- Email: [email protected]
On the magnetotransport of 3D systems in quantizing magnetic field
Solid State Communications 200 (2014) 48–50
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
On the magnetotransport of 3D systems in quantizing magnetic field M.V. Cheremisin n A.F. Ioffe Physical-Technical Institute, St. Petersburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 16 August 2014 Received in revised form 12 September 2014 Accepted 22 September 2014 by M. Grynberg Available online 28 September 2014
The resistivity components of 3D electron gas placed in quantizing magnetic field are calculated taking into account the correction caused by combined action of the Peltier and Seebeck thermoelectric effects. The longitudinal, transverse and the Hall magnetoresistivities exhibit familiar 1/B-period oscillations being universal functions of magnetic field and temperature. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Bulk semiconductors D. Magnetotransport
1. Introduction The electronic transport of 3D solids subjected to strong magnetic field has been intensively studied since the observation of the Schubnikov–de Haas oscillations [1]. Most of the interest concerned the transverse (i.e ? B) magnetotransport. The problem was considered semiclassically [2,3] and, then, using the rigorous quantum mechanical approach [4,5]. With the help of density matrix equation the components of the conductivity tensor associated with transverse magnetotransport were obtained in Ref. [4]. Despite this progress, the incorporation of the thermal effects faced the strong difficulties [6–9]. As it was demonstrated for the first time in Ref. [10], in quantizing magnetic field it is necessary to take into account the diamagnetic surface currents. Neglecting dissipation the transverse thermal current and the energy flux are given by the sum of bulk and surface components. With the diamagnetic currents accounted [10] the transport coefficients satisfies the phenomenological Einstein and Onsager relationships.
2. General formalism The main goal of the present paper concerns the peculiar thermodynamic approach [11–13] regarding 3D electron transport in strong magnetic fields. We will follow the argumentation first put forward by Kirby and Laubitz [11], and then modified in Refs. [12,13] for magnetotransport in 2D electron(hole) systems. This approach allows one to account for both the Schubnikov–de Haas Oscillations and n
Corresponding author. Tel.: þ 7 8122976132; fax: þ 7 8122971017. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ssc.2014.09.018 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
Integer Quantum Hall Effect [14] modes. Let us consider the isotropic sample subjected in strong magnetic field B ¼ Bz . Neglecting spin splitting, the 3D energy spectrum yields [15]
εn ¼ ℏωc ðn þ 1=2Þ þ ε ; J
ð1Þ
where n¼0,1… is the LL number, ωc ¼ eB=mc is the cyclotron frequency, m is the electron effective mass. Then, pz and ε J ¼ p2z =2m are the momentum and the kinetic energy respectively of an electron moving along the magnetic field. The electron motion in x y plane is quantized, thus results in discrete Landau level set. For simplicity, we further neglect the broadening of the Landau levels. The density of states associated with the certain LL obeys the textbook result Γ ¼ 1=2π l2B , where lB ¼ ðℏc=eBÞ1=2 is the magnetic length. In the present paper we are mostly interested in the strong quantum limit case, when ℏωc 4 4 kT. In general, the macroscopic current, j, and the energy flux, q, densities are given by [16] j ¼ σ^ ðE α∇TÞ; q ¼ ðαT ζ =eÞj ϰ^ ∇T:
ð2Þ
Here, E ¼ ∇ζ =e is the electric field, μ and ζ ¼ μ eφ are the chemical and electrochemical potential respectively, α is the thermopower. Then, σ^ and ϰ^ ¼ LT σ^ are the conductivity and 2 thermal conductivity tensors respectively, L ¼ π 2 k =3e2 is the Lorentz number. It will be recalled that Eq. (2) is valid for a confined-topology sample for which the diamagnetic surface currents [10] are taken into account. Both the Einstein and Onsager relationships are satisfied. The movement of the electron along the magnetic field is not quantized, thus results in Drude longitudinal component σ zz ¼ Ne2 τ=m of the conductivity tensor, where N is the 3D electron density. In what follows we assume the constant momentum relaxation time τ in the z-direction. Compared to
M.V. Cheremisin / Solid State Communications 200 (2014) 48–50
unperturbed movement of an electron along the magnetic field, the transverse drift of electrons in x y plane results in offdiagonal Hall components [4] σ yx ¼ σ xy ¼ Nec=B of the conductivity tensor. Then, in contrast to the previous studies [2–5] we suggest that in strong quantum limit σ xx ¼ σ yy ¼ 0. Actually, x y the fraction of Eq. (2), related to transport in x-y plane, can be viewed [7] as the current and energy dissipationless fluxes caused by electron drift in crossed fields. With the help of the above 1 notations, the resistivity tensor ρ^ ¼ σ^ obeys the same symmetry, namely ρyx ¼ ρxy ¼ 1=σ yx , ρzz ¼ 1=σ zz . The crucial idea of the present paper is that a nonzero transverse magnetoresistivity ρ can, nevertheless, arise due to combined action of the Peltier and Seebeck thermoelectric effects [11–13].
3. Results and discussion Let us discuss in detail the typical experimental setup allowed to measure the Hall resistivity ρyx and the transverse magnetoresistivity ρ. The standard Hall-bar geometry sample (Fig. 1, inset) is connected to the current source by means of two identical leads. The contacts “a” and “b” are ohmic. The longitudinal voltage is measured between the open ends (“c” and “d”), maintained at the ambient temperature. The sample is placed in a chamber (not shown in Fig. 1, inset) kept at the bath temperature T0. Note that the electron temperature may, in principle, exceed the bath temperature if the electron–phonon coupling is weak at low temperatures. Actually, the 3D electron gas cooling can occur predominantly through the contacts of the sample and the leads connected to them. However, the heat leakage via metal leads can be disregarded similar to that reported [17] for two-dimensional electron case. Finally, we will consider the adiabatic cooling of 3D electrons. Recall that the Peltier heat is generated by a current across the interface between two different conductors. At the contact (e.g. “a”
in Fig. 1, inset), the temperature, Ta, electrochemical potential ζ , normal components of the total current, I, and the total energy flux are continuous. There exists a difference Δα ¼ αm α between the thermoelectric powers of the metal conductor and 3D sample. Then, Q a ¼ I ΔαT a is the amount of Peltier heat released per unit time in contact “a”. For Δα 4 0 and current flow direction shown in Fig. 1, the contact “a” is heated and contact “b” is cooled. The contacts are at different temperatures, and ΔT ¼ T a T b 4 0. At small currents, the temperature gradient is small and T a;b T 0 . In this case the 3D sample thermopower α is nearly constant, hence, one can disregard the Thompson heating, IT∇α, in 3D bulk. Note that the amount of the Peltier heat evolved at the contact “a” is equal to that absorbed at the contact “b”. The energy flux is continuous at each contact, thus Eq. (2) yields κ yx ∇x Tja;b ¼ jΔαT a;b . Here, we take into account that the current is known [18] to enter and leave the sample at two diagonally opposite corners (Fig. 1, inset). One can fulminantly find the longitudinal temperature gradient ∇x T ¼ jΔα=Lσ yx , which is linear in current. Omitting the contribution of the conductor resistances, the voltage drop, U, measured between the ends “c” and “d” is equal to Seebeck thermoelectromotive force R R U ¼ Ex dx ¼ α dT ¼ ΔαðT a T b Þ, thus gives the transverse magnetoresistivity ρ ¼ U=jl ¼ ðΔαÞ2 ρyx =L. We now intend to find the longitudinal resistivity (i.e. || Bz) taking into account the Peltier–Seebeck thermoelectric effects. Since the longitudinal movement of an electron is unaffected by magnetic field, we make use of the result [11] valid for B ¼0. The longitudinal resistivity of 3D electron gas acquires the correction Δρzz ¼ ðΔαÞ2 ρzz =L. Finally, we summarize the actually measured values of the 3D electron gas resistivities as follows:
ρyx ¼
B ; Nec
ρ ¼ sρyx ;
ρzz ¼ ð1 þsÞ
ρyx ; ωc τ
ð3Þ
where we take into account that for the actual case of the metal leads s ¼ ðΔαÞ2 =L C α2 =L. Remind that in strong quantum limit, the thermopower of dissipationless 3D electron gas is a universal thermodynamic quantity proportional to the entropy per electron [10]:
α¼
Fig. 1. Dimensionless Hall resistivity ρyx, transverse magnetoresistivity ρ (both scaled by ρc) and the longitudinal resistivity ρzz =ρzz ð0Þ vs. magnetic field B ν 1 for certain value of the dimensionless temperature ξ ¼ 0:025. The dashed (dotted) line represents the classical result for Hall and longitudinal resistivity respectively. Inset: the experimental setup.
49
S ; eN
ð4Þ
where S ¼ is the 3D electron entropy, ∂Ω=∂T μ;B N ¼ ∂Ω=∂μ T;B is the 3D electron density, Ω ¼ 2kT Γ ∑n;pz lnð1 þ expððμ εn Þ=kTÞÞ is the thermodynamic potential which accounts for the spin degeneracy. Ii is instructive to discuss the applicability of the Gibbs statistics formalism for actual 3D electrons case. Utilizing thermodynamic potential presumes a variable number of 3D particles while the chemical potential is assumed to be a constant. We argue that this method is not prohibited since the chemical potential of 3D electrons can be, for example, pined by an external reservoir of carriers. The metal leads can play the role of such a reservoir. Indeed, in thermodynamic equilibrium the chemical potential of the multiple system “3D electrons-conductor” flattens out. Let us suppose a certain value of the chemical potential μ, and, thus 3D density N0 at B ¼ 0. The magnetic field growth may result in subsequent changes of 3D electron density N. In strong fields the only few Landau levels are filled, hence, the absolute change in N can be of the order of the zero-field carrier density, N0. Simultaneously, the number electrons in the external reservoir changes. The relative variation of the chemical potential of the external reservoir yields δμ=μ ¼ 23 N0 =N m 10 7 , where N m 1023 cm 3 is the electron density the metal leads, N 0 ¼ 1016 cm 3 is the typical density of 3D solid. We conclude that the chemical potential of metal leads, and, hence 3D electrons remains constant. Note that conventional reasoning of 3D electron plasma neutrality is rather doubtful since the Debye screening length can be of the order of
50
M.V. Cheremisin / Solid State Communications 200 (2014) 48–50
macroscopic sample length scale (see the Appendix) in quantizing magnetic field. It can be demonstrated that both the electron density N and the entropy S are universal functions of the reduced temperature ξ ¼ kT=μ and dimensionless magnetic field ℏωc =μ ¼ ν 1 , where ν is the so-called filling factor. Following the conventional Lifshitz– Kosevich formalism [19], we can easily derive asymptotic equations for N; S, and, hence, for α, valid in the low-temperature and low-field limit ν 1 ; ξ o 1: 0 1 π k ð 1Þ sin 2 π k ν 2 1 B π 4 C C; pffiffiffi ∑ N ¼ N0 B @1 þ A 1 k¼1 z sinh z 2F 1=2
ξ
N0 kπ π ∑ ð 1Þk ΦðzÞ cos 2π kν ; S ¼ S0 þ 1 k¼1 4 2F 1=2 3
1
ð5Þ
ξ
coth z 1 where ΦðzÞ ¼ zz3=2 is a form-factor, z ¼ 2π 2 ξνk kT=ℏωc is the U sinh z dimensionless parameter which roughly corresponds to quantum 3=2 limit criteria at k¼1. Then, N 0 ¼ n0 32 ξ F 1=2 ð1=ξÞ and ! 5F 3=2 ð1=ξÞ 1 S0 ¼ N 0 k 3F 1=2 ð1=ξÞ ξ
are the zero-field electron gas density and entropy respectively, Fn(y) is the Fermi integral and n0 ¼ ð2mμÞ3=2 =3π 2 ℏ3 is the density of strongly degenerate 3D electron gas at T¼0. In Fig. 1 we plot the magnetic field dependencies of the Hall resistivity ρyx ðν 1 Þ and transverse magnetoresistivity ρðν 1 Þ given by Eqs. (3)–(5). Both curves arepscaled ffiffiffiffiffiffiffiffiffiffi in units of ρc ¼ mμ=n0 e2 ℏ ¼ ðh=e2 Þð3π =4kF Þ, where kF ¼ 2mμ=ℏ is the Fermi vector. Then, the longitudinal resistivity ρzz ðν 1 Þ specified by Eq. (3) is scaled in units of zero-field resistivity ρzz ð0Þ ¼ m=n0 e2 τ ¼ 2ρc =kF l, where l ¼ ℏkF τ=m is the mean free path. The overall behavior of the resistivity curves shown in Fig. 1 can be easily understood in terms of the 3D energy spectrum, and, thus the related density of states. Indeed, apart from the discrete Landau levels, the 3D energy spectrum specified by Eq. (1) contains the continuous component ε J . At fixed μ the number of occupied states, and, hence ρyx ; ρzz N 1 are smooth functions of the magnetic field. Then, the density of states is continuous, hence provides the nonzero transverse magnetoresistivity ρ. It is worthwhile to mention that in 2D electron gas case the energy spectrum is purely discrete, thus leads to quantized Hall resistivity ρyx, with the longitudinal resistivity ρ vanishing within the Hall plateaus [14]. We now make an attempt to compare our results with the experimental observations. As expected, the longitudinal, transverse and the Hall components of magnetoresistivity exhibit familiar 1/B-period in-phase oscillations [20] associated with discrete LL energy spectrum. However, in contrast to our predictions the behavior of the longitudinal and the transverse magnetoresistivity components [20] becomes uncorrelated at high fields when the only few LLs are filled. The possible reason for the above discrepancy could be, for example, the constant momentum relaxation time, i.e. τ a τðBÞ, adopted in our simple model. For typical n-InSb sample n0 ¼ 1:2 1016 cm 3 used in Ref. [20] we obtain μ ¼ 170 K and, then ρc ¼ 0:1 Ω cm. At filling ν ¼ 3=2 our zero spin-split approach provides (see Fig. 1) the transverse magnetoresistivity amplitude ρ C 0:002 Ω cm at liquid helium temperature ξ ¼ 0:025. This value is consistent with that 0:015 Ω cm observed experimentally [20] at B ¼2.3 T.
4. Conclusions In conclusion, we have calculated the longitudinal, transverse and the Hall magnetoresistivities of a 3D solid placed in quantizing magnetic taking into account the contribution caused by combined action of Peltier and Seebeck thermoelectric effects. The transport coefficients demonstrate familiar 1/B-period oscillations being universal functions of the filling factor. The amplitude of the transverse resistivity is consistent with that observed in the experiment.
Acknowledgments The reported study was partially supported by RFBR, research project No.15-02-01165 A.
Appendix Let us examine the charge relaxation and screening for 3D solids placed in strong magnetic field. These effects can be described by continuity equation ∂ρe þ div j ¼ 0: ∂t
ð6Þ
In quantizing magnetic field the non-dissipative transverse current specified by Eq. (2) can be re-written [21] as follows: j ¼ ðcN=B2 Þ rot ðζ BÞ. Thus, the charge relaxation is absent since div j 0, therefore ∂ρe =∂t ¼ 0. The retardation effects, if accounted, could result in slow charge relaxation ðσ yx =cÞ2 analogous to that in 2D electron gas case [22]. We argue that the transverse length of the Debye screening could be comparable with the macroscopic (i.e. sample) length scale. We conclude that in quantizing magnetic field the electron plasma neutrality may be violated in 3D sample bulk. References [1] L.W. Schubnikov, W.I. de Haas, Commun.Kemerlingh Onnes Lab, vols. 207, 207a, 210a, 210b, (1930). [2] S. Titeica, Ann. Phys. 22 (1935) 128. [3] B. Davidov, N. Pomeranchuk, ZhETF 9 (1939) 1924. [4] E.N. Adams, T.D. Holdstein, J. Phys. Chem. Solids 10 (1959) 254. [5] R. Kubo, H. Hasegawa, N. Hashitsume, J. Phys. Soc. Jpn. 14 (1959) 56. [6] A.I. Anselm, B.M. Askerov, Sov. Phys. Solid State 2 (1960) 2330; A.I. Anselm, B.M. Askerov, Sov. Phys. Solid State 3 (1961) 3668. [7] P.S. Zyryanov, V.P. Silin, Sov. Phys. JETP 19 (1964) 366. [8] A.I. Akhiezer, V.G. Bar'yakhtar, S.V. Peletminskii, Sov. Phys. JETP 21 (1965) 136. [9] S.V. Peletminskii, V.G. Bar'yakhtar, Sov. Phys. Solid State 7 (1965) 446. [10] Yu.N. Obraztsov, Sov. Phys. Solid State 6 (1964) 331; Yu.N. Obraztsov, Sov. Phys. Solid State 7 (1965) 573. [11] C.G.M. Kirby, M.J. Laubitz, Metrologia 9 (1973) 103. [12] M.V. Cheremisin, Sov. Phys. JETP 92 (2001) 357. [13] M.V. Cheremisin, Physica E 28 (2005) 393; M.V. Cheremisin, Physica E 64 (2014) 15. [14] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [15] L. Landau, Z. Phys. 64 (1930) 629. [16] L.D. Landau, E.M. Lifshits, Electrodynamics of Continuous Media, Pergamon, New York, 1966. [17] A. Mittal, et al., Physica B 194–196 (1994) 167. [18] A.H. Thompson, G.S. Kino, J. Appl. Phys. 41 (1970) 3064. [19] I.M. Lifshitz, A.M. Kosevich, Sov. Phys. JETP 29 (1955) 730. [20] F.A. Egorov, S.S. Murzin, Sov. Phys. JETP 67 (1988) 1045. [21] P.S. Zyryanov, Fiz. Tverd. Tela 6 (1964) 3562. [22] A.O. Govorov, A.V. Chaplik, Sov. Phys. JETP 68 (1989) 1143.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
On the magnetotransport of 3D systems in quantizing magnetic field M.V. Cheremisin n A.F. Ioffe Physical-Technical Institute, St. Petersburg, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 16 August 2014 Received in revised form 12 September 2014 Accepted 22 September 2014 by M. Grynberg Available online 28 September 2014
The resistivity components of 3D electron gas placed in quantizing magnetic field are calculated taking into account the correction caused by combined action of the Peltier and Seebeck thermoelectric effects. The longitudinal, transverse and the Hall magnetoresistivities exhibit familiar 1/B-period oscillations being universal functions of magnetic field and temperature. & 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Bulk semiconductors D. Magnetotransport
1. Introduction The electronic transport of 3D solids subjected to strong magnetic field has been intensively studied since the observation of the Schubnikov–de Haas oscillations [1]. Most of the interest concerned the transverse (i.e ? B) magnetotransport. The problem was considered semiclassically [2,3] and, then, using the rigorous quantum mechanical approach [4,5]. With the help of density matrix equation the components of the conductivity tensor associated with transverse magnetotransport were obtained in Ref. [4]. Despite this progress, the incorporation of the thermal effects faced the strong difficulties [6–9]. As it was demonstrated for the first time in Ref. [10], in quantizing magnetic field it is necessary to take into account the diamagnetic surface currents. Neglecting dissipation the transverse thermal current and the energy flux are given by the sum of bulk and surface components. With the diamagnetic currents accounted [10] the transport coefficients satisfies the phenomenological Einstein and Onsager relationships.
2. General formalism The main goal of the present paper concerns the peculiar thermodynamic approach [11–13] regarding 3D electron transport in strong magnetic fields. We will follow the argumentation first put forward by Kirby and Laubitz [11], and then modified in Refs. [12,13] for magnetotransport in 2D electron(hole) systems. This approach allows one to account for both the Schubnikov–de Haas Oscillations and n
Corresponding author. Tel.: þ 7 8122976132; fax: þ 7 8122971017. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.ssc.2014.09.018 0038-1098/& 2014 Elsevier Ltd. All rights reserved.
Integer Quantum Hall Effect [14] modes. Let us consider the isotropic sample subjected in strong magnetic field B ¼ Bz . Neglecting spin splitting, the 3D energy spectrum yields [15]
εn ¼ ℏωc ðn þ 1=2Þ þ ε ; J
ð1Þ
where n¼0,1… is the LL number, ωc ¼ eB=mc is the cyclotron frequency, m is the electron effective mass. Then, pz and ε J ¼ p2z =2m are the momentum and the kinetic energy respectively of an electron moving along the magnetic field. The electron motion in x y plane is quantized, thus results in discrete Landau level set. For simplicity, we further neglect the broadening of the Landau levels. The density of states associated with the certain LL obeys the textbook result Γ ¼ 1=2π l2B , where lB ¼ ðℏc=eBÞ1=2 is the magnetic length. In the present paper we are mostly interested in the strong quantum limit case, when ℏωc 4 4 kT. In general, the macroscopic current, j, and the energy flux, q, densities are given by [16] j ¼ σ^ ðE α∇TÞ; q ¼ ðαT ζ =eÞj ϰ^ ∇T:
ð2Þ
Here, E ¼ ∇ζ =e is the electric field, μ and ζ ¼ μ eφ are the chemical and electrochemical potential respectively, α is the thermopower. Then, σ^ and ϰ^ ¼ LT σ^ are the conductivity and 2 thermal conductivity tensors respectively, L ¼ π 2 k =3e2 is the Lorentz number. It will be recalled that Eq. (2) is valid for a confined-topology sample for which the diamagnetic surface currents [10] are taken into account. Both the Einstein and Onsager relationships are satisfied. The movement of the electron along the magnetic field is not quantized, thus results in Drude longitudinal component σ zz ¼ Ne2 τ=m of the conductivity tensor, where N is the 3D electron density. In what follows we assume the constant momentum relaxation time τ in the z-direction. Compared to
M.V. Cheremisin / Solid State Communications 200 (2014) 48–50
unperturbed movement of an electron along the magnetic field, the transverse drift of electrons in x y plane results in offdiagonal Hall components [4] σ yx ¼ σ xy ¼ Nec=B of the conductivity tensor. Then, in contrast to the previous studies [2–5] we suggest that in strong quantum limit σ xx ¼ σ yy ¼ 0. Actually, x y the fraction of Eq. (2), related to transport in x-y plane, can be viewed [7] as the current and energy dissipationless fluxes caused by electron drift in crossed fields. With the help of the above 1 notations, the resistivity tensor ρ^ ¼ σ^ obeys the same symmetry, namely ρyx ¼ ρxy ¼ 1=σ yx , ρzz ¼ 1=σ zz . The crucial idea of the present paper is that a nonzero transverse magnetoresistivity ρ can, nevertheless, arise due to combined action of the Peltier and Seebeck thermoelectric effects [11–13].
3. Results and discussion Let us discuss in detail the typical experimental setup allowed to measure the Hall resistivity ρyx and the transverse magnetoresistivity ρ. The standard Hall-bar geometry sample (Fig. 1, inset) is connected to the current source by means of two identical leads. The contacts “a” and “b” are ohmic. The longitudinal voltage is measured between the open ends (“c” and “d”), maintained at the ambient temperature. The sample is placed in a chamber (not shown in Fig. 1, inset) kept at the bath temperature T0. Note that the electron temperature may, in principle, exceed the bath temperature if the electron–phonon coupling is weak at low temperatures. Actually, the 3D electron gas cooling can occur predominantly through the contacts of the sample and the leads connected to them. However, the heat leakage via metal leads can be disregarded similar to that reported [17] for two-dimensional electron case. Finally, we will consider the adiabatic cooling of 3D electrons. Recall that the Peltier heat is generated by a current across the interface between two different conductors. At the contact (e.g. “a”
in Fig. 1, inset), the temperature, Ta, electrochemical potential ζ , normal components of the total current, I, and the total energy flux are continuous. There exists a difference Δα ¼ αm α between the thermoelectric powers of the metal conductor and 3D sample. Then, Q a ¼ I ΔαT a is the amount of Peltier heat released per unit time in contact “a”. For Δα 4 0 and current flow direction shown in Fig. 1, the contact “a” is heated and contact “b” is cooled. The contacts are at different temperatures, and ΔT ¼ T a T b 4 0. At small currents, the temperature gradient is small and T a;b T 0 . In this case the 3D sample thermopower α is nearly constant, hence, one can disregard the Thompson heating, IT∇α, in 3D bulk. Note that the amount of the Peltier heat evolved at the contact “a” is equal to that absorbed at the contact “b”. The energy flux is continuous at each contact, thus Eq. (2) yields κ yx ∇x Tja;b ¼ jΔαT a;b . Here, we take into account that the current is known [18] to enter and leave the sample at two diagonally opposite corners (Fig. 1, inset). One can fulminantly find the longitudinal temperature gradient ∇x T ¼ jΔα=Lσ yx , which is linear in current. Omitting the contribution of the conductor resistances, the voltage drop, U, measured between the ends “c” and “d” is equal to Seebeck thermoelectromotive force R R U ¼ Ex dx ¼ α dT ¼ ΔαðT a T b Þ, thus gives the transverse magnetoresistivity ρ ¼ U=jl ¼ ðΔαÞ2 ρyx =L. We now intend to find the longitudinal resistivity (i.e. || Bz) taking into account the Peltier–Seebeck thermoelectric effects. Since the longitudinal movement of an electron is unaffected by magnetic field, we make use of the result [11] valid for B ¼0. The longitudinal resistivity of 3D electron gas acquires the correction Δρzz ¼ ðΔαÞ2 ρzz =L. Finally, we summarize the actually measured values of the 3D electron gas resistivities as follows:
ρyx ¼
B ; Nec
ρ ¼ sρyx ;
ρzz ¼ ð1 þsÞ
ρyx ; ωc τ
ð3Þ
where we take into account that for the actual case of the metal leads s ¼ ðΔαÞ2 =L C α2 =L. Remind that in strong quantum limit, the thermopower of dissipationless 3D electron gas is a universal thermodynamic quantity proportional to the entropy per electron [10]:
α¼
Fig. 1. Dimensionless Hall resistivity ρyx, transverse magnetoresistivity ρ (both scaled by ρc) and the longitudinal resistivity ρzz =ρzz ð0Þ vs. magnetic field B ν 1 for certain value of the dimensionless temperature ξ ¼ 0:025. The dashed (dotted) line represents the classical result for Hall and longitudinal resistivity respectively. Inset: the experimental setup.
49
S ; eN
ð4Þ
where S ¼ is the 3D electron entropy, ∂Ω=∂T μ;B N ¼ ∂Ω=∂μ T;B is the 3D electron density, Ω ¼ 2kT Γ ∑n;pz lnð1 þ expððμ εn Þ=kTÞÞ is the thermodynamic potential which accounts for the spin degeneracy. Ii is instructive to discuss the applicability of the Gibbs statistics formalism for actual 3D electrons case. Utilizing thermodynamic potential presumes a variable number of 3D particles while the chemical potential is assumed to be a constant. We argue that this method is not prohibited since the chemical potential of 3D electrons can be, for example, pined by an external reservoir of carriers. The metal leads can play the role of such a reservoir. Indeed, in thermodynamic equilibrium the chemical potential of the multiple system “3D electrons-conductor” flattens out. Let us suppose a certain value of the chemical potential μ, and, thus 3D density N0 at B ¼ 0. The magnetic field growth may result in subsequent changes of 3D electron density N. In strong fields the only few Landau levels are filled, hence, the absolute change in N can be of the order of the zero-field carrier density, N0. Simultaneously, the number electrons in the external reservoir changes. The relative variation of the chemical potential of the external reservoir yields δμ=μ ¼ 23 N0 =N m 10 7 , where N m 1023 cm 3 is the electron density the metal leads, N 0 ¼ 1016 cm 3 is the typical density of 3D solid. We conclude that the chemical potential of metal leads, and, hence 3D electrons remains constant. Note that conventional reasoning of 3D electron plasma neutrality is rather doubtful since the Debye screening length can be of the order of
50
M.V. Cheremisin / Solid State Communications 200 (2014) 48–50
macroscopic sample length scale (see the Appendix) in quantizing magnetic field. It can be demonstrated that both the electron density N and the entropy S are universal functions of the reduced temperature ξ ¼ kT=μ and dimensionless magnetic field ℏωc =μ ¼ ν 1 , where ν is the so-called filling factor. Following the conventional Lifshitz– Kosevich formalism [19], we can easily derive asymptotic equations for N; S, and, hence, for α, valid in the low-temperature and low-field limit ν 1 ; ξ o 1: 0 1 π k ð 1Þ sin 2 π k ν 2 1 B π 4 C C; pffiffiffi ∑ N ¼ N0 B @1 þ A 1 k¼1 z sinh z 2F 1=2
ξ
N0 kπ π ∑ ð 1Þk ΦðzÞ cos 2π kν ; S ¼ S0 þ 1 k¼1 4 2F 1=2 3
1
ð5Þ
ξ
coth z 1 where ΦðzÞ ¼ zz3=2 is a form-factor, z ¼ 2π 2 ξνk kT=ℏωc is the U sinh z dimensionless parameter which roughly corresponds to quantum 3=2 limit criteria at k¼1. Then, N 0 ¼ n0 32 ξ F 1=2 ð1=ξÞ and ! 5F 3=2 ð1=ξÞ 1 S0 ¼ N 0 k 3F 1=2 ð1=ξÞ ξ
are the zero-field electron gas density and entropy respectively, Fn(y) is the Fermi integral and n0 ¼ ð2mμÞ3=2 =3π 2 ℏ3 is the density of strongly degenerate 3D electron gas at T¼0. In Fig. 1 we plot the magnetic field dependencies of the Hall resistivity ρyx ðν 1 Þ and transverse magnetoresistivity ρðν 1 Þ given by Eqs. (3)–(5). Both curves arepscaled ffiffiffiffiffiffiffiffiffiffi in units of ρc ¼ mμ=n0 e2 ℏ ¼ ðh=e2 Þð3π =4kF Þ, where kF ¼ 2mμ=ℏ is the Fermi vector. Then, the longitudinal resistivity ρzz ðν 1 Þ specified by Eq. (3) is scaled in units of zero-field resistivity ρzz ð0Þ ¼ m=n0 e2 τ ¼ 2ρc =kF l, where l ¼ ℏkF τ=m is the mean free path. The overall behavior of the resistivity curves shown in Fig. 1 can be easily understood in terms of the 3D energy spectrum, and, thus the related density of states. Indeed, apart from the discrete Landau levels, the 3D energy spectrum specified by Eq. (1) contains the continuous component ε J . At fixed μ the number of occupied states, and, hence ρyx ; ρzz N 1 are smooth functions of the magnetic field. Then, the density of states is continuous, hence provides the nonzero transverse magnetoresistivity ρ. It is worthwhile to mention that in 2D electron gas case the energy spectrum is purely discrete, thus leads to quantized Hall resistivity ρyx, with the longitudinal resistivity ρ vanishing within the Hall plateaus [14]. We now make an attempt to compare our results with the experimental observations. As expected, the longitudinal, transverse and the Hall components of magnetoresistivity exhibit familiar 1/B-period in-phase oscillations [20] associated with discrete LL energy spectrum. However, in contrast to our predictions the behavior of the longitudinal and the transverse magnetoresistivity components [20] becomes uncorrelated at high fields when the only few LLs are filled. The possible reason for the above discrepancy could be, for example, the constant momentum relaxation time, i.e. τ a τðBÞ, adopted in our simple model. For typical n-InSb sample n0 ¼ 1:2 1016 cm 3 used in Ref. [20] we obtain μ ¼ 170 K and, then ρc ¼ 0:1 Ω cm. At filling ν ¼ 3=2 our zero spin-split approach provides (see Fig. 1) the transverse magnetoresistivity amplitude ρ C 0:002 Ω cm at liquid helium temperature ξ ¼ 0:025. This value is consistent with that 0:015 Ω cm observed experimentally [20] at B ¼2.3 T.
4. Conclusions In conclusion, we have calculated the longitudinal, transverse and the Hall magnetoresistivities of a 3D solid placed in quantizing magnetic taking into account the contribution caused by combined action of Peltier and Seebeck thermoelectric effects. The transport coefficients demonstrate familiar 1/B-period oscillations being universal functions of the filling factor. The amplitude of the transverse resistivity is consistent with that observed in the experiment.
Acknowledgments The reported study was partially supported by RFBR, research project No.15-02-01165 A.
Appendix Let us examine the charge relaxation and screening for 3D solids placed in strong magnetic field. These effects can be described by continuity equation ∂ρe þ div j ¼ 0: ∂t
ð6Þ
In quantizing magnetic field the non-dissipative transverse current specified by Eq. (2) can be re-written [21] as follows: j ¼ ðcN=B2 Þ rot ðζ BÞ. Thus, the charge relaxation is absent since div j 0, therefore ∂ρe =∂t ¼ 0. The retardation effects, if accounted, could result in slow charge relaxation ðσ yx =cÞ2 analogous to that in 2D electron gas case [22]. We argue that the transverse length of the Debye screening could be comparable with the macroscopic (i.e. sample) length scale. We conclude that in quantizing magnetic field the electron plasma neutrality may be violated in 3D sample bulk. References [1] L.W. Schubnikov, W.I. de Haas, Commun.Kemerlingh Onnes Lab, vols. 207, 207a, 210a, 210b, (1930). [2] S. Titeica, Ann. Phys. 22 (1935) 128. [3] B. Davidov, N. Pomeranchuk, ZhETF 9 (1939) 1924. [4] E.N. Adams, T.D. Holdstein, J. Phys. Chem. Solids 10 (1959) 254. [5] R. Kubo, H. Hasegawa, N. Hashitsume, J. Phys. Soc. Jpn. 14 (1959) 56. [6] A.I. Anselm, B.M. Askerov, Sov. Phys. Solid State 2 (1960) 2330; A.I. Anselm, B.M. Askerov, Sov. Phys. Solid State 3 (1961) 3668. [7] P.S. Zyryanov, V.P. Silin, Sov. Phys. JETP 19 (1964) 366. [8] A.I. Akhiezer, V.G. Bar'yakhtar, S.V. Peletminskii, Sov. Phys. JETP 21 (1965) 136. [9] S.V. Peletminskii, V.G. Bar'yakhtar, Sov. Phys. Solid State 7 (1965) 446. [10] Yu.N. Obraztsov, Sov. Phys. Solid State 6 (1964) 331; Yu.N. Obraztsov, Sov. Phys. Solid State 7 (1965) 573. [11] C.G.M. Kirby, M.J. Laubitz, Metrologia 9 (1973) 103. [12] M.V. Cheremisin, Sov. Phys. JETP 92 (2001) 357. [13] M.V. Cheremisin, Physica E 28 (2005) 393; M.V. Cheremisin, Physica E 64 (2014) 15. [14] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [15] L. Landau, Z. Phys. 64 (1930) 629. [16] L.D. Landau, E.M. Lifshits, Electrodynamics of Continuous Media, Pergamon, New York, 1966. [17] A. Mittal, et al., Physica B 194–196 (1994) 167. [18] A.H. Thompson, G.S. Kino, J. Appl. Phys. 41 (1970) 3064. [19] I.M. Lifshitz, A.M. Kosevich, Sov. Phys. JETP 29 (1955) 730. [20] F.A. Egorov, S.S. Murzin, Sov. Phys. JETP 67 (1988) 1045. [21] P.S. Zyryanov, Fiz. Tverd. Tela 6 (1964) 3562. [22] A.O. Govorov, A.V. Chaplik, Sov. Phys. JETP 68 (1989) 1143.