*Corresponding author. Tel.: #44 117 9288731; fax:#44 117 9255624; e-mail: m.springford@bristol.ac.uk.

Physica B 246—247 (1998) 73—77

On the origin of the de Haas—van Alphen effectin a superconductor

C. Haworth!, S.M. Hayden!, T.-J.B.M. Janssen!, P. Meeson!, M. Springford!,*,A. Wasserman"

! H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK" Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA

Abstract

A quantitative comparison is made between the dHvA effect in the mixed state of a superconductor and severaltheories which have been advanced to explain the nature and origin of the effect. The experiments were performed inNbSe

2, V

3Si and Nb

3Sn over a wide range of field and temperature in both normal and superconducting states. A critical

comparison is made with various microscopic theoretical models which have been advanced to explain this phenomenon.( 1998 Elsevier Science B.V. All rights reserved.

Keywords: Superconductor; Mixed state; Quantum oscillation; de Haas—van Alphen effect

1. Introduction

During the past several years, evidence hasaccumulated to indicate that magnetic quantumoscillations are a fundamental property of type-IIsuperconductors in the mixed state. The originalobservations of Graebner and Robbins in 2H—NbSe

2[1] have been confirmed and extended

[2—4] and reports of de Haas-van Alphen (dHvA)effect studies now include V

3Si [5,6], Nb

3Sn [7],

Ba(K)BiO3

[8], CeRu2

[9]. YNi2B2C [10—12],

the organic molecular metal i-(BEDT-TTF)2

Cu(NCS)2

[13] and even reported measurements

in the high-¹#

material YBCO [14—19]. At a timewhen a better understanding of unconventionalsuperconductors is both pressing and elusive, theprospect of dHvA studies in the mixed state, asa means of investigating the electronic structureand anisotropies of the many-body state, is anattractive one, particularly, as in some cases theymay be conducted in magnetic fields well belowB#2

. It is evident, however, that for such a probe tobe effective, a proper theoretical foundation of thephenomena is needed. But superconductivity in thepresence of a quantizing magnetic field poses theor-etical problems of considerable subtlety, so that,while a number of theories have been advanced toexplain the above experiments, no consensus hasyet emerged on the underlying physical mechanism.

0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reservedPII S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 0 0 2 8 - 3

Fig. 1. Variation of the measured damping, C4(B), of the dHvA

effect in the mixed state of NbSe2

with *2(B)K. The followingrelations are assumed: C

4(B)"+/2q

4(B), *2(B)"D2(0)(1!B/B

#2),

D(0)"1.76 kB¹

#and K"(2+eB)~1@2. The linear dependence

expressed by Eq. (1) is confirmed. ¹"20mK and results areshown for two different orientations of the magnetic field withrespect to the c-axis.

Our approach here is to make a quantitativecomparison between the predictions of these the-ories and new measurements of the dHvA effect inthe normal and mixed states of the three ‘conven-tional’ superconductors, 2H—NbSe

2, V

3Si and

Nb3Sn [20]. Whilst our ultimate objective is to

measure the superconducting order parameter andits anisotropy in unconventional type II materials,our emphasis here is to seek to clarify the underly-ing physical process which dominates the principalexperimental observation, namely that quantumoscillations suffer an additional damping in thepresence of superconducting order.

2. Key experimental results

Experimental measurements of the dHvA effectin the mixed state are now sufficiently extensivethat one may usefully summarise the key features asfollows.

1. Quantum oscillations observed in the normalstate persist for magnetic fields below B

#2down

to values of +u#/*(B,¹)&1. *(B,¹) is the field-

and temperature-dependent superconductingorder parameter and the cyclotron frequencyu

#"eB/m*.

2. The dHvA frequency in the superconductingstate is unchanged, to within &0.1%, from itsvalue in the normal state.

3. The oscillations maintain a constant phase, withrespect to infinite field, between normal andsuperconducting states.

4. Quantum oscillations suffer an additionaldamping in the superconducting state whencompared to their amplitudes in the normalstate. As shown in Fig. 1, the damping C

4(B) is

observed experimentally to vary with field ac-cording to

C4(B)J*2(B)K, (1)

in which the magnetic length K"(2+eB)~1@2.5. As in the normal state, the amplitude of quan-

tum oscillations in the mixed state is found tovary with temperature according to, X/sinhX,where X"2p2rk

B¹/+u

#. Such a temperature

dependence is the signature of the Fermi—Dirac

distribution function, and its persistence to milli-kelvin temperatures (i.e. &1 leV) demonstratesthat the quasiparticle excitation spectrum in themixed state is gapless in quantizing magneticfields. The value of effective mass determined inthis way, is unchanged between normal andsuperconducting states, to within the experi-mental uncertainty of a few percent. An excep-tion to this may be the case of heavy-fermionsuperconductors in which the mass is reportedto be diminished in fields (B

#2[21].

3. Aspects of the analysis of experiments

In experiments employing the field-modulationmethod of detection, a transition region is normallyobserved just below B

#2in which measurements are

hysteretic, harmonic generation is enhanced andthe dHvA effect is heavily damped. These responsesare correlated with the so-called “peak effect”,which has its origin in the fluid-like state of thevortex lattice just below B

#2. In certain cases [13]

these effects appear to be absent, presumably asa result of the particular flux-lattice dynamics but,

74 C. Haworth et al. / Physica B 246—247 (1998) 73—77

where present, dHvA amplitude measurements inthe transition region have been excluded in thepresent analysis. We note further that screening ofthe applied magnetic field by surface currents isassumed to be negligible in the strongly type IImaterials under discussion.

The damping of quantum oscillations in themixed state is presumed to include the influence ofimpurities and static defects, as observed in thenormal state, in addition to that arising from super-conducting order. Hence, we may write for the totalscattering rate, Sq~1T"Sq~1

0T#Sq~1

4T, in which

S T designates an orbitally averaged value. Thisadditive relation, or “orbital Mattheissen’s rule”will break down if q

0is influenced by the supercon-

ductivity or if the anisotropies q~10

(k) and q~14

(k)differ. In analysing the experiments, and by analogywith the “Dingle factor” in the normal state, weexpress the damping due to superconductivity,

R4"exp(!pr/u

#q4), (2)

even though q4is field dependent and a Lorentzian

broadening of Landau levels may not be an appro-priate description.

4. Physical models for the damping

The damping of quantum oscillations is conve-niently described in terms of either the scattering ofquasiparticles or of phase smearing, and several ofthe models proposed fall into the latter category.

We can dispose first of the idea that damping hasits origin in the presence of an inhomogeneousmagnetic field characteristic of the flux lattice. Dif-ferently centred cyclotron orbits will experiencedifferent magnetic field environments leading toa phase smearing of the dHvA effect. Methods foranalysing this problem are well established [22]and calculations based on the spatial variation ofthe field in the vortex lattice, indicate a level of damp-ing which is typically 2 orders of magnitude less thanthat observed [20]. We conclude therefore that fieldinhomogeneity is of negligible importance in deter-mining q~1

4under present experimental conditions.

However, the presence of an inhomogeneous or-der parameter is also expected to give rise to amodified quasiparticle band structure and, in par-

ticular, to a broadened distribution of Landaulevels. As before, this is not a scattering problem asindividual eigenstates are sharply defined in theordered vortex lattice. Norman et al. [23,24] havepursued this approach by solving the mean fieldBdG equations in the vortex state in the presence ofLandau quantization. For the damping of dHvAoscillations they find,

R4"expA!

C1p*

+u#n1@4k B (for D/n1@4k '2pk

B¹), (3a)

R4"expA!

C2p*2

(+u#)2n1@2k B (for D/n1@4k (2pk

B¹),

(3b)

in which nk"k/+u#+F/B,k is the chemical po-

tential, F the dHvA frequency, and C1

and C2

areconstants to be determined.

The prevailing experimental conditions forNbSe

2, V

3Si and Nb

3Sn [20] are such that Eq. (3a)

is the appropriate relation, but analysis of the ex-periments on this basis yields neither a good fit tothe data nor plausible values of D(0), as may be seenin Table 1.

An alternative approach which also rests uponnumerical solution of the BdG equations, but usesa real-space recursion method, has been used byMiller and Gyorffy [25]. Having demonstrated theexistence of a Landau-level-like structure to thedensity of states in the mixed state, they attributethe existence of quantum oscillations, as in thenormal state, to the passage of the Landau levelsthrough the Fermi energy with changing magneticfield. The damping has its origin in the mixedparticle-hole character of states within an energyrange &D of the Fermi energy. This results inphase smearing in which, at ¹"0, the Fermi func-tion is replaced by the energy dependence of theparticle density DuD2 yielding,

R4"aK

1(a), (4)

in which the Bessel function K1

has argumenta"2p2rD/+u

#. The similar functional form of

Eq. (4) for a<1 to the Dingle factor shows that, inits effect upon the amplitude of quantum oscilla-tions, the temperature ¹ has effectively been re-placed by D. However, Table 1 shows that Eq. (4)

C. Haworth et al. / Physica B 246—247 (1998) 73—77 75

Table 1Values of the superconducting order parameter, D(0) in (meV), obtained by fitting the several theories discussed in the text to theexperimental data on NbSe

2, V

3Si and Nb

3Sn33. The columns labelled (s2) give the quality of the fit. The errors listed for the order

parameters do not include the errors in the normal state parameters such as the effective mass, Dingle temperature, B#2

, and dHvAfrequency

NbSe2

(s2) V2Si (s2) Nb

3Sn (s2)

NMA 0.17$0.04 (4.19) 7.30$0.90 (1.3) 3.41$0.79 (5.81)MG 0.12$0.02 (1.62) 1.60$0.27 (1.1) 0.54$0.25 (4.51)DT 1.50$0.04 (2.49) 4.50$0.50 (1.2) 5.34$0.31 (6.53)MSWS 0.69$0.02 (1.04) 6.80$0.80 (5.7) 3.01$0.35 (3.12)

does not provide a reasonable fit to experimentaldata as the damping is strongly overestimated.

A different approach is provided by Maniv andco-workers [26—28], who focus on the normal andsuperconducting state free energies. They find anoscillatory term in the superconducting condensa-tion energy of similar magnitude, but of oppositesign, to the normal state oscillatory free energy.They associate damping of quantum oscillations inthe mixed state to the destructive interference be-tween these terms and predict an abrupt phasechange of p at a field below B

#2. They additionally

stress the importance of coherence in the scatteringof quasiparticles for the ordered vortex lattice, andfind for the damping of magnetic oscillations (tosecond order in D),

R4"1!

p3@2D2

(+u#)n1@2k

. (5)

Such a variation is consistent with Eq. (7) below, inthe limit of small D. However, we are unaware ofany experiments which have provided evidence fora phase shift of the dHvA effect at any field belowB#2

.Dukan and Tesanovic [29,30] note that, in the

quantum limit, where Landau levels can be treatedexplicitly, *(r) goes to zero at sites corresponding tothe positions of the vortex cores. The quasiparticleexcitation spectrum, calculated by analyticallysolving the BdG equations is found to be gapless ata set of points in the Brillouin zone of the vortexlattice. Their analysis yields for the damping

R4"2CC maxA

¹

D,CDBD

2expA

!pu

#q B (6)

with +q~1"2C(B) for D'2C0

and +q~1"

2C0"+q~1

0for D(2C

0. Comparison with experi-

ment shows that this functional form for R4does

not provide a good fit to the data, the quality of fitbeing the poorest of the several models discussedhere.

Finally, we turn to the theories of Maki [31],Stephen [32] and Wasserman and Springford[33,34], all of which yield for the damping,

R4"expA!

p3@2D2

(+u#)2 n1@2k B. (7)

The emphasis is on the self-energy of quasiparticlesarising from superconducting order and hence, viaits imaginary part, to Eq. (7) for the damping. Makiemploys an earlier result by Brandt et al. [35]appropriate to a clean type II superconductor atfields close to B

#2, in which only the spatially aver-

aged value of D is retained. Maki’s result is con-firmed both by Stephen in a quantum mechanicaltreatment, and by Wasserman and Springford whouse the same self-energy as Maki, but in a generalfield theoretic approach. Thus, in these theories, thedamping is determined by the spatially averagedvalue of the order parameter, a result which under-lines its insensitivity to the degree of order in thevortex lattice. The inclusion of higher-order termsrepresenting Fourier components of the 2D peri-odic order parameter associated with the orderedvortex lattice, is straightforward, but was found byWasserman and Springford not to add significantlyto the result expressed in Eq. (7).

As is evident from Table 1 and discussed in detailelsewhere [20], the most satisfactory theories are

76 C. Haworth et al. / Physica B 246—247 (1998) 73—77

those which incorporate the spatial variation of theorder parameter. However, none of the models atpresent consistently explain the data for each of thethree materials studied, indicating that more theor-etical work is required.

Acknowledgements

We acknowledge helpful correspondence anddiscussions with M. Norman and Z. Tesanovic andare grateful to J. Maniv, T. Terashima and Y.Onuki for informing us of their work prior topublication. The financial support of the EPSRC isgratefully acknowledged.

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