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Superconductivity: Tunneling
Superconductivity: Tunneling 105
zone of the triangular vortex lattice with area pk2BZ . In the London limit, one finds for isotropic superconductors, the elastic moduli % 0 =m0 B% 2 =m0 BF c11 ðkÞE ; c66 E ; 2 1 þ k2 l 16pl2 % 2 =m0 % 0 =m0 k2 B BF c44 ðkÞE þ ln ½18 2 2 1 þ k2 l 8pl 1 þ k2z l2
In this result, a large factor ½c44 ð0Þ=c44 ðkBZ Þ1=2 EkBZ lEpl=ac1 originates from the elastic nonlocality. In anisotropic superconductors with B||c (the crystalline c axis), the thermal fluctuations, eqn [19], are enhanced by an additional factor G ¼ lc =lab c1, where lab and lc are the two penetration depths of uniaxially anisotropic superconductors.
% The GL theory yields an additional factor ð1 B= % B % Bc2 Þ2 , and replaces l Bc2 Þ2 in c66, that is, c66 pBð % c2 Þ1=2. in c11 and c44 (first term) by l0 ¼ l=ð1 B=B The k dependence (dispersion) of the compression and tilt moduli c11(k) and c44(k) means that the elasticity of the vortex lattice is nonlocal, that is, strains with short wavelengths, 2p/k{2pl, have a much lower elastic energy than a homogeneous compression or tilt (corresponding to k-0) with the same amplitude. This elastic nonlocality comes from the fact that the magnetic interaction between the flux lines typically has a range l much longer than the flux-line spacing a0; therefore, each flux line interacts with many other flux lines. Note that a large l causes a small shear stiffness since c66 pl2 , and a smaller c11 ðkol1 Þ at short wavelengths, but the uniform compressibility c11 ðk ¼ 0Þ is independent of l. As a consequence of nonlocal elasticity, the vortex displacements un(z) caused by local pinning forces, and also the space- and time-averaged thermal fluctuations of the vortex positions, /un ðzÞ2 S, are much larger than they would be if c44(k) had no dispersion, that is, % a EB % 2 =m0. The maxif it were replaced by c44 ð0Þ ¼ BH imum vortex displacement uð0Þpf caused at r ¼ 0 by a point force of density fd3(r), and the thermal fluctuations /u2 SpkB T, are given by similar expressions, Z 2uð0Þ /u2 S d3 k 1 E E 3 ðk2 þ k2 Þc þ k2 c ðkÞ f kB T 8p 66 BZ x y z 44
See also: Magnetism, History of; Quantum Mechanics: Methods; Statistical Mechanics: Classical; Superconductivity: BCS Theory; Superconductivity: Flux Quantization; Superconductivity: General Aspects; Superconductivity: Tunneling; Superconductors, High Tc.
E
k2BZ l 8p½c66 c44 ð0Þ1=2
½19
PACS: 74.20.De; 74.25.Qt; 74.25.Op
Further Reading Brandt EH (1986) Elastic and plastic properties of the flux-line lattice in type-II superconductors. Physical Review B 34: 6514– 6517. Brandt EH (1995) The flux-line lattice in superconductors. Reports of Progress in Physics 58: 1465–1594. Brandt EH (2003) Properties of the ideal Ginzburg-Landau vortex lattice. Physical Review B 68(054506): 1–11. Brandt EH (2005) Ginzburg-Landau vortex lattice in superconducting films of finite thickness. Physical Review B 71(014521): 1–21. Brandt EH and Mikitik GP (2001) Meissner-London currents in superconductors with rectangular cross section. Physical Review Letters 85: 4164–4167. Carneiro GM and Brandt EH (2000) Vortex lines in films: fields and interactions. Physical Review B 61: 6370–6376. Clem JR (1991) Two-dimensional vortices in a stack of thin superconducting films: a model for high-temperature superconducting multilayers. Physical Review B 43: 7837–7846. DeGennes PG (1966) Superconductivity of Metals and Alloys. New York: Benjamin. Landau LD and Lifshitz EM (1960) Theoretical Physics, vol. 8 Electrodynamics of Continuous Media. New York: Pergamon. London F (1950) Superfluids, Vol. 1. Chichester: Wiley. Tinkham M (1975) Introduction to Superconductivity. NewYork: McGraw-Hill.
Superconductivity: Tunneling A N Cleland, University of California, Santa Barbara, CA, USA & 2005, Elsevier Ltd. All Rights Reserved.
Introduction The quantum mechanical tunneling of electrons and other charge carriers through an insulating or a
vacuum barrier is at the heart of many electronic devices. In superconductors, the controlled tunneling of Cooper-paired electrons forms the basis for the Josephson junction and the DC and RF super-conducting quantum interference devices (RF and DC SQUIDs), as well as a number of other related devices. The tunneling of a single particle through an energy barrier, such as an electron through an insulator or through vacuum, can be understood using a
zone of the triangular vortex lattice with area pk2BZ . In the London limit, one finds for isotropic superconductors, the elastic moduli % 0 =m0 B% 2 =m0 BF c11 ðkÞE ; c66 E ; 2 1 þ k2 l 16pl2 % 2 =m0 % 0 =m0 k2 B BF c44 ðkÞE þ ln ½18 2 2 1 þ k2 l 8pl 1 þ k2z l2
In this result, a large factor ½c44 ð0Þ=c44 ðkBZ Þ1=2 EkBZ lEpl=ac1 originates from the elastic nonlocality. In anisotropic superconductors with B||c (the crystalline c axis), the thermal fluctuations, eqn [19], are enhanced by an additional factor G ¼ lc =lab c1, where lab and lc are the two penetration depths of uniaxially anisotropic superconductors.
% The GL theory yields an additional factor ð1 B= % B % Bc2 Þ2 , and replaces l Bc2 Þ2 in c66, that is, c66 pBð % c2 Þ1=2. in c11 and c44 (first term) by l0 ¼ l=ð1 B=B The k dependence (dispersion) of the compression and tilt moduli c11(k) and c44(k) means that the elasticity of the vortex lattice is nonlocal, that is, strains with short wavelengths, 2p/k{2pl, have a much lower elastic energy than a homogeneous compression or tilt (corresponding to k-0) with the same amplitude. This elastic nonlocality comes from the fact that the magnetic interaction between the flux lines typically has a range l much longer than the flux-line spacing a0; therefore, each flux line interacts with many other flux lines. Note that a large l causes a small shear stiffness since c66 pl2 , and a smaller c11 ðkol1 Þ at short wavelengths, but the uniform compressibility c11 ðk ¼ 0Þ is independent of l. As a consequence of nonlocal elasticity, the vortex displacements un(z) caused by local pinning forces, and also the space- and time-averaged thermal fluctuations of the vortex positions, /un ðzÞ2 S, are much larger than they would be if c44(k) had no dispersion, that is, % a EB % 2 =m0. The maxif it were replaced by c44 ð0Þ ¼ BH imum vortex displacement uð0Þpf caused at r ¼ 0 by a point force of density fd3(r), and the thermal fluctuations /u2 SpkB T, are given by similar expressions, Z 2uð0Þ /u2 S d3 k 1 E E 3 ðk2 þ k2 Þc þ k2 c ðkÞ f kB T 8p 66 BZ x y z 44
See also: Magnetism, History of; Quantum Mechanics: Methods; Statistical Mechanics: Classical; Superconductivity: BCS Theory; Superconductivity: Flux Quantization; Superconductivity: General Aspects; Superconductivity: Tunneling; Superconductors, High Tc.
E
k2BZ l 8p½c66 c44 ð0Þ1=2
½19
PACS: 74.20.De; 74.25.Qt; 74.25.Op
Further Reading Brandt EH (1986) Elastic and plastic properties of the flux-line lattice in type-II superconductors. Physical Review B 34: 6514– 6517. Brandt EH (1995) The flux-line lattice in superconductors. Reports of Progress in Physics 58: 1465–1594. Brandt EH (2003) Properties of the ideal Ginzburg-Landau vortex lattice. Physical Review B 68(054506): 1–11. Brandt EH (2005) Ginzburg-Landau vortex lattice in superconducting films of finite thickness. Physical Review B 71(014521): 1–21. Brandt EH and Mikitik GP (2001) Meissner-London currents in superconductors with rectangular cross section. Physical Review Letters 85: 4164–4167. Carneiro GM and Brandt EH (2000) Vortex lines in films: fields and interactions. Physical Review B 61: 6370–6376. Clem JR (1991) Two-dimensional vortices in a stack of thin superconducting films: a model for high-temperature superconducting multilayers. Physical Review B 43: 7837–7846. DeGennes PG (1966) Superconductivity of Metals and Alloys. New York: Benjamin. Landau LD and Lifshitz EM (1960) Theoretical Physics, vol. 8 Electrodynamics of Continuous Media. New York: Pergamon. London F (1950) Superfluids, Vol. 1. Chichester: Wiley. Tinkham M (1975) Introduction to Superconductivity. NewYork: McGraw-Hill.
Superconductivity: Tunneling A N Cleland, University of California, Santa Barbara, CA, USA & 2005, Elsevier Ltd. All Rights Reserved.
Introduction The quantum mechanical tunneling of electrons and other charge carriers through an insulating or a
vacuum barrier is at the heart of many electronic devices. In superconductors, the controlled tunneling of Cooper-paired electrons forms the basis for the Josephson junction and the DC and RF super-conducting quantum interference devices (RF and DC SQUIDs), as well as a number of other related devices. The tunneling of a single particle through an energy barrier, such as an electron through an insulator or through vacuum, can be understood using a