We study the problem of the upper critical field (Hc2 ) for tight-binding electrons in a phase with stripes. Carrying out calculations for finite systems we analyze the influence of the external field in the commensurable
and incommensurable case on an equal footing. The upper critical field is discussed for anisotropic intersite pairing as a function of the width of stripe. We show that the upper critical field increases with a decrease of
the width of stripe. This effect is of particular importance close to the superconducting transition temperature

Maśka, Maciej M.

Mierzejewski, Marcin

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Upper critical field in a stripe-phase  Author: Marcin Mierzejewski, Maciej M. Maśka  Citation style: Mierzejewski Marcin, Maśka Maciej M. (2001). Upper critical field in a stripe-phase. "Acta Physica Polonica B" (Vol. 32, no. 2 (2001), s. 523-527).   U P P E R  CRITICA L FIELD IN A S T R IP E -P H A S E *M a r c in  M i e r z e j e w s k i  a n d  M a c i e j  M . M a s k aDepartment of Theoretical Physics Institute of Physics, University of Silesia Uniwersytecka 4, 40-007 Katowice, Poland(Received November I, 2000)We study the problem of the upper critical field (Hc2 ) for tight-binding electrons in a phase with stripes. Carrying out calculations for finite sys­tems we analyze the influence of the external field in the commensurable and incommensurable case on an equal footing. The upper critical field is discussed for anisotropic intersite pairing as a function of the width of stripe. We show that the upper critical field increases with a decrease of the width of stripe. This effect is of particular importance close to the superconducting transition temperature.PACS numbers: 74.25.Ha, 74.60.Ec, 71.70.Di1. In tro d u c tio nDespite enormous efforts the mechanism, which is responsible for the form ation of high-tem perature superconductivity, remains an open problem. Moreover, neither theoretical nor experimental results allow for a complete and unique description of the norm al-state. Above the superconducting tran ­sition tem perature one finds two m ajor phases: underdoped, and overdoped. The natural question which arises concerns the distribution of holes which enter the copper-oxvgen planes in the doping process. There is a convincing argum entation speaking in favor of spatially inhomogeneous distribution of holes. It results in a stripe-phase which consists of antiferrom agnetic do­mains separated by hole-reach domain walls [1-7]. In the present paper we investigate influence of the spatial distribution of carriers upon the upper* 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.critical field, H c2, for anisotropic superconductivity. W ithin a simple mean- field approach we dem onstrate th a t inhomogeneous distribution of holes af­fects the structure of Landau levels and leads to  a serious modification of the upper critical field.In order to  simulate the presence of a stripe-phase we investigate a two­dimensional rectangular-shape clusters th a t consist of iVj| lattice sites along the domain wall and N±  sites in the perpendicular direction, with N±  -C iVj|. Since the length of a stripe is assumed to  be infinite we make use of peri­odic boundary conditions in the parallel direction. The antiferromagnetic domains between stripes are insulating and thus we take fixed boundary conditions in the perpendicular direction.2. T h e  m odelIn order to  determine H c2 for anisotropic superconductivity we consider the nearest-neighbor pairing and introduce a uniform magnetic field, which is perpendicular to  the cluster. The Hamiltonian reads11 Y  %  4 t cj<t +  9Tb H z Y  ( 4 t cd  -  4-  ( 4 t cj-j-A b' +  c4 ci t^ * j j  • w( h i )Here, 4(7 icia) creates (annihilates) an electron with spin o  on site i; g and ¡ib  are the gvromagnetic ratio and the Bohr m agneton, respectively. V  stands for the m agnitude of the nearest-neighbor a ttraction  and A  is the vector potential corresponding to  the external magnetic field H .  We introduce anisotropic superconducting order param eterA i j  =  {ciWjj ~ ci i cj l ) ! (2 )which, in general, can be site-dependent. According to the Peierls substi­tu tion [8] the original hopping integral tjj is multiplied by a phase factor, which accounts for coupling of electrons to  the magnetic field(  R i  \t{j (A )  =  iex pl ] A ' dl\  Ri f( 3 )In order to  evaluate the upper critical field we make use of an unitary  trans­form ation th a t diagonalizes the kinetic part of the Hamiltonian. Then, in the new basis, equations of m otion for the Green function allow one to ob­tain  the upper critical field. The details of this approach can be found in Refs. [9,10].3. D iscussionIn order to visualize the role of magnetic field in the norm al-state we have calculated the resulting current distribution:Figures 1 and 2 show results obtained for a square cluster with N ± =  iVj|, which can simulate a homogeneous phase, and for N ±  -C iVj| th a t is relevant to  the stripe-phase. In the stripe-phase the geometry of the system does not allow for a formation of standard  Landau orbits. Then, one can expect th a t the diam agnetic pair-breaking mechanism is reduced and superconductivity can survive in the presence of much stronger m agnetic field than  in the case of a homogeneous phase.■ • ' *■ ^ ‘ '■ • * * * ¿y & ■ •Y- N/fi-Zy ¿JT * YX• * ’ • ^ * <]— <]— ft ft • « ■> ^i * • cf - 4 ' '  A 'A UV , - ' t ~ if4  i  A x' 4 ^  ^ <- «■ i  y y  i  fi  i  *  A T  & ■ ■ ■r / i i  * * O' 1 * i ^  ^  '  4  ^ q+7 ' 4 * if Zr a <f- ^ jj o- * *  ^ ^<] h * d ■ - h e f \i ■» Y " ^ ^ ^  ^  l< ft. • * ^  Y ? ^ ^ ? ft ' ’ *  * ft ■ * q  Y ^" Vl Vl ' f l  r 4 t fyy ' h t ■ y■qii 4 * -<> Vl Vl * A * * N f3 . , f glj ^ * -* * * ^  ■ ■ ■ - fl X7 » f ^ ?i  » - A *  •  -» * V j  Vl  -  t  -> < t  J 7i, - ■ Vl  ^ * > - 'AYYV+7■ * • • *1  (>-(> M ■*? -* * 3^ • '• • *  * *  ^  -A  » + 7  "£ >  '  -A7 *• * •• * » -6. -t> -p. ~c> J7 .Fig. 1. Current distribution originating from external magnetic field calculated for a 20 x 20 cluster with fixed boundary conditions. ea2Hz/(hc) = 0.1 was used.. . f t * * »  + ++ ♦ ► « t ♦ ♦ t  t  ♦ ♦ - - ° * t  ♦ ♦ ♦ ♦ » r f ♦ ♦ ♦ ♦ * ♦ * »  »Fig. 2. The same as above, but for 100x5 cluster.This observation can be confirmed by the calculation of the upper critical field for clusters of different geometry (see Fig. 3).Transition form the stripe to  homogeneous phase has been simulated by changing the ratio iVj_ / iVj | . Dependentlv on the geometry of cluster we have adjusted the m agnitude of the pairing potential to obtain the same Tc inkBT/tFig. 3. Comparison of the upper critical field for a homogeneous system (continuous line) and systems with stripes of different width (3, 5 and 10 lattice sites).the absence of m agnetic field. One can see, th a t the upper critical field in­creases when the width of the cluster decreases. This effect is of particular im portance for weak m agnetic fields and leads to  a dram atic change of the slope, dH c2/d T ,  calculated at T  = Tc. These results can, to  some extend, account for the tem perature dependence of the upper critical field observedin the copper-oxide superconductors. One should keep in mind th a t evenin overdoped and underdoped samples the measurements of the upper crit­ical field are restricted to  tem peratures close to  Tc [11,12]. In this region even very strong magnetic fields hardly affect the superconducting transition tem perature, leading to a large slope of H c2(T).This work has been supported by the Polish S tate Com m ittee for Scien­tific Research (KBN), G rant No. 2P03B 018 19.REFERENCES[1] S.-W. Cheong et al, Phys. Rev. Lett. 67, 1791 (1991).[2] T.E. Mason, G. Aeppli, H.A. Mook, Phys. Rev. Lett. 68, 1414 (1992).[3] T.R. Thurston et al, Phys. Rev. B46, 9128 (1992).[4] K. Yamada et al, Phys. Rev. Lett. 75, 1626 (1995).[5] S.N. Hayden et al, Phys. Rev. Lett. 76, 1344 (1996).[6] U. Low, V.J. Emery, K. Fabricius, S.A. Kivelson, Phys. Rev. Lett. 72, 1918(1994).[7] J. Zaanen, O. Gunnarsson, Phys. Rev. B40, 7391 (1989).[8] R.E. Peierls, Z. Phys. 80, 763 (1933); J.M. Luttinger, Phys. Rev. 84, 814 (1951).[9] M. Mierzejewski, M.M. Maśka, Phys. Rev. B60, 6300 (1999).[10] M.M. Maśka, M. Mierzejewski, cond-mat/0005142.[11] H.H. Wen, X.H. Chen, W.L. Yang, Z.X. Zhao, Phys. Rev. Lett. 85, 2805 (2000).[12] V.B. Geshkenbein, L.B. Ioffe, A.J. Millis, Phys. Rev. Lett. 80, 5778 (1999).

Upper critical field in a stripe-phase

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Upper critical field in a stripe-phase

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