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Order parameter induced phase shift in the dHvA oscillations in type-II superconductors
Solid St&e Communications, Vol. 101, No. 8, pp. 621-625. 1997
Pergamon
@ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved oow1098/97 f17.00+.00
PII:SOO38-1098(96)00610-2
ORDER PARAMETER
INDUCED
PHASE SHIFT IN THE DHVA OSCILLATIONS SUPERCONDUCTORS
IN TYPE-II
T. Maniv, a A.Y. Rom, a I.D. Vagner b.c and P. Wyder b*c a Department of Chemistry and Solid State Institute, Technion, Haifa 32000, Israel b Grenoble High Magnetic Field Laboratory, Max-Planck-Institute fur Festkorperforschung, Germany c Center National de la Recherche Scientific, 25 Avenue des Martyres. F-38042. Grenoble, Cedex 9. France (Received 22 September 1996; accepted 30 September 1996 by A. H. MacDonald)
In a pure type-II superconductor under a strong magnetic field every Landau level around the Fermi energy, which is significantly broadened by the pair potential at low temperatures, splits into two major magnetic subbands. Such a splitting reflects the fact that the magnetic flux threading a unit cell in the Abrikosov lattice is one half electronic flux quantum. This effect is predicted to be observed as a rapid 180” phase shift in the de Haas-van Alphen(dHvA) oscillations just below Hcz, and is expected to be sensitive to disorder and vortex lines motion. @ 1997 Elsevier Science Ltd. All rights reserved
The de Haas-van Alphen (dHvA) effect in the vortex state of type-11 superconductors has recently become a rapidly growing field of research; clear magnetization oscillations in the vortex state have been reported for half a dosen type-11 superconucting (SC) materials of different types, such as the old high T,, A - 15 compounds,fiSi [l] and Nb+Sn [2]. the new HTSC YBCO [3], the (ET)zCu(NCS)z organic superconductor [4], the borocarbide superconductor YNi2B2C [5], and the layered dichalcogenide NbSe2 [6]. In all these experiments, the only observable effect of the SC order parameter on the measured magnetization oscillations, so far, has been an additional damping of the signal in the vortex state. Several theoretical attempts have been made to account quantitatively for this attenuation [7-lo]. As usual1 in spectroscopy, however, a quantitative analysis of the signal intensity is difficult since many factors, not necessarily intrinsic, may influence it. In this communication we argue that the dHvA oscillations in the vortex state contain a significant spectroscopical information, which should be observable in sufficienly pure SC material. We will show in what follows that at a certain magnetic field. just below Hc2, the dHvA amplitude should change sign as a result of strong,1 80” out of phase oscillations in the SC condensation energy with respect to the normal electrons oscillations [I I]. This remarkable effect is shown to be
related to the opening of a deep hole in the quasi particle (QP) density of states (DOS) at the highest occupied Landau band by the increasing SC order parameter below Hr2. Since the considered effect is associated with a fundamental property of the QP DOS at high magnetic fields, we shall ignore, for the sake of simplicity, irrelevant features like Zeeman spin splitting and energy dispersion along the magnetic field direction ( the z-axis). In an isotropic 3D system of normal electrons, for example, where the extremal orbit corresponds to the momentum component k, = 0, the well known Lifshitz-Kosevich expression [12] for the oscillatory magnetization is, up to the proportionality factor Jw, where We is the cyclotron frequency, and EF the Fermi enrergy, identical to the corresponding formula in a 2D system [13]. i.e. :
Mn,,,.vc 0~
-z,&sin
Pnp(m - :)I
(1)
where the subscript n refers to the contribution of the normal electrons. Here X = 2\‘kn,T, and rrF = cF/AW,, which corresponds to the hzhest occupied Landau level index in the 3D case. It is well known that for a type-11 superconductor in the presence of a magnetic field the pairing scheme does not necessarily involve mates of normal electron states on the same energy shell [14-161, even though
621
622
DHVA OSCILLATIONS
IN TYPE-II SUPERCONDUCTORS
their magnetic wave vectors [I 5,161 (q, q’), are exactly equal in magnitude (and opposite in direction , i.e. q’ = -4). This is due to the fact that the order parameter mixes strongly different Landau levels. This phenomenon is known as off-diagonal Landau level pairing, and is the origin of a great technical difficulty. Thus. to make our analysis transparent, we shall begin our discussion by considering two limiting cases, where analytical results are known: (a) Small A,high temperatures (X 2 I), asymptotic (nF >
I) limit [ll, 141.
In this limiting case one assumes that H is sufficiently close to H,z so that the free energy can be expanded in the small order parameter A(r) near H,z. This is essentially a high temperature expansion. since in the absence of Zeeman splitting. the Gorkov’s expansion in small A diverges with I/X at very low temperatures (i.e. where X < 1). This divergence is due to the resonant pairing effect [14] occuring between two degenerate Landau levels at the Fermi energy. where the effective Cooper-pair density of states qCp= ex/[cosh X + cos(2rrrz~)] diverges like 1/X2. The method used takes also advantage of the large Larmor radius at the Fermi surface . rF = JmaH >> 1 (which is the case for all known superconductors) by applying an asymptotic (semiclassical) expansion. The expansion is kept up to fourth order in A(r), with the following variational form A(r) = Ace-il’l’f(z). In this variational form the complex function f(z) , z = (s + iy)/aH, is an arbitrary entire function with a definite number of zeros N in the complex plane, which is determined by the total flux 4 threading the superconductor, i.e. N = +/&, with 40 = hc/2e-the Cooper-pair elementary magnetic flux. More specifically. the entire function f has the form - zi), which means that each vortex f(z)- Hy=,(z carries a single Cooper-pair flux quantum 40. The stationary solution with respect to variation of the functional form of f(z) is the Abrikosov lattice, for which the SC free energy per unit area has the form of a Landau expansion: fi = ( 1/V - A)Ai + [B/ (rr/~gT,.)~]A~, where A0 is an arbitrary real amplitude, V is the effective BCS interaction constant, and the explicit expressions for A and B can be found in Ref( [14]). The final . . variational step za/, = 0 determines the equilibrium value of A,. In terms of this expression for the SC free energy the oscillatory magnetization is calculated in the high temperature limit X > 1 by adding to the SC part M, = -3 the zeroth order (i.e. the normal) part M, (Eq.( 1)). The final result for the first harmonic of the dHvA magnetization, up to fourth order in Ao. is:
+fi?~~‘~(~)
Vol. 101, No. 8
Ao
4
-I/Z nF
1
(2)
nF
where A0 3 Ao/ncu,.. This expression is, up to second order in &/nF4, identical to the exponential damping factor: _ M’!‘/M’i’, = exp[-rr3/2(a0)2] (3) WC n.o.\c 114 nF
obtained by Maki [7] and Stephen [8]. The fourth order term in Eq.(2) is , however, smaller by a factor of the order nF”2 than the corresponding term in the expansion of the above exponential. The Maki-Stephen damping factor , Eq.(3), is obtained within a highly simplified approach, where the vortex lattice is assumed to act like a random scattering potential for the quasi-particles [8]. Strictly speaking the ‘scattering’ of quasi-particles by the vortex lattice is a highly coherent process, similar to multiple Andreev reflections [ 171at a 2D periodic array of superconducting-normal phase boundaries. The averaging procedure used in the MS approach destroys this coherence and thus greatly overestimates the damping effect. A detailed analysis of this important point will be published elswhere [ 181. Eq.(2) indicates that the small expansion parameter in the asymptotic limit is - (A~/nff/~) rather than (&/ny4) , as suggested by the MS damping factor, Eq.(3). An examination of higher order terms in our expansion [18] confirms this conclusion. Thus we find that at the field where the quadratic term in Eq.(2) is of the order one, higher order terms are of the order r~i”~ or smaller, i.e. much smaller than one in the asymptotic limit. This pattern of the expansion indicates that the amplitude of the total magnetization oscillations can change sign at Hi,,,, below HC2, where A\o = (nF/+)“4, that is: Hi,, =: Kz[l
-
L/G/18HR41k/A~21
(4)
where WC*is the cyclotron frequency at H = H,I. A similar sign change has been obtained by Norman et al. [IO]. who calculated the 2D magnetization oscillations in the vortex state on the basis of an exact numerical solution of the corresponding BogoliubovdeGennes (BdG) equations. This effect is expected to be destroyed by disorder in the vortex lattice, as suggested by the strong monotonic damping, Eq.(3), obtained in the disordered lattice model used by Stephen [8].
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DHVA OSCILLATIONS
IN TYPE-II SUPERCONDUCTORS
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(b) Very low temperarues X << 1 , small A limit
As already discussed above, in this limit, and in the absence of spin splitting, the Gorkov’s expansion of the free energy diverges with 1/X - 1I T at flF = half integer, and the divergence is due to the diagonal n = n’ = nF - l/2 Landau level pairing. In this special case the contribution to the QP DOS near the Fermi energy from off diagonal matrix elements of the BdG Hamiltonaian can be neglected, and so the diagonal approximation of the the QP energies: En(q) =
[(n + l/2 -
nF)Aw,12 + IA,,,(q)l’
(5)
can be used at the n = nF - l/2 Landau band. The corresponding QP DOS vanishes with 1~1at the center of the n = np - l/2 Landau band due to the linear dispersion relation of the quasi-particle (or quasi-hole) energies:
q = qx + iq,, near the zeros 4 = & of An,n(4) [ 161. The n = nF - l/2 Landau band. thus splits into two major subbands. This splitting occures because of the magnetic translational symmetry [19], inherent to the Abrikosov lattice, and the fundamental condition of flux quantization in the vortex state, which implies that the magnetic flux through a unit cell in the Abrikosov lattice is exactly one elementary Cooper-pair flux (PO= ch/2e (or alternatively l/2 electronic flux quantum) This particular flux per unit cell ratio determines that the zeros of the order parameter in real space, and consequently the corresponding zeros, &, of the matrix elements A,,,(# [16], in momentum space, are of the first order. As a result the QP energy dispersion is linear near the band center, and a pseudogap is opened in the QP DOS there. Note that two Cooperpair flux quanta (or a single electronic flux quantum ) per unit cell, which correspond to second order zeros, would have led to a quadratic dispersion law and to a (nonzero) constant DOS near the band center. These relationships are most transparent in the limiting case considered here, where the Landau band under study contains the Fermi energy, and where offdiagonal Landau level pairing can be neglected. Their general nature , discussed above. indicates, however, that they hold also for off-diagonal pairing, and that the splitting is not restricted only to the highest occupied Landau band. but also occures in all Landau bands around the Fermi energy. This conjecture has been confirmed by the numerical calculations of Rom et al. [20]. The presence of prominent holes in the QP DOS of the Landau bands near the Fermi energy should gen-
Fig. 1. Local quasi particle density of states (at the center of a vortex core),for a model similar to that employed in Ref. [lO],calculated by a numerical solution of the corresponding (real) time dependent Gorkov’s equations [20]. The relevant paramters used are: A = 0.5, and nF = 10.5. Shown are two plots for two different propogation titneS: T = 20 X ~TT/W, (solid line), and T = 10 x 2rr/w,. (dashed line). The oscillations appearing around the band edges are artifacts due to the sharp cut-off in the QP propogation time. erate a beating frequency in the oscillatory magnetization M,,,.( 1/H). This effect is connected to the sign inversion discussed above. To see this let us examine in detail the dependence of the magnetization oscillations on the QP DOS: The relevant free energy density (i.e. per unit area of the 2D sample) in the vortex state below Hc2 may be written as a functional of the SC order parameter A(r) in a form due to Bardeen et al. [22]: f = -$ = sz + Nrrfr2 v Jd2rIA(r)12
(7)
H
where R = -2ksT
J
R’ED(E) In (2cosh ifle)
(8)
is the thermodynamical potential for noniteracting quasi particles with density of states 23(c) per unit area, and 8 = 1/keT. The second term on the RHS of Eq.(7) corrects for the double counting of the interaction energy in the BCS theory, as in the Hartree Fock approximation. Note that Ai = [ d2rlA(r) 12/Nrra$. In calculating the magnetization M = -$$ one af should be aware of the stationarity condition aa,, = 0, which implies that while differentiating the thermodynamical potentials with respect to H the magnetic field dependence of the equilibrium A0 should be omitted. ThusM= -afiaHlhrl = -ac21aHlh4,
624
DHVA OSCILLATIONS
IN TYPE-II SUPERCONDUCTORS
Assuming a spectrum of nonoverlaping Landau bands, an assumption which will be justified a little bit later, and using a single shape function to describe the QP DOS in many Landau bands around the Fermi energy, an assumption supported by nonperturbative numerical calculations [20]. one may write the QP DOS in the form: D(E) = _? DzoG(Z + no - n - l/2) n=O
(9)
where Z = E/AU, , G(Z) is a normalized shape function (i.e. ]_“mG(Z)& = l), like that shown in Fig.(l), and Dlo = m/d2 (i.e. the DOS per unit area for a 2D free electron gas in the absence of magnetic field). in the T - 0 limit , we get for the oscillatory part of the thermodynamical potential:
Qosc=
Xi
&6(2pn) p=,P =
$p mu,)2
cos2p?T(nF - l/2)
(10)
where cP(2pr-r) I I_“00G(,u) cos (2prrp)dp, and the particle-hole symmetry condition, G(-u) = G(p) , is used. If the magnetic field dpendence of the Fourier transforms +(2prr) is much weaker than that of cos 2prr(n,~ - l/2) we may write the oscillatory magnetization as a superposition of sine Fourier transforms:
where !DLF(E) is the QP DOS for the Landau band around the Fermi energy DLF(E) z &-,G(EIAw,). The origin of the sign change in the dHvA amplitude can be easily infered from Eq.(l 1): Considering the first harmonic , p = 1, and assuming that the splitting energy (in units of tZwc) of the doublet in ;I)LF(E), is 26, the integral in Eq.(l 1) can be estimated by: sin (2n~~ + 27-r&)+ sin (27rn~ - 2776). so that for 62 << 1: M,i,i,’- sin (Zrrn~)[l - 2(ns)‘]
(12)
A comparison to Eq.(2) yields: 6 = 0.53% HF
in agreement with the expression obtained by Norman et al. [IO] for the total width of the Landau band.
Vol. 101, No. 8
Note that at the point of inversion d = $ , so that at this field : 6 = .53/rr314 = 0.225, which means that the band width at the point of inversion is significantly smaller than the cyclotron energy Aw,.. This result provides a justification for the nonoverlaping Landau bands mode1 used above. The presence of magnetic pseudogaps at the centers of the Landau bands is expected to be affected by disorder in the vortex lattice, or by vortex lines motion. This type of symmetry breaking should act to ‘fill’ the holes in middle of the bands, in analogy with the conventional Bloch problem, where disorder tends to smear the sharp energy gaps characterizing the spectrum in an ideally periodic lattice. This smearing effect is consistent with the strong structureless damping of the dHvA amplitude obtained by Stephen [8] after averaging over realizations of a random vortex lattice. Another possible source of ‘hole filling’ is Landau levels broadening by impurity scattering. In this case the diagonal LL pairing resonance is smeared and the contribution from off diagonal elements of the pair potential is not negligible even at the n = RF - l/2 Landau band. The corresponding QP DOS at the Fermi energy, D(O), does not vanish in this case. A related situation, that is where the QP have finite life times, was recently considered by Rom et al. [20], who employed a numerical method for the solution of the time dependent Gorkov’s equations, based on a Chebychev polynomial expansion scheme of the evolution operator, developed by Kosloff [21]. An example of the local QP DOS, obtained in this nonperturbative scheme from the calculated one point Green’s function at the center of a vortex core, is shown in Fig.(l) for two different propogation times, T yp = ( 10,20) x g. The presence of a deep hole in the middle of the k = no - l/2 Landau band, and its ‘filling’ with the decreasing QP ‘life’ time TV,,,are apparent. In conclusion, we have shown here, that at least within mean field theory for a pure type-II superconductor, one expects a sign inversion of the dHvA amplitude at a certain field Hi”, below H(2. This sign change is related to the splitting of the highest occupied Landau band crossing the Fermi energy, into two major magnetic subbands by the SC order parameter. Note that since the splitting takes place also in many Landau bands around the Fermi energy, the sign change should not be weakened significantly in 3D systems. Disorder in the vortex lattice tends to destroy, however. the coherence in the QP ‘scattering’ amplitude responsible for this effect. The effect is expected to be actually obsereved in
Vol. 101, No. 8
DHVA OSCILLATIONS
dHvA experiment as a rapid 180” phase shift in the magnetization oscillations while sweeping the magnetic field accross the inversion field Hint,.The absence of this effect from experiment, so far, seems to be due to smearing by disorder or by vortex lines motion. Acknowledgements-We acknowledge valuable discussions with S. Dukan, S.Hayden, W. Joss, R.Kosloff, A.MacDonald, M.Norman, E. Steep, and Z.Tesanovic. This research was supported by a grant from the German-Israeli Foundation for Scientific Research and Development, No. I-0222- 1136.07/91, and by the fund for the promotion of research at the Technion. REFERENCES
1. Corcoran R.,Harrison N. , Hayden S. M., Meeson P., Springford P , and van der Wel P J., Phys. Rev. Lett. 72, 701( 1994) 2. Harrison N., Hayden S. M. ,Meeson P., Springford M., van der We1 P J., and Menovski A. A., Phys.Rev.B50,4208( 1994) 3. Haanappel E. G. et al..J.Phys.Chem.Solids 54, 1261(1993) 4. Van der We1 P J. et al., Physica C235-240 (1994) 5. Go11 G., Heinecke M. , Jansen A. G. M. , Joss W., Nguyen L., Steep E., Winzer K.. and Wyder P., Phys. Rev. B53, R887 1 (1996) 6. Corcoran R. et al.. J.Phys.Condens. Matter 6, 4479 (1994); Steep E., Rettenberger S., Meyer F.. Jansen A.G.M., Joss W., Biberacher W., Bucher E., Oglesby C. S., Physica B24l-247,2 11( 1995)
IN TYPE-II SUPERCONDUCTORS
625
7. Maki K., Phys.Rev.B 44,2861(1991) 8. Stephen M. J., Phys. Rev.B 45,5481(1992) 9. Dukan S. and Tesanovic Z.. PhysRev. Lett.74 2311(1995) 10. Norman M. R., Akera H., and MacDonald A. H., Phys. Rev.B 51.5927( 1995) 11. Maniv T.,Rom A. I.,Vagner I. D., and Wyder P., Physica C235-240,154l (1994) 12. Lifshitz I. M. and Kosevich A. M., Sov.Phys.JETP 2,636 (1956). 13. Shoenberg D., J.Low.Temp.Phys.,56,417( 1984) 14. Maniv T., Rom A. I. ,Vagner I. D. , and Wyder P., Phys.Rev.B 46,836O (1992); T.Maniv, R.S.Markiewicz, I.D.Vagner, and PWyder, ibid 45,13084( 1992) 15. Norman M. R., Akera H., and MacDonald A. H., Physica C196,43 (1992) 16. Dukan S., and Tesanovic Z., Phys.Rev. B49,13017 (1994); Dukan S., Andreev A. V, and Tesanovic Z. ,Physica C183,355(1991) 17. Andreev A. F. , Zh. Eksp. Teor. Fiz.46, 185.1823; 47, 2222( 1964) 18. Zhuravlev V., and Maniv T., unpublished. 19. Zak J., Phys. Rev.134, A1602 (1964); E. Brown, Phys. Rev.133, A1038 (1964). 20. Rom A. I., Kosloff R., and Maniv T., unpublished; Rom A. I., Dissertation , Technion (1995) 21. Kosloff R., J. Chem. Phys.92, 2087 (1988) 22. Bardeen J. , Kummel R. , Jacobs A. E., and Tewordt L., Phys. Rev., 187, 556 (1969)
Pergamon
@ 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved oow1098/97 f17.00+.00
PII:SOO38-1098(96)00610-2
ORDER PARAMETER
INDUCED
PHASE SHIFT IN THE DHVA OSCILLATIONS SUPERCONDUCTORS
IN TYPE-II
T. Maniv, a A.Y. Rom, a I.D. Vagner b.c and P. Wyder b*c a Department of Chemistry and Solid State Institute, Technion, Haifa 32000, Israel b Grenoble High Magnetic Field Laboratory, Max-Planck-Institute fur Festkorperforschung, Germany c Center National de la Recherche Scientific, 25 Avenue des Martyres. F-38042. Grenoble, Cedex 9. France (Received 22 September 1996; accepted 30 September 1996 by A. H. MacDonald)
In a pure type-II superconductor under a strong magnetic field every Landau level around the Fermi energy, which is significantly broadened by the pair potential at low temperatures, splits into two major magnetic subbands. Such a splitting reflects the fact that the magnetic flux threading a unit cell in the Abrikosov lattice is one half electronic flux quantum. This effect is predicted to be observed as a rapid 180” phase shift in the de Haas-van Alphen(dHvA) oscillations just below Hcz, and is expected to be sensitive to disorder and vortex lines motion. @ 1997 Elsevier Science Ltd. All rights reserved
The de Haas-van Alphen (dHvA) effect in the vortex state of type-11 superconductors has recently become a rapidly growing field of research; clear magnetization oscillations in the vortex state have been reported for half a dosen type-11 superconucting (SC) materials of different types, such as the old high T,, A - 15 compounds,fiSi [l] and Nb+Sn [2]. the new HTSC YBCO [3], the (ET)zCu(NCS)z organic superconductor [4], the borocarbide superconductor YNi2B2C [5], and the layered dichalcogenide NbSe2 [6]. In all these experiments, the only observable effect of the SC order parameter on the measured magnetization oscillations, so far, has been an additional damping of the signal in the vortex state. Several theoretical attempts have been made to account quantitatively for this attenuation [7-lo]. As usual1 in spectroscopy, however, a quantitative analysis of the signal intensity is difficult since many factors, not necessarily intrinsic, may influence it. In this communication we argue that the dHvA oscillations in the vortex state contain a significant spectroscopical information, which should be observable in sufficienly pure SC material. We will show in what follows that at a certain magnetic field. just below Hc2, the dHvA amplitude should change sign as a result of strong,1 80” out of phase oscillations in the SC condensation energy with respect to the normal electrons oscillations [I I]. This remarkable effect is shown to be
related to the opening of a deep hole in the quasi particle (QP) density of states (DOS) at the highest occupied Landau band by the increasing SC order parameter below Hr2. Since the considered effect is associated with a fundamental property of the QP DOS at high magnetic fields, we shall ignore, for the sake of simplicity, irrelevant features like Zeeman spin splitting and energy dispersion along the magnetic field direction ( the z-axis). In an isotropic 3D system of normal electrons, for example, where the extremal orbit corresponds to the momentum component k, = 0, the well known Lifshitz-Kosevich expression [12] for the oscillatory magnetization is, up to the proportionality factor Jw, where We is the cyclotron frequency, and EF the Fermi enrergy, identical to the corresponding formula in a 2D system [13]. i.e. :
Mn,,,.vc 0~
-z,&sin
Pnp(m - :)I
(1)
where the subscript n refers to the contribution of the normal electrons. Here X = 2\‘kn,T, and rrF = cF/AW,, which corresponds to the hzhest occupied Landau level index in the 3D case. It is well known that for a type-11 superconductor in the presence of a magnetic field the pairing scheme does not necessarily involve mates of normal electron states on the same energy shell [14-161, even though
621
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IN TYPE-II SUPERCONDUCTORS
their magnetic wave vectors [I 5,161 (q, q’), are exactly equal in magnitude (and opposite in direction , i.e. q’ = -4). This is due to the fact that the order parameter mixes strongly different Landau levels. This phenomenon is known as off-diagonal Landau level pairing, and is the origin of a great technical difficulty. Thus. to make our analysis transparent, we shall begin our discussion by considering two limiting cases, where analytical results are known: (a) Small A,high temperatures (X 2 I), asymptotic (nF >
I) limit [ll, 141.
In this limiting case one assumes that H is sufficiently close to H,z so that the free energy can be expanded in the small order parameter A(r) near H,z. This is essentially a high temperature expansion. since in the absence of Zeeman splitting. the Gorkov’s expansion in small A diverges with I/X at very low temperatures (i.e. where X < 1). This divergence is due to the resonant pairing effect [14] occuring between two degenerate Landau levels at the Fermi energy. where the effective Cooper-pair density of states qCp= ex/[cosh X + cos(2rrrz~)] diverges like 1/X2. The method used takes also advantage of the large Larmor radius at the Fermi surface . rF = JmaH >> 1 (which is the case for all known superconductors) by applying an asymptotic (semiclassical) expansion. The expansion is kept up to fourth order in A(r), with the following variational form A(r) = Ace-il’l’f(z). In this variational form the complex function f(z) , z = (s + iy)/aH, is an arbitrary entire function with a definite number of zeros N in the complex plane, which is determined by the total flux 4 threading the superconductor, i.e. N = +/&, with 40 = hc/2e-the Cooper-pair elementary magnetic flux. More specifically. the entire function f has the form - zi), which means that each vortex f(z)- Hy=,(z carries a single Cooper-pair flux quantum 40. The stationary solution with respect to variation of the functional form of f(z) is the Abrikosov lattice, for which the SC free energy per unit area has the form of a Landau expansion: fi = ( 1/V - A)Ai + [B/ (rr/~gT,.)~]A~, where A0 is an arbitrary real amplitude, V is the effective BCS interaction constant, and the explicit expressions for A and B can be found in Ref( [14]). The final . . variational step za/, = 0 determines the equilibrium value of A,. In terms of this expression for the SC free energy the oscillatory magnetization is calculated in the high temperature limit X > 1 by adding to the SC part M, = -3 the zeroth order (i.e. the normal) part M, (Eq.( 1)). The final result for the first harmonic of the dHvA magnetization, up to fourth order in Ao. is:
+fi?~~‘~(~)
Vol. 101, No. 8
Ao
4
-I/Z nF
1
(2)
nF
where A0 3 Ao/ncu,.. This expression is, up to second order in &/nF4, identical to the exponential damping factor: _ M’!‘/M’i’, = exp[-rr3/2(a0)2] (3) WC n.o.\c 114 nF
obtained by Maki [7] and Stephen [8]. The fourth order term in Eq.(2) is , however, smaller by a factor of the order nF”2 than the corresponding term in the expansion of the above exponential. The Maki-Stephen damping factor , Eq.(3), is obtained within a highly simplified approach, where the vortex lattice is assumed to act like a random scattering potential for the quasi-particles [8]. Strictly speaking the ‘scattering’ of quasi-particles by the vortex lattice is a highly coherent process, similar to multiple Andreev reflections [ 171at a 2D periodic array of superconducting-normal phase boundaries. The averaging procedure used in the MS approach destroys this coherence and thus greatly overestimates the damping effect. A detailed analysis of this important point will be published elswhere [ 181. Eq.(2) indicates that the small expansion parameter in the asymptotic limit is - (A~/nff/~) rather than (&/ny4) , as suggested by the MS damping factor, Eq.(3). An examination of higher order terms in our expansion [18] confirms this conclusion. Thus we find that at the field where the quadratic term in Eq.(2) is of the order one, higher order terms are of the order r~i”~ or smaller, i.e. much smaller than one in the asymptotic limit. This pattern of the expansion indicates that the amplitude of the total magnetization oscillations can change sign at Hi,,,, below HC2, where A\o = (nF/+)“4, that is: Hi,, =: Kz[l
-
L/G/18HR41k/A~21
(4)
where WC*is the cyclotron frequency at H = H,I. A similar sign change has been obtained by Norman et al. [IO]. who calculated the 2D magnetization oscillations in the vortex state on the basis of an exact numerical solution of the corresponding BogoliubovdeGennes (BdG) equations. This effect is expected to be destroyed by disorder in the vortex lattice, as suggested by the strong monotonic damping, Eq.(3), obtained in the disordered lattice model used by Stephen [8].
Vol. 101. No. 8
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IN TYPE-II SUPERCONDUCTORS
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(b) Very low temperarues X << 1 , small A limit
As already discussed above, in this limit, and in the absence of spin splitting, the Gorkov’s expansion of the free energy diverges with 1/X - 1I T at flF = half integer, and the divergence is due to the diagonal n = n’ = nF - l/2 Landau level pairing. In this special case the contribution to the QP DOS near the Fermi energy from off diagonal matrix elements of the BdG Hamiltonaian can be neglected, and so the diagonal approximation of the the QP energies: En(q) =
[(n + l/2 -
nF)Aw,12 + IA,,,(q)l’
(5)
can be used at the n = nF - l/2 Landau band. The corresponding QP DOS vanishes with 1~1at the center of the n = np - l/2 Landau band due to the linear dispersion relation of the quasi-particle (or quasi-hole) energies:
q = qx + iq,, near the zeros 4 = & of An,n(4) [ 161. The n = nF - l/2 Landau band. thus splits into two major subbands. This splitting occures because of the magnetic translational symmetry [19], inherent to the Abrikosov lattice, and the fundamental condition of flux quantization in the vortex state, which implies that the magnetic flux through a unit cell in the Abrikosov lattice is exactly one elementary Cooper-pair flux (PO= ch/2e (or alternatively l/2 electronic flux quantum) This particular flux per unit cell ratio determines that the zeros of the order parameter in real space, and consequently the corresponding zeros, &, of the matrix elements A,,,(# [16], in momentum space, are of the first order. As a result the QP energy dispersion is linear near the band center, and a pseudogap is opened in the QP DOS there. Note that two Cooperpair flux quanta (or a single electronic flux quantum ) per unit cell, which correspond to second order zeros, would have led to a quadratic dispersion law and to a (nonzero) constant DOS near the band center. These relationships are most transparent in the limiting case considered here, where the Landau band under study contains the Fermi energy, and where offdiagonal Landau level pairing can be neglected. Their general nature , discussed above. indicates, however, that they hold also for off-diagonal pairing, and that the splitting is not restricted only to the highest occupied Landau band. but also occures in all Landau bands around the Fermi energy. This conjecture has been confirmed by the numerical calculations of Rom et al. [20]. The presence of prominent holes in the QP DOS of the Landau bands near the Fermi energy should gen-
Fig. 1. Local quasi particle density of states (at the center of a vortex core),for a model similar to that employed in Ref. [lO],calculated by a numerical solution of the corresponding (real) time dependent Gorkov’s equations [20]. The relevant paramters used are: A = 0.5, and nF = 10.5. Shown are two plots for two different propogation titneS: T = 20 X ~TT/W, (solid line), and T = 10 x 2rr/w,. (dashed line). The oscillations appearing around the band edges are artifacts due to the sharp cut-off in the QP propogation time. erate a beating frequency in the oscillatory magnetization M,,,.( 1/H). This effect is connected to the sign inversion discussed above. To see this let us examine in detail the dependence of the magnetization oscillations on the QP DOS: The relevant free energy density (i.e. per unit area of the 2D sample) in the vortex state below Hc2 may be written as a functional of the SC order parameter A(r) in a form due to Bardeen et al. [22]: f = -$ = sz + Nrrfr2 v Jd2rIA(r)12
(7)
H
where R = -2ksT
J
R’ED(E) In (2cosh ifle)
(8)
is the thermodynamical potential for noniteracting quasi particles with density of states 23(c) per unit area, and 8 = 1/keT. The second term on the RHS of Eq.(7) corrects for the double counting of the interaction energy in the BCS theory, as in the Hartree Fock approximation. Note that Ai = [ d2rlA(r) 12/Nrra$. In calculating the magnetization M = -$$ one af should be aware of the stationarity condition aa,, = 0, which implies that while differentiating the thermodynamical potentials with respect to H the magnetic field dependence of the equilibrium A0 should be omitted. ThusM= -afiaHlhrl = -ac21aHlh4,
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DHVA OSCILLATIONS
IN TYPE-II SUPERCONDUCTORS
Assuming a spectrum of nonoverlaping Landau bands, an assumption which will be justified a little bit later, and using a single shape function to describe the QP DOS in many Landau bands around the Fermi energy, an assumption supported by nonperturbative numerical calculations [20]. one may write the QP DOS in the form: D(E) = _? DzoG(Z + no - n - l/2) n=O
(9)
where Z = E/AU, , G(Z) is a normalized shape function (i.e. ]_“mG(Z)& = l), like that shown in Fig.(l), and Dlo = m/d2 (i.e. the DOS per unit area for a 2D free electron gas in the absence of magnetic field). in the T - 0 limit , we get for the oscillatory part of the thermodynamical potential:
Qosc=
Xi
&6(2pn) p=,P =
$p mu,)2
cos2p?T(nF - l/2)
(10)
where cP(2pr-r) I I_“00G(,u) cos (2prrp)dp, and the particle-hole symmetry condition, G(-u) = G(p) , is used. If the magnetic field dpendence of the Fourier transforms +(2prr) is much weaker than that of cos 2prr(n,~ - l/2) we may write the oscillatory magnetization as a superposition of sine Fourier transforms:
where !DLF(E) is the QP DOS for the Landau band around the Fermi energy DLF(E) z &-,G(EIAw,). The origin of the sign change in the dHvA amplitude can be easily infered from Eq.(l 1): Considering the first harmonic , p = 1, and assuming that the splitting energy (in units of tZwc) of the doublet in ;I)LF(E), is 26, the integral in Eq.(l 1) can be estimated by: sin (2n~~ + 27-r&)+ sin (27rn~ - 2776). so that for 62 << 1: M,i,i,’- sin (Zrrn~)[l - 2(ns)‘]
(12)
A comparison to Eq.(2) yields: 6 = 0.53% HF
in agreement with the expression obtained by Norman et al. [IO] for the total width of the Landau band.
Vol. 101, No. 8
Note that at the point of inversion d = $ , so that at this field : 6 = .53/rr314 = 0.225, which means that the band width at the point of inversion is significantly smaller than the cyclotron energy Aw,.. This result provides a justification for the nonoverlaping Landau bands mode1 used above. The presence of magnetic pseudogaps at the centers of the Landau bands is expected to be affected by disorder in the vortex lattice, or by vortex lines motion. This type of symmetry breaking should act to ‘fill’ the holes in middle of the bands, in analogy with the conventional Bloch problem, where disorder tends to smear the sharp energy gaps characterizing the spectrum in an ideally periodic lattice. This smearing effect is consistent with the strong structureless damping of the dHvA amplitude obtained by Stephen [8] after averaging over realizations of a random vortex lattice. Another possible source of ‘hole filling’ is Landau levels broadening by impurity scattering. In this case the diagonal LL pairing resonance is smeared and the contribution from off diagonal elements of the pair potential is not negligible even at the n = RF - l/2 Landau band. The corresponding QP DOS at the Fermi energy, D(O), does not vanish in this case. A related situation, that is where the QP have finite life times, was recently considered by Rom et al. [20], who employed a numerical method for the solution of the time dependent Gorkov’s equations, based on a Chebychev polynomial expansion scheme of the evolution operator, developed by Kosloff [21]. An example of the local QP DOS, obtained in this nonperturbative scheme from the calculated one point Green’s function at the center of a vortex core, is shown in Fig.(l) for two different propogation times, T yp = ( 10,20) x g. The presence of a deep hole in the middle of the k = no - l/2 Landau band, and its ‘filling’ with the decreasing QP ‘life’ time TV,,,are apparent. In conclusion, we have shown here, that at least within mean field theory for a pure type-II superconductor, one expects a sign inversion of the dHvA amplitude at a certain field Hi”, below H(2. This sign change is related to the splitting of the highest occupied Landau band crossing the Fermi energy, into two major magnetic subbands by the SC order parameter. Note that since the splitting takes place also in many Landau bands around the Fermi energy, the sign change should not be weakened significantly in 3D systems. Disorder in the vortex lattice tends to destroy, however. the coherence in the QP ‘scattering’ amplitude responsible for this effect. The effect is expected to be actually obsereved in
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DHVA OSCILLATIONS
dHvA experiment as a rapid 180” phase shift in the magnetization oscillations while sweeping the magnetic field accross the inversion field Hint,.The absence of this effect from experiment, so far, seems to be due to smearing by disorder or by vortex lines motion. Acknowledgements-We acknowledge valuable discussions with S. Dukan, S.Hayden, W. Joss, R.Kosloff, A.MacDonald, M.Norman, E. Steep, and Z.Tesanovic. This research was supported by a grant from the German-Israeli Foundation for Scientific Research and Development, No. I-0222- 1136.07/91, and by the fund for the promotion of research at the Technion. REFERENCES
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