We study the influence of a strong magnetic field on a superconducting
state of electron gas in a two-dimensional square lattice. The Harper
equation is extended in order to include pairing interactions between electrons.
We examine the effects of superconductivity with different pairing
symmetries on the Hofstadter energy spectra

Maśka, Maciej M.

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Superconductivity in the presence of magnetic field  Author: Maciej M. Maśka  Citation style: Maśka Maciej M. (2001). Superconductivity in the presence of magnetic field. "Acta Physica Polonica. B" (2001, no. 2, s. 535-541) SU PE R C O N D U C T IV IT Y  IN T H E  P R E S E N C E  OF M A G N ETIC FIELD*M a c i e j  M . M a s k aInstitute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland(Received October 19, 2000)We study the influence of a strong magnetic field on a superconduct­ing state of electron gas in a two-dimensional square lattice. The Harper equation is extended in order to include pairing interactions between elec­trons. We examine the effects of superconductivity with different pairing symmetries on the Hofstadter energy spectra.PACS numbers: 7!.20..\In. 74.25.Ha, 71.70.Di1. In tro d u c tio nThe energy spectrum  of a system of electrons on a square lattice in a perpendicular uniform magnetic field exhibits m ultifractal properties and the band spectrum  for rational values of the magnetic flux is known as the “H ofstadter butterfly” [1]. This spectrum  can be modified by e.g., the inclusion of Coulomb interaction [2] or the antiferrom agnetic correlations [3]. The aim of this paper is to  study the effect of pairing interactions between electrons on the the H ofstadter energy spectra.We s ta rt with a mean-field Hamiltonian which describes electrons in a 2D square lattice in a perpendicular m agnetic field,H  =  ^  (A ) c\aCja — V  ^  +  h.c.^ . (1)* 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.In the pairing term  we put i = j  for on-site pairing, and ( i , j )  — near­est neighbor sites for inter-site pairing. The magnetic field appears in the hopping amplitudes,where t  is the nearest neighbor hopping am plitude in zero field and V x A ( r ) =  B(r) .  We consider an uniform magnetic field in ^-directioni.e., B  = (0,0, B ).  Using the Landau gauge the corresponding vector po­tential is given by A ( r ) =  (0 ,B x ,0 ) .  We neglect the Zeeman term .In the momentum  space the Hamiltonian (1) takes the formp /q  = (f>/(f)o is the number of flux quanta per plaquette ((j) = B a 2, (j)Q = hc/e). The energy gap is given byIntroducing multicom ponent Nambu notation the Hamiltonian (2) can be w ritten aswhere the prime denotes sum m ation over the reduced Brillouin zone:H  =  -  / ^  (2  cos k ;i<i ,Tek ..T +  e * * * 4 _ fl)(Tcfc)(T +  é kx4+g,<xcfc,<x)k,cr(2)fcwherefc'fc7T nandc k - ( q - l ) g , p  c - f c 4 ;  r  f c  • c - f c + 2 g 4 ;  • • • ;  c — f e - | - ( q r — l ) p , 4 _ . ) •  ( 3 )The Hamiltonian m atrix  n k  has a block structure:n k ( T k A-k“ I*the matrices T k and A k are:(  M.oQ —i k x Ml00T k = —t0 ikx M20 0 0 —0 0M n =  2 cos (ky + n g) , 9 = \90M,q - 2 &A h  =(  A k 00 A k g  \  0e-tkx M ç_ i /2irpq ’ 0 \o0 :■/ nr/ /( 4 )2. D en sity  o f s ta te sIn the case of noninteracting electrons (V  =  0) the spin-up and spin-down segments decouple and the eigenproblem of the Hamiltonian (3) reduces to the original H arper’s equation [4] in a form derived by Hasegawa et al. [5] The solutions constitute the famous “H ofstadter butterfly” [1] presented in Fig. 1. The calculations were carried out for a finite-size 60 x 60 square lattice with periodic boundary conditions.W hen turning on the pairing interaction the subbands become broader than  in the noninteracting case. We have numerically solved the eigen­problem of the Hamiltonian (3) for s-wave ( A k =  const) and d-wave ( A k =  A  (coskx — cosky )) sym m etry of the energy gap. The results are presented in Figs. 2 and 3.Due to  the fact th a t T _ k f  T k the subband containing the Fermi level (ef =  0 for half filled band) is split into two subbands only for certain values of the magnetic field. Therefore there is a finite density of states at the Fermi level, even in the case of s-wave pairing.Fig. 1. Density of states for noninteracting electrons in a perpendicular magnetic field. The horizontal variable is the qnasiparticlo energy, and the vertical variable is the flnx through a lattice cell divided by the flnx quantum.e^-. 0  h------------------1 1 1------------------1­-5.0-------------2.5-------------0.0 2.5------------- 5.0EFig. 3. Density of states for d-wave state.3. B C S eq u a tio n sWe have also carried out a sclfeonsistcnt cluster calculations for the or­der param eter in the presence of magnetic field. The fractal structure of the “H ofstadtcr butterfly” results in a very irregular dependence of A  on the external magnetic field, presented in Fig. 4. Moreover, when the de-Anom inator q (<f>/<f>o = p/<l) i® incommensurable with the size of the cluster additional, finite size errors appears (marked with horizontal bars). In spite of these irregularities one can note th a t the m axima of A ( f )  appear at cvcn- denom inator values of q in p/q, e.g., for <f> =  + 0; |/>0; • • • • Also forsmall odd values of q the m axima appear, but they are much less noticeable. The oscillatory behavior can be explained within a framework of a composite fermion picture, but it is out of the scope of this paper.The Hofstadter spectrum  corresponding to the sclfconsistcntly calculated A  is presented in Fig. 5.EFig. 5. Density of states for s-wavc state (A is calculated sclfconsistcntly for V = 1).4. S u m m aryThe proposed generalization of H arper’s equations allows to study the impact of the external m agnetic field on superconductivity with different symmetries. We have found oscillatory behavior of the energy gap as a func­tion of the external field. Here, we have presented results for the zero tem perature ease but relevant calculations were carried out also for finite tem perature. The values of the critical tem perature (not presented here) are the same as the results obtained from the use of much more CPU tim e­consuming real-space cluster calculations [6,7].This work has been supported by the Polish S tate Com m ittee for Scien­tific Research (KBN), Grant No 2PP03B 018 19.REFERENCES(1] D. R. Hofstadter, Phys. Rev. B 1 4 , 2239 (1976).(2] A. Barelli, J. Bellissard, P. Jacquod, L. Shepelyansky, Phys. Rev. Lett. 77, 4752 (1996); S.N. Evangelou, D.E. Katsanos, Phys. Rev. B 5 6 , 12797 (1997).(3] Seung-Pyo Bong, Sung-Bo Suck Salk, Phys. Rev. B 6 0 , 9550 (1999); Byeonjin Doh, Sung-Bo Suck Salk, Phys. Rev. B 5 7 , 1312 (1998).(4] RG. Barper, Proc. R. Soc. Lond. A 6 8 , 874 (1955).(5] Y. Basegawa, P. Lederer, T.M . Rice, P.B. W iegm ann, Phys. Rev. Lett. 63, 907 (1989).(6] M. Mierzejewski, M.M. Maśka, Phys. Rev. B 6 0 , 6300 (1999).(7] M.M. Maśka, M. Mierzejewski, cond-mat/0005142.

Superconductivity in the presence of magnetic field

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Superconductivity in the presence of magnetic field

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