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Magnetization oscillations of a superconducting disk
Volume 150, number I
Magnetization
22 October 1990
PHYSICS LETTERS A
oscillations of a superconducting
0. Buisson ‘, P. Gandit,
R. Rammal,
disk
Y.Y. Wang and B. Pannetier
Centre de Recherches SW les TrPs Basses Temp&atures, Laboratoire associP d I’Universitt J. Fourier, C.N.R.S., B.P. 166X, F38042 Grenoble Cedex. France
Received 5 June 1990; revised manuscript received 28 August 1990; accepted for publication 30 August 1990 Communicated by D. Bloch
We present magnetic measurements on micron-sized superconducting disks. The phase transition line r,(H) and magnetization exhibit spectacular undamped oscillations as function of magnetic field. These results reflect the contribution of the edge states on the energy spectrum of electron pairs in the disk. We discuss their sensitivity to boundary conditions.
1. Introduction
Technological improvements in microelectronics have motivated new studies in solid state physics such as quantum transport in mesoscopic circuits [ 11, two-dimensional electron gas in semiconductors [ 2 ] and superconducting networks [ 31. In these systems, the boundary conditions given by the shape of the circuit and the nature of the material to which contact is made are very important. For example, theoretical studies [ 4,5 ] of quantum point contacts with different confinement geometries have shown that the modification of the circuit-shape (disk to ellipse) may drastically affect the energy spectrum and therefore the transport and magnetic properties. Pioneer experimental studies on quantum transport in confined high mobility electron gas have been reported by Van Wees et al. [ 61. Except in narrow point contacts where conductance quantization is observed, the electronic wavelength in such low carrier density systems remains smaller than the lateral size of the system (typically micron scale). In contrast, quantum coherence in superconducting materials involves large electron pair wavelengths controlled by the superconducting coherence length. Recently, transport and magnetic properties of twodimen’ Present address: Laboratorio de Materia condensada, Dpt.
sional artificial superconducting networks [ 3,7,8] have been investigated as function of their topology (periodic, quasicrystalline or fractal arrays ) . The detailed study of these systems relies upon the remarkable fact that in the superconducting phase all Cooper pairs occupy coherently the same quantum state. In particuiar the equilibrium superconducting properties, such as critical temperature and magnetization, can be obtained from the energy spectrum of the pair wavefunction in the considered geometry. The appropriate boundary condition is given, in the wire network geometry, by Kirchoff’s current conservation condition at the nodes of the network [ lo]. To date only multiconnected superconducting systems have been considered. In this paper, we report on our investigation of the properties of superconducting thin film disks in the vicinity of the phase transition. This study involves both the low field regime where the magnetic length 1”= (&,/rcH) ‘I2 is large compared to the disk radius R,and the high field regime where Znis smaller than R. Our study provides a very simple determination of the ground state of a quantum charged particle confined into a two-dimensional disk under an external magnetic field. After a brief description of the energy spectrum of a pair confined in a disk, we discuss the experimental results on the superconducting critical temperature and the magnetization.
Fisica, P.U.C. 225 rua Marques de Sao Vicente, 22453 Rio de Janeiro, Brazil.
36
0375-9601/90/$
03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Volume 150, number 1
PHYSICS LETTERS A
22 October 1990
by the ground state energy E,,,(H) and depends upon the density of superconducting electrons I y,,, I 2,
2. Theoretical model In the vicinity of the mean field phase transition, the superconducting thin film disk behaves as a type II superconductor [ IO,1 11. Moreover, near the transition, the effective penetration length of the perpendicular magnetic field is much larger than the disk radius R. So the magnetic field inside the disk is equal to the applied external field. Solving the linearized Ginzburg-Landau (GL) equation is equivalent to finding the eigenvalue of the following Hamiltonian:
(1) where A = QrxH is the vector potential, m* the mass of an electron pair and @,=h/2e is the superconducting flux quantum. This Hamiltonian merely represents the kinetic energy of an electron pair in the presence of an external magnetic field. The geometrical shape dictates the confinement of the pair [ lo] through the boundary conditions:
(2) where b is a length which depends on the nature of the interface along the disk circumference; b- r2/a= 03 for an ideal superconducting-insulator interface and b=O for an interface with a normal metal or a magnetic material (a is the lattice parameter). Near the phase transition, Wang et al. [9] have derived simple relationships between the thermodynamic variables of the superconducting circuits such as the critical temperature T,(H) or the equilibrium magnetization M( T, H) and the ground state E,,,(H) of the energy spectrum. The superconducting critical temperature, T,(H), is directly determined by E,,,:
(4)
’ ‘-
T T,(O)
where r,~,( 0) is the order parameter at zero temperature, PAthe Abrikosov parameter taken as/3A= 1 for simplification. The second term in eq. (5) describes the variation versus magnetic field of the average pair current and depends only on the energy spectrum. It can be seen from eqs. (4) and ( 5) that the temperature derivative dM/dT (experimental quantity) directly measures ti/dH and is temperature independent. In order to obtain T,(H) and M(H) for the disk, we have solved the eigenvalue equation (eqs. ( 1) and (2)) with b=oo. Given the radial symmetry of the disk, the order parameter can be written as t~=f(r) ei” w he re 1.is an integer number. The radial function f( r) is expressed as function of degenerate hypergeometric functions [ 121. Fig. 1 shows the lowest energy levels as function of the reduced flux
30
$20 k : 10
0 0
where Tc( 0) is the zero field critical temperature and r(O) the zero temperature coherence length
(~(T)=~(0)[1-T/T(0)l-“2). The magnetization ii thermodynamic equilibrium close to the mean field transition is also controlled
5
10
15
Fig. 1. Lower energy levels, in units of f?/2m*R* of a charged particle (mass m* and charge 2e) confined in an isolated circular disk as function of the reduced applied magnetic flux @/&,= rR2H/@,,. Boundary conditions are those for a superconducting to insulator boundary (b=cq see text). The dashed line shotis the lowest bulk Landau level.
37
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22 October 1990
PHYSICS LETTERS A
$I/&, where Q)=nR ‘H. If we compare this energy spectrum to the Landau level spectrum of free charged particles (the dashed line of fig. 1 represents the first bulk Landau level), we notice quite interesting features. First, the confinement of the electron pair in the disk splits the degeneracy of the first Landau level. Electron states at the edge of the disk have a lower energy than electron states located near the center of the disk. This situation, specific to the superconducting systems, is at the origin of the well known H,, line in thin films of type II superconductors [ 13 ] and contrasts with the classical picture of Landau levels in confined geometries where the energy levels of edge states are enhanced due to the confining potential [ 141. An additional interesting feature results from the fact that the edge states form closed orbits around the disk. The family of curves which describes the spectrum of fig. 1 are indexed by the quantum number I related to the angular momentum. The ground state E,,,(H) is a singular curve made of a sequence of arcs corresponding successively to l=O, 1, 2, ... as the external field is increased. At high field values, the high angular momentum states are favoured. If b decreases, the ground state energy E increases and the intersection between successive 1numbers is shifted to higher and higher magnetic field values. For b = 0, one recovers the usual boundary condition in quantum mechanics (v/= 0 at the boundary): the ground state is given by the I=0 state at low magnetic fields and coincides with the first Landau level at high magnetic fields. This ground state exhibits no oscillation.
technique is the same as that described in ref. [ 71. The zero field critical temperature of the disk is 1.273 K. No hysteresis effect has been observed on increasing or decreasing the field. Fig. 2 shows the transition line in reduced units of magnetic field @/ go and temperature AT,/ T, = 1 - T,(H) / T, (0 ) . The upper curve a represents the experimental line and conversely, the bottom curve b corresponds to the theoretical transition, for b=m, i.e. for perfect superconducting-insulating interface. The coherence length, which is the only unknown parameter, was determined by fitting the two curves in the low field regime (see below). In the low field regime, i.e. for @/do< 1, the reduced critical temperature has a quadratic field dependence. This dependence can be understood as follows: since the magnetic length lH is larger than the disk radius the pair wave function has zero angular momentum I=0 and is nearly constant across the disk. The superfluid velocity increases linearly with magnetic field as V,= -cd/m* so that lT,(H) / T, (0) which directly measures the kinetic energy is quadratic in H and increases as [ t2(0)/2R2] ($/t#~~o)~.This quadratic dependence has been checked experimentally with accuracy. We have obtained r(O) ~0.20 urn which is a reasonable value for aluminum thin films. In the high field limit, i.e. for 9/g,, 3 1 ( lHcc R),
.02, 1.2
t
,
, ,
, tbl
02 A
3. Transition line The resistive measurements were made on an aluminum disk of radius 7.2 urn and thickness 800 8, (sample 1). The contact leads were made by evaporating two 1.3 urn wide gold wires prior to the Al evaporation. The sample was prepared by photolithography using a lift-off process. The transition line is obtained by measuring the variations of the critical temperature as a function of the magnetic field for a constant value of the sample resistance. The critical temperature is defined at the steepest point on the resistance curve R(T). The experimental 38
Fig. 2. (a) Experimental transition line of a disk in reduced units of temperature 1- T’,(H)/T,(O) and flux #/@a (vertical scale to the left). (b) Theoretical transition line for b=cc (vertical scale to the right). A vertical shift has been made to separate the two curves. The inset show a magnified view (vertical axis x 1000) of the high field oscillations once the linear envelope has been subtracted.
Volume 150, number
PHYSICS LETTERS A
1
the T,(H) curve shows a small amplitude oscillation superimposed onto a linear global dependence. The theoretical prediction (curve b) obtained using the coherence length as determined from the low field regime is in excellent agreement with the experimental results (curve a). The measured slope near o/&=50 leads to the experimental parameter ,u’= 0.56 + 0.02 in the asymptotic expression R2 p20 ar,_ TC -2r2(o)x. The theory predicts p2=0.564 at o/o,,= 50 and ~*=0.59 for infinite field. The p2 coefftcient represents the ratio of the ground state energy to the first Landau level and coincides with the ratio of the upper critical field H,, to the surface critical field H,, of a semi-infinite superconducting system [ 131. Along the transition line, superconductivity nucleates near the edge of the disk over a width given by I,. This state is similar to the giant vortex state described earlier in bulk cylinders [ 15 1. The observed period of the oscillatory component is a slightly decreasing function of the magnetic field which converges to a constant value in the high field limit (see inset of fig. 2). The high field period AH= 0.124 Oe corresponds within the accuracy of radius measurement (2%) to one flux quantum in the disk area nR2. The origin of this pseudoperiodic oscillation is the quantization of the angular momentum of the superconducting edge state. We may notice that the singly connected character of our circuit drastically modifies the superconducting properties as compared to Little and Parks’ one-dimensional loop [ 161. In the disk, the pair velocity is radially dependent, vs=---
fi
m*r (
I
---
rcHr
,
$0 >
and changes sign on a critical circle of effective radius Rem= I,4 along which the pair current is zero. As H increases with fixed 1, i.e. between two cusps in the E(H) curve, there is a gradual unbalance of these two opposite currents which results in a change of the total supercurrent from paramagnetic to diamagnetic. For high magnetic fields, Ii.,becomes much smaller than R and Rew approaches the disk radius I,_,. The magnetic period correas &-r x R - m sponds to one flux quantum in the area nR& and is
22 October 1990
expected to follow the asymptotic limit A+=@o( 1 +mI,lR)
.
(6)
This field dependence is well observed experimentally for @/&,o)5. The inset of fig. 2 shows a magnification of the high field oscillations above @/ &,= 50. One notices that the period is perfectly well defined and that there is no damping of the oscillations, in contrast to the single ring Little-Parks situation. Actually the observed amplitude is larger than expected by a factor of 2 (see curves a and b). However, we do not consider this observation to be significant since this amplitude is found to depend strongly upon our T, definition on the R(T) curve. The main significant difference between predicted and experimental curves is in the intermediate regime 1-c r$/@,,< 10 where I, is of the order of R. The l=O state persists over a field range larger than expected. The jump into the I = 1 state or, equivalently, the entrance of the first flux quantum takes place at G/e,,= 2.5 instead of the theoretical value $/do= 1.9. In addition, subsequent oscillation branches have irregular periods. These differences may be due to the presence of the two gold contacts which modify the boundary conditions. Taking account of the local perturbation is difficult because it destroys the radial symmetry. In fact, as shown below, the contactless magnetization measurements exhibit similar delay in flux entrance. The magnetic measurements will let us discuss with more details the actual boundary conditions.
4. Magnetization The magnetic measurements were made on an array of 250x250 indium disks (fig. 3), deposited on a 6 x 6 mm2 silicon substrate. A thin layer of titanium (60 A) was evaporated prior to the 1100 A indium deposition in order to improve the adherence onto the substrate. The silicon substrate was held at 77 K during the indium deposition. The center-tocenter distance between adjacent disks is 24 pm. This sample was realized by electron beam lithography using a two layer resist. The experimental technique consists in measuring the response of the magnetization to a low frequency (4 Hz) temperature oscillation. The experimental set-up and the details of the 39
Volume 150. number 1
PHYSICS LETTERS A
22 October 1990
Fig. 3. SEM micrograph of the array of 250 x 250 indium disks of diameter 12 pm and thickness 1100 A.
dM/dT measurement was described in ref. [ 17 1. Also described in ref. [ 171 are the results of coherence length measurements <( 0) ~0.18 urn on indium films prepared in the same way as those of fig. 3. The measurements were made in the vicinity of the transition temperature ( T,( 0) = 3.24 K) where the magnetization is reversible. The temperature modulation was about 10 mK. In our experimental situation, the dipolar interaction between disks is weak because of their large separation. We have estimated that this interaction leads to less than a few percent variation of the magnetic flux within each disk. In the following, every disk will be considered as isolated and exposed to the external magnetic field only. Fig. 4a shows the experimental derivative dM/dT as function of the applied magnetic field. It increases linearly at low field up to @/&,w 3.8, where a crossover to a high field regime takes place. Then a weak oscillation appears with an approximate period of 0.21 Oe ( ~1.2&,/~R*). Superimposed to this oscillation, one observes a smooth variation as function of the field. dM/dTreaches a maximum near @/ &,= 7.5 and then decreases for larger field. This behaviour can be explained qualitatively from the prediction of eq. (4). As discussed previously, for @/@oC-Z1, the ground state is the I=0 state. All electron pairs move in the clockwise direction (diamagnetic behaviour ). The 40
Fig. 4. (a) Experimental derivative dh4/dT as function of the reduced applied field in arbitrary units. (b) Predicted dM/dT using expression (7) and b=0.44R (a=50 mK,P=80 mK).
superfluid velocity and therefore dA4ldT increases linearly with field. As the field increases, one expects that the jump into the I= 1 state is accompanied by a sharp drop of dM/dT. Further increase of the magnetic field is expected to induce a sawtooth variation of the magnetization with successive discontinuous jumps of almost 100% amplitude accompanying the 1=2, 3, ... increments. This can be seen qualitatively by simply examining the slope dE/dH of the band edge of fig. 1. The oscillations experimentally observed in fig. 4a have a much lower amplitude than expected but show the expected period within a few percent. We notice however that the crossover from linear to oscillatory regime is observed at about twice the predicted field value ( $/&, = 1.9 ) . The influence of contact leads suggested in the above section cannot be invoked in the present situation. At present we have no definitive explanation for this discrepancy. We propose the following picture based upon a slight modification of the boundary conditions at the disk boundary. Due to both the
Volume 150, number 1
PHYSICS LETTERS A
limited resolution of the lithography, which may produce rough edges, and/or the decreasing indium thickness along the border, one expects that superconductivity is degraded along the disk contour. In the extreme situation where the superconductivity is destroyed at the disk edge, the boundary condition (cl/(R) =O) would be b=O and no oscillation is expected on the ground state. In fact, finite values of the parameter b in eq. (2 ) allow to characterize this boundary effect quantitatively. In order to compare these results to the theoretically predicted values, we must consider both the magnetization saturation of the disks when the effective penetration length is smaller than the disk radius, and the superconducting fluctuation near T,(H). These two effects which were neglected in the theoretical model, result in a peak shaped dM/dT curve which can be parametrized by the following empirical law: [T-T,(H)+cx]~
dM dT =exp
-
P2
dE, > dH ’
22 October 1990
5. Conclusions
In conclusion, we have measured the superconducting transition line T,(H) and he derivative dM/ dT of the magnetization of superconducting disks. Well-defined oscillations are observed in both quantities in a singly-connected circuit. The occurrence of these oscillations can be understood as a consequence of the quantization of the angular momentum of the macroscopic electron pair state in the disk. In the semi-classical high field limit, the superconducting state can be described as an edge state since Rx=-I,. In the intermediate regime, we observed a behaviour which is presumably strongly dependent upon the disk shape, and the quality of the disk boundary. We propose an interpretation which takes this into account via the b interface parameter. Further studies are needed to understand completely the intermediate regime.
(7)
where (Yand p are constants to be determined by fitting the magnetization curves with the experimental results at different fields and temperatures. Fig. 4b shows the theoretical derivative dM/dT for the best fitting value b=0.44R (2.64 urn). This b parameter was chosen in order to adjust the position of the first magnetization jump to the measured value @I@,,=3.8. This new boundary condition yields a satisfactory agreement between theory and experiment except that the amplitude of the measured oscillations is much smaller than the predicted ones. Within the above model, a small variation of the parameter b from one disk to the other may be at the origin of the observed small amplitude. However we have not yet any complete understanding of this anomalous amplitude. A detailed discussion of these effects should certainly include the possibility of trapping metastable states corresponding to high energy pair states not considered in the present study since we assume that, for each magnetic flux condition, the superconducting wavefunction occupies the ground state.
Acknowledgement We are grateful for stimulating conversations with Drs. J. Jose and J. Chaussy. We wish to thank J. Geneste for his help in sample preparation. The electron beam lithography was made in Centre National d’Etudes des Telecommunications, F38243 Meylan.
References [ I] B. Kramer ed., Proc. NATO Workshop on Quantum coherence in mesoscopic systems, Les Arcs (France), April 1990, to appear in NATO AS1 Series (Plenum, New York). [ 2 ] J.Y. Marzin, Y. Guldner and J.C. Maan, eds., Proc. 8th Conf. on Electronic properties of 2D electronic systems (EP2DS8), Surf. Sci. 229 (1990). [ 31 J.E. Mooij and G. Schon, eds., Coherence in superconducting networks, Physica B 152 ( 1988). [ 4 ] U. Sivan, Y. Imry and C. Hortzstein, Phys. Rev. B 39 ( 1989) 1242. [ 51K. Nakamura and H. Thomas, Phys. Rev. Lett. 61 ( 1988) 247. [ 6 ] B.J. Van Wees, L.P. Kouwenhoven, C.J.P.M. Harmous and J.G. Williamson, Phys. Rev. Lett. 62 (1989) 2523. [7] B. Pannetier, J. Chaussy, R. Rammal and J.C. Villegier, Phys. Rev. Lett. 53 (1984) 1845. [S] J. Gordon, A. Goldman, J. Maps, D. Costello, R. Tiberio and B. Whitehead, Phys. Rev. Lett. 56 (1986) 2280.
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[9] Y.Y. Wang, R. Rammal and B. Pannetier, J. Low Temp. Phys. 68 (1987) 301. IO] PG. de Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1964); C.R. Acad. Sci. B 292 ( 1981) 9,279. [ 111 G. Dolan, J. Low Temp. Phys. 15 ( 1974) 133. [ 121 M. Abramowitz, ed., Handbook of mathematical functions. [ 131 D. Saint-James and P.G. de Gennes, Phys. Lett. 7 ( 1963) 306.
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22 October 1990
[ 141 M. Btittiker, Phys. Rev. B 38 (1988) 9375. [15]H.J.FinkandA.G. Presson, Phys. Rev. 151 (1966) 219. [ 161 W.A. Little and R. Parks, Phys. Rev. A 13 (1964) 97. [ 171 P. Gandit, J. Chaussy, B. Pannetier, A. Vareille and T. Tissier, Europhys. Lett. 3 ( 1987) 624; Physica B 152 ( 1988) 32.
Magnetization
22 October 1990
PHYSICS LETTERS A
oscillations of a superconducting
0. Buisson ‘, P. Gandit,
R. Rammal,
disk
Y.Y. Wang and B. Pannetier
Centre de Recherches SW les TrPs Basses Temp&atures, Laboratoire associP d I’Universitt J. Fourier, C.N.R.S., B.P. 166X, F38042 Grenoble Cedex. France
Received 5 June 1990; revised manuscript received 28 August 1990; accepted for publication 30 August 1990 Communicated by D. Bloch
We present magnetic measurements on micron-sized superconducting disks. The phase transition line r,(H) and magnetization exhibit spectacular undamped oscillations as function of magnetic field. These results reflect the contribution of the edge states on the energy spectrum of electron pairs in the disk. We discuss their sensitivity to boundary conditions.
1. Introduction
Technological improvements in microelectronics have motivated new studies in solid state physics such as quantum transport in mesoscopic circuits [ 11, two-dimensional electron gas in semiconductors [ 2 ] and superconducting networks [ 31. In these systems, the boundary conditions given by the shape of the circuit and the nature of the material to which contact is made are very important. For example, theoretical studies [ 4,5 ] of quantum point contacts with different confinement geometries have shown that the modification of the circuit-shape (disk to ellipse) may drastically affect the energy spectrum and therefore the transport and magnetic properties. Pioneer experimental studies on quantum transport in confined high mobility electron gas have been reported by Van Wees et al. [ 61. Except in narrow point contacts where conductance quantization is observed, the electronic wavelength in such low carrier density systems remains smaller than the lateral size of the system (typically micron scale). In contrast, quantum coherence in superconducting materials involves large electron pair wavelengths controlled by the superconducting coherence length. Recently, transport and magnetic properties of twodimen’ Present address: Laboratorio de Materia condensada, Dpt.
sional artificial superconducting networks [ 3,7,8] have been investigated as function of their topology (periodic, quasicrystalline or fractal arrays ) . The detailed study of these systems relies upon the remarkable fact that in the superconducting phase all Cooper pairs occupy coherently the same quantum state. In particuiar the equilibrium superconducting properties, such as critical temperature and magnetization, can be obtained from the energy spectrum of the pair wavefunction in the considered geometry. The appropriate boundary condition is given, in the wire network geometry, by Kirchoff’s current conservation condition at the nodes of the network [ lo]. To date only multiconnected superconducting systems have been considered. In this paper, we report on our investigation of the properties of superconducting thin film disks in the vicinity of the phase transition. This study involves both the low field regime where the magnetic length 1”= (&,/rcH) ‘I2 is large compared to the disk radius R,and the high field regime where Znis smaller than R. Our study provides a very simple determination of the ground state of a quantum charged particle confined into a two-dimensional disk under an external magnetic field. After a brief description of the energy spectrum of a pair confined in a disk, we discuss the experimental results on the superconducting critical temperature and the magnetization.
Fisica, P.U.C. 225 rua Marques de Sao Vicente, 22453 Rio de Janeiro, Brazil.
36
0375-9601/90/$
03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)
Volume 150, number 1
PHYSICS LETTERS A
22 October 1990
by the ground state energy E,,,(H) and depends upon the density of superconducting electrons I y,,, I 2,
2. Theoretical model In the vicinity of the mean field phase transition, the superconducting thin film disk behaves as a type II superconductor [ IO,1 11. Moreover, near the transition, the effective penetration length of the perpendicular magnetic field is much larger than the disk radius R. So the magnetic field inside the disk is equal to the applied external field. Solving the linearized Ginzburg-Landau (GL) equation is equivalent to finding the eigenvalue of the following Hamiltonian:
(1) where A = QrxH is the vector potential, m* the mass of an electron pair and @,=h/2e is the superconducting flux quantum. This Hamiltonian merely represents the kinetic energy of an electron pair in the presence of an external magnetic field. The geometrical shape dictates the confinement of the pair [ lo] through the boundary conditions:
(2) where b is a length which depends on the nature of the interface along the disk circumference; b- r2/a= 03 for an ideal superconducting-insulator interface and b=O for an interface with a normal metal or a magnetic material (a is the lattice parameter). Near the phase transition, Wang et al. [9] have derived simple relationships between the thermodynamic variables of the superconducting circuits such as the critical temperature T,(H) or the equilibrium magnetization M( T, H) and the ground state E,,,(H) of the energy spectrum. The superconducting critical temperature, T,(H), is directly determined by E,,,:
(4)
’ ‘-
T T,(O)
where r,~,( 0) is the order parameter at zero temperature, PAthe Abrikosov parameter taken as/3A= 1 for simplification. The second term in eq. (5) describes the variation versus magnetic field of the average pair current and depends only on the energy spectrum. It can be seen from eqs. (4) and ( 5) that the temperature derivative dM/dT (experimental quantity) directly measures ti/dH and is temperature independent. In order to obtain T,(H) and M(H) for the disk, we have solved the eigenvalue equation (eqs. ( 1) and (2)) with b=oo. Given the radial symmetry of the disk, the order parameter can be written as t~=f(r) ei” w he re 1.is an integer number. The radial function f( r) is expressed as function of degenerate hypergeometric functions [ 121. Fig. 1 shows the lowest energy levels as function of the reduced flux
30
$20 k : 10
0 0
where Tc( 0) is the zero field critical temperature and r(O) the zero temperature coherence length
(~(T)=~(0)[1-T/T(0)l-“2). The magnetization ii thermodynamic equilibrium close to the mean field transition is also controlled
5
10
15
Fig. 1. Lower energy levels, in units of f?/2m*R* of a charged particle (mass m* and charge 2e) confined in an isolated circular disk as function of the reduced applied magnetic flux @/&,= rR2H/@,,. Boundary conditions are those for a superconducting to insulator boundary (b=cq see text). The dashed line shotis the lowest bulk Landau level.
37
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PHYSICS LETTERS A
$I/&, where Q)=nR ‘H. If we compare this energy spectrum to the Landau level spectrum of free charged particles (the dashed line of fig. 1 represents the first bulk Landau level), we notice quite interesting features. First, the confinement of the electron pair in the disk splits the degeneracy of the first Landau level. Electron states at the edge of the disk have a lower energy than electron states located near the center of the disk. This situation, specific to the superconducting systems, is at the origin of the well known H,, line in thin films of type II superconductors [ 13 ] and contrasts with the classical picture of Landau levels in confined geometries where the energy levels of edge states are enhanced due to the confining potential [ 141. An additional interesting feature results from the fact that the edge states form closed orbits around the disk. The family of curves which describes the spectrum of fig. 1 are indexed by the quantum number I related to the angular momentum. The ground state E,,,(H) is a singular curve made of a sequence of arcs corresponding successively to l=O, 1, 2, ... as the external field is increased. At high field values, the high angular momentum states are favoured. If b decreases, the ground state energy E increases and the intersection between successive 1numbers is shifted to higher and higher magnetic field values. For b = 0, one recovers the usual boundary condition in quantum mechanics (v/= 0 at the boundary): the ground state is given by the I=0 state at low magnetic fields and coincides with the first Landau level at high magnetic fields. This ground state exhibits no oscillation.
technique is the same as that described in ref. [ 71. The zero field critical temperature of the disk is 1.273 K. No hysteresis effect has been observed on increasing or decreasing the field. Fig. 2 shows the transition line in reduced units of magnetic field @/ go and temperature AT,/ T, = 1 - T,(H) / T, (0 ) . The upper curve a represents the experimental line and conversely, the bottom curve b corresponds to the theoretical transition, for b=m, i.e. for perfect superconducting-insulating interface. The coherence length, which is the only unknown parameter, was determined by fitting the two curves in the low field regime (see below). In the low field regime, i.e. for @/do< 1, the reduced critical temperature has a quadratic field dependence. This dependence can be understood as follows: since the magnetic length lH is larger than the disk radius the pair wave function has zero angular momentum I=0 and is nearly constant across the disk. The superfluid velocity increases linearly with magnetic field as V,= -cd/m* so that lT,(H) / T, (0) which directly measures the kinetic energy is quadratic in H and increases as [ t2(0)/2R2] ($/t#~~o)~.This quadratic dependence has been checked experimentally with accuracy. We have obtained r(O) ~0.20 urn which is a reasonable value for aluminum thin films. In the high field limit, i.e. for 9/g,, 3 1 ( lHcc R),
.02, 1.2
t
,
, ,
, tbl
02 A
3. Transition line The resistive measurements were made on an aluminum disk of radius 7.2 urn and thickness 800 8, (sample 1). The contact leads were made by evaporating two 1.3 urn wide gold wires prior to the Al evaporation. The sample was prepared by photolithography using a lift-off process. The transition line is obtained by measuring the variations of the critical temperature as a function of the magnetic field for a constant value of the sample resistance. The critical temperature is defined at the steepest point on the resistance curve R(T). The experimental 38
Fig. 2. (a) Experimental transition line of a disk in reduced units of temperature 1- T’,(H)/T,(O) and flux #/@a (vertical scale to the left). (b) Theoretical transition line for b=cc (vertical scale to the right). A vertical shift has been made to separate the two curves. The inset show a magnified view (vertical axis x 1000) of the high field oscillations once the linear envelope has been subtracted.
Volume 150, number
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1
the T,(H) curve shows a small amplitude oscillation superimposed onto a linear global dependence. The theoretical prediction (curve b) obtained using the coherence length as determined from the low field regime is in excellent agreement with the experimental results (curve a). The measured slope near o/&=50 leads to the experimental parameter ,u’= 0.56 + 0.02 in the asymptotic expression R2 p20 ar,_ TC -2r2(o)x. The theory predicts p2=0.564 at o/o,,= 50 and ~*=0.59 for infinite field. The p2 coefftcient represents the ratio of the ground state energy to the first Landau level and coincides with the ratio of the upper critical field H,, to the surface critical field H,, of a semi-infinite superconducting system [ 131. Along the transition line, superconductivity nucleates near the edge of the disk over a width given by I,. This state is similar to the giant vortex state described earlier in bulk cylinders [ 15 1. The observed period of the oscillatory component is a slightly decreasing function of the magnetic field which converges to a constant value in the high field limit (see inset of fig. 2). The high field period AH= 0.124 Oe corresponds within the accuracy of radius measurement (2%) to one flux quantum in the disk area nR2. The origin of this pseudoperiodic oscillation is the quantization of the angular momentum of the superconducting edge state. We may notice that the singly connected character of our circuit drastically modifies the superconducting properties as compared to Little and Parks’ one-dimensional loop [ 161. In the disk, the pair velocity is radially dependent, vs=---
fi
m*r (
I
---
rcHr
,
$0 >
and changes sign on a critical circle of effective radius Rem= I,4 along which the pair current is zero. As H increases with fixed 1, i.e. between two cusps in the E(H) curve, there is a gradual unbalance of these two opposite currents which results in a change of the total supercurrent from paramagnetic to diamagnetic. For high magnetic fields, Ii.,becomes much smaller than R and Rew approaches the disk radius I,_,. The magnetic period correas &-r x R - m sponds to one flux quantum in the area nR& and is
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expected to follow the asymptotic limit A+=@o( 1 +mI,lR)
.
(6)
This field dependence is well observed experimentally for @/&,o)5. The inset of fig. 2 shows a magnification of the high field oscillations above @/ &,= 50. One notices that the period is perfectly well defined and that there is no damping of the oscillations, in contrast to the single ring Little-Parks situation. Actually the observed amplitude is larger than expected by a factor of 2 (see curves a and b). However, we do not consider this observation to be significant since this amplitude is found to depend strongly upon our T, definition on the R(T) curve. The main significant difference between predicted and experimental curves is in the intermediate regime 1-c r$/@,,< 10 where I, is of the order of R. The l=O state persists over a field range larger than expected. The jump into the I = 1 state or, equivalently, the entrance of the first flux quantum takes place at G/e,,= 2.5 instead of the theoretical value $/do= 1.9. In addition, subsequent oscillation branches have irregular periods. These differences may be due to the presence of the two gold contacts which modify the boundary conditions. Taking account of the local perturbation is difficult because it destroys the radial symmetry. In fact, as shown below, the contactless magnetization measurements exhibit similar delay in flux entrance. The magnetic measurements will let us discuss with more details the actual boundary conditions.
4. Magnetization The magnetic measurements were made on an array of 250x250 indium disks (fig. 3), deposited on a 6 x 6 mm2 silicon substrate. A thin layer of titanium (60 A) was evaporated prior to the 1100 A indium deposition in order to improve the adherence onto the substrate. The silicon substrate was held at 77 K during the indium deposition. The center-tocenter distance between adjacent disks is 24 pm. This sample was realized by electron beam lithography using a two layer resist. The experimental technique consists in measuring the response of the magnetization to a low frequency (4 Hz) temperature oscillation. The experimental set-up and the details of the 39
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Fig. 3. SEM micrograph of the array of 250 x 250 indium disks of diameter 12 pm and thickness 1100 A.
dM/dT measurement was described in ref. [ 17 1. Also described in ref. [ 171 are the results of coherence length measurements <( 0) ~0.18 urn on indium films prepared in the same way as those of fig. 3. The measurements were made in the vicinity of the transition temperature ( T,( 0) = 3.24 K) where the magnetization is reversible. The temperature modulation was about 10 mK. In our experimental situation, the dipolar interaction between disks is weak because of their large separation. We have estimated that this interaction leads to less than a few percent variation of the magnetic flux within each disk. In the following, every disk will be considered as isolated and exposed to the external magnetic field only. Fig. 4a shows the experimental derivative dM/dT as function of the applied magnetic field. It increases linearly at low field up to @/&,w 3.8, where a crossover to a high field regime takes place. Then a weak oscillation appears with an approximate period of 0.21 Oe ( ~1.2&,/~R*). Superimposed to this oscillation, one observes a smooth variation as function of the field. dM/dTreaches a maximum near @/ &,= 7.5 and then decreases for larger field. This behaviour can be explained qualitatively from the prediction of eq. (4). As discussed previously, for @/@oC-Z1, the ground state is the I=0 state. All electron pairs move in the clockwise direction (diamagnetic behaviour ). The 40
Fig. 4. (a) Experimental derivative dh4/dT as function of the reduced applied field in arbitrary units. (b) Predicted dM/dT using expression (7) and b=0.44R (a=50 mK,P=80 mK).
superfluid velocity and therefore dA4ldT increases linearly with field. As the field increases, one expects that the jump into the I= 1 state is accompanied by a sharp drop of dM/dT. Further increase of the magnetic field is expected to induce a sawtooth variation of the magnetization with successive discontinuous jumps of almost 100% amplitude accompanying the 1=2, 3, ... increments. This can be seen qualitatively by simply examining the slope dE/dH of the band edge of fig. 1. The oscillations experimentally observed in fig. 4a have a much lower amplitude than expected but show the expected period within a few percent. We notice however that the crossover from linear to oscillatory regime is observed at about twice the predicted field value ( $/&, = 1.9 ) . The influence of contact leads suggested in the above section cannot be invoked in the present situation. At present we have no definitive explanation for this discrepancy. We propose the following picture based upon a slight modification of the boundary conditions at the disk boundary. Due to both the
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limited resolution of the lithography, which may produce rough edges, and/or the decreasing indium thickness along the border, one expects that superconductivity is degraded along the disk contour. In the extreme situation where the superconductivity is destroyed at the disk edge, the boundary condition (cl/(R) =O) would be b=O and no oscillation is expected on the ground state. In fact, finite values of the parameter b in eq. (2 ) allow to characterize this boundary effect quantitatively. In order to compare these results to the theoretically predicted values, we must consider both the magnetization saturation of the disks when the effective penetration length is smaller than the disk radius, and the superconducting fluctuation near T,(H). These two effects which were neglected in the theoretical model, result in a peak shaped dM/dT curve which can be parametrized by the following empirical law: [T-T,(H)+cx]~
dM dT =exp
-
P2
dE, > dH ’
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5. Conclusions
In conclusion, we have measured the superconducting transition line T,(H) and he derivative dM/ dT of the magnetization of superconducting disks. Well-defined oscillations are observed in both quantities in a singly-connected circuit. The occurrence of these oscillations can be understood as a consequence of the quantization of the angular momentum of the macroscopic electron pair state in the disk. In the semi-classical high field limit, the superconducting state can be described as an edge state since Rx=-I,. In the intermediate regime, we observed a behaviour which is presumably strongly dependent upon the disk shape, and the quality of the disk boundary. We propose an interpretation which takes this into account via the b interface parameter. Further studies are needed to understand completely the intermediate regime.
(7)
where (Yand p are constants to be determined by fitting the magnetization curves with the experimental results at different fields and temperatures. Fig. 4b shows the theoretical derivative dM/dT for the best fitting value b=0.44R (2.64 urn). This b parameter was chosen in order to adjust the position of the first magnetization jump to the measured value @I@,,=3.8. This new boundary condition yields a satisfactory agreement between theory and experiment except that the amplitude of the measured oscillations is much smaller than the predicted ones. Within the above model, a small variation of the parameter b from one disk to the other may be at the origin of the observed small amplitude. However we have not yet any complete understanding of this anomalous amplitude. A detailed discussion of these effects should certainly include the possibility of trapping metastable states corresponding to high energy pair states not considered in the present study since we assume that, for each magnetic flux condition, the superconducting wavefunction occupies the ground state.
Acknowledgement We are grateful for stimulating conversations with Drs. J. Jose and J. Chaussy. We wish to thank J. Geneste for his help in sample preparation. The electron beam lithography was made in Centre National d’Etudes des Telecommunications, F38243 Meylan.
References [ I] B. Kramer ed., Proc. NATO Workshop on Quantum coherence in mesoscopic systems, Les Arcs (France), April 1990, to appear in NATO AS1 Series (Plenum, New York). [ 2 ] J.Y. Marzin, Y. Guldner and J.C. Maan, eds., Proc. 8th Conf. on Electronic properties of 2D electronic systems (EP2DS8), Surf. Sci. 229 (1990). [ 31 J.E. Mooij and G. Schon, eds., Coherence in superconducting networks, Physica B 152 ( 1988). [ 4 ] U. Sivan, Y. Imry and C. Hortzstein, Phys. Rev. B 39 ( 1989) 1242. [ 51K. Nakamura and H. Thomas, Phys. Rev. Lett. 61 ( 1988) 247. [ 6 ] B.J. Van Wees, L.P. Kouwenhoven, C.J.P.M. Harmous and J.G. Williamson, Phys. Rev. Lett. 62 (1989) 2523. [7] B. Pannetier, J. Chaussy, R. Rammal and J.C. Villegier, Phys. Rev. Lett. 53 (1984) 1845. [S] J. Gordon, A. Goldman, J. Maps, D. Costello, R. Tiberio and B. Whitehead, Phys. Rev. Lett. 56 (1986) 2280.
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[9] Y.Y. Wang, R. Rammal and B. Pannetier, J. Low Temp. Phys. 68 (1987) 301. IO] PG. de Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1964); C.R. Acad. Sci. B 292 ( 1981) 9,279. [ 111 G. Dolan, J. Low Temp. Phys. 15 ( 1974) 133. [ 121 M. Abramowitz, ed., Handbook of mathematical functions. [ 131 D. Saint-James and P.G. de Gennes, Phys. Lett. 7 ( 1963) 306.
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[ 141 M. Btittiker, Phys. Rev. B 38 (1988) 9375. [15]H.J.FinkandA.G. Presson, Phys. Rev. 151 (1966) 219. [ 161 W.A. Little and R. Parks, Phys. Rev. A 13 (1964) 97. [ 171 P. Gandit, J. Chaussy, B. Pannetier, A. Vareille and T. Tissier, Europhys. Lett. 3 ( 1987) 624; Physica B 152 ( 1988) 32.