semiconductor Physics Quantum Electronics & Optoelectronics
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
format Article
description The Seebeck coefficient α and Hall constant RH are calculated for monopolar crystal as based of quantum kinetic equation. It is shown that α and RH in the case of simple isotropic band do not depend on relaxation characteristics in contrast to the result obtained for the same crystals by calculations founded on relaxation time approximation.
issn 1560-8034
url https://nasplib.isofts.kiev.ua/handle/123456789/118609
citation_txt Specific thermoemf and Hall-effect in crystals with monopolar conductivity / I.I. Boiko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 1. — С. 47-52. — Бібліогр.: 6 назв. — англ.
work_keys_str_mv AT boikoii specificthermoemfandhalleffectincrystalswithmonopolarconductivity
first_indexed 2025-11-25T20:35:27Z
last_indexed 2025-11-25T20:35:27Z
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fulltext Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 47 PACS 71.20.Nr, 71.55. Eq, 72.20.My Specific thermoemf and Hall-effect in crystals with monopolar conductivity I.I. Boiko V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine, prospect Nauky, 03028 Kyiv, Ukraine E-mail: igorboiko@yandex.ru Phone: (044)236-5422 Abstract. The Seebeck coefficient α and Hall constant RH are calculated for monopolar crystal as based of quantum kinetic equation. It is shown that α and RH in the case of simple isotropic band do not depend on relaxation characteristics in contrast to the result obtained for the same crystals by calculations founded on relaxation time approximation. Keywords: dispersion law, quantum kinetic equation, Seebeck coefficient, Hall constant. Manuscript received 03.12.08; accepted for publication 18.12.08; published online 03.02.09. 1. Introduction If mobile carriers in crystal belong to several different groups there is no practical possibility to relate measured values with space flows of separate groups. The system that contains several substantially different groups of carriers has specific properties absent for crystals where unique group of carriers is actual. For instance, in a multy-group situation the state of zero total current can be provided in crystal with nonzero density of current for every group. Therefore, distribution of carriers over space of momentum in this crystal can be nonequilibrium in absence of total current. Quite another situation appears if only one group is actual: the measured voltage drop on the total crystal length is evidently tied to the density of current for this unique group. Below we consider two problems. The first one concerns monopolar semiconductor where carriers belong to one group (simple band) and their space distribution is nonuniform due to a small gradient of temperature T and Fermi-energy εF . Phenomenological relation between the density of current j r and microscopic dynamical and statistic forces in the stationary case has the form jTeE F rrrr 1~~)/1( −σ=∇α−ε∇− . (1) Here e is the charge of band carriers, E r is the vector of electrical field, σ~ is the tensor of conductivity and α~ is the so-called Seebeck coefficient; the latter in general is a tensor value, too. For simplicity, we restrict our consideration in this paper by the case of band with spherical symmetry and for small divergence of equilibrium. Then the tensor values in Eq. (1) degenerate in scalars, and we obtain jTeE F rrrr 1)/1( −σ=∇α−ε∇− . (2) Note that coefficient α is not a kinetic coefficient because it does not ensure a direct relation between some force and the density of current flow. We can call it as a structural coefficient, that depends on the band structure. Another problem relates to an uniform crystal being in the stationary uniform external magnetic field H r of a classical value. Here, we consider again one group of carriers with the isotropic dispersion law and calculate the Hall-constant RH . Calculation of α and RH can be performed using by several different methods. Usual method starts from obtaining the nonequilibrium distribution function f by the way of approximate solution of the kinetic Bolzmann equation, where the collision integral is replaced by a simple form containing relaxation time τ(ε) = 1/ν(ε) (see, for example, Refs. [1] and [2]). Here ε is the band carrier energy. During the next step one calculates the current density based on the obtained f and marks out the values interesting for us (for instant, the coefficient α in Eq. (2) ) . Another method applied here uses a set of balance equations which are moments of quantum kinetic equa- tion in the space of wave vector (see Refs. [3] and [4]). Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 48 We will compare together α and RH obtained by different methods and will compare the calculated values of α with some experimental data. 2. Specific thermo-emf 2. 1. Seebeck coefficient α in τ-approximation The Seebeck coefficient calculated in Refs. [1] and [2] can be presented in the form ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 〉ετ〈 〉ετε〈 −ε=α τ )( )(1 )( FeT . (3) Here, angle brackets denote the statistic average with equilibrium distribution function (see below Eqs. (5), (7) and (14)). Consider the isotropic dispersion law of the following form: )(2 )( 22 ε =ε m kk r hr . (4) For this case, ∫ ∫ ∂ε∂ε∂ε∂ετ ∂ε∂ε∂ε∂εετ = 〉ετ〈 〉ετε〈 kdkf kdkf i i r r 32 0 32 0 ]/[]/)()[( ]/[]/)()[( )( )( . (5) 2.1.1. Parabolic dispersion law If the function )(εm in Eq. 4 does not depend on energy (that is dispersion law is parabolic), we have the standard simple form *2/)( 22 mkk r h r =ε . (6) Then, it is possible to rewrite Eq. 5 in the following manner (see Ref. [1], formula (7.2) of Chapter IX): ( ) ( ) =εεε∂ε∂εεεε∂ε∂=〉ε〈 ∫∫ ∞∞ 0 2/3 0 0 2/3 0 /)()(/)()( dfdCfC = [ ] ∫∫ ∞∞ εεεεεε ε∂ ∂ ε 0 2/1 0 0 2/3 0 )()()( 3 2 dfdCf . (7) Here, f0(ε) = {1+exp[(ε − εF) / kBT]} − 1. In Ref. [1] one relates the expression (3) to an opened circuit scheme ( 0=j r ). The expression (2) does not depend on the mode of electrical circuit and relates to an arbitrary current density in the region of small divergence of equilibrium. So, it relates to the case 0=j r as well as to the case 0≠j r . If one uses the model form r BTk ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ε τ=ετ 0)( (8) and dispersion law (6), then he obtains from Eqs (3) and (8) the folowing expression for the Seebeck coefficient: ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ε ε ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ +−ε=α + + τ )/( )/( 2 51 2/1 2/3 )( TkF TkFrTk eT BFr BFr BF , (9) Here, 2/1−=r for scattering by longitudinal acoustic phonons, 2/3=r for scattering by charged impurities. The Fermi integral is introduced by the following definition: ∫ ∞ η−++Γ =η 0 )exp(1)1( 1)( w dww r F r r . (10) In this formula Γ is the gamma-function. It is very strange: Seebeck coefficients (3) and (9) don’t depend absolutely on the intensity of carriers interaction with the scattering system. But it evidently depends on the energetic parameter of relaxation time r even if this interaction is very weak. One more significant objection concerns the formal record of value α. If we consider monopolar crystal with the simple band in the case 0=j r (then the distribution of carriers in the space of velocities is equilibrium), the expression for thermo-emf should not contain any relaxation characteristics. We consider these misunderstandings as a consequence of employing of the τ-approximation for rough solution of the Boltzmann equation. We think that this approach can give results of quality value only (see, for example, Ref.[4]). 2.1.2. Nonparabolic dispersion law Consider now an other dispersion law. Let in Eq.(4) )/1)(0()( Gmm εε+=ε . (11) This form is suitable for n-InSb at ε << 20εG (here, ;014.0)0( 0mm = εG = 0.17 eV). For this case, one obtains using Eq. (8) the following expression: . ])21()1()[/()/1974( ])21()1()[/()/1974( )( )( 0 12/32/30 0 0 12/32/50 0 ∫ ∫ ∞ −+ ∞ −+ ++ ++ × ×= 〉ετ〈 〉ετε〈 dzzzzdzdKTzf dzzzzdzdKTzf Tk r r B (12) Results of calculations performed with the formulae (3), (9) and (12) are shown below. 2.2. Seebeck coefficient obtained from the balance equation The quantum kinetic equation for small deviation of equilibrium for the system of uniform carriers has the form (see Ref. [3]) Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 49 k k f k f Ee r r r h r St1 0 = ∂ ∂ . (13) Here kf r is the nonequilibrium distribution function, )(0 0 ε= ffk r is the equilibrium distribution function, e is charge of bend carriers. For a slightly nonuniform system, the quantum kinetic equation can be presented in the form (stationary case) kk k f k f rv r r r h St)(1 0 =εΛ ∂ ∂ − , (14) where .)()( )())(()( )( rT T rEe rT T er Fk F FkE Fk rrrrr rrrrr v v v ∇ ε−ε +ε∇+−= =∇ ε−ε +ϕ+ε∇=εΛ (15) Applying the operator ∫π kdk rr 3 3)2( 2 (16) to Eq. (14), we obtain the equation that represents balance of dynamical and statistical forces: 0)( =++ε∇− resTF FFrEe rrrrr . (17) Here, ∫ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∇ ⋅ ∂ ∂ ε−ε π = kd T rT k f kk n F k FT rrr r rrr r 3 0 3 )(])([ )2( 2 , (18) n is the density of carriers. The force resF r is the first moment of the scattering integral and can be presented in the form σ= /jeFres rr . As a result (see Eqs. (17) and (2)), we obtain =α – ∫ ε−ε ∂ ∂ π kdk k f k eTn F j k i rrr 3 0 3 ])([ )2( 2 . (19) Note that for the linear theory this expression (in contrary to Eqs (3) and (8)) does not depend on relaxation characteristics. It contains only equilibrium values and depends only on parameters of the dispersion law. 2.2.1. Parabolic dispersion law Introducing the law (6) in Eq. (19) and performing integration, we find (see Eq. (10)): ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ε ε − ε −=α )/( )/( 2 5 2/1 2/3 TkF TkF Tke k BF BF B FB (20) (this formula, underline it again, does not depend on the mechanism of scattering). Pay attention that expression (20) coincides with the expression (9), formally accepted at 0=r . For nondegenerate carriers ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ε −=α 2 5 Tke k B FB ; (21) For high degeneration F B e Tk ε π =α 2 22 . (22) Results of numerical calculations performed for n- GaAs shown in Figs 1 , 2 and 3. Here and further curves 1 relate to the formula (20), curves 2 and 3 − to the formulae (9) at 2/3=r and 2/1−=r consequently (see Ref. [1]). Fig. 1 relates to density n = 3.5×10 17 c -3, Fig. 2 relates to the density n = 7.7×10 18 cm -3 . Black points in Fig. 3 represent the experimental data at KT 300= (see Ref. [5]). Fig. 1. Fig. 2. Fig. 3. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 50 2.2.2. Nonparabolic dispersion law Consider now the dispersion law (11) and apply it to n-InSb. Then it follows from (19): .)/21()/1()( )( 3 )0(22 2/1 0 0 2/3 32 3 εεε+εε+ε−ε ε∂ ε∂ ε× × π =α ∫ ∞ d f eTn m GGF h (23) Fig. 3 represents the dependences α(n) calculated for different temperatures as based on the formula (23) (see curves 1) and on formulae (8), (12) at 2/3=r (see curves 2). The curves (a) relate to temperature KT 50= , the curves (b) relate to temperature KT 100= , the curves (c) relate to temperature KT 150= , the curves (d) relate to temperature KT 200= , the curves (e) relate to temperature KT 300= , the curves (f) relate to temperature KT 400= . Distinction between the curves 1 and 2 calculated with the different approaches is quite evident. Fig. 4. Fig. 5. Fig. 5 represents the dependences α(T) calculated at 31410 −= cmn . Here and father the curve 1 relates to the formula (23), the curve 2 − to the formulae (9) and (12) at 2/3=r . One can see that the model which uses τ-approximation is not suitable for electron indium antimonide. 3. Hall-effect In the presence of uniform magnetic field H r , the quantum kinetic equation for band carriers has the form (stationary case) .St)(,1 )()( 0 kk k ffkv k H c e f kv rr r rr r r h rrr = ⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ ×+ +εΛ ε∂ ∂ − + (24) Here, k v kr h r r ∂ ε∂ = 1 ; ))(2/1(],[ BAABBA +=+ ; )(εΛ is represented by Eq. (15). The magnetic field is supposed to be classical: 〉ε〈<<cmeH */h . Apply to Eq. (24) the operator (16). Then, we obtain the equation that represents balance of forces (see Eq. (18)): 0)()])(/1([ =++ε∇−×+ resTF FFruHcEe rrrrrrr . (25) Here, σ−=σ−= //2 jeuneFres rrr (we denote the average drift velocity of carriers as ur ). In Eq. (25), we have omitted the term ,)( )2( )( 3 0 3 ∫ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⋅⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ × ε∂ ∂ ∂ ε∂ × × π −= kdkv k Hk f v u cn eHC k rrr r rr r r rr r because further we use the simple dispersion law (6). For this case 0)( =HC rr . Note that Eq. (25) was obtained from the quantum kinetic equation (24) without any approximation. Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 51 Fig. 6 (a). Fig. 6 (b). Fig. 6 (c, d, e). Fig. 6 (f, g, h). Consider here uniform crystal with n-conductivity and simple band. For this case 0)( =+ε∇− TF Fr rrr and Eq. (25) has the form (here and further 0>e ) 0)/1())(/1( =σ−×+ jjHencE rrrr . (26) Let z-axis is directed along the vector j r and vector H r is disposed in {xy}-plane: ),0,0( zjj = r , )0,,( yx HHH = r . (27) It follows from Eqs (26) and (27): .0)/1( ;0)/(;0)/( =σ− =−=+ zz zxyzyx jE jencHEjencHE (28) We consider here the vector zz Ee r as applied electric field and the vector yyxxH EeEeE rrr += as Hall- field. The following relation determines the Hall constant RH: jHRjHHREEE HyxHyx ⊥⊥ =+=+= 2222 ; zz Ejj ||σ== . (29) As a result, one obtains: cen RH 1 = . (30) We can see that in the classical limit for magnetic field Hall the constant RH does not depend on magnetic field. We can see also that the Hall constant does not depend on the form of dispersion law )(k r ε , if this law is isotropic. These results completely differ from results obtained in Ref. [1] for −τ approximation. There 2 2 22 2 2 2 )*/)((1 */)( )*/)((1 )( )*/)((1 )( 1 cmHe cmHe cmHe cmHe enc RH ετ+ ετ + ετ+ ετ ετ+ ετ = . (31) Results of numerical calculations performed using of Eqs (31) − (34) and (8) at different densities and temperatures are shown in Fig. 6 . Here T = 300 K and n = 10 15 cm –3 for (a) , T = 300 K and n = 10 18 cm - 3 for (b) , T = 300 K and eτ0 H / m*c → 0 for (c) , T = 300 K and eτ0 H / m*c = 1 for (d) , T = 300 K and eτ0 H / m*c = 4 for (e) , n = 3.5×10 17 cm –3 and eτ0 H / m*c → 0 for (f ), n = 3.5×10 17 cm –3 and eτ0 H / m*c = 1 for (g), n = 3.5×10 17 cm –3 and eτ0 H / m*c = 4 for (h) . APPENDIX Nernst-Ettingsgauzen effect in nonuniform crystal Consider the case when external magnetic field and constant gradient of temperature are applied to monopolar crystal with a simple isotropic band. Then, Eq. (26) changes to (here and below 0>e ) 0)/1()()()/1(* =σ−∇α+×+ jrTjHencE rrrrrr . (A1) Here (see Eqs. (18), (10) and (6)), FeEE ε∇+= rrr )/1(* ; (A2) . )/( )/( 2 5 ])([ )( )( )2(3 4 2/1 2/3 3 0 3 TkF TkF Tk kdk k f k eTn BF BF B F F k ε ε + ε −= =ε−ε ε∂ ∂ ε π −=α ∫ rr r r r (A3) Using Eq. (A1) consider Nernst-Ettingsgauzen effect showing relation of the measured voltage drop with the crossed gradient of temperature T∇ r and Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 1. P. 47-52. © 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 52 magnetic field H r . Choose the following orientation of fields and current: )0,0,( xHH = r ; ),0,0( TT z∇=∇ r ; ),0,0( zjj = r . (A4) It follows from (A1) − (A4): TecnHE zxy ∇ασ= )/(* . (A.5) Determine the Nernst-constant Нернста Q by the relation: THEQ zxy ∇= /* . (A.6) Then, ecnQ /ασ= . (A.7) Using Eq. (20) one obtains: )/( )/( 2 5)0( 2/1 2/3 2 TkF TkF Tknce HkQ BF BF B FB ε ε − ε=σ = . (A.8) The value σ/Q does not depend on relaxation characteristic (in contrary to subsequent result obtained in Ref. [1]). References 1. A.I.Anselm: Introduction to the Theory of Semiconductors, Nauka, M.: 1978 (in Russian). 2. E.M. Lifshits and L.P. Pitaevskiy: Physical kinetics, Nauka, M.: 1979 (in Russian). 3. I.I. Boiko: Kinetics of Electron Gas Interacting with Fluctuating Potential. Naukova dumka, Kyiv, 1993 (in Russian). 4. I.I. Boiko and S.I.Kozlovskiy: Investigation of conductivity and piezoresistance of n-type silicon on basis of quantum kinetic equation and model distribution function.// Sens. Actuators, A147, p.17, 2008. 5. Gallium Arsenide. Production and Properties. Editors F.P. Kesamanly and D.N. Nasledov. M.: Nauka, (in Russian) 1973. 6. S. M. Puri ,T. H. Geballe. Phonon Drag in n-type InSb.// Phys. Rev. A136, p.1767, 1964.

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