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Magnus force and acoustic Stewart-Tolman effect in type II superconductors V.D.Fil1, D.V.Fil2, A.N.Zholobenko1, N.G.Burma1, Yu.A.Avramenko1, J.D.Kim3, S.M.Choi3, and S.I.Lee3,4 1B.Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine, Lenin av. 47 Kharkov 61103, Ukraine 2Institute for Single Crystals, National Academy of Sciences of Ukraine, Lenin av. 60, Kharkov 61001, Ukraine 3Pohang University of Science and Technology, Pohang, 794784, Korea 6 0 4Korea Basic Science Institute, Daejeon, 305333, Korea 0 2 Atzero magnetic fieldwehaveobservedan electromagnetic radiation from superconductorssub- jected by a transverse elastic wave. This radiation has an inertial origin, and is a manifestation of n the acoustic Stewart-Tolman effect. The effect is used for implementing a method of measurement a ofan effectiveMagnusforce intypeII superconductors. Themethod doesnot requirethefluxflow J regime andallows toinvestigate thisforce for almost thewhole rangeof theexistenceof themixed 6 state. Wehavestudied behavior of thegyroscopic force in nonmagnetic borocarbides and Nb. It is found that in borocarbides thesign of the gyroscopic force in themixed state is thesame as in the ] n normal state, and its value (counted for one vortex of unit length) has only a weak dependenceon o the magnetic field. In Nb the change of sign of the gyroscopic force under the transition from the c normal to themixed state is observed. - r PACSnumbers: 74.25.Qt,74.70.Ad p u s . The Stewart-Tolman (ST) effect, the emergence of an t a electrical current in a coil under an accelerate or decel- (cid:23)(cid:19) m erate rotation, is a famous experimental proof for the (cid:20)(cid:17)(cid:19) < 7E 1L % & - electron nature of conductivity of metals. The inertial + (cid:19)(cid:17)(cid:28)(cid:24) (cid:19)(cid:17)(cid:19)(cid:24) (cid:21) (cid:21) ond feoxrccietesdhobuyldagtrivaensavecrosnetraicbouutsiotinctwoatvheeperloepctargiacatilncguirnretnhtes _(_(cid:3)(cid:18)\ (cid:22)(cid:19) HJ(cid:12) c metal (the acoustic ST effect). Nevertheless, it was be- (cid:19)(cid:17)(cid:24) '/ (cid:21)(cid:19) (cid:11)G [ lievedthatthis effect is negligible anddoes nothaveany S T V D a( F 1 practicaluse. Asfarasweknow,uptonowtherewasno X(cid:19) \ (cid:20)(cid:19) v efforts to register inertial force in acoustic experiments. 6 InourstudywehaveobservedtheacousticSTeffectand + [ ] 1 use it for a developmentof a powerfulmethod ofinvesti- (cid:19) 1 (cid:3)(cid:19) (cid:20) (cid:21) (cid:22) (cid:23) gationofthegyroscopicforcesactingonamovingvortex 1 +(cid:11)7(cid:12) in type II superconductors. 0 6 Our study was motivated by the present situation in 0 understanding of the nature of the gyroscopic force (the FIG. 1: The modulus (solid line) and the phase (dots) of / t effective Magnus force[1]) in superconductors. Most of Ey(H) in Y0.95Tb0.05Ni2B2C at T = 1.7 K (Hc2 = 3.8 T). a m all, it concerns the so called Hall anomaly - the change Theratio|Ey|/H isnormalizedasdescribedinthetext. Inset of sign of the Hall voltage at the superconducting tran- - the sketch of the experiment (p, the piezotransducer, DL, - thedelay line, s, the sample, and a, theantenna) sition. Accordingtocurrenttheoreticalconceptions,this d n effect may have both the microscopic origin, connected o with peculiarities of the electron spectrum [2], and the c macroscopic one: the appearance of the transverse force range of the existence of the mixed state. : v underlarge(muchlargethanthecoresize)displacements The method is based on the measurements of the am- i of the vorticesin the pinning potential [3]. The flux flow plitude and the phase of an electromagnetic field (EF) X experimentsdidnotallowtoseparatethesetwocontribu- radiatedfromthesuperconductorduetothevortexoscil- r a tions. The macroscopic effects may remove the genuine lations excited by a transverse elastic wave. The scheme (microscopic) Hall anomaly or, on the contrary, mimic of the experiment and the frame of reference are shown it. Forinstance,inNbthe Hallanomalywasobservedor in Fig.1(inset). Technically, the method is close to the not, depending on the sample quality [4]. one used in [5]. A key new point is the registration not To overcome this difficulty we provide measurements only large Ey, but also small Ex component of the EF of the Magnus force under an oscillatory motion of the field. The Ex component in the Meissner state is caused vortices with a small (of the atomic scale) amplitude of solely by the acoustic ST effect. the displacements. It allows to exclude the influence of In our study we used the single crystals of non- the pinning forces. Another advantage of the method is magnetic borocarbides (YNi B C, Y Tb Ni B C 2 2 0.95 0.05 2 2 the ability to fulfill measurements in almost the whole and LuNi B C) and Nb as the objects of the investiga- 2 2 2 tions. The borocarbide single crystals were prepared by connectsthe currentwiththe fieldE andthe extraneous thesamemethodasin[6]. Thesampleshavetheshapeof forces, and Eq. (5) is the Galilean invariant equation plateletswiththethickness∼0.5mmandthetransverse of motion of the vortex lattice, normalized to the unit size ∼3 mm. In all samples the C axis was orientedor- volume [9]. The boundary condition for the system (3)- 4 thogonalto the platelet plane, whichwas usedas the ra- (5) is dE/dz| ≈0 [8]. z=0 diatedsurface. Theworkingfrequenciesare54÷55MHz, Taking into account that for typical superconductors and the intensity of the exciting signal is ∼10 W/cm2. α ≪ η and q2λ2 ≪ 1 (λ is the London penetration M L L The details of the measuring technics are given in [7]. depth) we obtain Inthe normalstatethe E andE componentscanbe x y found from the joint solution of the Maxwell equations Ex(m) =[1−iβdm(B)]X(B)uST+ and the kinetic equation. With accounting for the gy- iωαM [1−X(B)]X(B) u , (6) ind roscopic forces this problem was solved in [8]. Let the (cid:20)iωη+αL(cid:21) vector of the elastic displacements is aligned along the E(m) =X(B)u . (7) x axis (u(z,t)=(u cos(qz)eiωt,0,0)). In the local limit y ind 0 (ql ≪ 1, q is the wave number, and l is the mean free Here u = iωBu /c, and X(B) = k2 /(1+k2 ) with ind 0 m m path) and for |Ω|τ ≪ 1 (Ω is the cyclotron frequency, k2 = 4π(iωη +α )/(q2B2). The deformation interac- m L and τ is the relaxationtime) the electrical field radiated tion is not included into Eqs. (3)-(5). But since the from the sample at z =0 has the form: structure of Eqs. (6), (7) is similar to one of Eqs. (1), (2), we add the deformationcorrectioninto Eq. (6) phe- E(n) =[1−iβ ]Xu +[1−X]XΩτu , (1) x d ST ind nomenologically. One can assume that the quantity β dm Ey(n) =Xuind, (2) is approximated as βdm(B) ≈ βdB/Hc2. Going ahead, weemphasizethatforthemeasurementofα wedonot where X = k2/(1 + k2) with the dimensionless pa- M n n need the exact form of β (B) rameter k2 = 4πiωσ /q2c2 (σ = ne2τ/m, the static dm n 0 0 The experimental procedure consists in the measuring conductivity), u and u are the extraneous forces: ST ind of the relative changes of the fields E and E under u =m ω2u /|e| is the ST inertial force (m and e are x y ST e 0 e a variation of H. To obtain the dynamical parameters the mass and the charge of free electron, respectively) from the experimental data one should set the reference and u = iωHu /c is the inductive force, H is the ex- ind 0 points for the amplitude and the phase of E and E . ternal magnetic field. The sign of charge of the carriers x y In other words, at certain H we should determine the is included into the definition of Ω. The term in Eq.(1) relation between E , E and u . ∝ iβ is caused by the deformation interaction. For the x y 0 d According to the Bardin-Stephen estimate the viscos- quadratic spectrum of the carriers β = (1/5)(v /s)ql, d F ity η ∝ B. Therefore, at H close to H the quan- where v is the Fermi velocity, and s is the sound ve- c1 F tity |k2 | ≫ 1, and the modulus and the phase of X locity. For the magnetic field used in the experiment m approaches to 1 and 0, correspondingly. Let us count |u /u | = Ω/ω ≫ 1. This strong inequality makes ind ST the phase of E from its value at H → H . Then the magnitude of the second (gyroscopic) term in Eq. y c1 argE = argX. On the other hand, at H = H the (1) comparable or larger than that of the ST term, and quantyity |X(H )| = [1+tan2argX(H )]−1/2 andc2it is allows to neglect the term ∝u in Eq. (2). c2 c2 ST convenient to choose such units for |E | that satisfy the In the mixed state the EF can be evaluated from the y relation |E (H )|/H =|X(H )|. following system of equations y c2 c2 c2 If the Ginzburg-Landau parameter κ ≫ 1, one can d2E 4πiω neglect the difference between B and H in almost the = j, (3) dz2 c2 whole range of H where the mixed state exists. Then, at H ∼ (5÷10)H (in which |k2 | ≫ 1) the quantity iω c1 m j=σ E+uST + c [w×B] , (4) |Ey(H)|/H =|X(H)|(intheunitsof|Ey|chosen)should (cid:18) (cid:19) approach unity. The fulfillment of this condition can be 1 [j×B]+(iωη+α )(u−w) consideredasanindependenttestforthe measuringpro- L c B cedureproposed. Ifthis conditionfailsitcanbe duetoa +iωαM[(u−w)× ]=0, (5) strong non-uniformity of the conductivity near the sur- |B| face of the sample (see [5]), and in this case one cannot where B is the magnetic induction, j is the current, w, use the relations(6), (7) for the analysisofthe results of the displacement of the vortex lattice, σ = n e2/iωm, the measurements. In the experiments presented in this s the a.c. conductivity in the superconducting state (n paper the above mentioned condition was fulfilled with s is the superfluid density), the dynamical parameters η, the accuracy ∼ 5%. The typical example of the experi- α , α , are the viscosity, the Labush parameter, and mental data is presented in Fig.1. From these data one L M the Magnus coefficient, correspondingly. Eq. (3) is the can find the dependencies η(H) and α (H) [10]. L Maxwell equation, Eq. (4) is the matter equation that At H = 0 and T ≪ T the deformation correction is c 3 (cid:3) (cid:3) (cid:24)(cid:19) (cid:12) (cid:11)D(cid:12) J (cid:21) (cid:22)(cid:19) H (cid:20)(cid:17)(cid:19) G (cid:19) (cid:20) <(cid:19)(cid:17)(cid:28)(cid:24)7E(cid:19)(cid:17)(cid:19)(cid:24)1L(cid:21)%(cid:21)& (cid:12)(cid:11) (cid:12)7 (_(cid:18)X[67(cid:19)(cid:19)(cid:17)(cid:17)(cid:27)(cid:28) 1E (cid:20)(cid:21)(cid:19)(cid:19) ((cid:18)X[67 (cid:11)((cid:18)X[6(cid:16)(cid:24)(cid:19) (cid:3) (cid:21) (cid:20) (cid:11) P _ J (cid:19)(cid:17)(cid:26) (cid:19) $U (cid:15)(cid:3), (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:3) (cid:12)7 (cid:21) 6 (cid:25) (cid:27) (cid:20)(cid:19) (cid:20)(cid:21) (cid:20)(cid:23) (cid:20)(cid:25) (cid:20)(cid:27) (cid:21)(cid:19) (cid:21)(cid:21) X (cid:11)E(cid:12) (cid:18) 7(cid:11).(cid:12) ([ (cid:20) (cid:11) H 5 FIG. 2: The modulus (solid lines) and the phase (dashed lines)oftheEFcausedbytheacousticStewart-Tolmaneffect (cid:19) (H=0) (cid:19) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:23) (cid:19)(cid:17)(cid:25) +(cid:11)7(cid:12) frozen, and, as follows from Eqs. (3), (4), E = u . It x ST means that under such conditions the inertial force be- FIG.3: Thereal(solidlines)andtheimaginary(dottedlines) comesthe only sourceforthe EFradiatedfromthe sam- parts of Ex for the borocarbides (1 - Y0.95Tb0.05Ni2B2C at ple. Therefore, it is convenient to choose the amplitude T = 1.7 K, 2 - LuNi2B2C at T = 9 K) (a) and for Nb at and the phase of the E signal at H =0 and T ≪T as T =1.7 K (b). Thearrows indicate Hc2. x c the reference points for the complex quantity E (H,T). x At T > T and H = 0 the component E is given by c x with the transport data [11]. At the same time, in Nb the first term in the r.h.s. of Eq. (1). In this case it the carriers are the holes (as well known, see. e.g. [12]). contains the contribution of the deformation force. But inratherdirtysamples(|k−2|>β )theE fieldisdeter- Eq. (6) can be presented in the form n d x mined in the main part by the ST effect. In such sam- E (H) [1−X(H)]2 ωα |e|H ples the amplitude of the radiated EF should increase, x =1−iβ (H)− M . while its phase should decrease under transition to the uSTX(H) dm X(H) "q2H2#meωc 4π superconducting state. The experimental dependencies (8) |E (T)| and argE (T) presented in Fig.2 demonstrate One can see that for real valued β and α the func- x x dm M such behavior. We consider these results as the first ex- tion α (H) can be found from the real part of Eq. (8) M perimental observation of this effect. We would like to without the knowledge of the exact form of β (H). dm emphasize that for our method the registration of the Eq. (8) is applicable for the normal state, as well, EFradiationcausedbythe STeffectplaysthevitalrole, under the replacements ofthe ratio 4πωα /q2H2 in the M sinceonlyinthiscaseonecanobtaintrueamplitudeand r.h.s. of Eq. (8) to the quantity |k2|Ωτ. To present n phase characteristics of the E (H) dependence. the data in a compact form it is convenient to define x Let us mention one technical detail important for formally the parameter ωα at H > H as ωα = M c2 M the measurement of E . When the receiving antenna q2|k2|H2 Ωτ/4π. x n c2 is oriented along the weak component E , the mea- The obtained dependencies of the Magnus coefficient x sured field E (H) contains the admixture of the E on the magnetic field for the borocarbides are shown in meas y component caused by the edge effects: E (H) = Fig.4. OnecanseefromFig.4thatfortheY-basedboro- meas E (H)+ γE (H), where γ is the complex valued geo- carbidesthedependenceωα (H)isroughlylinearinthe x y M metrical factor that remains nonzero at any orientation whole range of the magnetic field. At H > H one can c2 of the antenna. The extraction of the E (H) compo- expect the linear increase of ωα (H) just from the defi- x M nent can be done using different parity of the E (H) nitionofthisquantityinthenormalstate(seeabove). At x and E (H) components with respect to H: E (H) = H < H the linear dependence is no more than the re- y x c2 [E (H)+E (−H)]/2. The examples of the real flection ofthe fact that the Magnusforce is proportional meas meas (in-phase with u ) and the imaginary (quadrature) to the number of vortices. In LuNi B C the dependence ST 2 2 parts of the E field are shown in Fig.3. ωα (H) is almost saturated in the normal state. The x M Onecaneasilyfindthatinthenormalstatethesignof latter is connected with a strong non-linearity of Ωτ, re- theimaginarypartofthesecondterminther.h.s. ofEq. ported, for the first time, in Ref. [11]. We note that the (1) coincides with the sign of Ω. Thus, in borocarbides coefficient α obtained is in two orders of magnitude M the carriers are of the electron type that is in agreement smaller than η (Ref. [10]) at the same H. 4 neglect the difference between B and H. To apply the (cid:12) (cid:23)P (cid:19) procedure of finding of αM described above we should (cid:20) F know the magnetic permeability µ(H)=B/H. To eval- Q(cid:18)(cid:16)(cid:21) (cid:23) uate this function we use the phenomenological expres- \ (cid:25) (cid:21)G (cid:21) sionforη: iωη=kn2(BHc2/4π)[1+δ(1−B2/Hc22)],where (cid:20) δ is the fitting parameter. This expression is in better (cid:19)(cid:16)(cid:23) (cid:3) (cid:20) agreement with the theoretical estimates [2] and with (cid:11)M (cid:24) the experiment [5] than the Bardin-Stephen formula. If a (cid:16)(cid:25) w the quantity iωη is given, the dependencies µ(H) and (cid:22) α (H) can be determined from the experimental data. L (cid:19) (cid:21) (cid:23) Theresultforµ(H)shouldbeasmoothboundedfunction +(cid:11)7(cid:12) (|E (H)|/H ≤ µ(H) ≤ 1, where E is in units defined y y above). This requirement is satisfied for δ ≈ 0.6÷0.7. Thefurtherprocedureoftheobtainingofα isthesame FIG. 4: Field dependencies of the Magnus coefficient. Solid M lines 1,2, and 3 - Y0.95Tb0.05Ni2B2 at T = 8 K (Hc2 = 1.7 as before. The result for Nb is also presented in Fig. 5. T); 4 K (3.3 T); and 1.7 K (3.8 T), correspondingly; dotted The mainqualitativeconclusionis thatinNb the signof lines 4 and 5 - YNi2B2C at 8 K (2 T) and 5 K (3.8 T), the Magnus force in the mixed state is opposite to the correspondingly; dashed line 6 - LuNi2B2C at 9 K (2.15 T). sign of the gyroscopic force in the normal state. In conclusion, we have used the acoustic ST effect for realizationofthe methodofmeasurementofthe Magnus (cid:23) (cid:21) force in type II superconductors that does not require P (cid:24) (cid:21) the overthresholdflow of vortices. The measurements in F (cid:22)(cid:12) (cid:19) non-magneticborocarbideswithconductivityoftheelec- (cid:16)(cid:20)(cid:19) (cid:21) tron type show that the gyroscopicforce remains almost (cid:20)(cid:16)(cid:21) (cid:22) (cid:3) unchanged under the transition from the normal to the (cid:11) M a (cid:16)(cid:23) (cid:20) mixed state. At the same time, in Nb this force changes w its sign below the transition point. (cid:16)(cid:25) (cid:23) This study is supported in part by CRDF Foundation (cid:16)(cid:27) (cid:19) (Grant No UP1-2566-KH-03) and by INTAS (Grant No (cid:19)(cid:17)(cid:24) (cid:20)(cid:17)(cid:19) (cid:20)(cid:17)(cid:24) +(cid:18)+ 03-51-3036). We would like to thank N.B.Kopnin for &(cid:21) helpful discussions. FIG.5: TheMagnuscoefficientforonevortexofunitlength. 1, 2-Y0.95Tb0.05Ni2B2 at 8K and1.7 K,correspondingly; 3 - YNi2B2C at 8 K; 4 - LuNi2B2C at 9 K; 5 - Nb at 1.7 K. [1] E.B.Sonin, Phys. Rev.B 55,485 (1997). [2] N. B. Kopnin, Rep.Progr. Phys. 65, 1633 (2002). ItisinterestingtocomparethevalueoftheHallcoeffi- [3] N.B.Kopnin,V.M.Vinokur,Phys.Rev.Lett.83,4864 cient R =Ωτ/(σ H) that follows from our experimen- H 0 (1999). tal data with the results obtained by the transportmea- [4] K. Noto, S.Shinzawa, Y. Muto, Sol. St. Commun. 18, surements. For the Y-based compounds at H =4 T and 1081 (1976). T =5KwehaveR ≈2.8·10−12Ω·cm·Oe−1thatalmost [5] V. D. Fil, D. V. Fil, Yu. A. Avramenko, A. L. Gaiduk, H coincides with the results of Ref. [11]. For LuNi B C W. L. Johnson, Phys.Rev.B 71, 092504 (2005). 2 2 at H = 4 T and T = 9 K we find R ≈ 5.3· 10−12 [6] M. O. Mun, S. I. Lee, W. C. Lee, P. C. Canfield, B. K. Ω·cm·Oe−1 that is also close to the valHue given in [11]. Cho, D.C. Johnston, Phys. Rev.Lett 76, 2790 (1996). [7] E. A. Masalitin, V. D. Fil, K. N. Zhekov, A. N. For Nb we have RH ≈ 6.5·10−13 Ω·cm·Oe−1 (compare Zholobenko, T. V. Ignatova, Low Temp. Phys. 29, 72 with [4]). (2003). The quantity ωα =ωα (4π/q2|k2|(H2 )(φ /B) (φ [8] V. D. Fil, Low Temp.Phys. 27, 993 (2001). M M n c2 0 0 is the flux quantum) is shown in Fig.5. For the mixed [9] E. B. Sonin, Phys.Rev.Lett. 76, 2794 (1996). state thesedata yieldthe Magnuscoefficientforonevor- [10] A. N. Zholobenko, G. P. Mikitik, V. D. Fil, D. V. Fil, tex of unit length (in units of q2|k2|H2 /4π). J. D. Kim, E. M. Choi, S. I. Lee, Low. Temp. Phys., in n c2 press. Thequalitativedifferenceinbehaviorofthegyroscopic [11] V.N.Narozhnyi,J.Freudenberger,V.N.Kochetkov,K. force in Nb is apparent already from Fig.3b. The in- A.Nenkov,G.Fuchs,A.Handstein,K.H.Mu¨ller, Phys. phase with uST component of Ex/uST becomes smaller Rev B 59, 14762 (1999). than unity, i.e. the gyroscopic term in E changes its [12] A.P.Cracknell,K.C.Wong,TheFermiSurface,Claren- x sign. In Nb the parameter κ is small and one cannot don Press, Oxford , 1973.