CALCULATION OF 2D-THERMOMAGNETIC CURRENT AND ITS FLUCTUATIONS USING THE METHOD OF EFFECTIVE HAMILTONIAN. R. G. Aghayeva

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1 CALCULATION OF D-THERMOMAGNETIC CURRENT AND ITS FLUCTUATIONS USING THE METHOD OF EFFECTIVE HAMILTONIAN H. M. Abullaev Institute of Phsics, National Acaem of Sciences of Azerbaijan, H. Javi ave. 33, Baku, Az-43 Azerbaijan (Receive 4 April ) Abstract The metho of effective Hamiltonian (MEH) is propose for the calculation of the nonissipative thermomagnetic current in samples of an imension. This metho inclues the temperature in the Hamiltonian as oppose to the previous researches. The noniagonal component of the thermomagnetic tensor calculate b the MEH is in full accor with results obtaine previousl. The calculations of the thermomagnetic current an its ispersion are performe in the coherent-state representation for the D-sample an nonegenerate statistics. The results obtaine can be use, for eample, to increase the sensitivit of temperature sensors.. Introuction The avances in electronics, especiall in nanoscale electronics, have attracte a wie interest in the stu of noise in the electron evices, since noise restricts the sensitivit of evices operating with weak signals []. A noise is create b fluctuations. Fluctuations are generall small. However, the pla the ecisive role in the precise measurements an communication techniques []. In aition to the equilibrium fluctuations, there is a variet of the fluctuation mechanisms etecte onl in the process of a current flow. The simplest measure of the fluctuations, for eample, of the current, is the current ispersion. Currentl known works contain, as a rule, a general analsis of noise (see, for eample, []). The principal object of this paper is the calculation an estimation of the ispersion of nonissipative thermomagnetic current in the concrete D-sample. Namel, we consiere a nonegenerate electron gas in a quantum well with the potential U m / an in a quantizing magnetic fiel H II z. Here m is the effective mass of electrons an characterizes the parabolic potential. We suppose that the temperature graient in the sample is irecte along the ais: T = T(). To calculate the current ispersion equal to ˆ ˆ j j, () it is necessar to estimate the current ensit ĵ create b the electron. Man authors trie to solve this problem via etermining the thermoelectromotive force in a quantizing magnetic fiel []. However, in those papers, temperature was not inclue in the Hamiltonian, because temperature is relate to the statistical force. The metho that inclues temperature in the Hamiltonian (the metho of effective Hamiltonian (MEH)) is propose in [3] for 3D-sample. MEH makes it possible to calculate the thermomagnetic current an its ispersion immeiatel.

2 It is shown in present paper that MEH can be applie to calculate the thermomagnetic current in samples of an imension, not onl in 3D-samples. The results obtaine for can be use, for eample, to increase the sensitivit of temperature sensors. It shoul be note that the ispersion of thermomagnetic current in the 3D-sample was calculate in [3].. The metho of effective Hamiltonian Let us suppose that the sstem uner consieration is weakl nonuniform an T=T(). Then, the temperature eviation from its equilibrium magnitue is small an, in the simplest case of constant temperature graient, Here T( ) T L () L is the sample length along the ais, L / ) ( L / ), is the parameter of smallness, <<. ( In the presence of an electric fiel E, a temperature graient T, an a magnetic fiel H the current ensit has the form Here E l il an j i l ilel illt (3) are the components of the conuctivit tensors, i,l =,, z, il e, is the electrostatic potential, -e is the electron charge, is the chemical potential. On the assumption T = T() an oes not epen on or z. Then, in the absence of the eternal electric fiel (=) an =const, we erive from (3) an () j T T / L. (4) / We calculate this current, starting from the well known epression j entrˆ, (5) vˆ where vˆ is the velocit operator, n is the ensit of conuctivit electrons, an ˆ is the nonequilibrium electron ensit matri: ˆ Z ep ( ) / kt. (6) Here Z is the partition function an k is the Boltzmann constant. In epression (6) T T (7) L is epane in <<. Henceforth, we restrict ourselves to the first-orer terms in. The Hamiltonian of the sstem can be represente as H ˆ kr grat. (8) The secon term in equation (8) is connecte with the inclusion of the temperature graient effect 3

3 Molavian Journal of the Phsical Sciences, Vol., N, on the Hamiltonian of the equilibrium sstem Ĥ. B analog with the electric fiel, we assume that the temperature is the potential of a certain eternal fiel with the intensit gra T. The kr grat corresponing potential energ takes the form, which in the case uner consieration is reuce to kt /L. Consequentl, in constructing Hamiltonian (8), we procee from the formal corresponence of the electrostatic potential with temperature T an the absolute value of the electron charge e with Boltzmann constant k. Finall, instea of equation (6), we obtain ˆ ˆ Z ep ( H ) /, (9) where kt ), Vˆ, () ( ˆ ˆ V H ˆ ˆ H H L L L L. () The hermiticit of the operator Vˆ is realize b the smmetrization of the prouct of the operators Ĥ an. Equation (9) shows that, in the presence of a small an uniform temperature graient, the ensit matri of the sstem is similar to that of the same sstem in the absence of the temperature graient, but epose to an eternal fiel whose contribution to the Hamiltonian of the sstem is given b the operator Vˆ. It is clear from () that Vˆ is a small perturbation, as it is proportional to the parameter of smallness. Hence, we epan ensit matri (9) in a series using perturbation theor an restrict ourselves to a linear approimation of the parameter of smallness: ˆ ˆ ˆ. () Here ˆ is the equilibrium ensit matri: ˆ Z ep ( Ĥ ), (3) Z is the corresponing statistical sum, an ˆ is the nonequilibrium aition to the ensit matri [4]: ˆ ˆ ep( Ĥ )Vˆ ep( Ĥ ). (4) Starting from the well known epression for the velocit operator (i / ) Ĥ,, (5) vˆ we write equation (5) to first orer in :, ˆ V ˆ, ˆ j ien / ) Tr ˆ,. (6) ( Comparing equations (6) an (4), we can obtain the epression for the noniagonal component of the thermomagnetic tensor. 4

4 Epression (6) is a general formula for the calculation of the transverse thermomagnetic current in the quantizing magnetic fiel. The Hamiltonian of the equilibrium sstem Ĥ can be written as H ˆ h ˆ U, (7) where hˆ pˆ ( pˆ m ) pˆ c z, c =eh/mc is the cclotron frequenc. m For the D-sample U m /. This case was consiere in [5]. z For the D-sample U m /. For the 3D-sample U. This case was consiere in [3]. Calculate b the MEH, the noniagonal component of the thermomagnetic tensor in [5, 3] is in full accor with results obtaine previousl [6, 7]. It shoul be note that the MEH nees no account of the electron iamagnetism [7, ]. It is eas to verif that the conition of = const oes not affect the final result (6). 3. D-Thermomagnetic current To calculate the D-thermomagnetic current j (see (6)) we aopt the scheme from [3]. All the calculations in this paper are mae on the basis of coherent states. Following [3] we construct coherent states of Ĥ. For this we write Ĥ in the form where ˆ h ( m )/ H (8) pˆ m ˆ Aˆ Aˆ ˆ H m, (9) pˆ p, ˆ z, m m m ip Aˆ ˆ m, k, k z t ˆ ep i t c, ˆ c pˆ, () m, () are the quantum numbers. Then the solution of the wave equation ˆ H i can be presente as Here k k z. () c 5

5 Molavian Journal of the Phsical Sciences, Vol., N, k ep ik, k z ep ik zz, p k. (3) satisfies all the necessar requirements of the coherent states [3] an equals 4 m ep m ep it ep it To calculate the current, we aopt the scheme from [3]: epress an. (4) pˆ of the operators, arrange the operators, replace the operators b their eigenvalues; integrate over, k, k z s using the stanar integrals. We omit the terms proportional to the prouct A ˆ ( ) p ( A ˆ ) at ps, to k,an k at l=,3,5,, as the give zero when integrate over, k, k, respectivel. l l z The first term in (6) equals to zero when integrate over k. Making the cclic permutation, we transform the secon term in (6) to the form Vˆ, Tr ˆ Ĥ, Tr ˆ. (5) L Making use of equations () can be epresse in terms ˆ A ˆ, ˆ A : e m it e it z. (6) The integration over in thir term is reuce to Tr Tr ˆ, ˆ ( ) ˆ L,. (7) It can be easil shown that, in the coherent-state representation, Finall, from equation (6) we erive j c nck H i, pˆ kt m T coth kt kt L. (8). (9) This epression is in complete agreement with the results of [8] obtaine without the introuction of temperature in the Hamiltonian. 4. D-Thermomagnetic current ispersion Let us calculate the ispersion of the D-thermomagnetic current in the linear approimation on the basis of equation (). Consiering that the current ĵ is proportional to (see (9)), we omit ˆj in the 6

6 linear approimation. Then, base on epression (), we write the ispersion of our sstem in the form n e Tr( ˆˆ ). (3) v Using the ensit matri ˆ from ()-(4) an velocit operator (5) an (8), we fin ne Tr ˆ m Tr ˆ 4 Tr ˆ pˆ ˆ pˆ ˆ pˆ 7 L. (3) The member appears in the first term of the statistical sum epansion in the perturbation. The equilibrium statistical sum is wiel use in equation (3). The secon term in the epression for the ispersion tens to zero when integrate over. It can easil be shown that In this case, where ˆ is the equilibrium part of the ispersion, while 4 ˆ p m Tr. (3) n e m ne Tr ˆ ˆ ˆ p Tr m (33) (34) is the aitional nonequilibrium part. It is evient from epression (3) that we restrict ourselves to the ispersion ue to the electron velocit fluctuation. The epression for can be obtaine from at =, which correspons to the equilibrium state of the sstem characterize b the temperature T. Consequentl, is relate to the fluctuation ue to the chaotic thermal motion of the particles forming the sstem (thermal noise). If the conuctor has a constant temperature graient, this gives rise to an aitional contribution to the thermomagnetic current ispersion connecte with the ranom character of the charge carrier iffusion. Let us consier. Subsequent to arranging the operators, using the eigenvalues an integrating over α, k, k z (see [5]) we obtain Tr ˆ pˆ Tr ˆ coth, m coth ( 35) (36)

7 Molavian Journal of the Phsical Sciences, Vol., N, Substituting (36) into (35) an using (34), we finall obtain the relation of the nonequilibrium part of the ispersion to the equilibrium part in the form Consier the longituinal isothermal Nernst-Ettingshausen effect, i.e., the appearance of an electric fiel E along the temperature graient in a soli conuctor place in a magnetic fiel which is perpenicular to the temperature graient. It is known that, if the temperature graient is irecte along the ais an the strong magnetic fiel along the z ais, then j nceh (37) E, (38) is the epression for the noniagonal component of the conuctivit tensor. The quantit E is convenientl epresse in terms of the voltage u: u j L. (39) Taking into account the thermomagnetic current ispersion, we obtain the mean-square noise voltage in the form u / L. (4) This noise limits the sensitivit of the temperature sensor base on the longituinal isothermal Nernst-Ettingshausen effect. The input signal is reall inistinguishable from the backgroun of u ranom vibrations if it is less than. It is eas to verif that the ratio of noise voltage (4) to the noise voltage (4) at = is [ / 4], an the term / 4is associate with correction. It is clear that the correction ecreases the noise voltage. Therefore, we shoul consier the contribution from to the temperature sensors while performing measurements in a small temperature range but with sufficientl high accurac. I wish to epress m sincere thanks to Prof. F.M. Hashimzae for helpful iscussion. References [] M.Di. Ventra, Electrical Transport in Nanoscale Sstems, Cambrige Universit Press, N-Y., 8. [] B.M. Askerov, Electron Transport Phenomena in Semiconuctors, Worl Scientific, 994. [3] R.G. Agaeva, J. Phs. C: Soli State Phs. 8, 584(985). [4] R.P. Fenman, Statistical Mechanics. W.A.Benjamin, Inc., Massachusetts, 97. [5] R.G. Agaeva, Fizika XV, nos. 4, 3 (9). [6] M.D. Bloch, Fiz. Tver. Tela 7, 896 (975). [7] Yu.N. Obraztsov, Fiz. Tver. Tela 7, 573 (965). [8] F.M. Hashimzae an Kh.A. Hasanov, Izvestia NANA, seria FMTN, XXVII, nos. 5, 3 (7). 8