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Superconductivity 1
Fichas del conjunto (101)
- 1Which properties are characteristic for superconductors?- perfect conductivity (rho = 0)
- Meissner-Ochsenfeld Effect (perfect diamagnetism) - 2Discuss the difference between a perfect conductor and a superconductor.- perfect conductor: B > 0 ... T < Tcrit ... field remains in perfect conductor (is preserved)
- superconductor: B > 0... T< Tcrit ... field is repelled from within (perfect diamagnet)
[if T both repell] - 3How can a persistent current be generated and frozen in a superconducting ring?ring of NC in B field, cool under Tc, switch off B field-> trapped flux in ring with persistent current
- 4How can we measure a vanishingly small resistivity?Measure decay of magnetic moment of frozen current in SC ring.
I(t) = I0 exp(-R/L *t) - 5How can the magnetization of a superconductor be measured quantitatively?In a superconductor the magnetization is equal to the external field with opposite sign, because the Magnetic field inside has to be 0 due to perfect diamagnetism.
- 6How was flux quantization measured in the experiment by Doll and Näbauer?- SC cylinder with trapped flux hanging on quartz thread
- apply B field perpendicular to trapped B field. (->Torque)
- measure the twisting of the thread by interference - 7Describe the magnetic field H and the magnetic flux density B inside and outside a superconductor. Discuss possible limiting cases.Outside B and H equal.
Inside B zero and H = Hext. (Due to magnetization)
Edge of SC: penetration of the B field as exponential decay on Landau penetration depth - 8Describe the B-field inside a superconducting sphereWhen the field is repelled, the magnetic field density on the equator of the sphere is increased due to the field line bending. This can result in a Bequ > Bcrit and partial penetration of the SC at the equator
- 9Which elementary superconductors do you know?- 80Hg with 4.15 K
-50 Sn with 3.7 K
- Nb with 9.2 K - 10Do you know superconducting materials with Tc > 30 K?High Tc oxide SCs:
-up to 135 K
- e.g. Hg-1223 (HgBa2Ca2Cu3O9) - 11At which temparature is the liquification of nitrogen and helium? What are the pricesnitrogen: 77K (2$ per liter),
4K (20$ per liter) - 12Different types of SC- elemental: e.g. Nb (9.2K)
- alloys and intermetallic compounds:
e.g. NbTi (10K), Nb3Sn (18K)
- oxide SC: e.g. Hg-1223 (135 K)
- heavy Fermion, organic, fullerides, iron pnictide - 13How big is the difference of the free enthalpies in the superconducting and normal state?Gn-Gs = Vs(fn-fs) - 1/2 mu0 Hext^2*Vs
for Hext = Hcth this is 0 - 14Free energy FdFI = -S dT - p dV - m dBext
dFII = -S dT - p dV + Bext dm
minimize FI for T,V,Bext constants
minimize FII for T,V,m constants - 15Free enthalpy GG = U - TS + pV - mBext
dG = -SdT + Vdp - mdBext
minimize G for T,p,Bext const
delta g = Bcth^2(T)/2mu0 - 16Describe the change of the entropy S and the electronic heat capacity cp at the transition temperature Tc- delta s = 0 at Tc
- delta cp < 0 at Tc - 17How does the temperature dependence of the entropy S in the superconducting state look like and how does it compare to that of the normal conducting material?s(SC) prop to T^3
delta s = 0 for T -> Tc,
delta s = 0 for T -> 0,
delta s > 0 for 0 - 18Sketch the temperature dependence cp(T) for a conventional superconductor with an isotropic gap.- cp first lower, then higher than in the normal state.
-cp has delta cp < 0 at Tc - 19Discuss the implications of the exponential temperature dependence of cp at low temperatures.Thermodynamics only predicts power law dependence. BCS theory predicts exponential * T^3/2 behaviour which is accurate in experiments
- 20What is the meaning of the London equations?1st: the conductance is preserved
2nd: Meißner Ochsenfeld effect
assumes: local relation between Js, E and B, but mean free path infinite -> fields have to be constant for correct result - 211st London equationfrom: equation of motion:
ms dv/dt + ms/tau vs = qs E (tau ->inf)
-> del(Lambda Js)/delt = E
London coefficient: Lambda = ms/ns qs^2
(or from: current-phase/energy-phase relation and time derivative) - 222nd London equationfrom:curl of 1st London, rot(E) =-delB/delt
-> rot(Lambda Js) + B = 0
also: lap(B) - B/lamda_L^2 = 0
lambda_L = sqrt(Lambda/mu0) = sqrt(ms/mu0 ns qs^2)
(or from: current phase/energy phase relation and curl) - 23How can the London penetration depth λL be derived? Which quantities determine λL?from: second London equation and Maxwell equation: rot(B) = - mu0 Js
lambda_L = sqrt(Lambda/mu0) = sqrt(ms/mu0 ns qs^2) - 24Discuss the meaning of the macroscopic wave function of a superconductorpsi^2 is the superconducting electron density ns
- 25current-phase relationLambda Js = [h'/qs *grad(theta(r,t))- A(r,t)]
[] = vs * qs/ms - 26energy-phase relationh' del(theta(r,t))/delt = -[1/2ns*Lambda Js^2(r,t) + qs phi_el(r,t) + mu(r,t)]
[] = total energy - 27Discuss the implications of current-phase and energy phase relation1st and 2nd London equations can be derived for ns(r,t) = ns(t)
-> Meissner Ochsenfeld effect
-> dissipation-less dc current - 28How can we derive the London equations if we assume that a superconductor can be described by a macroscopic wave function?- Get current-phase and energy-phase relation from Schrödinger equation of macroscopic wave-function superconductor.
- Get London equations from time derivative & curl of current-phase relation and usage of energy-phase relation - 29Derive the fluxoid quantization from the general expression of the supercurrent density derived from the macroscopic quantum model.Integrate current-phase relation over closed contour
-> over A = int(area s) B*n^ dS = phi
-> over grad theta = 2pi n
-> curve(Lambda Js) + area(B*n^) = n*h/qs= n*phi0 - 30What is the difference between flux and fluxoid quantization? What are the experimental requirements for measuring flux quantization?flux quantization happens if B is present while cooling. If superconduction cylinder has thickness d >> lambda_L we can take curve over middle where Js = 0 and get flux quantization from fluxoid quantization
- 31Sketch the basic ideas of the Ginzburg-Landau (GL) theorygradual change of wavefunction on characteristic lenght scale
but no time-dependence
free enthalpy gs as expansion in psi^2 up to second order
alpha = alpha'(T/Tc -1) alpha'>0
beta = const > 0
minimize gs by derivation - 32How can we describe the phase transition into the superconducting state (homogeneous superconductor in zero applied magnetic field) within the Landau theory of phase transitions? How does the free enthalpy density look like?gs-gn = alpha psi^2+1/2 beta psi^4
alpha'/2 = -Bcth(0)^2/2mu0 ns(0)
beta = Bcth(T)^2/2mu0 * 2/ns(T)^2
-> T>Tc -> alpha >0 -> U shape
T alpha <0 -> W shape with height Bcth^2/2mu0
2nd order transition in Ehrenfest theory - 33How can we classify phase transitions?discontinuity in first or second derivative of the order parameter in the given variable
- 34Comment on the order of the phase transition into the superconducting state at various point in the phase diagram.For Bext = 0 we have a second order phase transition in the type 1 SC. For applied B-Field it may turn into a 1st order transition.
- 35Which additional terms have to be taken into account when we discuss a spatially inhomogeneous superconductor in an external magnetic field? What is the meaning of these additional terms?kin energy:1/2 psi^2 ms(h'/ms *grad(theta)- qs/ms *A)^2
stiffness: ns h'^2k^2/2ms = h'^2(grad(|psi|)^2/2ms
field energy density: [b(r) - Bext]^2/2mu0 = 1/2 mu0 M^2(r) - 36How does the kinetic energy and the stiffness of the SC look combined?1/2m|h'/i grad(psi(r)) - qsA(r)psi(r)|^2
psi is the complete wavefunction not the absolute value - 37Discuss qualitatively how we can derive the GL equations from the expression for the free enthalpy density.minimize free new enthalpy
gauss theo.-> surface and volume term
- minimize respect to del psi, del psi*
-> volume term give 1st Ginz-Landau
->surface gives interface between SC and
minimize with respect to A
expand gs in delA and neglect delA^2
Maxwell equations -> 2nd Ginz-Landau - 381st Ginz. Landau1/2ms [h'/i grad -qsA]^2 psi(r) + alpha psi(r) + 1/2 beta |psi(r)|^2*psi(r) = 0
1st part is the kin. energy + stiffness
second part is normal terms with one psi less - 392nd Ginz. LandauJs = qs h'/2ms i *(psi*grad(psi)-psi*grad(psi*)) - qs^2/ms |psi|^2 A
- 40Compare the GL equations with the current-phase and energy-phase relation derived from the macroscopic quantum model. What are the analogies and differences?GL 2 <-> current-phase relation (but instead of grad(theta): grad(psi))
/ <-> energy-phase relation (no correspondance because delpsi/delt = 0)
GL1 <-> / (no correspondance because ns(r) = const assumed in macr. quant. model) - 41Which characteristic length scales are associated with the GL equations? What is the order of magnitude of these length scales?- penetration depth: 2nd GL and ns const.
-> ns = psi^2 = |alpha|/beta
lambda_GL = lambda_L (40 - 300 nm)
- coherence lenght: 1st GL -> xi_GL in front of spatial variance term
xi_GL = sqrt(h'^2/2ms|alpha|)
(2.6-1600 nm) - 42Explain the meaning of the GL coherence length ξGLFor A = 0 the spatial variance of psi decays exponentially on char. scale of xi_GL
->e.g. on interfaces |psi|^2 changes singificantly on lenghtscales of xi_GL or above - 43How do ξGL and λGL vary close to Tc? Can the temperature dependence be derived from the GL theory?GL predicts x_GL = x_GL(0)/sqrt(1-T/Tc)
experimentally: sqrt(1-T/Tc^4) but close to Tc this is equivalent with x_GL = x_L/2 - 44Discuss the GL parameter κkappa = lambda_GL/xi_GL
high kappa -> psi^2 can vary on lenght scale of B field penetration -> type 2 SC
low kappa -> psi^2 can not vary on lenght scale of B field penetration -> type 1 SC - 45Discuss the range of validity of the GL theory.- can explain all London results
- can explain type-2 SC
- can explain behaviour at SC interfaces
- not applicable for T< - 46thermodynamical critical field in GL theoryBcth(T) = phi0/[2pi sqrt(2) xi_GL(T) lambda_GL(T)]
- 47Describe the differences between London and GL theory.London equations follow out of the second GL equation for constant |psi|^2 -> Landau theory follows out of spatially homogenous GL theory
- 48What types of superconductors do we have to distinguish?Type 1 and type 2 SCs.
Type 1 has 1 critical field with Meißner state below and normal state above
Type 2 has 2 critical fields with Meißner below Bc1 and Shubnikov between Bc1 and 2 (above is normal conductor) - 49Sketch the magnetization curves of type-I and type-II superconductors having the same condensation energy?Type 1: linear increase until Bcth then sudden drop to 0
Type 2: linear increase until Bc1 - 50Explain the origin of the fact that the one superconductor behaves as type-I and the other as type-II. Discuss the energy density of an superconductor/normal metal interface.Boundary energy: Delta E_C/F + Delta E_B/F
= Bcth^2/2mu0 (xi_GL - (Bext/Bcth)^2*lambda_L)
if negative (xi_GL << lambda_L) then the interface is maximized -> Type 2 SC
kappa <= 1/sqrt(2) -> type 1 otherw. type 2 - 51Which parameter can be used to distinguish between different type-I and type-II of superconductors?The GL parameter kappa = lambda_GL/xi_GL
- 52How does the Ginzburg-Landau parameter change if we add impurities to a clean metallic superconductor?similiar to type 2 SC
constant order parameter until impurity. Sharp drop to 0 then sharp rise to 1 again. On the lenght scale of xi_GL - 53Discuss the meaning of the lower and upper critical field.- lower critical field: transition of Meißner into Shubnikov phase in the Type 2 SC
- upper critical field: transition from Shubnikov into normal conducting phase - 54How can we derive the lower and upper critical field? How are the fields related to the characteristic length scales ξGL and λGL?Lower: argument flux by Bc1 in area pilambda_L^2 must have at least phi0
-> Bc1 = phi0/2pi lambda_L^2 *(ln(kappa)+0.08)
Upper: linearize 1st GL eq. -> diff. eq. corr. to Landau levels
-> Bc2 = phi0/2pi xi_GL^2 *(1- T/Tc) - 55scetch the normal conducter/super conducter interface with xi_GLquickly dropping eps_C
slowly rising eps_B
combined gives minimum
|psi|^2 slow quick slow rising on xi_GL
Bz exponentially falling on Lambda_L - 56Describe the isotope effect and possible conclusions.With the isotope effect one can predict, that the exchange bosons for some SCs are phonons due to the dependence of Tc prop 1/sqrt(M) prop omega_phon
works for some but not all elemental SCs - 57Sketch the Feynman diagram representing the interaction between two conduction electrons mediated by a phonon. Which other exchange bosons could do the job?e- : lu->ml->lo
e- : ru->mr->ro
phonon: ml->mr
possible candidates: magnons, plasmons, spin fluctuations,... - 58How can we qualitatively describe the attractive interaction between two conduction electrons via the exchange of virtual phonons? Which conditions have to be satisfied to make the mechanism work?e- deforms ion lattice. If omega_pl is smaller than omega_ph the lattice moves retarded -> retarded potential can be felt by other e- which is attracted
- 59Make an estimate of the interaction range for the retarded coulomb interaction. What would change if you go from a metal to a semiconductorsvF * tau = 0.1 mu m
tau = 1/omega_ph
metal -> semiconductor
drastic reduction of conduction e-
maybe too low interaction range left? - 60Cooper pairs: Provide a qualitative argument why the attractive interaction between two conduction electrons is most efficient for electrons with opposite momentum. Which electrons can participate in the interaction process?Momentum cons. energy cons. within energy interval h' omega_ph
-> phase space for interaction much higher for k1 = -k2 because whole shell overlaps.
Electrons that have more energy than the fermi sea can interact. - 61How can we estimate the interaction energy? Which assumptions are usually made to keep the problem simple?E ~ 2EF - 2h'omega_D exp(-4/V0 D(EF))
Assumptions: two electrons on top of Fermi sea, no interactions between Sea and e-
Vkk' = -V0 for k,k' in kf + Delta k interval
0 else
DOS constant around E_F - 62What is the order of magnitude of the attractive interaction energy? Which characteristic parameters determine the interaction energy?E_bind ~ 2h'omega_D exp(-4/V0 D(EF))
E_bind << 2h'omega_D = 6 meV - 63Discuss the symmetry of the pair wave function. Which combinations of orbital and spin wave functions are possible?Total function has to be anti-symmetric
-> s = 1 -> triplet pairing, anti-symmetric orbital function (L = 1,3,5 (p,f,..))
-> s = 0 -> singlet pairing, symmetric orbital function (L = 0,2,4 (s,d,...)) - 64Qualitatively discuss how we can construct a coherent many body state from bosons and fermions. What are the fundamental differences?Bosons: linear combination of number states |alpha> = e^(-|alpha|^2/2)* e^(alpha a+)|0>
Fermions: analogous |psi_BCS> = c1 exp(sum(k)[alpha_k Pk+])|>
Pauli principle: |psi_BCS> = c1 prod(k)[1+alpha_k Pk+]|0>, c1 = 1/sqrt(1+alphk^2) - 65Which form has the BCS ground state?|psi_BCS> = prod(k)[uk + vk ck_up+ c -k_d+ ]|0>
= |vk|^2 - 66What is the condensation energy density at T = 0? How is it related to the energy gap? Provide an intuitive argument for the relation.Delta E = E_cond(0) = -1/4 D(E_F)Delta^2(0)
D(E_F)/2 Delta(0) -> number of cooper pairs
-Delta(0)/2 -> average energy gain per pair - 67Gap equationDelta_k = -sum(k')[Vkk' Delta_k'/2sqrt(xi^2k'+|Delta_k'|^2)]
simplified:
Delta = h'omega_D/sinh(2/V0 D(E_F)) ~ 2h'omega_D exp(-2/V0 D(E_F)) - 68What is the relation between the zero temperature energy gap and the critical temperature according to BCS theory?Delta(0)/kbTc = pi/e^gamma = 1.764
- 69Sketch the temperature dependence of the energy gap.Delta(T)/Delta(0) = 1.74(1-T/Tc)^1/2
1 at T= 0
0 at T=Tc - 70How can we measure the energy gap of a superconductor?Tunneling spectroscopy with NIS and SIS tunnel junctions
- 71Sketch the excitation spectrum of SCEk = E-k = sqrt(xi^2k + |Delta_k|^2)
Delta k for xi = 0 then asymptotic behaviour on linear increasing function - 72Discuss the energy dependence of the so-called coherence factors uk and vk close to the Fermi energy for T = 0. What is the meaning of the coherence factors?|uk|^2 = 1/2[1+xik/Ek]
|vk|^2 = 1/2[1-xik/Ek]
|vk|^2 is probability that k_up, -k_down is occupied. Even at T = 0 the fermi distribution is smeared out in order to allow pairing interaction - 73Sketch the dispersion of the Bogoliubov quasiparticles. Elaborate on particle-hole mixing.two crossing lines. energy gap* 2 in the middle -> asymptote on the crossing lines above and below
Bogoliubov particles are superpositions of particle hole pairs. (When cooperpair is broken, two quasi particles are produced) - 74Discuss the density of states of the Bogoliubov quasiparticles in the superconducting state and compare it to the normal state density of states.D_s(Ek) = Dn(xi_k) dxi_k/dEk =
-> Dn(E_F) Ek/sqrt(Ek^2-Delta^2) for Ek>Delta
-> 0 for Ek - 75How can we derive the Josephson equation from the expression for the supercurrent density in a bulk superconductor?The supercurrent density in a bulk SC is prop. to the invariant phase gradient
gamma = grad(theta)-qs/h' *A
replace gamma by phase difference phi
periodicity (J(0) = J(n*2pi))->1st Josephson
time derivative of phi & energy-phase relation -> 2nd Josephson - 761st Josephson equationcurrent-phase relation:
Js(phi) = Jc sin(phi) + sum(m=2) Jc,m sin(mphi)
most cases:
Js(phi) = Jc sin(phi) - 772nd Josephson equationvoltage-phase relation
del phi/del t = 2*pi/phi0 V(t) = qs V(t)/h' - 78discuss the Josephson equations.For constant voltage V across junction:
integrate 2nd Josephson, plug into 1st josephson:
Js(phi(t)) = Jc sin(2pi/phi0 *V*t)
voltage controlled oscillator - 79What is the Josephson coupling energy, how can it be derived and what is its meaning?Ej/A = phi0 Jc/2pi *(1-cos(phi))
from integration over Js*V and 1st and 2nd Josephson equation
To separate two coupled SCs one will need to overcome the coupling Energy - 80What are potential applications of the Josephson effect?microwave radiation, Josephson standard for Voltage (because through oscillation the Voltage can be measured)
Josephson switching devise (V= 0 Jos. current, V not 0 qp current)
SC quantum bits
SQUIDs (magnetic field sensors) - 81Discuss how the critical current Ic and normal resistance Rn of a superonductor/insulator/superconductor Josephson junction depend on the thickness of the tunneling barrier.Jc prop 1/sinh(2kd) -> Jc ~ exp(-2kd)
normal resistance????? - 82What is the relation of the product IcRn to the energy gap?IcRn = pi/2e *Delta(T) tanh(Delta(T)/2kbT)
- 83Sketch current-voltage charachteristics of SIS junctionJosephson current oscillates if voltage is applied. -> only contribution at V = 0
at eV >> 2Delta(T)
NIN junction (linear increase)
at eV > 0 T = 0 -> no current til 2Delta(0)
at eV>0 T>0 -> low constant current til 2Delta(T) than asymptote on NIN junction - 84Josephson junction with applied current: how does the potential look like?Epot(phi) = Ej0 (1-cos(phi)-I/Ic phi)
- 85What are typical power applications of superconductors?MRI, high energy physics, fusion, fault current limiters, energy transport and storage
- 86What are the relevant materials parameters for power applications of superconductors?critical temperature (critical magnetic field)
critical current density,
Bulk pinning force,
Whole wire critical current density - 87What is the critical current density of a type-I superconductor expected from London and GL theory? Are there differences? If yes, why?reduction of |psi|^2 prop to kinetic/condensation energy
Jc ~= 8*10^11 A/m^2 in Pb
Jc = 0.544 Hcth/Lambda_L
Jc_L = Hcth/lambda_L because reduction of psi with increasing Js isnt part of Landau theory - 88Why can’t we use type-I superconductors for superconducting wires and magnetic field coils?Jc only in thin film around the SC
-> technical current density very small 10A/mm^2
-> very small filaments are hard to produce
-> type 2 SC are needed - 89How does the force acting on a flux line in the presence of a finite transport current density look like in a type-II superconductor?F_L = L int(A)(Jt x b dA) = LJt x phi0
f_L = Jt x b - 90How would the current-voltage characteristics of an ideal (no defects) type II superconductor look like?E = [(Jt x B) * v_L] Jt
E parallel to J -> flux-flow resistance
Jt prop E -> Ohms law -> zero critical current density - 91How can we pin magnetic flux lines in a type-II superconductor? What is the relation between the pinning force and the shape of the pinning potential?pinning via defects (N conducting volumes)
-> flux line is pinned to defect because otherwise condensation energy in fluxline area has to be paid
Fp ~= -Econd/rp
only for rp>~ xi - 92How should an optimum pinning center (maximum pinning force) look like?optimum is a cylinder defect with radius xi_GL
but thermal activation can happen because total energy needed is very small compared to the Force acting - 93Sketch the magnetization curve for a type-II superconductor with pinning.hysteresis loop!
Square pulled on upper right and lower left vertex, lower right and upper left are rounded. Bc2 fixes the vertices to the right and left, virgin curve increases at Bc1 within the loop. - 94Sketch the magnetization and the magnetic flux density inside a type-I and type-II superconductor as a function of the applied magnetic field.type 1: zero till Bcth then external flux density, full magnitization then 0
type 2: same for B - 95How does the magnetic flux distribution look like in the mixed state of a type-II superconductor?without pinning:
Bext, Bext reduced by value, Bext for B>Bc1
Bext, 0, Bext for B - 96Why has the discovery of HTS generate so much excitement?Because it is possible to operate these SCs with much cheaper Nitrogen cooling.
Maybe there will be even higher Temp. SCs
-> SCs without cooling at all
long time people thought there wouldnt be SCs above 40 K - 97How does the typical phase diagram of a cuprate superconductor look like?Axes: temperature, doping
n (electron doping): AFM insulator, small blob of Superliquid at 0.14
p(hole doping): narrow AFM insulator, nothing, bigger blob of SL at 0.16 +-0.1 - 98How does the band structure of cuprate superconductors look like?I, II, I, II, I
I is doped (high fermi level)
II is insulator but with lower bands and low fermi level
-> Bending of bands and equal fermi level
-> high mobility 2D electron gas on II sites - 99How does the typical electronic structure of a cuprate superconductor look like?EA - alpha - 2beta[cos(kx a) + cos(ky a)]
through hopping the E = 0 square gets stretched out to the sides
doping lowers the Fermilevel - 100What is the symmetry of the order parameter of YBaCuOhole doped couprates: d-wave s = 0 order parameter
makes sense because free electron gas can seperate and lower Coulomb repulsion through L=2 d wave - 101How can we measure the symmetry of the order parameter of YBaCuOLondon penetration depth is different for s and d wave
ARPES measurements of Delta resolved for angles (needs clean surface)
tri-crystal experiments (measurement of spontanious magnetic flux at intersection)
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- Physik Fichas sobre Superconductivity 1, creado por Markus Schmidt el 14/03/2022.
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- Markus Schmidt
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