We calculate and discuss orbital magnetic susceptibilities in mesoscopic
cylinders made of a normal metal or a semiconductor for different shapes
of the Fermi surfaces and for different circumferences of the cylinders

Lisowski, Mariusz

Zipper, Elżbieta

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Magnetic susceptibilities in mesoscopic cylinders  Author: Mariusz Lisowski, Elżbieta Zipper  Citation style: Lisowski Mariusz, Zipper Elżbieta. (2001). Magnetic susceptibilities in mesoscopic cylinders. "Acta Physica Polonica. B" (2001, no. 2, s. 451-457). M A G N ETIC SU SC EPTIB ILITIES IN M ESO SCO PIC  CYLINDERS*M. L i s o w s k i  a n d  E .  Z i p p e rInstitute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland(Received October 20, 2000)We calculate and discuss orbital magnetic susceptibilities in mesoscopic cylinders made of a normal metal or a semiconductor for different shapes of the Fermi surfaces and for different circumferences of the cylinders.PACS numbers: 72.10.-d, 72.90.+yIt is well known th a t param agnetic or diam agnetic persistent currents, run by coherent electrons, can be induced in mesoscopic systems by a static m agnetic field [1]. Persistent currents in mesoscopic systems have been the subject of many investigations [1-3]. For recent reviews on persistent cur­rents in normal m etals see Ref. [4].In this paper we present some model calculations of orbital magnetic sus­ceptibilities in thinwalled mesoscopic cylinders made of a normal m etal or a semiconductor. We assume th a t electrons interact via the m agnetostatic (current-current) interaction taken here in the self-consistent mean field ap­proximation. As a result the current is calculated in a self-consistent way and one obtains spontaneous current solutions at low tem peratures [3]. Both the current and the susceptibility are enhanced by the presence of the inter­actions. The m agnitude of persistent current and hence of the susceptibility depends also strongly on the coherence of currents from different channels, i.e. on the shape of the Fermi Surface (FS). We show below th a t strong m agnetic response can be obtained for systems with the FS exhibiting some departure from the spherical shape.It is well known [5] th a t mesoscopic systems can exhibit strong orbital m agnetism  for tem perature T  < A q^ tx2^  = T '  ( A q =  h v p / L x is the* Presented a t the XXIV International School of Theoretical Physics “Transport Phenom ena from Q uantum  to  Classical Regimes”, Ustron, Poland, September 25- October 1, 2000.“inverse tim e of flight” or in other words, the quantum  size energy gap for an electron going around the circumference of the cylinder, L x is the circum­ference of the cylinder). The orbital m agnetic susceptibility of the electron gas can be then much larger than  the Landau value and can have either sign [5].In this paper we neglect spin because the orbital quantum  numbers close to  the Fermi energy are large, and the inclusion of spin does not change qual­itatively the considered phenomena. Systems with spin have been analysed in Ref. [6].In the system of cylindrical geometry with the m agnetic field applied along the cylinder axis, the to ta l flux <f> which drives the current I  is the sum of the external flux <f>e and the flux f  j  coming from the current itself41 =  +  <i>I, <f>I =  F I , (1)where £  is the inductance coefficient, for a long cylinder £  =  pqhR 2/ L z , R  is the radius of the cylinder.In normal m etal cylinders of mesoscopic size coherent and normal elec­trons coexist. The persistent current response is given by coherent electrons and it is reduced by the presence of disorder and fluctuations of different kinds. In our recent papers [3,7] we derived the formula for persistent cur­rents as a function of tem perature and magnetic flux in the presence of fluctuations of the to ta l momentum  and disorder,Mz oo/(</>, 7 , T) =to /1  q / l4 1 0 ( m )7Tk.r2^27r lk BT \Acexp Î L X . 27r2fcRT Y  q \  7  +  A 0 ( l - p ) J _1 — exp [ ( L x  , 47r2fcBT \ ]  _ q \ N  ¿ 0 ( 1  - P ) ) _x cos(qkpx ( m ) L x ) sin ( 27rq-j-90(2 )where l / y  is a disorder param eter, related to the mean free path  le by the formula [7],r  /  t \  1 - 4 / 3l e -  a ( l / 7 r 4/3 3.5a In ( —  a ( 3 )P (0 <  p < 1) is the relative fluctuation param eter, p is proportional to the number of normal (non-coherent) electrons in the system; Io(m) = ehkFx (m ) / ( m eL x)-, M z is the number of channels in the kz direction, M r is the number of channels in the y  direction, a is the lattice constant.Persistent currents given by Eq. (2) can be param agnetic or diamagnetic. In this paper we discuss only the param agnetic current solutions.In order to discuss the influence of (j)j on the coherent properties of mesoscopic cylinders we calculate the energy gap at the FS for electrons going around the circumference of the cylinder (and for rf> < ef>o/2):A t E: E n =  Ar r n<i>e n£\I \1 — 2 -+  +  2—!—L90  90( 4 )where =  h2[(am — 4>/4>q)2 +  k 2( m ) R 2)) / (2meR 2). Ap  contains a termmAa = Ar</>o(5)Ad  is the dynamic part of the energy gap. A^  strongly depends on the relative fluctuation p, what is presented in Fig. 1.T [K]Fig. 1. The dynamic energy gap as a function of temperature T  for different values of the relative fluctuaction p.Thus on the top of the quantum  size energy gap there appears in the system the dynamic gap which comes from the m agnetostatic interaction and which increases the coherence in the sample.The to ta l current depends on the correlation of currents from different channels which is related to  the shape of the F S . In the fo llw ing we study the magnetic response of the normal m etal for different shapes of the FS . To simulate different shapes of the 2D FS  [3,8] we used the relationk U = \kFJU + k u ^ (6 )kpx (m) = kpkpz ( m ) kp(1/u)kFz(m) =m  is a positive integer.For u =  2 we get the semicircular FS, for u > 12 we get the nearly rectangular FS, and for 2 <  u < 12 we get the FS interm ediate between semicircular and rectangular, see Fig. 2. For u =  2 the currents from different channels add almost without correlation and the to ta l current is the smallest. The correlation and the m agnitude of the current increases with increasing u, i.e. with increasing curvature of the FS.Fig. 2. Shapes of the Fermi surfaces for different values of parameter u.M ^ 1/u) is introduced here to  simulate the shape of the FS in the ky direction. In order to calculate the current I  in real 3D systems exactly (i.e., where kpx depends as on kpz as on kpy), very fast computers would be necessary.Eqs. (1) and (2) form the two self-consistent equations for the current. Using them  we can calculate the susceptibility x(T ):X =d MdHl.(7)where the m agnetization M  is given by the formulam . L . (8)IT _e 90 rrR2(9)H e is the external magnetic field. Inserting Eqs. (8) and (9) into (7) and making use of Eq. (1) we obtain the final formula for the susceptibility in interacting system:X =  T ^ - >  (10)1 XowhereXo =  AdI(cf>,7,T)d<f)( 11 )We can see from Eq. (10) th a t if yo 1 we get an instability towards an ordered state.In Figs. 3 and 4 the susceptibilities as functions of tem perature T  for different FS (different u ) and different circumferences L x of the cylinder are presented. We see th a t at a given tem perature T  the magnetic response increases with increasing curvature of the FS and with decreasing L x . The crossover tem perature Tc from para- to ferromagnetic sta te  also increases with increasing curvature of the FS (increasing u) and decreasing circumfer­ence L x .T [K]Fig. 3. Susceptibility y as a function of temperature T  for different shapes of the Fermi surfaces (different u) and le = 2 pm, M r = 120.Now let us compare the orbital magnetism of a mesoscopic system in­teracting via the m agnetostatic interaction with the magnetism  of a set of spins coupled by the exchange interaction. Let us assume we have N s spins. In a param agnetic statet [K]Fig. 4. Susceptibility y as a function of temperature T  for different circumferences Lx of the cylinder and le = 2 pm, M r = 120.where M e  is the m agnetisation, andXl kBT  ~  T  ’ ^p i  =  M e / N s is the m agnetic moment.In the case of a spin ferromagnet, according to Weiss theory, there is a local field coming from spin interactions A M e , and henceM H =  X i(He +  AM H) = x H e , (14)where A =  2 J / N s , J  is the interaction constant.From Eq. (14) we find the formula for the magnetic susceptibilityX =  1 , (15)1 -  W iIf iVs 7S> 1 then A is very small. According to  Eq. (13)g  _  CX T - C X  T  -  Tew  ’where Tew  is the Curie-W eiss tem perature, i.e. the tem perature at which a phase transition from the param agnetic to the ferromagnetic sta te  occurs.Summarizing, in the presented paper we have discussed the orbital mag­netic susceptibility in a mesoscopic cylinder made of a normal m etal or asemiconductor. We have investigated the dependence of the m agnetic re­sponse on the shapes of the Fermi surfaces and on the circumferences of the cylinders. The crossover tem perature Tc from para- to  ferromagnetic state  increases with increasing curvature of the FS (increasing u) and decreasing circumference L x .The formula for the orbital m agnetic susceptibility can be compared with a susceptibility of a set of interacting spins. In these two cases we observea different behavior in the external magnetic field. We can also notice th a tthe “orbital ferromagnetism” is a phenomenon opposite to  superconductivity where H e +  M h  =  0 and hence x  =  — 1, *-e- M l expulsion of the magnetic flux from the sample occurs.Work supported by the Polish S tate Com m ittee for Scientific Research (KBN) G rant 5P03B 033 20.REFERENCES[1] H.-F. Cheung, Y. Gefen, E.K. Riedel, IBM J. Res. Develop. 32, 359 (1988);H.-F. Cheung, E.K. Riedel, W.-H. Shih, Phys. Rev. B37, 6050 (1988).[2] M. Stebelski, M. Szopa, E. Zipper, Z. Phys. B103, 79 (1997).[3] M. Lisowski, E. Zipper, M. Stebelski, Phys. Rev. B59, 8305 (1999).[4] B.L. Altshuler, Nanostructures and Mesoscopic Systems, eds. W.P. Kirk and M.A. Reed, Academic, New York 1992; U. Eckern, P. Schwab, Adv. Phys. 44, 387 (1995); P. Mohanty, Arm. Phys. 8, 549 (1999).[5] B. Shapiro, Physica A200, 498 (1993).[6] E. Zipper, M. Szopa, Acta Phys. Pol. A87, 79 (1995).[7] M. Lisowski, E. Zipper, J. Magn. Magn. Mat. 189, 225 (1998).[8] A.P. Cracknell, K.C. Wong, The Fermi Surface, Oxford, Clarendon Press 1973.

Magnetic susceptibilities in mesoscopic cylinders

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Magnetic susceptibilities in mesoscopic cylinders

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