Spin fluctuations and their effect
on superconductivity in
Titanium-Vanadium alloys
Joel Barker
Superconductivity & Magnetism
University of Warwick
Supervised by Don Paul (Warwick) & Adrian Hillier (ISIS)
Outline
•
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•
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Introduction
Superconductivity
μSR
Spin fluctuations in TiV alloys
Outlook
• Type I
– Meissner state: H < HC
– Normal state: H > HC
• Type II
– Meissner state: H < HC1
– Mixed state: HC1 < H < HC2
– Normal State: H > HC
The vortex lattice
Normal core – contains
one quantum of flux
a
Superconducting region
2ξ
Magnetic
field
Microscopic mechanism behind
superconductivity
• Cooper pair of electrons: conventionally
a spin singlet state
– No net angular momentum
Microscopic mechanism behind
superconductivity
• Cooper pair of electrons: conventionally
a spin singlet state
– No net angular momentum
• Unconventional pairing: spin triplet
– Net angular momentum
– Time-reversal symmetry broken
μSR is sensitive enough to detect the low fields (~0.1 G)
caused by triplet pairing in the superconducting state
Unusual superconductivity in TiV
• TC of TiV alloys much lower
than predicted
• Hints of spin fluctuations in
the presence of
superconducting state
• Can we see them directly?
– Yes, with muons
• How are they tied to the
superconductivity?
– …not really sure
Matin et al., Eur. Phys. J. B, 87 131 (2014)
Fundamentals: the positive muon
• Microscopic particle
– Local probe
• Spin ½
– Sensitive to magnetism
• τ1/2 = 2.2 μs
– Independent of material
+
μ
μSR provides a picture of a the local magnetic
environment that may be missed by susceptibility
measurements
The Muon Source at ISIS
𝑊 𝜃 = 1 + 𝑎. cos(𝜃)
1.0
1.0
0.5
a = 1/3
0.5
a=1
0.5
1.0
1.5
2.0
0.5
0.5
1.0
0.5
0.5
1.0
1.0
Highest energy positrons
Average over all positron energies
F
H
B
Sample
MuSR instrument
at ISIS
H
F
Sample
μ+
B
After muon pulse
implanted start
clock and begin
counting
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
F
H
B
𝑁𝐹 − 𝛼𝑁𝐵
𝐴 𝑡 =
𝑁𝐹 + 𝛼𝑁𝐵
Zero-field studies of Ti55V45
Asymmetry
• Exponential term in
relaxation function
is evidence for the
presence of spin
fluctuations
0.30
0.25
0.20
2K, ZF
140K, ZF
0.15
0
2
4
6
Time (s)
8
10
𝛬 𝑇 = 𝛬0
𝑇
1−
𝑇𝑠𝑓
𝑁
N = 1.54±0.09 ≈ 3/2
+ 𝛬𝐵𝐺
0.16
Relaxation rate / s-1
• Spin fluctuations appear to
have an onset temperature
of ~140 K
Spin fluctuation rate
Two-fluid fit
0.12
0.08
0.04
0.00
0
50
100
150
Temperature / K
200
Superconducting state in Ti55V45
• Below TC: characteristic
depolarization rate
caused by the vortex
lattice
• Above TC: small
residual depolarization
rate from randomly
oriented nuclear spins
• Magnetic field distribution of the lattice leads to
depolarization of the muon ensemble
Decay of
oscillations
1
𝑛𝑠
𝜎∝ 2∝ ∗
𝜆
𝑚
Magnetic
penetration
depth
Superconducting carrier
density
Effective mass
• Isotropic energy gap of
2.1±0.2 meV
• Expected:
2∆𝐵𝐶𝑆
= 3.528
𝑘𝐵 𝑇𝐶
• For TiV
2∆ 𝑇𝑖𝑉
= 7.2±0.4
𝑘𝐵 𝑇𝐶
λ = 536±2 nm
• Predicted TC,pr = 13.7 K (using the McMillan formula)
– 𝑇𝐶,𝑝𝑟 =
𝜃𝐷
exp
1.45
−1.04
1+𝜆
𝜆−𝜇 ∗ −0.62𝜆𝜇 ∗
2∆ 𝑇𝑖𝑉
= 3.56
𝑘𝐵 𝑇𝐶,𝑝𝑟
• Much closer to BCS value
• Predicted TC,pr = 13.7 K (using the McMillan formula)
– 𝑇𝐶,𝑝𝑟 =
𝜃𝐷
exp
1.45
−1.04
1+𝜆
𝜆−𝜇 ∗ −0.62𝜆𝜇 ∗
2∆ 𝑇𝑖𝑉
= 3.56
𝑘𝐵 𝑇𝐶,𝑝𝑟
• Much closer to BCS value
2∆𝐵𝐶𝑆
= 3.528
𝑘𝐵 𝑇𝐶
Conclusions
• Spin fluctuations coexist with superconductivity, and
have an onset temperature of ~140 K
• Temperature dependence of the penetration depth
fits an s-wave BCS model
• Magnitude of energy gap is too large for measured
TC
• Are spin fluctuations unconventionally pairing the
Cooper pairs? Or are they suppressing the
conventional singlet superconductivity?
Outlook
• Single crystals + SANS
study
– Koss, 1976
• Stoichiometry variation
study with μSR
Thanks for listening.
Growth & characterization
• GSAS: refined to single
bcc phase with
expected Im-3m
spacegroup
5
-0.2
0
-0.4
x 10-4
0.0

-5
-0.6
-10
3
2
-0.8
4
5
6
7
8
ZFCW
FCC
-1.0 B = 10 mT
a
2
3
6
5
4
Temperature / K
7
8
Ti55V45
a = 3.1761(1) Å
Growth & characterization
200
1.8K
2K
3K
4K
5K
6K
6.5K
-10
6K
5.5 K
6.5 K
4.5 K
5.0 K
4.0 K
3.0 K
2.0 K
6.75 K
150
100
M / emu cm-3
M / emu cm-3
0
-20
50
0
-50
-100
-150
-30
0
200
400
600
Field / Oe
HC1(0)= 86(2) G
800
-200
0
20
40
60
80
H / kOe
HC2(0)= 110(1) kG
100
Muon production
Carbon target
p+
280 MeV
n
Muon production
n
π+
n
p+ + n → n + n + π+
Muon production
π+
Pions that
decay at rest
at the surface
of the target
Muon production
μ+
νμ
Sν = ½
π+ → μ+ + νμ
Sμ = ½
100% spin polarized
muon beam thanks
to parity violation
Example spectra (LF)
• Static order
1.0
μ+
μ+
PZ(t)
0.5
0.0
-0.5
𝑃𝑧 𝑡 =
μ+
μ+
1 2
+ cos(𝛾𝜇 𝐵𝑡)
3 3
-1.0
0
2
4
Time (s)
6
8
• Static disorder
1.0
μ+
μ+
μ+
μ+
PZ(t)
∆2 𝑡 2
1 2
−
𝑃𝑧 𝑡 = + (1 − ∆2 𝑡 2 )𝑒 2
3 3
Kubo-Toyabe
relaxation function
0.5
0.0
0
2
4
Time (s)
6
8