Physica C 196 (1992) 43-47 North-Holland PHYSICA

Mean-field superconductivity in a strong magnetic field

M.R. N o r m a n Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

H. Akera ~ and A.H. MacDona ld Department of Physics, Indiana University, Bloomington, IN 47405, USA

Received 3 February 1992

We have solved the microscopic mean-field equations for a BCS superconductor in a strong magnetic field. Calculations were performed for fields both above and below the semiclassical upper critical field. In the low-field regime we find small quantum corrections to the results of semiclassical theory, whereas in the reentrant strong-field regime we find that the vortex lattice state changes qualitatively.

Thirty-five years ago Bardeen, Cooper and Schrieffer (BCS) [ 1 ] derived their microscopic the- ory of the homogeneous superconducting state. Soon after the publication of the BCS theory, Gor 'kov [ 2 ] developed a description of the inhomogeneous su- perconducting state of type-II superconductors based on a generalization of the BCS mean-field theory. Gor 'kov also showed that for temperatures near Tc the phenomenological Ginzburg-Landau equations, used by Abrikosov [3] to predict the vortex-lattice state of a type-II superconductor in a magnetic field, could be derived from microscopic theory. This de- scription of the inhomogeneous superconducting state, though, is based on a semiclassical approxi- mation for the magnetic field. It is not equivalent to solving the full mean-field equations. Later, de Gennes [4] detailed the generalization of the BCS theory required to treat inhomogeneous states in non- zero magnetic fields. The usual gap equations are re- placed by a set of integro-differential equations, known as the Bogoliubov-de Gennes (BdG) equa- tions. As we discuss below, the equations for the in- homogeneous case are more complicated than the homogeneous-state gap equations, principally be-

On leave from Institute for Solid State Physics, University of Tokyo, Tokyo 106, Japan. Current address: Faculty of Engi- neering, Hokkaido University, Sapporo 060, Japan.

cause of the nature of extended electronic states in a magnetic field. The BdG equations have been used principally to describe bound states in an isolated vortex core [5] where these fundamental compli- cations can be avoided. Recently, several groups have shown how to solve the BdG equations for the case of pairing of electrons within one Landau level [6,7 ]. Although this work clarified the nature of pairing in a strong magnetic field, the case of pairing within a single Landau level is in a quantum regime where no experimental data yet exist. In this paper, we go be- yond the methods used in those calculations to treat the case of pairing in more than one Landau level, thus providing the first fully microscopic picture of the inhomogeneous superconducting state. We are able to reach magnetic fields weak enough that quan- tum corrections to Gor 'kov theory become small and can thus treat the entire field range from the extreme quantum regime will into the semiclassical regime.

The effect of a magnetic field on electronic states in quantum mechanics appears in the vector poten- tial (VXA=B) . For a non-zero average magnetic field A is ~ L where L is the system size and, unless the electronic states are localized, even a weak mag- netic field cannot be treated perturbatively. A quan- tum treatment of the mixed superconducting state must be performed in a basis where the average mag- netic field (Bo) is accounted for exactly; it is this re-

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44 M.R. Norman ctal. /Mean-field superconductivity

quirement which fundamentally changes the nature of the mean-field equations. To describe the vortex- lattice state it is convenient to use a Landau gauge (A= (0, Bc~v, 0) ) in which the kinetic energy eigen- states are

1 i ~,.,v_~ ( x - X ) ~ - o~,,,(~)= ~ e o,, , (1)

where 0x(x) = (2xN! \ : r d ) ~/~'e - <'/ett,~(x) and H:~ is a Hermite polynomial of order N (12=fw/eB) . The BdG equations are obtained by making a generalized Hartree-Fock factorization of the equation of mo- tion for electron creation and annihilation operators. The resulting equation can be diagonalized by a Bo- goliubov transformation and the Bogoliubov coef- ficients relating quasiparticle and electron creation/ annihilation operators and the quasiparticle energies are determined by solving the BdG equations. To ex- ploit the translational symmetry of the vortex-lattice state we first form magnetic Bloch states from the

~ N "l :

O,,', aN/~7 ~ " ' ""e'~'2/4"~ = e • v x . ,,,t~-+ .... ( r ) (2 )

( a , a , . - = r f f 2 and the vor tex- la t t ice aspect rat io is R = 2a,./a,,; R = 1 for a square lat t ice and R = 3 ~ / ~ for a triangular lattice). In this representation, the pairs are formed from combinations of OxkO,V-k, and the BdG equations for singlet pairing become

( {x--E~')u!4,k + Y] F,x.,,v!6k = 0 . (3) , 1 /

- -~,~ ' - -Ek)t . .~k+ ~ F * "~' = 0 (4) M

Here u ~ is the coefficient of ¢,~,, and v~,, is the coef- f * ficient o ¢,~._~ in the real-space Bogoliubov ampli-

tudes u ( r ) and v(r). The sums over Landau levels are restricted to states

whose kinetic energies are within ooD of the chemical potential level. For a a-function attractive interac- tion the off-diagonal self-energies are given by

l:~xat - 2 h ~ ¢ . "k" D ' ~ l X A - 2 Y , z , , + , , , t ~ , , , ( 5 )

/

wi th

z i ( k ) = Y', ei~-~""'e -i~"~/2Z~(2k~.12+2ta~) , (6) t

Z, = g6 , f 7 i

. [ I

j! ( A + M - j )! N!M! ~ : 1)? ~'* = 2,,,+ , , ]

( - 1 ) ~ '- x (8)

. . . . o ( j - m ) ! ( 3 : + m - j ) ! ( ? ~ / - m ) ! m ! "

In these equations, ~., is the single-particle energy of a Landau orbital with index N measured from the chemical potential, the quasiparticle energy E J,' is the #th positive eigenvalue of the BdG equations, ~o~ is the cyclotron frequency (h~o~. = e B / m c ) , and 2 is the BCS coupling constant (2he4= I ' / ( 2 M 2 ) where I ' is the strength of the attractive interaction ).

The sum over j in eq. ( 5 ) is over the possible par- titionings of the total quantized kinetic energy of the pair, h o ) ~ ( N + M + l ) , into contributions from the pair center of mass motion, ho)c(j+ ½ ), and the pair relative motion, h { o c ( N + . ' f f - / + ~ ). (DI '~) e is the probability that a pair of electrons in Landau levels N and M will have center-of-mass kinetic energy ho&.(j+ { ). J, is the amplitude for electron pairing with center-of-mass kinetic energy hoJ~(j+~) and these parameters which completely speci~ the mean- field solution must be determined self-consistently:

4rd /a~ • ,,,Y D'" Z

f iz x Y. ( 1 - . k, -/~,'~ )uf<., '*" . , t . , ( 9 ) tt

where f is the Fermi function evaluated at the ap- propriate quasiparticle energy. When the Zeeman energy is included, this is

Ef,~ = El, - g*ho)~ a~ 4 , ( I 0 )

where a = +_ 1 is the spin index and g* the effective g-factor ( g m * / m ) . The diagonal order parameter in a position representation is related to the J, by

J ( r ) - (x" 2L"l)L"e 4rd 2

x Y. ( - l ) 'J , ~ e'~'-'/-'0,., e,., ( . / : 2 r ) . (11 ) f !

The free energy of the solution is

F = ~, ~,v~N~,~ + Ev - "FS, ( 1 2 )

M.R. Norman et al. / Mean-field superconductivity 45

where

E P = - & h o g c ~ 1 4 1 2 , (13)

is the pairing energy, NN~ the occupation number of the Na Landau level, and S is the entropy.

We note that the above equations are for the two- dimensional (2D) case; one can include three-di- mensional (3D) effects by adding the z-axis disper- sion to the single-particle energies and performing the appropriate sums over k~ in the various equa- tions [ 8 ]. In addition these equations do not include the effect of screening currents on the magnetic flux density (B=-Bo) and hence are appropriate over a broad range of magnetic fields only for large K su- perconductors. The effect of screening currents ap- pears in the upper and lower diagonal blocks of the BdG equations which are no longer diagonal in N [8].

In fig. 1 we show a microscopically calculated Hc2 phase line ( j=0) . This curve was calculated from the linearized version of the above equations [9], which are diagonal in j, using 2=0.54, tnD/ka= 500 K, Er/kB= 1000 K and m*=2m; these parameters provide a crude model for a copper oxide supercon- ductor. (The generalj linearized equations have been derived independently by Rajagopal and Ryan [ 10 ] and by us [ 11 ].) The phase boundary line at weak fields looks similar to what is calculated using Gor'kov theory, except that magnetic oscillations in H~2 show up at stronger fields. The magnetic oscil- lations are due to the thermal potential jumping from

T u

2 0 0

1 5 0

1 0 0 -

5 0 -

i i

310 i 60 9 0

T (K)

Fig. 1. H¢2 (T) vs. T (K) for the case 2=0.54, Ev= 1000 K, tOo= 500 K, and m*=2.

one Landau level to the next as the field varies. (In realistic systems, the magnetooscillations will be fur- ther damped by 3D dispersion, impurity broaden- ing, and spin-splitting effects [ 12 ]. ) At strong fields and low temperatures, a new regime is reached where To2 increases with H. This regime was first studied by Tesanovic et al. [13]. (To2 in this reentrant re- gime will also be reduced by 3D dispersion, impurity broadening, and spin-splitting effects.) The super- conductivity in the reentrant regime is driven pri- marily by electron pairs formed within a single Lan- dau level. Hc2 increases with B because the Landau level degeneracy increases and because the proba- bility of finding a pair of electrons with the mini- mum quantized center-of-mass kinetic energy in- creases as the Landau level index at the Fermi level decreases. In the limit where pairing occurs between up-spin electrons with Landau index N and down- spin electrons with Landau index M, Hc2 is identical for the j = 0 and j = N + M channels [ 6 ]. Surprisingly, the condensation free energy associated with the j = N + M channel is always larger than for the j = 0 channel. The Abrikosov ( j=0 ) solution is obtained for purely intra-Landau level pairing only in the N= M = 0 case. Because of this qualitative change in the purely intra-Landau level pairing limit, we ex- pect that the order parameter in the reentrant regime will differ from that in the semiclassical regime, even when Landau level mixing becomes important. The crossover from the semiclassical regime to the reen- trant regime occurs at a field strength (Hc2(0)) somewhat weaker [ 11 ] than the semiclassical H¢2 at zero temperature in Gor'kov theory.

In the calculations described below we have set g to zero and present results only for the case where the chemical potential lines up with one of the Lan- dau levels whose index we denote by N,. With these restrictions the BCS model is defined by Nmi n - N u- Nma x and 2 where Nmi n and Nmax are the minimum and maximum Landau indices for levels participating in the pairing. In fig. 2, we show the energy (at zero temperature) versus aspect ratio, R, of the 4-8-12 case for three different values of 2: 0.54, 1.0, and 2.0. The 2=0.54 case corresponds to a field of about 175 T in fig. 1, which is about a factor of four larger than He2 (0) and thus is in the reentrant regime. For 2= 1.0, the field is just below H~2(0), and for 2=2.0, it is well below He2(0). In all three

4 6 M.R. Norman et al, ~Mean-field superconductivity

- 0 . 4 ~ L ~ • j . . . . .

F- m

[ (a) - - z ] ILl

d, !

i - ~ - ' . . . . ( c ) i

i 0

r t t !

1 2 3 4 5 6 R

Fig. 2. Energy vs. aspects ratio (R) for the 4-8-12 case with (a) 2=0.54, (b) 2= 1.0, and (c) 2=2.0. Case (a) is in the reentrant region, cases (b) and (c) are in the semiclassical region. (The energy for case (c) has been divided by 2 for the purposes of the graph. )

cases, the order paramete r is pr imar i ly j = 0 in char- acter. We contrast this with the case of pair ing purely within the N = 8 Landau level (the 8 - 8 - 8 case). As discussed above the solut ion is then j = 16 in char- acter. Despi te the recovery of the semiclassically ex- pected j = 0 form of the pairing, we see from the fig- ure that for the 2 = 0 . 5 4 case there appears to be no preferred vortex-lat t ice arrangement . This result is in agreement with our expectat ion that the vortex lattice energetics in the reentrant regime are gener- ally quite different from the energetics in the semi- classical regime. For the 2 = 1.0 case, a global mini- mum in the condensat ion energy occurs at a t r iangular aspect ratio, but there is still structure in the energy curve at larger aspect ratio. For the 2 = 2.0 case, the dependence of energy on aspect ratio has essentially recovered the smooth dependence of G i n z b u r g - L a n d a u theory [ 14 ].

In fig. 3, we trace the evolut ion of the order pa- rameter for the 2 = 1.0 case as a function of the field. For the 5 -10 -15 case ( H = 0 . 7 5 H c 2 ( 0 ) ) there is a dis t inct ive star shape to the vortex cores. Similar pat terns for the magnet iza t ion have been calculated from Gor ' kov theory [15] , but not, to our knowl- edge, for the order pa ramete r itself. As the field de- creases, the vortex core becomes more symmetric . (The same effect can be seen at f ixed magnet ic field by increasing 2. ) The dis tor t ions in the vortex-lat t ice are due to non-zero contributions to zl(r) from terms with j - - 6 n which can be interpreted as being due to

} o i

0

.O %

Fig. 3. Order parameter for (a) the 5-10-15 (b) the 8-15-22 and (c) 10-20-30 cases with 2= 1.0. The ratio of the field to He2 (0) for these cases are 0.75, 0.51, and 0.38, respectively. The box sides are a, (vertical) and a, (horizontal).

core-core interactions. (Because of the tr iangular symmetry only channels with j differing by 6 are mixed by the gap equations. ) The core distort ion can also be reduced by increasing the tempera ture to- wards Tcz as shown for the 4 - 8 - 1 2 case in fig. 4.

We have also examined the quasipart icle energies and local density-of-states for all the above cases. For pair ing in a single Landau level, we find that zeros occur in E(k), but in most other cases we have looked at we find a non-zero gap. This gap is quite small in the reentrant regime ( << fi~oc), but usually fairly large in the semiclassical region (~0 .1 -1ho)c ) . This in- dicates that type-II superconductors are not intrin- sically gapless in the clean limit. For the local den- sity-of-states, we find dis tor t ions s imilar to those of the order parameter . We note that s tar-shaped pal- terns in the local density-of-states have been ob- served by STM experiments [ 16 ], although these ex- per iments were at weak fields and may have a

M.R. Norman et al. I Mean-field superconductivity 47

I b

Fig. 4. Order parameter for the 4-8-12 case (2 = 1.0 ) for temper- atures (T/Tc2) of(a) 0.00 (b) 0.68 and (c) 0.95.

different or igin [ 17 ]. We also f ind that large differ- ences be tween posi t ive bias an d negat ive bias can oc- cur at s t rong fields because o f par t i c le -ho le asymmetry .

In conclus ion , we have der ived a s imple form for the microscopic q u a n t u m mean-f ie ld equat ions of the i nhomogeneous supe rconduc t ing state. We f ind that i m p o r t a n t quan t i t a t i ve correct ions to semiclassical theories may become impor t an t , especially in lay-

ered superconductors , at fields nea r He2 (0) . In the r een t ran t regime which occurs at s t ronger fields the mixed state may differ qua l i ta t ive ly f rom expecta-

t ions based on semiclassical theory.

Acknowledgements

We thank K u r t Scharnberg for i n fo rm ing us of the difference o f ou r results f rom those of ref. [ 15 ]. This

work was suppor ted in par t by the DOE, Basic En- ergy Sciences, u n d e r Cont rac t No. W-31-109-ENG- 38, a n d in part by the Midwest Superconduc t iv i ty C o n s o r t i u m through D O E grant no. DE-F G-02 - 90ER45427.

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