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Reentrance of the superconducting state in a strong magnetic field
Physica B 1658~166 (1990) North-Holland
RgENTRANcE
OF TFIE
361-362
SUPNRCONDUCTING STATE IN A STRONG MAGNNTIC
FIELD
T.Maniva* ,R.S.Markiewiczb, I.D.Vagnefl and P.Wydefl and Astronomy,The Johns HopkinsUniversity, Baltimore,Marya) Departmentof Physics land 21218;b) PhysicsDepartment,NortheasternUniversity, BostonMass. 02115,USA: c) Max-Planck-Institut fur Festkikpsrforschung, Hochfeld-Magnetlabor, F-38042,166X, GrenobleCedex,France.
Strongguantummagneto-oscillations in the order parameterwill result in the reentrance of the superconducting state at low temperaturesin extreme type II highly two-dimensional superconductors.In systemswith criticalfieldsas high as in the high T, cuprates,an additionaloscillatorystructureassociatedwith the pairing correlation may be observable.The averageorder parameterover a singlesuperconductingcore region in the vortex phase in guantizingmagnetic field near Hcz(T)is calculatedand applicationis made to the experimentalobservationof quantumoscillationsin the layereddichalcogenide compounds. 1. INTRODUCITON.
More than a decadeago,Graebnerand Robbins (1)observedsignificantmagnetotherma1 and dHvA oscillations well below sl in investigating the quasi-two-dimensional dichalcogenide SH-Nb Se,. The new high T, oxide superconductors,with their quasitwo-dimensionalstructureand their enormously highuppercriticalfields(2)are, in principle,very promising candidatesfor this type of investigation. The observation of quantum oscillationsin the superconductingstate would be of great importance as a Fermi surfaceprobe and also as a way of sortingout the very nature of the pairingmeohanismin this classof materials. -& (3) we have presented an analytical calculationof the linear kernel in the Gorkov equation,which providesthe information on the guantum oscillationsin Qz. Here we calculatethe nonlinearterm in the Gorkov equationand study,analyticallyand numerically,the guantum oscillationsin the orderparameter. 2. THEORY. We considera two-dimensional free electron gas model with a simple B.C.S.pairing interactionin the vicinityof He,(T)for an
* Permanentaddress:Department IlT,Haifa32000,Israel. 0921-4526/90/$03.50 @ 1990 -
arbitrarytemperature0 < T c TF.-Weassume that many Landau levels are fflled, i.e.we workwith the semiclassical electronic:wave functions. The extreme guantum case was consideredrecently by Tesanovicet al. (5).The Helfand-Werthamer ansatz,Ref. (4),which is an exacr solution of the linearizedGorkov equationin guantizingmagnetic field,is used as a variational solution,AL(r) = A0 exp ( - l/2 (r/aH)2), for the nonlinearequation.The Gaussianform of AD(r) is consistentwith the semiclassical structureof the kernelR and themultipleintegralassociatedwith it can be reducedto a singleintegralby employinga combinationof Gaussian (steepest descent)and contour integrationmethods. The only approximation made throughoutthe analysis is the semiclassicalapproximation. This provides a simple quadratic equationfor A0 , which is solved to yield: A2(H,T) =2
(2rkBTc)zA - t ( I
r11
with:
of Chemistryand the SolidState Institute, Technion-
Elsevier Science Publishers B.V. (North-Holland1
r21
362
T. Maniv,
2
00
e_2Q,P- P2
6, = 4r
R.S.Markiewicz, I.D.Vagner, P.Wyder
ap (31
0 The expressions for A and e, are presented in Ref. (3). Here: + = (2~+ 1) F
and X I
N(O)V is the effective B.C.S coupling constant with N(0) the density of states at the Fermi energy,Tc = 2TD 1.781e-1/x the tranSitiOn tSInpSratUre, c,, = fiVp/rk$‘c, TD - the cutoff temperature and VD s (TD/T - 1)/2 . fiwe = geH/mO c is the Zeeman splitting of a Landau level. Note that our imaginary (Matzubara)frequency formalism is eguivalent to the real frequency one only at discrete values of the temperature T for which VD is an exact integer. We therefore restrictthetemperature in ourcalculationto these values. 3.DISCUSSION Eguating Az(H,T) to zero in the entire temperature range 0 < T < T,, yields an equation for I&(T), which can be easily solved numerically. Strong QO will smear dramatically the transition to the superconducting state as a result of multiple reentrance transition.This phenomenon may account for the anomalously broad transition to the superconducting mixed state, reported in Ref. (l),for 2H-Nb Se, . For instance, by using m, = 0.26 m0 , which is a fairly good estimate of the experimentally observed effective mass, and fitting the upper edge of the reentrance regime to the experimental upper edge of the anomaly, we automatically reproduce the width of the transition.Another example is the deviation from the conventional theory of the measured +*(T) curves in the layered dichalcogenide compounds Nb,_x Tax Se,, reported in Ref. (6). Another interestingfeature of the present theory is revealed in model systems with much higher critical fields. Fig. (1) shows the calculated Az(H,T) in the range of the relevant parameters which may characterize some of the high TF oxides. Depending on the value of the (mplane) cyclotron mass, the dHvA-like peaks, associated with the field enhancement of the single electron density of states,may split at sufficiently low (_ 0.5 K) temperatures into doublets. This additionalstructure is due to a feedback effect associated with the enhanced pair-pairrepulsion at the peaks of the density of states, which tends to reduce the superconductingorder.
Fig.1 Quantum oscillationsin the orderparameter. Here: TD = 978.8X, Tc = 88.8K, $ = 12.86A, EF = 0,039 eV. The temperatureis T = 0.5K. The values of m, are: (a) m,/m, = 2 (dottedcurve), (b) m,/m, = 3 (solidcurve). We acknowledge valuable discussionswith W. Joss, W. Biberacher,D. Shoe&erg and Z. Tesanovic.This research was supported by a grant from the German-Israeli Foundation for Scientific Research and Development, No. G-112-279.7/88 .
(1) (2)
(3)
(4)
(5) (6)
J.E.Graebner and M.Robbins, Phys. Rev. Lett. 36 (1976)422. T.K.Worthington, W.J.Gallagher, D.L.Kaiser, F.H.Holtzberg and T.R.Dinger, Physica C 153-155 (1988) 32. T.Maniv,R.S.Markiewicz,I.D.Vagnerand P.Wyder, Physica C 153-155, (1988) 1179 ; R.S.Markiewicz, I.D.Vagner, P.Wyder and T.Maniv, Solid State Comm 67 (1988)43. E.Iielfandand R.Werthamer, Phys. Rev. Lett. 13 (1964)686; L.W.Gruenbergand L.Gunther, Phys. Rev. I&t. 16 (1966) 996;Phys.Rev. 176 (1968)606 . Z. Tesanovic, M. Rasolt and L. xing, Phys.Rev.Iett.63(1989)2425. M.Ikebe, KXatagiri, K.Noto and Y.Muto, Physica 998 (1980)209.
RgENTRANcE
OF TFIE
361-362
SUPNRCONDUCTING STATE IN A STRONG MAGNNTIC
FIELD
T.Maniva* ,R.S.Markiewiczb, I.D.Vagnefl and P.Wydefl and Astronomy,The Johns HopkinsUniversity, Baltimore,Marya) Departmentof Physics land 21218;b) PhysicsDepartment,NortheasternUniversity, BostonMass. 02115,USA: c) Max-Planck-Institut fur Festkikpsrforschung, Hochfeld-Magnetlabor, F-38042,166X, GrenobleCedex,France.
Strongguantummagneto-oscillations in the order parameterwill result in the reentrance of the superconducting state at low temperaturesin extreme type II highly two-dimensional superconductors.In systemswith criticalfieldsas high as in the high T, cuprates,an additionaloscillatorystructureassociatedwith the pairing correlation may be observable.The averageorder parameterover a singlesuperconductingcore region in the vortex phase in guantizingmagnetic field near Hcz(T)is calculatedand applicationis made to the experimentalobservationof quantumoscillationsin the layereddichalcogenide compounds. 1. INTRODUCITON.
More than a decadeago,Graebnerand Robbins (1)observedsignificantmagnetotherma1 and dHvA oscillations well below sl in investigating the quasi-two-dimensional dichalcogenide SH-Nb Se,. The new high T, oxide superconductors,with their quasitwo-dimensionalstructureand their enormously highuppercriticalfields(2)are, in principle,very promising candidatesfor this type of investigation. The observation of quantum oscillationsin the superconductingstate would be of great importance as a Fermi surfaceprobe and also as a way of sortingout the very nature of the pairingmeohanismin this classof materials. -& (3) we have presented an analytical calculationof the linear kernel in the Gorkov equation,which providesthe information on the guantum oscillationsin Qz. Here we calculatethe nonlinearterm in the Gorkov equationand study,analyticallyand numerically,the guantum oscillationsin the orderparameter. 2. THEORY. We considera two-dimensional free electron gas model with a simple B.C.S.pairing interactionin the vicinityof He,(T)for an
* Permanentaddress:Department IlT,Haifa32000,Israel. 0921-4526/90/$03.50 @ 1990 -
arbitrarytemperature0 < T c TF.-Weassume that many Landau levels are fflled, i.e.we workwith the semiclassical electronic:wave functions. The extreme guantum case was consideredrecently by Tesanovicet al. (5).The Helfand-Werthamer ansatz,Ref. (4),which is an exacr solution of the linearizedGorkov equationin guantizingmagnetic field,is used as a variational solution,AL(r) = A0 exp ( - l/2 (r/aH)2), for the nonlinearequation.The Gaussianform of AD(r) is consistentwith the semiclassical structureof the kernelR and themultipleintegralassociatedwith it can be reducedto a singleintegralby employinga combinationof Gaussian (steepest descent)and contour integrationmethods. The only approximation made throughoutthe analysis is the semiclassicalapproximation. This provides a simple quadratic equationfor A0 , which is solved to yield: A2(H,T) =2
(2rkBTc)zA - t ( I
r11
with:
of Chemistryand the SolidState Institute, Technion-
Elsevier Science Publishers B.V. (North-Holland1
r21
362
T. Maniv,
2
00
e_2Q,P- P2
6, = 4r
R.S.Markiewicz, I.D.Vagner, P.Wyder
ap (31
0 The expressions for A and e, are presented in Ref. (3). Here: + = (2~+ 1) F
and X I
N(O)V is the effective B.C.S coupling constant with N(0) the density of states at the Fermi energy,Tc = 2TD 1.781e-1/x the tranSitiOn tSInpSratUre, c,, = fiVp/rk$‘c, TD - the cutoff temperature and VD s (TD/T - 1)/2 . fiwe = geH/mO c is the Zeeman splitting of a Landau level. Note that our imaginary (Matzubara)frequency formalism is eguivalent to the real frequency one only at discrete values of the temperature T for which VD is an exact integer. We therefore restrictthetemperature in ourcalculationto these values. 3.DISCUSSION Eguating Az(H,T) to zero in the entire temperature range 0 < T < T,, yields an equation for I&(T), which can be easily solved numerically. Strong QO will smear dramatically the transition to the superconducting state as a result of multiple reentrance transition.This phenomenon may account for the anomalously broad transition to the superconducting mixed state, reported in Ref. (l),for 2H-Nb Se, . For instance, by using m, = 0.26 m0 , which is a fairly good estimate of the experimentally observed effective mass, and fitting the upper edge of the reentrance regime to the experimental upper edge of the anomaly, we automatically reproduce the width of the transition.Another example is the deviation from the conventional theory of the measured +*(T) curves in the layered dichalcogenide compounds Nb,_x Tax Se,, reported in Ref. (6). Another interestingfeature of the present theory is revealed in model systems with much higher critical fields. Fig. (1) shows the calculated Az(H,T) in the range of the relevant parameters which may characterize some of the high TF oxides. Depending on the value of the (mplane) cyclotron mass, the dHvA-like peaks, associated with the field enhancement of the single electron density of states,may split at sufficiently low (_ 0.5 K) temperatures into doublets. This additionalstructure is due to a feedback effect associated with the enhanced pair-pairrepulsion at the peaks of the density of states, which tends to reduce the superconductingorder.
Fig.1 Quantum oscillationsin the orderparameter. Here: TD = 978.8X, Tc = 88.8K, $ = 12.86A, EF = 0,039 eV. The temperatureis T = 0.5K. The values of m, are: (a) m,/m, = 2 (dottedcurve), (b) m,/m, = 3 (solidcurve). We acknowledge valuable discussionswith W. Joss, W. Biberacher,D. Shoe&erg and Z. Tesanovic.This research was supported by a grant from the German-Israeli Foundation for Scientific Research and Development, No. G-112-279.7/88 .
(1) (2)
(3)
(4)
(5) (6)
J.E.Graebner and M.Robbins, Phys. Rev. Lett. 36 (1976)422. T.K.Worthington, W.J.Gallagher, D.L.Kaiser, F.H.Holtzberg and T.R.Dinger, Physica C 153-155 (1988) 32. T.Maniv,R.S.Markiewicz,I.D.Vagnerand P.Wyder, Physica C 153-155, (1988) 1179 ; R.S.Markiewicz, I.D.Vagner, P.Wyder and T.Maniv, Solid State Comm 67 (1988)43. E.Iielfandand R.Werthamer, Phys. Rev. Lett. 13 (1964)686; L.W.Gruenbergand L.Gunther, Phys. Rev. I&t. 16 (1966) 996;Phys.Rev. 176 (1968)606 . Z. Tesanovic, M. Rasolt and L. xing, Phys.Rev.Iett.63(1989)2425. M.Ikebe, KXatagiri, K.Noto and Y.Muto, Physica 998 (1980)209.