Metody doświadczalne w badaniach nadprzewodnikó · 2008. 6. 3. · Superconductor-ferromagnet...
Transcript of Metody doświadczalne w badaniach nadprzewodnikó · 2008. 6. 3. · Superconductor-ferromagnet...
Teoria i praktyka w badania ciał stałych –rzut oka na badania prowadzone w IFPAN
Metody doświadczalne w badaniach nadprzewodników(oraz nienadprzewodzących układów silnie skorelowanych)
Outline1. Introduction
a) type I and type II superconductorsb) phase diagram of type II superconductors
2. Basic measurements testing phase diagrama) magnetic measurements
m-H (T=const)critical state models, hysteresis loop in real superconductors: fishtail anomaly, thin films
m-T(H=const)diamagnetic and Meissner effects, irreversibility line, paramagnetic Meissner effect
ac susceptibilityb) transport and magnetotransport
technical details measurement geometry, the role of microstructure
vortex lattice meltingvortex liquid-vortex glass transition
scaling predictions and methodology 3. Superconductor-ferromagnet heterostructures4. Superconductor-insulator transition
Meissner effect, 1933Meissner effect, 1933
Walther Meissner Robert Ochsenfeld
F.& H. London, 1935: shielding currents
2s
sn eJ Bm
∇× = −
0µ∇× =B J}
2
2
1 0zz
d B Bdx λ
− =Maxwell’s law
penetration depth
20
λµ
=Ls
mn e
λ x
B0/−= 0
xB(x) B e λ
www.physics.leidenuniv.nl paintings by nephew Harm
Heike Kamerlingh Onnes, 1911, Leiden
Zero resistanceZero resistance
0.7ξλ >0.7ξλ <
3.41 KIndium (In)3.72 KTin (Sn)4.15 KMercury (Hg)4.47 KTantalum (Ta)4.88 KLanthanum (La)
7.196 KLead (Pb)
<κ 1/ 2
λκ=ξ
15 K (nanotubes)9.25 K7.80 K5.40 K
CNbTcV
Nb3Ge 19 KNb3Si 18.1 KNb3Sn 18 K
MgB2 39 KOver 50 different high-Tc superconductors, Tc: 30-187K extreme type II
YBa2Cu3O7: ξ(0)~0.4-3 nm; λ(0)~150nm
Nb: ξ(0)~ 38nm λ(0)~ 39 nm
Pairing mechanism still unknown
>κ 1/ 2
List of superconductors: superconductors.org
Phenomenological Ginzburg-Landau theory,1950
( )
( )
2 2
Free energy expansion in powers of
expansion coefficients
-1 2
2
2
ψ
→α β
∇ − ψ +β ψ ψ = −αψ
⎡ ⎤= ψ − ∇ − ψ +⎣ ⎦s
,
,
i eAm
eJ i eA c.c .m
Complex order parameter: ( )( ) ( ) ie ϕΨ = Ψ rr r2 ( )( ) sn
nΨ =
rra measure of the superfluid density
theory of phase transitions: macroscopic properties of superconductors
2
0c
c0c
Φ TH = =H (0 ) 1 -T2π µ 2 ξ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟
λ ⎢ ⎥⎝ ⎠⎣ ⎦( )2
0 c n s
Thermodynamic critical field: 1µ H T = f-f2given by a free-energy difference between the normal and superconducting states(i.e. condensation energy)
c
ccT
dH 4π∆CSpecific heat jump ∆C: dT T
⎛ ⎞ =⎜ ⎟⎝ ⎠
λκ=ξ
GL parameter:
F0
cB
BCS coherence length, mean-free pathhvξ 0.18 -k T
= l
1/2c
0 0c
1/2c
0 0c
(clean limit)
(dirty limit)
Tξ(T)=0.74ξ , ξT-T
Tξ(T)=0.85 ξ , ξ T -T
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎝ ⎠
l
l l
coherence length penetration depth,ξ(T) - (T) - λ
1/2c
0c
1/2c0
0c
(clean limit)
(dirty limit)
1 T(T)= , ξT-T2
ξ T(T)=0.64 , ξ T -T
⎛ ⎞λ λ ⎜ ⎟
⎝ ⎠
⎛ ⎞λ λ ⎜ ⎟
⎝ ⎠
L
L
l
ll
cand diverge asξ T Tλ →
0.7ξλ >0.7ξλ <
Triangular (Abrikosov) vortex lattice (1957)
ξλ
3.41 KIndium (In)3.72 KTin (Sn)4.15 KMercury (Hg)4.47 KTantalum (Ta)4.88 KLanthanum (La)
7.196 KLead (Pb)
<κ 1/ 2
⎛ ⎞⎜ ⎟⎝ ⎠
1/20
02Φa =3B
λκ=ξ
-15 20Φ = =2.07x10 Tm
2he
15 K (nanotubes)9.25 K7.80 K5.40 K
CNbTcV
Nb3Ge 19 KNb3Si 18.1 KNb3Sn 18 K
MgB2 39 KOver 50 different high-Tc superconductors, Tc: 30-187K extreme type II
YBa2Cu3O7: ξ(0)~0.4-3 nm; λ(0)~150nm
Nb: ξ(0)~ 38nm λ(0)~ 39 nm
Pairing mechanism still unknown
>κ 1/ 2
02
0c2
c1
c2
ΦH = lnκ4πΦH = = 2κH
2π ξ
λ
cc2
cc2
T Y PE I
T Y PE I I
H H H H
<
>
List of superconductors: superconductors.org M. Tinkham, “Introduction to superconductivity”
Thermal fluctuations
Ginzburg number:
relative size of the thermal energy and the condensadion energy within the coherence volume; ε – anisotropy parameter
HighTc, small ξ : Gi~10-2 (low-Tc: Gi~10-8)
2cB
i 2 3c
εk TG / 2H (0)ξ(0)
⎛ ⎞= ⎜ ⎟
⎝ ⎠
Pinning of the vortex by disorder (defects)
Vortex of length L removes condensation energy ~
It is energetically favorable to place vortex in the defect site, with locally suppressed order parameter.
Small ξ weak pinning - but artificial pinning sites may be created (irradiation, doping, thickness modulation,…).
( )2 2c0µ H π ξ L / 2
Melting of the vortex latticeMelting of the vortex lattice
Lindeman criterion: , cL ~ 0.1-0.4
u - thermal displacement of the vortices from equilibrium positions
Clean superconductor: melting is the 1-st order transition
Disorder: vortex glass instead of lattice; transition is continuous
Note: in liquid vortices may remain pinned !!!
24L
m c2i c
5.6 c TB (T ) H (0 ) 1G T
⎛ ⎞ ⎛ ⎞≈ −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2 2 2m L 0th
u (T ) ~ c a
large Gi small Bm: large region of the vortex liquid
1-st order
continuous
Thermal depinning of vortices (crossover)
u - thermal displacement of the vortices from equilibrium positions
Correlated disorder (twin boundaries, screw dislocations, artificial columnar defects) with mean distance dr < a0:
depinning in case of collective pinning
Vortex dynamics Experiments:
Flux creep at j<jc: disordered landscape hops of vortex segments and bundles over the energy barriers U
FF Flux flow: U << kT (unpinned liquid) ohmic resistance (j-independent)
TAFF Thermally assisted flux flow: U >> kT (pinned liquid)small j: ohmic ; large j: power-law
Vortex glass: U diverges as j 0 ρ = 0 in the low-j limit
2
i c2dpc
TB (T ) 8 G H (0 )T
⎛ ⎞≈ ⎜ ⎟
⎝ ⎠
2 2m th
u (T ) ~ ξ
Theoretical descriptions:
TAFF - Anderson-Kim model: wrong large-j limit (exponential)
Vortex glass-vortex liquid transition model (M.P.A.Fisher, PRL 62 (1989); M.P.A.Fisher, D.S.Fisher, and D.A.Huse, PRB 43 (1991)).
2 2m rth
u (T ) ~ d
G. Blatter et al., Rev. Mod. Phys. 66 (’94); Y. Yeshurun et al., RMP 68 (’96)
TAFF
FF
Vortex glass
High T
Low T
isotherms
0
BM= -Hµ
Magnetic measurements in type II superconductor:
Thermodynamic evidence of the 1st order magnetic phase transitions
But: challenging to interpret (sometimes)
Magnetometers:
SQUID, VSM, etc…
local magnetometry: magnetooptics, Hall (effect) probe
Ac susceptibility
-4 -2 0 2 4-1
0
1
CA
4πM
Ha
M vs H, T=const
Hysteresis loop
Tc*T
M vs T, H=const
Meissner and diamagnetic effects
Shielding current in type II superconductor: critical state
B0
B0
2a
z 0
0
y0 0
B (x) = n(x) Φ ; n(x): vortex per unit area Φ : flux per vortex
dΦ n(x) = µ J (x)dx
Maxwell Equation: ∇ 0×B = µ J → z y0d B (x) = µ J (x)dx
Lorentz Force density acting on vortex: F=J×B
Pinning Force: pF = F = J×B
zpx y z z
0
1 dBF = J B = Bµ dx
z y pB (x), J (x), F (x)
=cy z c
z
JJ (B ) = ; J(B )
constfCritical state models:
Ref.: Charles P. Poole, Jr., Superconductivity, Farach, Creswick, Academic Press, 1995
Shielding current in type II superconductor: critical state
+
⎡ ⎤+⎣ ⎦
c
cp
k
c
k
c
k
J(B) = JJJ(B) = (F=const);
B(x) / BJJ(
Bean model, Bean (1962,1964)
Fixed pinning model, Ji et al. 1989
Kim modeB) = 1 B(x) / B
JJ(B) = 1 B
l, Kim et al., 1962,1963
Generalized model, Lam(x) / B
et alβ ., Xu et al.,1990
c0
0
0
0BJ =µ (a- )
B=' µa a
When a’=0
Magnetization in the Bean model
a0µ M= B -B
→ =aac1 0
0
B <H B 0, µ M=-Bµ
→* *a a a0
1 1 3 3B = B B = B µ M=- B =- B2 4 4 8
→* * *a 0
1 1B =B B = B µ M=- B2 2→ →c c
0
+ -B* M -Ma'=0 J = high B: J =µ a a
0c
0
BJ =µ (a-a')
High field range: B-dependent Jc
-4 -2 0 2 4-1
0
1
4πM
Ha
Hysteresis loop in real bulk type II superconductor
Fietz et al., Phys. Rev.136 (’64): Kim-like model description of niobium
M. F. Schmidt et al., PRB48 (1993)
Nb film 500 nm thick
c1H (T)
Peak effect and fishtail anomaly
Peak effect: amorphization of the vortex matter at high and low H
Banerjee et al., PRB62 (2000)
Bi2Cr2CaCu2O8 -local (Hall probe)
magnetization measurementssample center
Bsp : second-order transition from vortex lattice (low H) into a strongly pinned state (high H)
Kayakovich et al., PRL76 (1996)
enhanced pinning in the glassy state due to disorder –better adaptation of the vortices to disordered landscape
Angst et al., PRB65 (2002)
Fishtail anomaly:
-100 -50 0 50 100
-1.0x10-3
0.0
1.0x10-3
Sample: Si(3nm)/Nb(78nm) Tc=8.55K
M[e
mu]
H[Oe]
T=8K; T/Tc=0.94
Film is almost always in the mixed state - flux enters the film at the edge as soon as the field is applied.
The maximum of M is always in close vicinity of H=0.
λ~200nm d < λ
Thin films
( )
0
z x xy
0 0
×B = µ J1 H H 1 HJ x =- - - -µ x z µ z
∇
∂ ∂ ∂⎛ ⎞ ⎛ ⎞≈⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠
Critical state differs from the bulk
Films: Jy depends on ∂Hx/∂z which has a large change across the film thicknessBulk: H=(0,0,Hz)
In thin films currents flow all the way to the middle of the film;
The B(x) in always nonlinear, even for the Bean model.
Zeldov et al., Phys. Rev. B 49 (‘94)Bean model for slab & solution for thin film (Bz=0 inside)
Slab
Film
c1edge appliedWB B H<H not accesibled
W-sample width, d-thickness
≈ →
W~1 mm, and d~100 nm x100
Films: large demagnetization factor increase of the external
magnetic field at the film edge
Meissner effect and diamagnetic effect
Tc
Ceramic YBa2Cu3O7 +Au
G. Xiao et al., PRB38, 776 (1988)
FC = Meissner effectmeasurement during cooling in the field
FC less than ZFC :
flux pinning by voids between grains
FC ZFCM < M
→B=0 M=-H
ZFC = diamagnetic (shielding) effect
1) cooling sample without field2) measurement during warming
Ideal superconductor:
≈
ZFCVM =- H
1-n1 r
1-n t
V – sample volumen - demagnetization factor
2r
tB