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Non-linear oscillations of the vortices in a hollow superconducting YBaCuO cylinder
Physica C 405 (2004) 163–172 www.elsevier.com/locate/physc
Non-linear oscillations of the vortices in a hollow superconducting YBaCuO cylinder M.T. Ayvazyan, A.A. Kteyan, R.A. Vardanian
*
Solid-State Division, Institute of Radiophysics and Electronics, Armenian National Academy of Sciences, Ashtarak-2 378410, Armenia Received 21 March 2003; received in revised form 29 May 2003; accepted 21 January 2004 Available online 5 March 2004
Abstract A response of a hollow YBa2 Cu3 O7d ceramic superconducting cylinder (at a mixed state) to the 100 kHz frequency alternating field was investigated. It was found that the response is non-linear and contains the odd harmonics of the excitation frequency. The amplitudes of the first, third and fifth harmonics are measured depending on an alternating field value h 6 1 Oe, applied to the cylinder external surface, and on a stationary magnetic field H 6 500 Oe driving the sample to the mixed state. For the results analysis a theoretical model of the vortices non-linear oscillations is proposed based on a non-linear vortex–vortex and vortex–surface interaction. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.60.Ge Keywords: Ceramic YBaCuO; Vortex; Non-linear oscillations
1. Introduction The non-linear phenomena of the vortices dynamics in the superconductors were first studied in details in [1] where a response of the superconducting alloy Pb–In cylinder being in the mixed state was investigated for a low-frequency alternating field. The authors had found that at the alternating field values above some ‘‘threshold’’ the sample response contains the odd harmonics of the excitation signal frequency. The non-linear effects in this work were interpreted as a non-linear skin effect attributed to the magnetic flux creep.
*
Corresponding author. E-mail address: [email protected] (R.A. Vardanian).
Namely the activation character of the creep process led to the threshold phenomena. The vortices study method [1] was later developed by Campbell [2] for the study of the magnetic flux penetration into the second-type superconductor Pb–Bi. In this method (now known as the Campbell method) the cylindrical superconducting sample is converted into the mixed state by an external magnetic field H . On the field H an alternating field h H is applied and the response (attributed by the flux variation) is registered by a receiving coil wound on the cylinder. The method allows to determine the flux penetration profile and the local current densities in the superconductor. For the mixed states study in the high-temperature superconductors (HTSC) the method was
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.028
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first used by D€ aumling et al. [3,4] at the DyBa2 Cu3 O7 ceramic samples investigations. In the present work, the Campbell method modification [5] is used whereby the hollow Pb–In alloy cylinder was investigated at T ¼ 4:2 K and the response to the external alternating field was measured by the receiving coil placed into the hollow. Since the signal is transmitted into the hollow by the vortices oscillations one can measure the fields in and out of the cylinder and get the oscillation amplitudes relation for the vortices placed at the inner and external surfaces, calculating thus the elastic modulus of the vortex lattice. Moreover by this method in [5] the critical value of the external alternating field amplitude was determined when the reversible character of the vortices shift is changed by the flux flow. The method was used also for the HTSC mixed state in [6] where the hollow cylindrical ceramic YBa2 Cu3 O7d samples were investigated at 77 K. In this work, the signal induced in hollow by the vortices oscillations was studied at the stationary magnetic field H 6 1200 Oe (producing the mixed state) and the external alternating field h 6 1 Oe. The experimental geometry allowed to study the signal phase shift in its passage through the cylinder wall and to evaluate the vortices viscous properties. An essential phenomenon at the study of the response in the hollow to an external harmonic signal is a signal form distortion that is indicative of the vortex oscillations non-linear nature. This non-linearity in the ceramic YBa2 Cu3 O7d material is the present paper subject. It must be noted that the non-linear response to the alternating magnetic field in the crystalline YBa2 Cu3 O7d samples was studied in [7] and explained by the superconductor non-linear magnetization. Contrary to the results of [7], were the response contained both the even and odd harmonics, our experiments revealed only the odd ones (as in [1]), that indicates the different mechanisms of the phenomena. Our measurements, unlike to [1], showed that the origination of the exciting frequency harmonics has not a threshold nature. Besides, it is essential that in our experiment the samples had the normal state resistance q 103 X cm and for the used frequency 100 kHz the
skin-layer depth k 5 mm. The hollow cylinders wall thickness was d ¼ 1 mm (hence ksk d) and the external field screening was defined by the superconducting currents only. Therefore the nonlinear skin-effect model used in [1] (where the sample thickness was more over ksk ) is unsuitable in our case. A complete theory of HTSC response to an alternating field for various vortex states (vortex liquid, vortex glass) was given in [14]. According to this work, the ac response in the superconducting mixed state is non-linear only in the flux creep regime, while for reversible vortex oscillations the response is linear. However, to account for the result of our work, a simple theory is proposed in which the non-linear oscillating vortex interacts with the hollow cylinder surface. The model describes the experimental curves adequately. The further development of the method can give new possibilities for the study of the HTSC vortex lattice dynamics, which has a very complicated character [8,9].
2. Experimental 2.1. Sample preparation and experimental setup The investigated samples are in the form of the hollow cylinder with the height 30 mm, external diameter 7 mm and inner diameter 5 mm. The samples were prepared by the well separated (50 lm) superconducting ceramic powder pressing in the special press-form under a pressure 172 MPa. The cylinders were sintered at 950 °C during 12 h in the oxygen atmosphere. The temperature changing rate on heating and cooling of the oven was equal 100 deg/h. For the inner stress reduction during the cooling the samples were additionally conditioned at 350 °C during 2 h. The samples obtained had a specific resistance q ¼ 2:16 103 X cm at 300 K and a mean increment in temperature oq=oT 5 106 X cm/deg. The superconducting transition temperature Tc is equal 89.8 K. On the central part of the cylinder external surface the excitation coil (with length 9 mm, 55 turns per cm) is wound creating the alternating magnetic field h on the sample external surface. At
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165
Fig. 1. The sketch of the experimental setup.
the middle part of the cylinder hollow there is the receiving coil (with length 5 mm, 350 turns per cm), which registers the magnetic field change into the hollow (see Fig. 1). The sample with the fitted coils was placed into a cryostat with liquid nitrogen. The cryostat in its turn is placed into a solenoid (with length 200 mm and inner diameter 20 mm), which converts the sample in the mixed state. The solenoid was powered by a high-stabilized current source and created the stationary magnetic field H parallel to the cylinder axis. At the excitation coil the alternating voltage with the frequency 105 Hz was fed from the highly precise generator of the sinusoid signals. The receiving coil signal was observed on the oscilloscope and studied by a spectrum analyzer. The spectrum analysis had shown that the response into the hollow contains (apart from the main harmonic) the frequencies corresponding to the odd harmonics of the excitation frequency. The measurements were carried out in the magnetic fields range H ¼ 0–500 Oe with the alternating magnetic field h amplitudes up to 1 Oe. The samples were frozen in the absence of the stationary external magnetic field H . For the every H value the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x correspondingly) were measured at the different values of h. The values of H were increased up to maximum of the mentioned range (Hmax ¼ 500 Oe) and then reduced to zero. It was the first cycle. After this the direction of H was changed to the opposite and the experiment was repeated, that was the second cy-
cle. The third cycle consists in the measurements with the repeat change of the H field direction. 2.2. Experimental results In Figs. 2–4 the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x ) are shown, depending on the value and direction of the stationary magnetic field 0 < H 6 500 Oe at the different values of the alternating magnetic field h 6 1 Oe. The dependences Ux ðH Þ (Fig. 2) are similar to the dependences for the total signal amplitude
.
.
Fig. 2. The dependence of the amplitude of the first harmonic Ux on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, (j) h ¼ 0:5 Oe, ( ) h ¼ 0:25 Oe.
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. .
Fig. 3. The dependence of the amplitude of the third harmonic U3x on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, (j) h ¼ 0:5 Oe, ( ) h ¼ 0; 25 Oe.
U ðH Þ obtained in [6]. At the initial region (first cycle, up-course) the first harmonic amplitude Ux rapidly rises with the H rise. At H 100–120 Oe the dependence weakens and at some value H 0 , connected with h (for h ¼ 1 Oe one has H 0 300 Oe) saturates. With reduction of H (third cycle, down-course) the Ux ðH Þ dependence demonstrates a hysteresis. When the field decreases to H 50– 40 Oe at down-course there is a minimum of Ux and at H ¼ 0 some residual signal Uxres remains.
. . . . . . . . .
Fig. 4. The dependence of the amplitude of the fifth harmonic U5x on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, ( ) h ¼ 0:19 Oe.
The form of the Ux ðH Þ curve depends only weekly on the alternating magnetic field h value. In Fig. 3 it is seen that the form of U3x ðH Þ curve is more sensitive to the alternating magnetic field h value and the dependence is more complicated at the low values of h. At the initial stage for the U3x ðH Þ function (first cycle, up-course) the rapid increase of the third harmonic U3x is observed when H rises, but at low values of the alternating magnetic field h (h 6 0:625 Oe) there is a minimum of U3x at H ¼ 80 Oe. When the alternating magnetic field h rises the valley depth (i.e. the height of the minimum point) decreases and at h ¼ 0:625 Oe the minimum transforms to the inflection point. On further rise of h the U3x ðH Þ dependence becomes monotonic increasing. The U3x amplitude rise with H continues up to some H 00 value that depends on the alternating magnetic field h value (for h ¼ 1 Oe we have H 00 200 Oe and for h ¼ 0:5 Oe H 00 280 Oe). At the further rise of the stationary magnetic field (from H 00 to the maximum value Hmax ) the third harmonic amplitude monotonically decreases. With the H decrease (first cycle, down-course) the third harmonic amplitude U3x firstly rises becoming greater than at up-course. The U3x ðH Þ rise continues to the some value Hrev ðhÞ (for h ¼ 1 Oe we have Hrev 290 Oe, for h ¼ 0:5 Oe Hrev 400 Oe) after which the U3x ðH Þ dependence monotonically decreases and crosses the up-course curve at value Hcr ðhÞ (for h ¼ 1 Oe we have Hcr 275 Oe, for h ¼ 0:5 Oe Hcr 360 Oe). The minimum of the down-course curve depends also on the alternating magnetic field h and with its rise is shifted to the less values of H (for h ¼ 1 Oe we have Hmin 80 Oe, for h ¼ 0:5 Oe Hmin 120 Oe). There is some res residual signal U3x for the switched off field (H ¼ 0) after the complete cycle. The U5x ðH Þ curve form depends heavily on the alternating magnetic field h values (Fig. 4). The fifth harmonic amplitude U5x (first cycle, upcourse) rapidly rises as H increases. At H ¼ 40 Oe the amplitude U5x begins to decrease rapidly and at H depending on h (at h ¼ 1 Oe we have H 120 Oe) there is a minimum at U5x ðH Þ curve. On further increase of H the U5x amplitude again begins to rise. At great values of the alternating magnetic field h (h > 0:625 Oe) there is a plateau at
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the U5x ðH Þ curve in the fields range H ¼ 160–200 Oe. The plateau height decreases as the alternating magnetic field h decreases, and for h ¼ 0:625 Oe in the mentioned interval 160–200 Oe the dependence becomes monotonically increasing. The U5x amplitude rise with H continues to the some H 000 value depending on h (for h ¼ 1 Oe we have H 000 360 Oe, for h ¼ 0:625 Oe H 000 500 Oe). During the reverse motion of the first cycle with the H decrease there is a hysteresis of the U5x curve. A minimum observed at the down-course depends on h also and for bigger h shifts to the H low values (for h ¼ 1 Oe we have Hmin 280 Oe, for h ¼ 0:5 Oe Hmin 360 Oe). At H ¼ 0 there is a res residual signal U5x as for the dependencies Ux ðH Þ and U3x ðH Þ. The up-course of the second cycle for all three dependencies (Ux ðH Þ, U3x ðH Þ and U5x ðH Þ) differs from the up-course of the first cycle. At the next cycles the up-course of these three dependences repeated the up-course of the second cycle. The main difference between the up-course of the first cycle and the up-course of the next cycles (second, third, etc.) is that the H 00 position in the U3x ðH Þ dependence shift to the lower side of H at 80 Oe. For the U5x ðH Þ dependences the main difference between the cycles is that during the next cycles there is not plateau observed at the range 160–200 Oe during the first cycle for the great values of the alternating magnetic field h. The down-courses of Ux ðH Þ, U3x ðH Þ and U5x ðH Þ for all cycles are very much like to the down-course of the first cycle. At Fig. 5, the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x ) are presented in the relative computing units (the two later amplitudes are given in fivefold) as functions of the stationary magnetic field H value and direction at the alternating magnetic field h value 1 Oe. It can be seen that the harmonics ratio is about Ux : U3x : U5x 50:4:1. The first harmonic amplitude Ux dependence on the alternating magnetic field value h is similar to the Ux ðhÞ dependence obtained in [6]. The third harmonic amplitude U3x dependence on h at first cycle up-course is presented at Fig. 6 for different values of H .
167
.
.
.
.
.
.
Fig. 5. The amplitude of the first and fivefold amplitudes of the third and fifth harmonics (Ux , 5U3x and 5U5x ) (in the relative units) as functions of the stationary magnetic field H value and direction at the alternating magnetic field h ¼ 1 Oe.
Fig. 6. The dependence of the amplitude of the third harmonic U3x on the value of the alternating magnetic field h at different values of the stationary magnetic field H : (j) H ¼ 40 Oe, ( ) H ¼ 200 Oe, (N) H ¼ 280 Oe, (.) H ¼ 500 Oe.
3. Discussion 3.1. Electric field induced by vortex oscillations in the cylinder hollow Since the external electric field in the superconducting cylinder is screened at the London the excitation coil penetration depth k 5000 A, signal can be transferred to the hollow only due to the vortices oscillations [5,6]. The behavior of the hollow superconducting cylinder in the mixed state
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was theoretically treated in [9]. The magnetic field ð1Þ h0 in the hollow is the sum of the fields h0 (conð2Þ nected with the trapped flux) and h0 (created by the vortices). For a single vortex located at a distance x from the inner surface the field in the hollow is equal [10,11] U0 2pkR K0 ½ðR þ xÞ=k ; K1 ðR=kÞ þ ðR=2kÞK0 ðR=kÞ ð1Þ
h0 ðrÞ ¼ h0 þ
ð1Þ
where R is the hollow radius, r––the distance to the center, K0;1 ðxÞ––Hankel functions of imaginary argument. It is obvious that at the vortices oscilð1Þ lation the h0 field does not change, so we do not ð2Þ need to determine its value. The h0 field (described by the second term in (1)) in the case when the hollow radius is much more than the London penetration depth (R k) and x k is equal ð2Þ
h0 ¼
U0 ex=k : pk R
ð2Þ
The vortex shift on the Dx distance changes the field on the extent ð2Þ
Dh0 ¼
x=k
U0 e Dx: pk2 R
ð3Þ
According to Faraday law such alternating field induces an electric field which is registered by the receiving coil in the cylinder hollow. ð2Þ Since the h0 field is exponentially decreased with the vortex distance from the surface the main contribution in this field is due to the vortices placed immediate at the hollow. Therefore for the theoretical explanation of the electric signal induced by the vortex lattice oscillations in the hollow it is enough to study the single vortex oscillations DxðtÞ near the cylinder inner surface. 3.2. Theory of the vortex non-linear oscillations into the hollow superconducting cylinder We do not consider the oscillations propagation along the vortex thread and will determine the forces acting on the vortex unit length and consider the vortex as an oscillator with mass m. The vortex located on the distance x from the hollow is subjected to the interaction force f 1 with the sur-
face and the interaction force f 2 with the nearest vortex. In the equilibrium state (when the vortex coordinate is x ¼ x0 ) these forces are balanced, i.e. f 01 ¼ f 02 . In the case under consideration (when R k) one can regard that the vortex is near the planar border by which the magnetization current flows [5] c Im ¼ ðh0 BÞ; ð4Þ 4p where h0 is the field in hollow, B ¼ nU0 ––the field in superconductor. The current density at the distance x from the surface is equal Im x=k e : ð5Þ k Let us consider first the low values region of the stationary magnetic field H when the vortex distance from the surface x and the inter-vortex distance a are far more than the penetration depth (x k and a k). At equilibrium the force of the vortex–surface interaction is equal jm ¼
1 h0 B f10 ¼ jm ðx0 ÞU0 ¼ U0 ex0 =k : c 4pk
ð6Þ
The interaction force between the vortices is 1 f2 ¼ jv U0 : c The vortex current jv at the distance x from the core is [12] x c U0 K jv ¼ 1 4p 2pk3 k and therefore f20 ¼
a U20 K : 1 k 8p2 k3
ð7Þ
At a k using the Hankel function asymptotic we obtain 1=2 U20 pk f20 ¼ ea=k : ð8Þ 3 2 8p k 2a Eqs. (6), (8) and the relation f10 ¼ f20 f 0 give the relationships between H , a and x0 at equilibrium. Lets consider now that an alternating current (with frequency x) flows along the cylinder external surface. In this case the vortices near the sur-
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face have the vibrations Dxex ¼ Xex cos xt, which transferred to the vortices at the hollow. For to derive an equation for the vortex coordinate near the inner surface x ¼ x0 þ Dx lets consider that the nearest vortex vibrations are known: Dxin ¼ Xin cos xt. A relation between Dxin and Dxex is defined by the elasticity modulus of the vortex lattice and obtained earlier [5]. We shall consider the small oscillations Dx k. Thus the problem reduces to the task of the two connected oscillators. The interaction force between vortex and inner surface is equal f1 ¼
h0 B U0 eðx0 þDxÞ=k : 4pk
ð9Þ
The variation of h0 due to vortex oscillations is neglected in (9). The inter-vortex distance in this case is a þ Dxin Dx and therefore according to (8) the interaction force between them is equal 1=2 U20 pk f2 ¼ eðaþDxin DxÞ=k ð10Þ 3 2 8p k 2a (if we neglect the change of a in the pre-exponent term). Taking into account (6) and (8) the force acting on the vortex at the hollow f ¼ f1 f2 can be written as Dxin : f ¼ f 0 eDx=k eDx=k þ f 0 eDx=k k In the second term (proportional to Dxin ) we can consider eDx=k 1 and obtain for the external force acting on the vortex an equation 0
f f~ ¼ Dxin : k
ð11Þ
In the first term the exponent can be expanded in series to the cubic term which leads to equation " 3 # 1 Dx 0 Dx f ¼ 2f þ f~ : k 3 k 2 The equation for the vortex oscillation m o otðDxÞ ¼ f~ 2 gives
m
o2 ðDxÞ 2f 0 2f 0 3 Dx 3 ðDxÞ ¼ F cos xt; þ 2 ot k 3k
ð12Þ
where m is the vortex mass, F ––the acting force amplitude (i.e. f~ ¼ F cos xt). Namely the cubic term presence in (12) leads to the emergence of the
169
oscillations high harmonics. It must be noted that we neglected the friction forces for ease of further analysis. Now we will demonstrate that a non-linear equation for the vortex oscillation can be obtained for the strong fields H too (i.e. when the intervortex distance is a k). In this case the vortex lattice is sufficiently tough and we can consider that all vortices vibrate with the same amplitude Dx. Because the vortex vibrations are only radial we can consider one vortex row along the cylinder radius (below we use the term ‘‘vortex lattice’’ namely for such one-dimensional lattice). When the vortex concentration is sufficiently great this lattice can be considered as a one oscillator with mass M ¼ mdn (d––the thickness of the sample wall, n––the vortices linear concentration) affected by two forces: f1 (for an interaction of the nearhollow vortex with the hollow surface) and f2 (for an interaction of the vortex near to the external surface with this surface). At equilibrium (when the first coordinate is x0 and the latter vortex coordinate is xex ) the force f10 is given by Eq. (6) and the force f20 is equal f20 ¼
H B U0 exex =k : 4pk
ð13Þ
When an alternating field hðtÞ with frequency x is applied to the field H the vortices begin to vibrate and Dx ¼ X cos xt. In this case the force f1 is given by (9) and f2 is equal f2 ¼
H þhB U0 eðxex DxÞ=k : 4pk
ð14Þ
Although h H we keep up in (14) the force h for to obtain the acting force in an explicit form. Taking into account Eqs. (6), (13) and the condition f10 ¼ f20 f 0 we have f ¼ f 0 eDx=k eDx=k þ f~ ; ð15Þ hU0 xex =k f~ ¼ e : 4pk Once more expanding the exponent correct to third order we obtain the force " 3 # 1 Dx 0 Dx f ¼ 2f þ f~ k 3 k
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for which oscillation equation coincides with Eq. (12), except that one has the mass M instead of m, the force f 0 is determined by (13) and the external force by Eq. (15). 3.3. Results analysis Eq. (12) is the well-known Duffing equation widely used in the non-linear oscillation theory. The cubic term presence in the expression for the external force leads to the solutions presented by the odd harmonics of the driving frequency [13] Dx ¼ Ax cos xt þ A3x cos 3xt þ A5x cos 5xt þ The fact that our experiment revealed the odd harmonics only testifies that Eq. (12) really can be used for the experimental data explanation. Here we will analyze the amplitudes of the first and third harmonics only. The cubic term is small thus the highest harmonics amplitudes are small too (and it is proved experimentally). The first harmonic amplitude Ax differs little from the first approximation amplitude, i.e. from the solution of (12) when the cubic term is absent. The solution has a form F ; ð16Þ mjx2 x20 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ 2f0 =km is the proper frequency of the harmonic oscillations. According to (11) on the vortex at the hollow surface acts the force F proportional to the oscillation amplitude Dxin of the neighboring vortex whereas the external alternating field h determines the vortex oscillations amplitude Dxex . Thus at given h the force F is determined by the relation Dxin =Dxex . It is shown in [5] that A¼
Dxin c11 ; Dxex d
B2 1 : 4p 1 þ ðqkÞ2
A
F F ¼ Mx20 2f0 =k
and if the force F is defined by (15) the amplitude does not depend on H . If the cubic term in (12) is remained the first harmonic amplitude only little differs from the first approximation amplitude (16) which for this case is determined by the exciting signal frequency [13] 3 F ; x2 ¼ x20 þ bA2 4 mA
ð17Þ
ð18Þ
where b ¼ 2f 0 =3k3 m is the relation of the cubic term coefficient to the mass. The third harmonic amplitude is equal A3x ¼
where d is the cylinder wall thickness and c11 ––the elastic modulus of the vortex lattice compression depending on the wavelength q1 of the lattice deformation [8,12] c11
For the low stationary fields H ða kÞ and low alternating fields h the deformation vector q is small (qk 1) and the elastic modulus c11 sharply increases when H rises. As a consequence both the force F and amplitude A rise (16). But for the great fields H and h the deformation vector q rises with H that leads, as follows from (17), to the slower rise of c11 and A. As indicated by Fig. 2 in our experiment it occurs at H 100 Oe (for h ¼ 1 Oe) and at H 160 Oe (for h ¼ 0:5 Oe). As it was shown in the previous section for the great values of H in the region a k the vortex lattice can be considered as a one oscillator having the mass M ¼ mdn. It means that for the strong fields the oscillator mass rises with H (due to the concentration n rise) whereas the force F , defined in this case by (15), practically does not depend on H . In Fig. 2 it is shown that for strong fields (H > 300 Oe at h ¼ 1 Oe) the first harmonic amplitude does not depend on H (i.e. on the vortex lattice mass M). It is possible only if the exciting frequency x is well below the proper frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ 2f0 =km. Indeed at x < x0 the oscillation amplitude is equal
bA3 : 32x2
ð19Þ
In the low fields H region the first harmonic amplitude Ax A increase leads to the rise of A3x . In Figs. 2 and 3 is shown that the first and third harmonics amplitudes sharply rise in the range H < 100 Oe (for h ¼ 1 Oe). When the dependence Ux ðH Þ weakens the dependence U3x ðH Þ weakens
M.T. Ayvazyan et al. / Physica C 405 (2004) 163–172
too. In this region the coefficient b in (19) does not depend on H . For the strong stationary fields (when the oscillator mass, defined by the lattice mass M ¼ mnd, increase with H ) the coefficient b ¼ 2f 0 =3k3 M decreases, therefore in this region when Ax ðH Þ const (i.e. when the dependence Ux ðH Þ becomes saturated) the amplitude A3x , as it follows from (19), begins to decrease with the H rise. In Fig. 3 it can be seen that the signal third harmonics U3x at h ¼ 1 Oe decreases in the region H > 160. The U3x dependence on the alternating field h is presented in Fig. 6. It is convenient to analyze the relation U3x =Ux as function of h. In Fig. 7 such dependence for the weak stationary fields is presented. As it follows from (18) and (19) the third and first harmonics amplitudes relation is equal A3x 1 : 2 ðF =mAÞ x Ax 32 34 þ 0 bA2
ð20Þ
At great values of h (when the force F increases) the difference (x20 F =mA) decreases: ðx20 F = mAÞ 34 bA2 and from (20) we obtain
171
. . . . . . . . . .
.
.
.
.
.
Fig. 8. The dependence of third and first harmonics amplitudes ratio U3x =Ux on the value of the alternating magnetic field h at strong stationary magnetic field H: (j) H ¼ 360 Oe, ( ) H ¼ 440 Oe, (N) H ¼ 500 Oe.
range 0.07–0.1. For the strong stationary fields the relation U3x =Ux has more complicated character of the dependence on h (Fig. 8). The investigations of the amplitude dependence on the alternating field require the study of the amplitude–frequency characteristics that will be carried out later.
A3x 1 : 24 Ax This value correlates well with the experimental results (Fig. 7): at the fields h 1 Oe the relation U3x =Ux weakly changes with h and fall within the .
.
.
.
.
.
. .
.
.
.
.
.
Fig. 7. The dependence of third and first harmonics amplitudes ratio U3x =Ux on the value of the alternating magnetic field h at weak stationary magnetic field H : (j) H ¼ 40 Oe, (r) H ¼ 80 Oe, (N) H ¼ 120 Oe, ( ) H ¼ 160 Oe.
4. Concluding remarks The aim of the present paper was the examination of the vortex non-linear dynamics and the revealing of the possible mechanism of the odd harmonics induction. Our analysis did not include the investigation of the hysteresis curves. The similar hysteresis phenomena in a hollow ceramic cylinder were studied in [6]. In present work, the non-linear response of superconductor in a mixed state was studied at the low frequencies and small amplitudes of the alternating magnetic fields when the vortex reversible oscillations are dominate over the flux creep process. In a flux creep mode the non-linear response and the HTSC resistive state were studied in details in [14]. The simple proposed model in point of fact explains the superconductor non-liner response by the non-linear character of the intervortex interaction and the vortex–surface interaction. The solutions of the obtained Eq. (12) give a
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sufficiently good description of the experimental results. But for a more accurate description a damping (connected with the normal currents arising due to the vortex motion) must be included in Eq. (12). It should be noted also that the analysis of the receiving coil signal revealed the odd sub-harmonic frequencies too (i.e. x=3, x=5, etc). In the fields range under consideration the amplitudes of these vibrations were negligibly small as compared with U3x and U5x . Nevertheless their presence gives an additional proof to our model because it is known that the Duffing equation allows to get subharmonic solutions too. References [1] P. Alais, Y. Simon, Phys. Rev. 158 (1967) 426. [2] A.M. Campbell, J. Phys. C 2 (1969) 1492.
[3] M. D€aumling, J. Seuntjens, D.C. Larbalestier, Physica C 153–155 (1988) 318. [4] M. D€aumling, J. Seuntjens, D.C. Larbalestier, Appl. Phys. Lett. 55 (1988) 590. [5] A.T. Abramian, V.V. Shmidt, Zh. Eksp. Teor. Fiz. 73 (1977) 2264. [6] M.T. Ayvazyan, K.Z. Papazian, R.A. Vardanian, Physica C 390 (2003) 31. [7] D.G. Xenikos, T.R. Lemberger, Phys. Rev. B 41 (1990) 869. [8] G. Blatter, M.V. FeigelÕman, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [9] J.E. Evetts, Adv. Phys. 50 (2001) 1249. [10] G.S. Mkrtchyan, V.V. Schmidt, Sov. Phys. JETP 34 (1972) 195. [11] V.V. Shmidt, G.C. Mkrtchian, Usp. Fiz. Nauk. 112 (1974) 459. [12] V.V. Shmidt, The Physics of Superconductors, SpringerVerlag, Berlin, Heidelberg, 1997. [13] J.J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley Classics, New York, 1992. [14] C.J. Van der Beak, V.B. Geshkenbein, V.M. Vinokur, Phys. Rev. B 48 (1993) 3393.
Non-linear oscillations of the vortices in a hollow superconducting YBaCuO cylinder M.T. Ayvazyan, A.A. Kteyan, R.A. Vardanian
*
Solid-State Division, Institute of Radiophysics and Electronics, Armenian National Academy of Sciences, Ashtarak-2 378410, Armenia Received 21 March 2003; received in revised form 29 May 2003; accepted 21 January 2004 Available online 5 March 2004
Abstract A response of a hollow YBa2 Cu3 O7d ceramic superconducting cylinder (at a mixed state) to the 100 kHz frequency alternating field was investigated. It was found that the response is non-linear and contains the odd harmonics of the excitation frequency. The amplitudes of the first, third and fifth harmonics are measured depending on an alternating field value h 6 1 Oe, applied to the cylinder external surface, and on a stationary magnetic field H 6 500 Oe driving the sample to the mixed state. For the results analysis a theoretical model of the vortices non-linear oscillations is proposed based on a non-linear vortex–vortex and vortex–surface interaction. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.60.Ge Keywords: Ceramic YBaCuO; Vortex; Non-linear oscillations
1. Introduction The non-linear phenomena of the vortices dynamics in the superconductors were first studied in details in [1] where a response of the superconducting alloy Pb–In cylinder being in the mixed state was investigated for a low-frequency alternating field. The authors had found that at the alternating field values above some ‘‘threshold’’ the sample response contains the odd harmonics of the excitation signal frequency. The non-linear effects in this work were interpreted as a non-linear skin effect attributed to the magnetic flux creep.
*
Corresponding author. E-mail address: [email protected] (R.A. Vardanian).
Namely the activation character of the creep process led to the threshold phenomena. The vortices study method [1] was later developed by Campbell [2] for the study of the magnetic flux penetration into the second-type superconductor Pb–Bi. In this method (now known as the Campbell method) the cylindrical superconducting sample is converted into the mixed state by an external magnetic field H . On the field H an alternating field h H is applied and the response (attributed by the flux variation) is registered by a receiving coil wound on the cylinder. The method allows to determine the flux penetration profile and the local current densities in the superconductor. For the mixed states study in the high-temperature superconductors (HTSC) the method was
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.028
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first used by D€ aumling et al. [3,4] at the DyBa2 Cu3 O7 ceramic samples investigations. In the present work, the Campbell method modification [5] is used whereby the hollow Pb–In alloy cylinder was investigated at T ¼ 4:2 K and the response to the external alternating field was measured by the receiving coil placed into the hollow. Since the signal is transmitted into the hollow by the vortices oscillations one can measure the fields in and out of the cylinder and get the oscillation amplitudes relation for the vortices placed at the inner and external surfaces, calculating thus the elastic modulus of the vortex lattice. Moreover by this method in [5] the critical value of the external alternating field amplitude was determined when the reversible character of the vortices shift is changed by the flux flow. The method was used also for the HTSC mixed state in [6] where the hollow cylindrical ceramic YBa2 Cu3 O7d samples were investigated at 77 K. In this work, the signal induced in hollow by the vortices oscillations was studied at the stationary magnetic field H 6 1200 Oe (producing the mixed state) and the external alternating field h 6 1 Oe. The experimental geometry allowed to study the signal phase shift in its passage through the cylinder wall and to evaluate the vortices viscous properties. An essential phenomenon at the study of the response in the hollow to an external harmonic signal is a signal form distortion that is indicative of the vortex oscillations non-linear nature. This non-linearity in the ceramic YBa2 Cu3 O7d material is the present paper subject. It must be noted that the non-linear response to the alternating magnetic field in the crystalline YBa2 Cu3 O7d samples was studied in [7] and explained by the superconductor non-linear magnetization. Contrary to the results of [7], were the response contained both the even and odd harmonics, our experiments revealed only the odd ones (as in [1]), that indicates the different mechanisms of the phenomena. Our measurements, unlike to [1], showed that the origination of the exciting frequency harmonics has not a threshold nature. Besides, it is essential that in our experiment the samples had the normal state resistance q 103 X cm and for the used frequency 100 kHz the
skin-layer depth k 5 mm. The hollow cylinders wall thickness was d ¼ 1 mm (hence ksk d) and the external field screening was defined by the superconducting currents only. Therefore the nonlinear skin-effect model used in [1] (where the sample thickness was more over ksk ) is unsuitable in our case. A complete theory of HTSC response to an alternating field for various vortex states (vortex liquid, vortex glass) was given in [14]. According to this work, the ac response in the superconducting mixed state is non-linear only in the flux creep regime, while for reversible vortex oscillations the response is linear. However, to account for the result of our work, a simple theory is proposed in which the non-linear oscillating vortex interacts with the hollow cylinder surface. The model describes the experimental curves adequately. The further development of the method can give new possibilities for the study of the HTSC vortex lattice dynamics, which has a very complicated character [8,9].
2. Experimental 2.1. Sample preparation and experimental setup The investigated samples are in the form of the hollow cylinder with the height 30 mm, external diameter 7 mm and inner diameter 5 mm. The samples were prepared by the well separated (50 lm) superconducting ceramic powder pressing in the special press-form under a pressure 172 MPa. The cylinders were sintered at 950 °C during 12 h in the oxygen atmosphere. The temperature changing rate on heating and cooling of the oven was equal 100 deg/h. For the inner stress reduction during the cooling the samples were additionally conditioned at 350 °C during 2 h. The samples obtained had a specific resistance q ¼ 2:16 103 X cm at 300 K and a mean increment in temperature oq=oT 5 106 X cm/deg. The superconducting transition temperature Tc is equal 89.8 K. On the central part of the cylinder external surface the excitation coil (with length 9 mm, 55 turns per cm) is wound creating the alternating magnetic field h on the sample external surface. At
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165
Fig. 1. The sketch of the experimental setup.
the middle part of the cylinder hollow there is the receiving coil (with length 5 mm, 350 turns per cm), which registers the magnetic field change into the hollow (see Fig. 1). The sample with the fitted coils was placed into a cryostat with liquid nitrogen. The cryostat in its turn is placed into a solenoid (with length 200 mm and inner diameter 20 mm), which converts the sample in the mixed state. The solenoid was powered by a high-stabilized current source and created the stationary magnetic field H parallel to the cylinder axis. At the excitation coil the alternating voltage with the frequency 105 Hz was fed from the highly precise generator of the sinusoid signals. The receiving coil signal was observed on the oscilloscope and studied by a spectrum analyzer. The spectrum analysis had shown that the response into the hollow contains (apart from the main harmonic) the frequencies corresponding to the odd harmonics of the excitation frequency. The measurements were carried out in the magnetic fields range H ¼ 0–500 Oe with the alternating magnetic field h amplitudes up to 1 Oe. The samples were frozen in the absence of the stationary external magnetic field H . For the every H value the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x correspondingly) were measured at the different values of h. The values of H were increased up to maximum of the mentioned range (Hmax ¼ 500 Oe) and then reduced to zero. It was the first cycle. After this the direction of H was changed to the opposite and the experiment was repeated, that was the second cy-
cle. The third cycle consists in the measurements with the repeat change of the H field direction. 2.2. Experimental results In Figs. 2–4 the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x ) are shown, depending on the value and direction of the stationary magnetic field 0 < H 6 500 Oe at the different values of the alternating magnetic field h 6 1 Oe. The dependences Ux ðH Þ (Fig. 2) are similar to the dependences for the total signal amplitude
.
.
Fig. 2. The dependence of the amplitude of the first harmonic Ux on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, (j) h ¼ 0:5 Oe, ( ) h ¼ 0:25 Oe.
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. .
Fig. 3. The dependence of the amplitude of the third harmonic U3x on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, (j) h ¼ 0:5 Oe, ( ) h ¼ 0; 25 Oe.
U ðH Þ obtained in [6]. At the initial region (first cycle, up-course) the first harmonic amplitude Ux rapidly rises with the H rise. At H 100–120 Oe the dependence weakens and at some value H 0 , connected with h (for h ¼ 1 Oe one has H 0 300 Oe) saturates. With reduction of H (third cycle, down-course) the Ux ðH Þ dependence demonstrates a hysteresis. When the field decreases to H 50– 40 Oe at down-course there is a minimum of Ux and at H ¼ 0 some residual signal Uxres remains.
. . . . . . . . .
Fig. 4. The dependence of the amplitude of the fifth harmonic U5x on the value and direction of the stationary magnetic field H at different values of the alternating magnetic field h: (N) h ¼ 1 Oe, ( ) h ¼ 0:19 Oe.
The form of the Ux ðH Þ curve depends only weekly on the alternating magnetic field h value. In Fig. 3 it is seen that the form of U3x ðH Þ curve is more sensitive to the alternating magnetic field h value and the dependence is more complicated at the low values of h. At the initial stage for the U3x ðH Þ function (first cycle, up-course) the rapid increase of the third harmonic U3x is observed when H rises, but at low values of the alternating magnetic field h (h 6 0:625 Oe) there is a minimum of U3x at H ¼ 80 Oe. When the alternating magnetic field h rises the valley depth (i.e. the height of the minimum point) decreases and at h ¼ 0:625 Oe the minimum transforms to the inflection point. On further rise of h the U3x ðH Þ dependence becomes monotonic increasing. The U3x amplitude rise with H continues up to some H 00 value that depends on the alternating magnetic field h value (for h ¼ 1 Oe we have H 00 200 Oe and for h ¼ 0:5 Oe H 00 280 Oe). At the further rise of the stationary magnetic field (from H 00 to the maximum value Hmax ) the third harmonic amplitude monotonically decreases. With the H decrease (first cycle, down-course) the third harmonic amplitude U3x firstly rises becoming greater than at up-course. The U3x ðH Þ rise continues to the some value Hrev ðhÞ (for h ¼ 1 Oe we have Hrev 290 Oe, for h ¼ 0:5 Oe Hrev 400 Oe) after which the U3x ðH Þ dependence monotonically decreases and crosses the up-course curve at value Hcr ðhÞ (for h ¼ 1 Oe we have Hcr 275 Oe, for h ¼ 0:5 Oe Hcr 360 Oe). The minimum of the down-course curve depends also on the alternating magnetic field h and with its rise is shifted to the less values of H (for h ¼ 1 Oe we have Hmin 80 Oe, for h ¼ 0:5 Oe Hmin 120 Oe). There is some res residual signal U3x for the switched off field (H ¼ 0) after the complete cycle. The U5x ðH Þ curve form depends heavily on the alternating magnetic field h values (Fig. 4). The fifth harmonic amplitude U5x (first cycle, upcourse) rapidly rises as H increases. At H ¼ 40 Oe the amplitude U5x begins to decrease rapidly and at H depending on h (at h ¼ 1 Oe we have H 120 Oe) there is a minimum at U5x ðH Þ curve. On further increase of H the U5x amplitude again begins to rise. At great values of the alternating magnetic field h (h > 0:625 Oe) there is a plateau at
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the U5x ðH Þ curve in the fields range H ¼ 160–200 Oe. The plateau height decreases as the alternating magnetic field h decreases, and for h ¼ 0:625 Oe in the mentioned interval 160–200 Oe the dependence becomes monotonically increasing. The U5x amplitude rise with H continues to the some H 000 value depending on h (for h ¼ 1 Oe we have H 000 360 Oe, for h ¼ 0:625 Oe H 000 500 Oe). During the reverse motion of the first cycle with the H decrease there is a hysteresis of the U5x curve. A minimum observed at the down-course depends on h also and for bigger h shifts to the H low values (for h ¼ 1 Oe we have Hmin 280 Oe, for h ¼ 0:5 Oe Hmin 360 Oe). At H ¼ 0 there is a res residual signal U5x as for the dependencies Ux ðH Þ and U3x ðH Þ. The up-course of the second cycle for all three dependencies (Ux ðH Þ, U3x ðH Þ and U5x ðH Þ) differs from the up-course of the first cycle. At the next cycles the up-course of these three dependences repeated the up-course of the second cycle. The main difference between the up-course of the first cycle and the up-course of the next cycles (second, third, etc.) is that the H 00 position in the U3x ðH Þ dependence shift to the lower side of H at 80 Oe. For the U5x ðH Þ dependences the main difference between the cycles is that during the next cycles there is not plateau observed at the range 160–200 Oe during the first cycle for the great values of the alternating magnetic field h. The down-courses of Ux ðH Þ, U3x ðH Þ and U5x ðH Þ for all cycles are very much like to the down-course of the first cycle. At Fig. 5, the amplitudes of the first, third and fifth harmonics (Ux , U3x and U5x ) are presented in the relative computing units (the two later amplitudes are given in fivefold) as functions of the stationary magnetic field H value and direction at the alternating magnetic field h value 1 Oe. It can be seen that the harmonics ratio is about Ux : U3x : U5x 50:4:1. The first harmonic amplitude Ux dependence on the alternating magnetic field value h is similar to the Ux ðhÞ dependence obtained in [6]. The third harmonic amplitude U3x dependence on h at first cycle up-course is presented at Fig. 6 for different values of H .
167
.
.
.
.
.
.
Fig. 5. The amplitude of the first and fivefold amplitudes of the third and fifth harmonics (Ux , 5U3x and 5U5x ) (in the relative units) as functions of the stationary magnetic field H value and direction at the alternating magnetic field h ¼ 1 Oe.
Fig. 6. The dependence of the amplitude of the third harmonic U3x on the value of the alternating magnetic field h at different values of the stationary magnetic field H : (j) H ¼ 40 Oe, ( ) H ¼ 200 Oe, (N) H ¼ 280 Oe, (.) H ¼ 500 Oe.
3. Discussion 3.1. Electric field induced by vortex oscillations in the cylinder hollow Since the external electric field in the superconducting cylinder is screened at the London the excitation coil penetration depth k 5000 A, signal can be transferred to the hollow only due to the vortices oscillations [5,6]. The behavior of the hollow superconducting cylinder in the mixed state
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was theoretically treated in [9]. The magnetic field ð1Þ h0 in the hollow is the sum of the fields h0 (conð2Þ nected with the trapped flux) and h0 (created by the vortices). For a single vortex located at a distance x from the inner surface the field in the hollow is equal [10,11] U0 2pkR K0 ½ðR þ xÞ=k ; K1 ðR=kÞ þ ðR=2kÞK0 ðR=kÞ ð1Þ
h0 ðrÞ ¼ h0 þ
ð1Þ
where R is the hollow radius, r––the distance to the center, K0;1 ðxÞ––Hankel functions of imaginary argument. It is obvious that at the vortices oscilð1Þ lation the h0 field does not change, so we do not ð2Þ need to determine its value. The h0 field (described by the second term in (1)) in the case when the hollow radius is much more than the London penetration depth (R k) and x k is equal ð2Þ
h0 ¼
U0 ex=k : pk R
ð2Þ
The vortex shift on the Dx distance changes the field on the extent ð2Þ
Dh0 ¼
x=k
U0 e Dx: pk2 R
ð3Þ
According to Faraday law such alternating field induces an electric field which is registered by the receiving coil in the cylinder hollow. ð2Þ Since the h0 field is exponentially decreased with the vortex distance from the surface the main contribution in this field is due to the vortices placed immediate at the hollow. Therefore for the theoretical explanation of the electric signal induced by the vortex lattice oscillations in the hollow it is enough to study the single vortex oscillations DxðtÞ near the cylinder inner surface. 3.2. Theory of the vortex non-linear oscillations into the hollow superconducting cylinder We do not consider the oscillations propagation along the vortex thread and will determine the forces acting on the vortex unit length and consider the vortex as an oscillator with mass m. The vortex located on the distance x from the hollow is subjected to the interaction force f 1 with the sur-
face and the interaction force f 2 with the nearest vortex. In the equilibrium state (when the vortex coordinate is x ¼ x0 ) these forces are balanced, i.e. f 01 ¼ f 02 . In the case under consideration (when R k) one can regard that the vortex is near the planar border by which the magnetization current flows [5] c Im ¼ ðh0 BÞ; ð4Þ 4p where h0 is the field in hollow, B ¼ nU0 ––the field in superconductor. The current density at the distance x from the surface is equal Im x=k e : ð5Þ k Let us consider first the low values region of the stationary magnetic field H when the vortex distance from the surface x and the inter-vortex distance a are far more than the penetration depth (x k and a k). At equilibrium the force of the vortex–surface interaction is equal jm ¼
1 h0 B f10 ¼ jm ðx0 ÞU0 ¼ U0 ex0 =k : c 4pk
ð6Þ
The interaction force between the vortices is 1 f2 ¼ jv U0 : c The vortex current jv at the distance x from the core is [12] x c U0 K jv ¼ 1 4p 2pk3 k and therefore f20 ¼
a U20 K : 1 k 8p2 k3
ð7Þ
At a k using the Hankel function asymptotic we obtain 1=2 U20 pk f20 ¼ ea=k : ð8Þ 3 2 8p k 2a Eqs. (6), (8) and the relation f10 ¼ f20 f 0 give the relationships between H , a and x0 at equilibrium. Lets consider now that an alternating current (with frequency x) flows along the cylinder external surface. In this case the vortices near the sur-
M.T. Ayvazyan et al. / Physica C 405 (2004) 163–172
face have the vibrations Dxex ¼ Xex cos xt, which transferred to the vortices at the hollow. For to derive an equation for the vortex coordinate near the inner surface x ¼ x0 þ Dx lets consider that the nearest vortex vibrations are known: Dxin ¼ Xin cos xt. A relation between Dxin and Dxex is defined by the elasticity modulus of the vortex lattice and obtained earlier [5]. We shall consider the small oscillations Dx k. Thus the problem reduces to the task of the two connected oscillators. The interaction force between vortex and inner surface is equal f1 ¼
h0 B U0 eðx0 þDxÞ=k : 4pk
ð9Þ
The variation of h0 due to vortex oscillations is neglected in (9). The inter-vortex distance in this case is a þ Dxin Dx and therefore according to (8) the interaction force between them is equal 1=2 U20 pk f2 ¼ eðaþDxin DxÞ=k ð10Þ 3 2 8p k 2a (if we neglect the change of a in the pre-exponent term). Taking into account (6) and (8) the force acting on the vortex at the hollow f ¼ f1 f2 can be written as Dxin : f ¼ f 0 eDx=k eDx=k þ f 0 eDx=k k In the second term (proportional to Dxin ) we can consider eDx=k 1 and obtain for the external force acting on the vortex an equation 0
f f~ ¼ Dxin : k
ð11Þ
In the first term the exponent can be expanded in series to the cubic term which leads to equation " 3 # 1 Dx 0 Dx f ¼ 2f þ f~ : k 3 k 2 The equation for the vortex oscillation m o otðDxÞ ¼ f~ 2 gives
m
o2 ðDxÞ 2f 0 2f 0 3 Dx 3 ðDxÞ ¼ F cos xt; þ 2 ot k 3k
ð12Þ
where m is the vortex mass, F ––the acting force amplitude (i.e. f~ ¼ F cos xt). Namely the cubic term presence in (12) leads to the emergence of the
169
oscillations high harmonics. It must be noted that we neglected the friction forces for ease of further analysis. Now we will demonstrate that a non-linear equation for the vortex oscillation can be obtained for the strong fields H too (i.e. when the intervortex distance is a k). In this case the vortex lattice is sufficiently tough and we can consider that all vortices vibrate with the same amplitude Dx. Because the vortex vibrations are only radial we can consider one vortex row along the cylinder radius (below we use the term ‘‘vortex lattice’’ namely for such one-dimensional lattice). When the vortex concentration is sufficiently great this lattice can be considered as a one oscillator with mass M ¼ mdn (d––the thickness of the sample wall, n––the vortices linear concentration) affected by two forces: f1 (for an interaction of the nearhollow vortex with the hollow surface) and f2 (for an interaction of the vortex near to the external surface with this surface). At equilibrium (when the first coordinate is x0 and the latter vortex coordinate is xex ) the force f10 is given by Eq. (6) and the force f20 is equal f20 ¼
H B U0 exex =k : 4pk
ð13Þ
When an alternating field hðtÞ with frequency x is applied to the field H the vortices begin to vibrate and Dx ¼ X cos xt. In this case the force f1 is given by (9) and f2 is equal f2 ¼
H þhB U0 eðxex DxÞ=k : 4pk
ð14Þ
Although h H we keep up in (14) the force h for to obtain the acting force in an explicit form. Taking into account Eqs. (6), (13) and the condition f10 ¼ f20 f 0 we have f ¼ f 0 eDx=k eDx=k þ f~ ; ð15Þ hU0 xex =k f~ ¼ e : 4pk Once more expanding the exponent correct to third order we obtain the force " 3 # 1 Dx 0 Dx f ¼ 2f þ f~ k 3 k
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for which oscillation equation coincides with Eq. (12), except that one has the mass M instead of m, the force f 0 is determined by (13) and the external force by Eq. (15). 3.3. Results analysis Eq. (12) is the well-known Duffing equation widely used in the non-linear oscillation theory. The cubic term presence in the expression for the external force leads to the solutions presented by the odd harmonics of the driving frequency [13] Dx ¼ Ax cos xt þ A3x cos 3xt þ A5x cos 5xt þ The fact that our experiment revealed the odd harmonics only testifies that Eq. (12) really can be used for the experimental data explanation. Here we will analyze the amplitudes of the first and third harmonics only. The cubic term is small thus the highest harmonics amplitudes are small too (and it is proved experimentally). The first harmonic amplitude Ax differs little from the first approximation amplitude, i.e. from the solution of (12) when the cubic term is absent. The solution has a form F ; ð16Þ mjx2 x20 j pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ 2f0 =km is the proper frequency of the harmonic oscillations. According to (11) on the vortex at the hollow surface acts the force F proportional to the oscillation amplitude Dxin of the neighboring vortex whereas the external alternating field h determines the vortex oscillations amplitude Dxex . Thus at given h the force F is determined by the relation Dxin =Dxex . It is shown in [5] that A¼
Dxin c11 ; Dxex d
B2 1 : 4p 1 þ ðqkÞ2
A
F F ¼ Mx20 2f0 =k
and if the force F is defined by (15) the amplitude does not depend on H . If the cubic term in (12) is remained the first harmonic amplitude only little differs from the first approximation amplitude (16) which for this case is determined by the exciting signal frequency [13] 3 F ; x2 ¼ x20 þ bA2 4 mA
ð17Þ
ð18Þ
where b ¼ 2f 0 =3k3 m is the relation of the cubic term coefficient to the mass. The third harmonic amplitude is equal A3x ¼
where d is the cylinder wall thickness and c11 ––the elastic modulus of the vortex lattice compression depending on the wavelength q1 of the lattice deformation [8,12] c11
For the low stationary fields H ða kÞ and low alternating fields h the deformation vector q is small (qk 1) and the elastic modulus c11 sharply increases when H rises. As a consequence both the force F and amplitude A rise (16). But for the great fields H and h the deformation vector q rises with H that leads, as follows from (17), to the slower rise of c11 and A. As indicated by Fig. 2 in our experiment it occurs at H 100 Oe (for h ¼ 1 Oe) and at H 160 Oe (for h ¼ 0:5 Oe). As it was shown in the previous section for the great values of H in the region a k the vortex lattice can be considered as a one oscillator having the mass M ¼ mdn. It means that for the strong fields the oscillator mass rises with H (due to the concentration n rise) whereas the force F , defined in this case by (15), practically does not depend on H . In Fig. 2 it is shown that for strong fields (H > 300 Oe at h ¼ 1 Oe) the first harmonic amplitude does not depend on H (i.e. on the vortex lattice mass M). It is possible only if the exciting frequency x is well below the proper frequency pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x0 ¼ 2f0 =km. Indeed at x < x0 the oscillation amplitude is equal
bA3 : 32x2
ð19Þ
In the low fields H region the first harmonic amplitude Ax A increase leads to the rise of A3x . In Figs. 2 and 3 is shown that the first and third harmonics amplitudes sharply rise in the range H < 100 Oe (for h ¼ 1 Oe). When the dependence Ux ðH Þ weakens the dependence U3x ðH Þ weakens
M.T. Ayvazyan et al. / Physica C 405 (2004) 163–172
too. In this region the coefficient b in (19) does not depend on H . For the strong stationary fields (when the oscillator mass, defined by the lattice mass M ¼ mnd, increase with H ) the coefficient b ¼ 2f 0 =3k3 M decreases, therefore in this region when Ax ðH Þ const (i.e. when the dependence Ux ðH Þ becomes saturated) the amplitude A3x , as it follows from (19), begins to decrease with the H rise. In Fig. 3 it can be seen that the signal third harmonics U3x at h ¼ 1 Oe decreases in the region H > 160. The U3x dependence on the alternating field h is presented in Fig. 6. It is convenient to analyze the relation U3x =Ux as function of h. In Fig. 7 such dependence for the weak stationary fields is presented. As it follows from (18) and (19) the third and first harmonics amplitudes relation is equal A3x 1 : 2 ðF =mAÞ x Ax 32 34 þ 0 bA2
ð20Þ
At great values of h (when the force F increases) the difference (x20 F =mA) decreases: ðx20 F = mAÞ 34 bA2 and from (20) we obtain
171
. . . . . . . . . .
.
.
.
.
.
Fig. 8. The dependence of third and first harmonics amplitudes ratio U3x =Ux on the value of the alternating magnetic field h at strong stationary magnetic field H: (j) H ¼ 360 Oe, ( ) H ¼ 440 Oe, (N) H ¼ 500 Oe.
range 0.07–0.1. For the strong stationary fields the relation U3x =Ux has more complicated character of the dependence on h (Fig. 8). The investigations of the amplitude dependence on the alternating field require the study of the amplitude–frequency characteristics that will be carried out later.
A3x 1 : 24 Ax This value correlates well with the experimental results (Fig. 7): at the fields h 1 Oe the relation U3x =Ux weakly changes with h and fall within the .
.
.
.
.
.
. .
.
.
.
.
.
Fig. 7. The dependence of third and first harmonics amplitudes ratio U3x =Ux on the value of the alternating magnetic field h at weak stationary magnetic field H : (j) H ¼ 40 Oe, (r) H ¼ 80 Oe, (N) H ¼ 120 Oe, ( ) H ¼ 160 Oe.
4. Concluding remarks The aim of the present paper was the examination of the vortex non-linear dynamics and the revealing of the possible mechanism of the odd harmonics induction. Our analysis did not include the investigation of the hysteresis curves. The similar hysteresis phenomena in a hollow ceramic cylinder were studied in [6]. In present work, the non-linear response of superconductor in a mixed state was studied at the low frequencies and small amplitudes of the alternating magnetic fields when the vortex reversible oscillations are dominate over the flux creep process. In a flux creep mode the non-linear response and the HTSC resistive state were studied in details in [14]. The simple proposed model in point of fact explains the superconductor non-liner response by the non-linear character of the intervortex interaction and the vortex–surface interaction. The solutions of the obtained Eq. (12) give a
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sufficiently good description of the experimental results. But for a more accurate description a damping (connected with the normal currents arising due to the vortex motion) must be included in Eq. (12). It should be noted also that the analysis of the receiving coil signal revealed the odd sub-harmonic frequencies too (i.e. x=3, x=5, etc). In the fields range under consideration the amplitudes of these vibrations were negligibly small as compared with U3x and U5x . Nevertheless their presence gives an additional proof to our model because it is known that the Duffing equation allows to get subharmonic solutions too. References [1] P. Alais, Y. Simon, Phys. Rev. 158 (1967) 426. [2] A.M. Campbell, J. Phys. C 2 (1969) 1492.
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