Low-energy quasiparticle excitations in the vortex state of the superconductor V3Si have been investigated using the de Haas-van Alphen effect. Quantum oscillations persist to surprisingly low values of B0/B(c2) is similar to 0.6 and T/T(c) is similar to 0.001. The superconducting state introduces a field-dependent quasiparticle damping which has a value HBAR tau-1 almost-equal-to 0.25 DELTA at the lowest fields investigated, considerably less than the superconducting gap DELTA. Quantum oscillations are attributed to the presence of a gapless excitation spectrum and may be a universal characteristic of superconductors in the vortex state

CORCORAN, R

HARRISON, N

HAYDEN, S M

MEESON, P

SPRINGFORD, M

VANDERWEL, P J

Royal Holloway Research Online

Quasiparticles in the vortex state of V3Si

Royal Holloway - Pure

VOLUME 72, NUMBER 5 PH YSICAL REVI EW LETTERSQuasiparticles in the Vortex State of V3Si31 JANUARY 1994R. Corcoran, N. Harrison, S. M. Hayden, P. Meeson, M. Springford, and P. J. van der WelH H . Wi. lls Physics Laboratory, University of Bristol, Tyndall Avenue. Bristol, BS8 I TL, United Kingdom(Received 2 August 1993)Low-energy quasiparticle excitations in the vortex state of the superconductor V3Si have been investi-gated using the de Haas-van Alphen effect. Quantum oscillations persist to surprisingly low values ofBo/B, 2 —0.6 and T/T, —0.001. The superconducting state introduces a field-dependent quasiparticledamping which has a value h r ' = 0.25k at the lowest fields investigated, considerably less than the su-perconducting gap h. Quantum oscillations are attributed to the presence of a gapless excitation spec-trum and may be a universal characteristic of superconductors in the vortex state.PACS numbers: 71.25.Hc, 74.70.AdThe nature of the electronic excitations in the vortexstate remains a fundamental problem in superconductivi-ty [1]. The existence of low-energy excitations is sup-ported by the observation of a field-dependent linear con-tribution [C=y(Bn)T] to the low-temperature specificheat [2], and by scanning tunneling microscopy measure-ments which reveal the presence of low-energy states(E & 2h) associated with the vortex cores [3]. However,measurements of Landau quantum oscillatory phenome-na, such as the de Haas-van Alphen (dHvA) effect, pro-vide, perhaps, the most direct probe of quasiparticle exci-tations in metals. It is nearly 20 years since the first ob-servation by Graebner and Robbins [4], who reportedboth dHvA and magnetothermal oscillations in the layercompound 2H-NbSe2 at magnetic fields below B,2. Theinfluence of Landau level quantization on the propertiesof superconductors has commanded much interest recent-ly [5-11]. This has, to a large extent, been motivated bythe prospect of observing the dHvA effect in the vortexstate of strong type-II superconductors, and hence, inprinciple, of probing rather directly the magnitude andsymmetry of the order parameter. Furthermore, dHvAoscillations have been reported in the A15 superconduc-tor V3Si [12] and in the high temperature superconductorYBa2Cu307 s [13-15].We report here the unambiguous (high signal to noise)observation of the dHvA effect in the vortex state of V3Si(X 106 nm, )=6.3 nm, tc= 17 [16]). Results obtainedby two experimental techniques on several crystals areinternally consistent, but disagree with previously report-ed data [12]. We find that quantum oscillations persistdeep into the superconducting state, at least to fields of-0.6B,2 and to temperatures of -0.001T„althoughwith an additional damping of the fermion quasiparticlescompared to the normal state. In contrast, recent mea-surements [17] on 2H-NbSe2 (A.,t, = 250 nm, g, =8 nm,tc =31 [18])also show oscillations deep into the supercon-ducting state, down to fields of -0.3B,2 and to tempera-tures of -0.003T„but with little additional damping inthe vortex state. We attribute the origin of the quantumoscillations to the existence of a gapless quasiparticle ex-citation spectrum for wave vectors in planes normal to theapplied magnetic field.Single crystals of V3Si were confirmed by x-rayanalysis to be single phase and, from resistivity measure-ments, to have a RRRR =47 (extrapolated to T 0) andT, =17 K. High field experiments in V3Si were per-formed using a pulsed magnet system described elsewhere[19], with the sample temperature in the range 1.3-2.2K. By employing magnetic fields extending to -30 T,the dHvA effect could be investigated both above andbelow B,2, which was at 18.5 T in these samples. Theeffects of eddy current heating of the samples in the rap-idly changing magnetic fields have been extensively stud-ied by us, but were effectively eliminated in the presentexperiments by using crystals of submillimeter dimen-sions. To make more sensitive measurements in the vor-tex state, a second experimental technique, using the fieldmodulation method at low frequency (5 Hz), was em-ployed in a dilution refrigerator, at temperatures down to20 mK and magnetic fields up to 13.5 T.The dHvA effect consists of an oscillatory variation ofthe magnetization M which is periodic in the inverse ap-plied magnetic field Bp ', where Bp=ppH. The frequen-cy and amplitude of the oscillations provide informationabout the low-energy quasiparticles. In the standardLifshitz-Kosevich theory [20], each extremal cross-sectional area A of the Fermi surface in the plane normalto the applied field Bn gives a fundamental oscillatorycomponent of M with frequency F= hA/2tre given byM =a(B,T) sin +p2'BpT . 2' + xgm *tx: Brj . sin p cossin hX Bp 2mxexpR'Nc&pwhere X=2tt2kaT/hta„ ta, is the cyclotron frequencyeBn/m*, and m* is the appropriately averaged effectivemass associated with the orbit around area A. Experi-mental records of the dHvA effect in V3Si are shown in0031-9OO'7/94/72 (5)/701 (4)$06.001994 The American Physical SocietyEVIE~ LETTER SpH&SICAL 31 jANUAR~VOLUME 72, NUMBER 525 20800 I700O 600500=- 4oo -ICL300EB {T)15 10P. 15— 0. 10.05 )CDC3CL-o.o5 E807060) 50CD 4030E 2O-10vortexrtormat state200 — -0. 1I I II I1000.04 0.05 O. 06~' P")-0.150. 10in V3Si, for the field along a direc-FIG.th above and beowtiondi oftll d ts obtained in athe.a-() a. ~h i I oIa 8' n 'gh -h nd scales re era' 'a a d(b), t I„.dng t athe pickup coils in ( n, rs are seen in a fromantum oscillationsw thegfiid d 1upper critical fie a, ', eh fl"'1""" '"dsu remenf i io i Fo gat 1als the p resence o a si'nt. For 80 parallel todHvA frequency.ith an average Fermi wo a od 1o o di0.1% b t thy gnorma an1 d vortex states. The resu speplicate1 to [001], it is surpnstnguency for Bo parallel to t isb f dfoave only een'n (cu-ah t calculationsartensitic hase transitiof V Si, above the mwhich occurs at T—bic to tetragonal iced angle-reso ve1 d measurements upa iio ypm extremal orbiwhich showitto +a maximum70 T derives froMueller et al. [12]s mm ' t [001]. In contrast, uetrical abouave reporte o980 T at this orientation.ntum oscillations inpFi . 1 varies wi th temperature as a0000 3000 400 1000 2F (T). 2. Fourier spectrum oove (dotted line an ei n ig.um reveals a sing eOh io fand below B,2. t erbl d' h' O. lfo above att ibuted to noise.the spectrum are'pirre ro ucire imes in accor withteno rmal and sup erconducting gimments in ewe find efi'ective mFrom measuremasses of .6andective m~hin the experimentaoriginateL d 1 1 d, o a0 g of the an ate thise,' b impurities. ea sinh(X)gbroadening from p e mxao T ] agaitlst Bg, as s 00 cz pocorrespovariation wit 0t or broadening roing rate6 I IJDG$3Q3XC:rDI8 0.09 0.15 006 007 000.03 0.04 0.0—i (r—1asinh(X)8tI T '] ttt V&Stpen enceis 1.6 K. Solid circ escond ct g aad fi ld experiments an old Eine the re-d lation technique i awit t eh latter arise mainly from t e nesuits of both experiments.702VOLUME 72, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JANUARY 1994into the vortex state, an additional attenuation of thedHvA signal is observed relative to an extrapolation ofEq. (1) into the vortex state, which is -2 at 17.4 T in-creasing to —10 at 14.3 T, and is not approximately con-stant below 8,2 as claimed by Mueller er al. [12]. Theadditional attenuation evident in Fig. 3 may be expressedas a field-dependent amplitude reduction factor, R,=exp( —z/ai, r, ), where r, '(Bti) is an additional effec-tive scattering rate experienced by the fermion quasipar-ticles in the superconducting state. Figure 4 shows thisscattering rate as a function of the reduced magneticfield, Bti/8, 2, for both V3Si and, from experiments, on2H-NbSe2 [17]. Over the field range studied, r, ' is seento be greater in V3Si than in 2H-NbSe2 by an order ofmagnitude.The persistence of quantum oscillations into the mixedstate could, in principle, be due to a small portion of thesample remaining normal. However, heat capacity mea-surements on the sample show a sharp superconductinganomaly at T, of width d, T=0.1 K and an extrapolatedlinear intercept y C/T (T 0) of less than 0.5% of thenormal state value. These values confirm that our sampleis fully transformed.In discussing the amplitude of quantum oscillations inthe vortex state, an important consideration is the extentto which the inhomogeneous field induced by the spatiallyvarying order parameter produces broadening of the Lan-dau levels. Under the experimental conditions (e.g. ,80 10.5 T), r, =130 nm and the separation of the fluxlines a = 10 nm. An estimate of the Landau levelbroadening [22] is r =vFbkF, where bkF is variationin radius of orbits cutting the same flux and t F is the Fer-mi velocity. An upper bound for the variation in area 8Aof orbits cutting the same flux is 8pbbs =Bp2xr, 8r,= b82r, a, where 88 is the extreme variation of the localfield magnetic field B(r) in the sample due to the flux lat-tice. Thus bkF hr, (eBri/ft ) = aebB/zh .Using the esti-mate [23] bB -(8,2 B—ti)/ir and vF = (2heF) ' /m*=1.6x10 ms ', we find, for Bp 10.5 T, z =2x 10' s '. %e conclude that damping arising from theinhomogeneous B field is unimportant compared with theobserved additional damping. This conclusion differsfrom that of Mueller et al. [12].It is interesting to compare our results with calcula-tions of the effect of superconductivity on the quasiparti-cles close to 8,2. In an extreme type-II superconductor,the resulting vortex lattice consists of highly overlappingflux tubes with an almost uniform magnetic field (seeabove). Within certain approximations [24], it is foundthat the spatial variation of the order parameter h(r)causes the quasiparticle excitation spectrum to vary fromBCS-like to "gapless' as the quasiparticle momentumvaries from being parallel to being perpendicular to themagnetic field. It is precisely the perpendicular momentaof quasiparticles that are probed in a dHvA experiment.Of the several recent theories relating to quantum oscilla-tions in the vortex state, those by Maki [6] and Wasser-man and Springford [10] are appropriate to the case of afully three-dimensional superconductor such as V3Si.Applicable, in principle, only for fields close to 8,2, boththeories find that the main effect of superconductivity isto introduce an additional term into the dHvA amplitudewhich can be expressed as an additional field dependent-relaxation rate,(2)in which 6 is the spatially averaged order parameter(6 =~6~ ) appropriate to the particular extremal orbitand A = (2he80) '/ . Assuming that 6 evolves accord-ing to the expression [6](3)'4 0.9I-0.6—0.30 0 0.2oppphappp p q0.4 0.6 0.8B /'BFIG. 4. Effective quasiparticle relaxation rate, r, , in thevortex state of V&Si (circles) and 2H-NbSez (squares), based onthe results of dHvA effect experiments. Filled and open sym-bols refer to pulsed field and field modulation experiments, re-spectively. The theoretical variation of r, ', based on Eqs. (2)and (3) in the text, is shown dotted for V3Si [h(0)/tt =2.6meV/fi 4.0x lO' s 'I and dashed for 2H-NbSe2 [h(0)/ti=l. l meV/h 1.7x lO' s '].the relaxation scattering rates are also plotted in Fig. 4.Given that the theory as expressed by Eqs. (2) and (3) isonly expected to apply close to 8,2, the result is surpris-ingly good, and it reflects the difference between the twomaterials approximately correctly. This theory also pre-dicts that the effective mass of the quasiparticle statesprobed by dHvA will be essentially unchanged on passingthrough 8,2, in agreement with our observations.In summary, we have observed de Haas-van Alphenoscillations deep in the mixed state of V3Si. The ex-istence of quantum oscillations in the mixed state is thusnot confined to strongly anisotropic systems such as 2H-NbSe2. An additional suppression of the dHvA oscilla-tions is observed in the mixed state, which cannot be ex-plained by the eA'ect of the inhomogeneous B field insidethe superconductor associated with the spatially varyingorder parameter in the vortex state. %e attribute the ad-ditional suppression to an eN'ective scattering of quasipar-ticles in the superconductor due to the direct influence of703YOLUME 72, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JANUARY 1994the (orbitally averaged) order parameter. We note thatthe largest measured values of br ' (0.63 meV for V3Siand 0.06 meV for 2H-NbSez) are considerably less thanthe gap parameters [d(0) =2.6 meV for V3Si [25] andI. l meV for 2H-NbSez [26]]. Finally, we note that theresults in Fig. 4 refer to a particular region (extremal or-bit) on the Fermi surface and so illustrate, in principle,how such experiments, made for a range of orientations,could be used to obtain information on the magnitudeand symmetry of the order parameter rather directly.Such experimental work is in progress, but it is clear thatwe require a deeper theoretical understanding of quan-turn oscillations in the vortex state in the limit far fromB,2 before they can be interpreted in detail.We are grateful to M. de Podestra and C. Lester forperforming the specific heat measurements and to A.Menovsky for providing V3Si crystals. Helpful discus-sions with B. Gyorffy, A. Junod, and A. Wasserman arealso acknowledged.[I] P. G. de Gennes, Superconductivity of Metals and Alloys(Addison-Wesley, Reading, MA, 1989).[2] G. R. Stewart and B. L. Brandt, Phys. Rev. 8 29, 3908(1984).[3] H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles,and J. V. Waszczak, Phys. Rev. Lett. 62, 214 (1989).[4] J. E. Graebner and M. Robbins, Phys. Rev. Lett. 36, 422(1976).[5] R. S. Markiewicz, I. D. Wagner, P. Wyder and T. Maniv,Solid State Commun. 67, 43 (1988).[6] K. Maki, Phys. Rev. 8 44, 2861 (1991); Physica (Am-sterdam) 8 (to be published).[7] M. J. Stephen, Phys. Rev. 8 43, 1212 (1991).[8] S. Dukan, A. V. Andreev and Z. Tesanovic, Physica (Am-sterdam) l$3C, 355 (1991).[9] T. Maniv, A. I. Rom, I. D. Vagner and P. Wyder, Phys.Rev. 8 46, 8360 (1992).[10] A. Wasserman and M. Springford, Physica (Amsterdam)8 (to be published).[11]K. Miyake, Physica (Amsterdam) 8 (to be published).[12] F. M. Mueller, D. H. Lowndes, Y. K. Chang, A. J. Arkoand R. S. List, Phys. Rev. Lett. 6$, 3928 (1992).[13]C. M. Fowler, B. L. Freeman, W. L. Hults, J. C. King, F.M. Mueller and 3. L. Smith, Phys. Rev. Lett. 68, 534(1992).[14] G. Kido, K. Komorita, H. Katayama-Yoshida and T.Takahashi, J. Phys. Chem. Solids 52, 465 (1991).[15] M. Springford, N. Harrison, P. Meeson and P.-A. Probst,Phys. Rev. Lett. 69, 2453 (1993).[16] D. K. Christen, H. R. Kerchner, S. T. Sekula, and Y. K.Chang, Physica (Amsterdam) 135B, 369 (1985).[17] R. Corcoran, P. Meeson, Y. Onuki, M. Springford, andK. Takita, Physica (Amsterdam) 8 (to be published).[18] L. P. Le, G. M. Luke, B. J. Sternlieb, W. D. Wu, Y. J.Uemura, J. W. Brill, and H. Drulis, Physica (Amster-dam) 1$5-1$9C, 2715 (1991).[19]N. Harrison, P. Meeson, P.-A. Probst and M. Springford,Meas. Sci. Technol. 4, 4210 (1993).[20] I. M. Lifshitz and A. M. Kosevich, Zh. Eksp. Teor. Fiz.29, 730 (1953).[21] T. Jarlborg, A. A. Manuel, and M. Peter, Phys. Rev. 827, 4210 (1983).[22] R. Corcoran, N. Harrison, S. M. Hayden, P. Meeson, M.Springford, and P. J. van der Wel (to be published).[23] E. H. Brandt, Phys. Rev. 8 1$, 6022 (1978).[24] U. Brandt, W. Pesch, and L. Tewordt, Z. Phys. 201, 209(1967).[25] R. Hackl and R. Kaiser, J. Phys. C 21, 453 (1988).[26] B. P. Clayman and R. F. Frindt, Solid State Commun. 9,1881 (1971).704

VOLUME 72, NUMBER 5 PH YSICAL REVI EW LETTERSQuasiparticles in the Vortex State of V3Si31 JANUARY 1994R. Corcoran, N. Harrison, S. M. Hayden, P. Meeson, M. Springford, and P. J. van der WelH H . Wi. lls Physics Laboratory, University of Bristol, Tyndall Avenue. Bristol, BS8 I TL, United Kingdom(Received 2 August 1993)Low-energy quasiparticle excitations in the vortex state of the superconductor V3Si have been investi-gated using the de Haas-van Alphen effect. Quantum oscillations persist to surprisingly low values ofBo/B, 2 —0.6 and T/T, —0.001. The superconducting state introduces a field-dependent quasiparticledamping which has a value h r ' = 0.25k at the lowest fields investigated, considerably less than the su-perconducting gap h. Quantum oscillations are attributed to the presence of a gapless excitation spec-trum and may be a universal characteristic of superconductors in the vortex state.PACS numbers: 71.25.Hc, 74.70.AdThe nature of the electronic excitations in the vortexstate remains a fundamental problem in superconductivi-ty [1]. The existence of low-energy excitations is sup-ported by the observation of a field-dependent linear con-tribution [C=y(Bn)T] to the low-temperature specificheat [2], and by scanning tunneling microscopy measure-ments which reveal the presence of low-energy states(E & 2h) associated with the vortex cores [3]. However,measurements of Landau quantum oscillatory phenome-na, such as the de Haas-van Alphen (dHvA) effect, pro-vide, perhaps, the most direct probe of quasiparticle exci-tations in metals. It is nearly 20 years since the first ob-servation by Graebner and Robbins [4], who reportedboth dHvA and magnetothermal oscillations in the layercompound 2H-NbSe2 at magnetic fields below B,2. Theinfluence of Landau level quantization on the propertiesof superconductors has commanded much interest recent-ly [5-11]. This has, to a large extent, been motivated bythe prospect of observing the dHvA effect in the vortexstate of strong type-II superconductors, and hence, inprinciple, of probing rather directly the magnitude andsymmetry of the order parameter. Furthermore, dHvAoscillations have been reported in the A15 superconduc-tor V3Si [12] and in the high temperature superconductorYBa2Cu307 s [13-15].We report here the unambiguous (high signal to noise)observation of the dHvA effect in the vortex state of V3Si(X 106 nm, )=6.3 nm, tc= 17 [16]). Results obtainedby two experimental techniques on several crystals areinternally consistent, but disagree with previously report-ed data [12]. We find that quantum oscillations persistdeep into the superconducting state, at least to fields of-0.6B,2 and to temperatures of -0.001T„althoughwith an additional damping of the fermion quasiparticlescompared to the normal state. In contrast, recent mea-surements [17] on 2H-NbSe2 (A.,t, = 250 nm, g, =8 nm,tc =31 [18])also show oscillations deep into the supercon-ducting state, down to fields of -0.3B,2 and to tempera-tures of -0.003T„but with little additional damping inthe vortex state. We attribute the origin of the quantumoscillations to the existence of a gapless quasiparticle ex-citation spectrum for wave vectors in planes normal to theapplied magnetic field.Single crystals of V3Si were confirmed by x-rayanalysis to be single phase and, from resistivity measure-ments, to have a RRRR =47 (extrapolated to T 0) andT, =17 K. High field experiments in V3Si were per-formed using a pulsed magnet system described elsewhere[19], with the sample temperature in the range 1.3-2.2K. By employing magnetic fields extending to -30 T,the dHvA effect could be investigated both above andbelow B,2, which was at 18.5 T in these samples. Theeffects of eddy current heating of the samples in the rap-idly changing magnetic fields have been extensively stud-ied by us, but were effectively eliminated in the presentexperiments by using crystals of submillimeter dimen-sions. To make more sensitive measurements in the vor-tex state, a second experimental technique, using the fieldmodulation method at low frequency (5 Hz), was em-ployed in a dilution refrigerator, at temperatures down to20 mK and magnetic fields up to 13.5 T.The dHvA effect consists of an oscillatory variation ofthe magnetization M which is periodic in the inverse ap-plied magnetic field Bp ', where Bp=ppH. The frequen-cy and amplitude of the oscillations provide informationabout the low-energy quasiparticles. In the standardLifshitz-Kosevich theory [20], each extremal cross-sectional area A of the Fermi surface in the plane normalto the applied field Bn gives a fundamental oscillatorycomponent of M with frequency F= hA/2tre given byM =a(B,T) sin +p2'BpT . 2' + xgm *tx: Brj . sin p cossin hX Bp 2mxexpR'Nc&pwhere X=2tt2kaT/hta„ ta, is the cyclotron frequencyeBn/m*, and m* is the appropriately averaged effectivemass associated with the orbit around area A. Experi-mental records of the dHvA effect in V3Si are shown in0031-9OO'7/94/72 (5)/701 (4)$06.001994 The American Physical SocietyEVIE~ LETTER SpH&SICAL 31 jANUAR~VOLUME 72, NUMBER 525 20800I700O 600500=- 4oo -ICL300EB {T)15 10P. 15— 0. 10.05 )CDC3CL-o.o5 E807060)50CD4030E2O-10vortexrtormat state200 — -0. 1I I II I1000.04 0.05 O. 06~ ' P")-0.150. 10in V3Si, for the field along a direc-FIG.th above and beowtiondi oftll d ts obtained in athe.a-() a. ~h i I oIa 8' n 'gh -h nd scales re era' 'a a d(b), t I„.dng t athe pickup coils in ( n, rs are seen in a fromantum oscillationsw thegfiid d 1upper critical fie a, ', eh fl"'1""" '"dsu remenf i io i Fo gat 1als the p resence o a si'nt. For 80 parallel todHvA frequency.ith an average Fermi wo a od 1o o di0.1% b t thy gnorma an1 d vortex states. The resu speplicate1 to [001], it is surpnstnguency for Bo parallel to t isb f dfoave only een'n (cu-ah t calculationsartensitic hase transitiof V Si, above the mwhich occurs at T—bic to tetragonal iced angle-reso ve1 d measurements upa iio ypm extremal orbiwhich showitto +a maximum70 T derives froMueller et al. [12]s mm't [001]. In contrast, uetrical abouave reporte o980 T at this orientation.ntum oscillations inpFi . 1 varies wi th temperature as a0000 3000 400 1000 2F (T). 2. Fourier spectrum oove (dotted line an ei n ig.um reveals a sing eOh io fand below B,2. t erbl d' h' O. lfo above att ibuted to noise.the spectrum are'pirre ro ucire imes in accor withteno rmal and sup erconducting gimments in ewe find efi'ective mFrom measuremasses of .6andective m ~ hin the experimentaoriginateL d 1 1 d, o a0 g of the an ate thise,'b impurities. ea sinh(X)gbroadening from p e mxao T ] agaitlst Bg, as s 00 cz pocorrespovariation wit 0t or broadening roing rate6 I IJDG$3Q3XC:rDI8 0.09 0.15 006 007 000.03 0.04 0.0—i (r—1asinh(X)8tI T '] ttt V&Stpen enceis 1.6 K. Solid circ escond ct g aad fi ld experiments an old Eine the re-d lation technique i awit t eh latter arise mainly from t e nesuits of both experiments.702VOLUME 72, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JANUARY 1994into the vortex state, an additional attenuation of thedHvA signal is observed relative to an extrapolation ofEq. (1) into the vortex state, which is -2 at 17.4 T in-creasing to —10 at 14.3 T, and is not approximately con-stant below 8,2 as claimed by Mueller er al. [12]. Theadditional attenuation evident in Fig. 3 may be expressedas a field-dependent amplitude reduction factor, R,=exp( —z/ai, r, ), where r, '(Bti) is an additional effec-tive scattering rate experienced by the fermion quasipar-ticles in the superconducting state. Figure 4 shows thisscattering rate as a function of the reduced magneticfield, Bti/8, 2, for both V3Si and, from experiments, on2H-NbSe2 [17]. Over the field range studied, r, ' is seento be greater in V3Si than in 2H-NbSe2 by an order ofmagnitude.The persistence of quantum oscillations into the mixedstate could, in principle, be due to a small portion of thesample remaining normal. However, heat capacity mea-surements on the sample show a sharp superconductinganomaly at T, of width d, T=0.1 K and an extrapolatedlinear intercept y C/T (T 0) of less than 0.5% of thenormal state value. These values confirm that our sampleis fully transformed.In discussing the amplitude of quantum oscillations inthe vortex state, an important consideration is the extentto which the inhomogeneous field induced by the spatiallyvarying order parameter produces broadening of the Lan-dau levels. Under the experimental conditions (e.g. ,80 10.5 T), r, =130 nm and the separation of the fluxlines a = 10 nm. An estimate of the Landau levelbroadening [22] is r =vFbkF, where bkF is variationin radius of orbits cutting the same flux and t F is the Fer-mi velocity. An upper bound for the variation in area 8Aof orbits cutting the same flux is 8pbbs =Bp2xr, 8r,= b82r, a, where 88 is the extreme variation of the localfield magnetic field B(r) in the sample due to the flux lat-tice. Thus bkF hr, (eBri/ft ) = aebB/zh .Using the esti-mate [23] bB -(8,2 B—ti)/ir and vF = (2heF) ' /m*=1.6x10 ms ', we find, for Bp 10.5 T, z =2x 10' s '. %e conclude that damping arising from theinhomogeneous B field is unimportant compared with theobserved additional damping. This conclusion differsfrom that of Mueller et al. [12].It is interesting to compare our results with calcula-tions of the effect of superconductivity on the quasiparti-cles close to 8,2. In an extreme type-II superconductor,the resulting vortex lattice consists of highly overlappingflux tubes with an almost uniform magnetic field (seeabove). Within certain approximations [24], it is foundthat the spatial variation of the order parameter h(r)causes the quasiparticle excitation spectrum to vary fromBCS-like to "gapless' as the quasiparticle momentumvaries from being parallel to being perpendicular to themagnetic field. It is precisely the perpendicular momentaof quasiparticles that are probed in a dHvA experiment.Of the several recent theories relating to quantum oscilla-tions in the vortex state, those by Maki [6] and Wasser-man and Springford [10] are appropriate to the case of afully three-dimensional superconductor such as V3Si.Applicable, in principle, only for fields close to 8,2, boththeories find that the main effect of superconductivity isto introduce an additional term into the dHvA amplitudewhich can be expressed as an additional field dependent-relaxation rate,(2)in which 6 is the spatially averaged order parameter(6 =~6~ ) appropriate to the particular extremal orbitand A = (2he80) '/ . Assuming that 6 evolves accord-ing to the expression [6](3)'4 0.9I-0.6—0.300 0.2oppphappp p q0.4 0.6 0.8B /'BFIG. 4. Effective quasiparticle relaxation rate, r, , in thevortex state of V&Si (circles) and 2H-NbSez (squares), based onthe results of dHvA effect experiments. Filled and open sym-bols refer to pulsed field and field modulation experiments, re-spectively. The theoretical variation of r, ', based on Eqs. (2)and (3) in the text, is shown dotted for V3Si [h(0)/tt =2.6meV/fi 4.0x lO' s 'I and dashed for 2H-NbSe2 [h(0)/ti=l. l meV/h 1.7x lO' s '].the relaxation scattering rates are also plotted in Fig. 4.Given that the theory as expressed by Eqs. (2) and (3) isonly expected to apply close to 8,2, the result is surpris-ingly good, and it reflects the difference between the twomaterials approximately correctly. This theory also pre-dicts that the effective mass of the quasiparticle statesprobed by dHvA will be essentially unchanged on passingthrough 8,2, in agreement with our observations.In summary, we have observed de Haas-van Alphenoscillations deep in the mixed state of V3Si. The ex-istence of quantum oscillations in the mixed state is thusnot confined to strongly anisotropic systems such as 2H-NbSe2. An additional suppression of the dHvA oscilla-tions is observed in the mixed state, which cannot be ex-plained by the eA'ect of the inhomogeneous B field insidethe superconductor associated with the spatially varyingorder parameter in the vortex state. %e attribute the ad-ditional suppression to an eN'ective scattering of quasipar-ticles in the superconductor due to the direct influence of703YOLUME 72, NUMBER 5 PHYSICAL REVIEW LETTERS 31 JANUARY 1994the (orbitally averaged) order parameter. We note thatthe largest measured values of br ' (0.63 meV for V3Siand 0.06 meV for 2H-NbSez) are considerably less thanthe gap parameters [d(0) =2.6 meV for V3Si [25] andI. l meV for 2H-NbSez [26]]. 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Quasiparticles in the vortex state of V3Si

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