Lecture - laboratory of theoretical physics

Lecture - laboratory of theoretical physics

ГЕНЕРАЦИЯ ЭЛЕКТРОМАГНИТНОГО ИЗЛУЧЕНИЯ
ВНУТРЕННИМИ ДЖОЗЕФСОНОВСКИМИ КОНТАКТАМИ В
ВЫСОКОТЕМПЕРАТУРНЫХ СВЕРХПРОВОДНИКАХ
Theory: A. Koshelev, Materials Science Division, Argonne National Laboratory, USA
S. Lin and L. Bulaevskii, Los Alamos National Laboratory, USA
Experiment: U. Welp, T. Benseman, C. Kurter¶, K. Gray, W. Kwok
Materials Science Division, Argonne National Laboratory, USA
L. Ozyuzer
Department of Physics, Izmir Institute of Technology, Turkey
H. Yamaguchi, T. Yamamoto§, H. Minami, K. Kadowaki
Institute for Materials Science, University of Tsukuba, Japan
_____________________________________
¶Current: University of Illinois at Urbana-Champaign, USA
§Current: Beam Science Directorate, Japan Atomic Energy Agency
Another
R. Kleiner, Universität Tübingen, Germany
experimental H. Wang, National Institute for Materials Science, Tsukuba , Japan
+ coworkers
group:
Международная зимняя школа физиков-теоретиков
«Коуровка - XXXV»
«Гранатовая бухта», Верхняя Сысерть,
23 февраля — 1 марта 2014 года
Outline
• History
– Josephson effects and generation of em waves
– Josephson junctions arrays, synchronization
– Intrinsic JJ stacks in layered superconductors (Bi2Sr2CaCu2O8+δ)
• Terahertz radiation from high-temperature superconductors
– Review of experiments
• Synchronization by internal resonance mode
–
–
–
–
Mechanisms of coupling & structure of coherent state
Radiated power, mode damping, features in IV
Synchronization in inhomogeneous junction stacks
Mechanism of line width
• Mesa arrays
Josephson effects in superconducting
tunneling junctions
B. D. Josephson, Phys. Lett. 1, 251 (1962)
I
φ2
A
φ1
2e 2
• phase difference θ = φ2 − φ1 −
Adl
∫
1
hc
• dc Josephson effect I ∝ sin θ
• ac Josephson effect V∝ dθ/dt
• alternating tunnel current I ∝ sin(2π f t)
f = 2eV/h
f/V = 0.483 THz/mV
Nobel prize 1973 (with Giaever and Esaki)
V
Phase dynamics in a single junction
Resistively Shunted Junction Model
Sine-Gordon equation
2
jext
1 ∂ 2θ ν ∂θ
∂
θ
2
+
− λJ 2 + sinθ =
2
2
ω p ∂t
jJ
ω p ∂t
∂x
Current
jJ
Parameters:
Finite T
λJ Josephson length
Voltage
ωp plasma frequency
cs = λJωp Swihart velocity
ν damping parameter
β = 1/ν2 McCumber parameter
underdamped (ν
1) vs overdamped (ν
1)
Vg
First observations of radiation out of
Josephson junctions
Also: D. N. Langenberg et al., PRL 15, 294 (1965)
Radiation power is very small: < 10-12 W
Dynamic Josephson effects:
Irradiate Jos. junction →
steps at Vm = mhfext/2e voltage
Shapiro steps
current
Dynamic Josephson effects:
Fiske resonances
= cavity modes excited by oscillating Josephson current
Coon and Fiske, Phys. Rev., 138, A744 (1965)
Kulik, Pis'ma ZhETF, 2, 134 (1965)
ωn=csπn/l
H0
l
n=1
n=2
n=3
Coupling can be tuned
by magnetic field
D. N. Langenberg et al.,
PRL 15, 294 (1965)
Field evolution of IVs
l=4λJ, ν=0.1
Josephson junction arrays
T. D. Clark, Phys. Lett. A 27, 585 (1968)
…
One-dimensional array
Reviews:
A. K. Jain et al., Phys. Rep. 109, 309 (1984)
M. Darula et al., Sup. Sci. Tech. 12, R1 (1999)
Two-dimensional array
Achievement:
S. Han et al., APL 64, 1425 (1994):
500 Nb/AlOx/Nb junctions, 47 µW at 394 GHz
Synchronized oscillations: P ∝ N2
Synchronization
adjustment of rhythms of oscillating objects due to their weak interaction
Christiaan Huygens, 1665
Large Number of oscillators:
Synchronization transition
Kuramoto, 1975
Synchronization problem
Linear array of point junctions
Wiesenfeld, Colet, Strogatz, Phys. Rev. Lett., 76, 404 (1996)
Kuramoto model (1975)
Self-consistent analysis:
Synchronization order parameter
E.g., for α = 0 and g( ω j ) =
γ /π
( ω j − ω )2 + γ 2
Synchronization transition at Kc = 2γ
0≤σ≤1
σ = 1 − 2γ / K
for K > Kc
Synchronization via coupling to
common resonance
P. Barbara et al., PRL 82, 1963 (1999)
Nb/Al/AlOx/Nb-junctions, 150 GHz
3×36
Coherent emission from large arrays of
discrete Josephson junctions
1D array of Nb-Al2O3-Al-Al2O3-Nb
SINIS junctions
Frequency is limited by the superconducting gap:
~ 0.7 THz for Nb
Layered High-Tc superconductors
Bi2Sr2CaCu2O8+δ (BSCCO) Tc up to 90 K
H. Maeda et al. 1988
anisotropy 400-1000
s =1.56 nm 640 junctions/µm
∆ = 30-60 meV
f < 15 THz
Terahertz em waves
Potential applications
• new spectroscopy
• medical imaging
• security screening
• quality control
~0.3THz
~20THz
Tonouchi, “Cutting-edge THz technology”,
Nature Photonics 1, 97 (2007)
No commercial efficient compact
continuous coherent THz sources
QCL: Quantum Cascade Lasers
DFG: Difference-Frequency Generation
IMPATT: impact ionization avalanche transit-time diode
MMIC: microwave monolithic integrated circuit
TUNNET: tunnel injection transit-time diode
RTD: resonant tunnel diode
Electronic devices
Optical devices
Intrinsic Josephson effect
BSCCO
R. Kleiner
al.,etPRL,
1992
S. M. et
Kim
al., 2005
~2000 hysteretic junctions
s
• Multiple branches
1.8 µm×10µm×120 junctions
Kleiner et al., 1992 …
• Fiske resonances
Irie et al., 1998; … Kim et al., 2005
H. B. Wang et al., 2001
• Shapiro steps
Wang et al., 2000, 2001;
Latyshev et al., 2001 …
• Detection of Josephson radiation
Hechtfischer et al., 1997 (6-120GHz);
Batov et al., 2006 (0.5THz)…
•…
Challenge:
Synchronize oscillations in all junctions
Concept: Using internal resonance
as a synchronizator
Batov et al., 2006
Hp= 0.765 T
Phase dynamics in stacks
Maxwell equations
+ material relations
for superconductor
Reduced equations for phases ϕn
and fields hn
∂ 2ϕn
jz = σcEz + jJ sin ϕn
jx = σabE x +
Ez ≈
cΦ 0
pn
2 2
8π λab
Φ0 ∂ϕn
Φ
; Ex ≈ 0
∂t 2
(l
2
Parameters:
λab, λc London penetration depths
γ = λc/λab anisotropy
εc λ c
∂ϕn
∂h
+ sin ϕn − l2 n = 0
∂t
∂x
)
⎞
∂ϕn
∂ ⎛ ∂ϕn
− 1 hn +
+ νab ⎜
− hn ⎟ = 0
∂x
∂t ⎝ ∂ x
⎠
∂pn
2πc ∂t Sakai et. al., 1993; Bulaevskii et. al., 1994
2πcs ∂t
ωp = c
∇n2
+ νc
plasma frequency
σab, σc quasiparticle conductivities
s interlayer spacing
λJ= γs Josephson length
Reduced parameters:
2πγs2B
h=
Φ0
4πσc
νc =
εc ωp
νab =
t → ωp t
(
4πσab
εc γ 2ωp
x→ x
λJ
0.002 )
(
0.1)
l=
λ ab
s
S. M. Kim et al., 2005
Plasma-waves
spectrum and Fiske modes
ω2p (k, q)
=
ω2p0
+
c02 k 2
c0 = c/ εc
1 + 2 ( λ ab / s ) (1 − cos q )
2
ωp0 Josephson plasma frequency, λab London penetration depth
q = 0: in-phase mode, c0
q = π: anti-phase mode, cπ= (s/2λab)c0
Finite-size stack → cavity (Fiske) modes
km=mπ/L
In-phase mode, ωm,0=c0km
ω1,0/2π = 1THz at L=43 µm
• Can synchronize oscillations in many junctions
• Generates outgoing radiation
~1/300
Anti-phase, ωm,π=cπkm
• Excited by Josephson vortex lattice
(Fiske steps in magnetic field)
Homogeneous oscillations at H=0 → no direct coupling
Coherent THz radiation from Bi2Sr2CaCu2O8 mesas
Ozyuzer, et al., Science 318, 1291 (2007)
Recent Review: Welp, Kadowaki and Kleiner, NATURE PHOTONICS 7, 702(2013)
Ar-ion milling, photolithography;
w = 40-100 µm, ~1µm high, 300 µm long
Transport and radiation (bolometer)
Radiation Power [a.u.]
L. Ozyuzer, Mesa KK03: 100×300×1
µm 3
T=40K
1
30
4
I [mA]
I [mA]
5
2
Heating:
larger T
↓
smaller R
↓
smaller V
20
10
3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
U [V]
0
0.0
0.5
1.0
U [V]
Radiation frequency
fcut(THz)
Parallel-plate filters:
TE: cut-off for waves
TM: no cut-off
with f < fcut= c/2d
E
E
Frequency:
1. Satisfies Josephson relation
2. Increases with decreasing width,
roughly ~1/w:
cavity resonance f = c0/2nw ;
f = 0.52 THz for w = 80 µm, n ≈ 3.5
Emitted power:
Initial: ~0.5 µW
Now: up to 50 µW
linewidth ~ 9 GHz, instrument resolution is 7.5 GHz
Coherent emission
Variable number of emitters n (= resistive junctions)
Radiation frequency does
not depend on n
Coherent: P ~ n2
Direct observation of standing waves in mesas
Emissions in low-bias and high-bias regimes
R
Hot-spot instability
61K
Benseman et al.,
J. of Appl. Phys. 113, 133902 (2013)
Measurements of line shape
Unexpected T dependence
Detector:
Nb/AlN/NbN integrated receiver
(Koshelets group)
Scientific issues
•
•
•
•
•
Coupling to the resonance mode
Structure and stability of coherent states
Mechanisms of damping of cavity mode
Limits of radiation power
Mechanisms of line width
Excitation of in-phase cavity mode
Homogeneous state + External modulation, AEK and L. Bulaevskii, PR B 77, 014530 (2008)
Lateral modulation of the Josephson critical current, jJ(x)=jJ g(x)
Pleft
Equation for c-axis homogeneous phase (reduced form)
∂ 2θ
∂θ
∂ 2θ
+ν c
+ g ( x)sin θ − 2 = 0
2
∂t
∂t
∂x
1/ωp unit of time z
0
Resistive state near resonance ω1 = c0π/L
θ
θ(x,t) = ω t + Re[ψ exp(-iω t)] cos(π x/L)
Mode amplitude:
Damping parameters:
νc =
4πσ c
ε cω p
y
λc unit of length
+ radiative boundary conditions
ψ≈
Pright
ig1
ω 2 − ω12 + iνω
ν = νc+ νr + νb …
x
Coupling parameter
2 L
g1 = ∫ cos(π x/L) g ( x)dx
L 0
quasiparticle radiation from edges radiation to crystal
~ 10-2 -10-3
Cavity quality factor
L
Qc =
ω
νω p
Alternating kink state
S.-Z. Lin and X. Hu, PRL 100, 247006 (2008); AEK, Phys. Rev. B 78, 174509 (2008)
ϕn ( x, t ) ≈ ω t + (−1)n ϕkink ( x)+Re [ψ exp(− iω t ) ] cos(π x/L) + ...
Static soliton (kink) at x = L/2
ϕkink (0) ≈ 0; ϕkink (L ) ≈ π
⎛ Lλ ω − ω
kink width ls ≈ ⎜
⎜ 16 ω 2
p
⎝
2
J
2
1
2
1/ 3
⎞
⎟⎟
⎠
Lx
Effective modulation
g ( x ) = cos [ ϕkink ( x )] ≈ −sign( x − L / 2)
ψ=
ig1
ω 2 − ω 12 + iνω
2 L
g1 = ∫ cos(π x/L) cos(ϕkink )dx ≈ 4 / π
L 0
Maximum possible coupling
• exists without external modulations
• provides efficient pumping of energy into the cavity mode
Short-scale instability
A.E.K., Phys. Rev. B 82, 174512 2010
Local plasma frequency ω p ( x ) ∝
C ( x ) ≡ cos (ωt + θ ( x, t ) )
t
≈
g ( x )C ( x )
g1(ω12 − ω 2 )
cos(π x / L )
2
(ω12 − ω 2 )2 + ν 2ω 2
For decreasing positive g(x) g ( x )C ( x ) < 0 at x > L/2
→ source of instability !!
Homogeneous state is only stable if g(x) = 0 at x > L/2
For alternating-kink state g(x) ≈ -sgn(x - L/2) and
(ω12 − ω 2 )
| cos(π x / L ) |
→ stable
≥
g ( x )C ( x ) = 2
0
2
(ω1 − ω 2 )2 + ν 2ω 2
Radiative boundary conditions for oscillating phase
A.E.K. and Bulaevskii, Phys. Rev. B 77, 014530 (2008)
Phase in resistive state
ϕn(x,t) ≈ ωt + Re[θn,ω(x) exp(-iωt)]
Boundary conditions for homogeneous oscillating phase θω ( x) = θω ,n ( x)
Long symmetric mesa
∂θω ( L)
% (0)
= iζθω ( L) + iζθ
ω
∂x
∂θω (0)
% ( L)
= −iζθω (0) − iζθ
ω
∂x
For isolated mesa on metallic plate
with thin metallic contact on the top
⎡ 2i
C ⎤
−
1
ln
⎢
⎥
⎣ π kω Lx ⎦
2
k
Lz (1)
ω
H 0 ( kω L)
ζ% ≈ −
2
k L
ζ ≈ ω z
2
2
kω=ω/c
Lz
n
Ly
L
Radiation dampings
1. Radiation into free space
(sensitive to mode and geometry)
νr=
4ω p λ 2 c
ωL
2ω Lz
Re ⎡⎣ζ − ζ% ⎤⎦ =
[1 + J 0 (kω L)] ∝
εω L
c
p
Lz
λω
2. Radiation into crystal
Journ. of Phys., 150, 052124 (2009)
Power flow to the crystal:
Pbottom =
νb =
νr
νb
CabΦ 02 λabω
2
64π s L
λab λc
Lx Lz
L2z
ε c λabLx
ψ
2
Stand alone mesa
Kashiwagi et al. JJAP, 2012
An et al. APL, 2013
0.5
0.06
Dominating mechanism of damping!
Stand-alone mesa
87×383 × 1.3 µm3
Transport and radiation near resonance
Similar to: M. Russo and R. Vaglio, Phys. Rev. B 17, 2171 (1978)
δj
δjrad
j
Excess current
(units of jJ)
g12νω
1
δj=
4 [ω 2 − ω12 ]2 + ν 2ω 2
(single junction, no radiation)
Energy balance
IV = V 2 /R + E% 2 L2z /R + P
δI V
V
Vr
Φ 02ω 3 N 2
νr
2
Radiated power Pedge = ALy
|
ψ
|
A~1
Pedge = δ IV
3 2
32π c
ν
In resonance, for νr á ν Pedge∝ N2
π Ly L2 g12 jJ2
for ν ≈ νr
Does not depend on N !
Pmax ≈
2ω
g1=0.3, jJ=500 A/cm2, Ly=300µm
P = 1.5 mW
ω/2π=1THz, L=43 µm
achieved ~ 50 µW
Synchronization in inhomogeneous mesas
Emission Power
Number of synchronized
junctions
Power of
the mode
Mesas with inhomogeneous cross section
LN
Feature in IV
Natural frequencies:
α = (L1 – LN)/LN/2
L1
Full synchronization
Partial synchronization
No synchronization
0.14
ωn ∝ Vn ∝ jn ∝ 1/Ln
0.24
α = 0.1
α = 0.0625
α = 0.08
0.3
0.25
0.2
0.2
0.13
0.16
0.15
0.12
12
0.12
13
N=50
νc=0.01,νab=0.2
l=50, L1=12.5
V
14
12
15
14
16
18
V
20
22
12
24
13.6
13.4
16
13
15
12.8
13.2
V
13.5
14
j=0.15
14
13
12.8
10
20
n
30
40
50
0
10
20
n
30
j=0.124
12.4
j=0.12
12
0
12.6
j=0.15
13
12.6
12.4
13
13.2
17
13.8
12.5
40
50
0
10
20
n
30
40
50
Visualization of electric field
0.2
α = 0.08
0.19
0.18
0.17
0.16
0.15
0.14
0.13
0.12
12
14
16
18
V
20
17
16
15
14
13
12
0
10
20
n
30
40
50
P
I
Line width of Josephson radiation
I0
V
V0
f0=2eV0/h
J ∝ sin(ω0t)
f
Current noise
P
I
Line width of Josephson radiation
I0
∆f ∝ Dθ
V
V0
f
J ∝ sin[ω0t + θ(t)]
θ2(t)=Dθt
Single junction:
Line width due to quasiparticle-current fluctuations, kBT > ℏω
Larkin and Ovchinnikov, Zh. Eksp. Teor. Fiz., 1967
Dahm et al., Phys. Rev. Lett. , 1969
Dθ ∝
j
2
f
ν2
j
2
f
kT
∝ B
R
ν ∝ 1/ Rd
2
R = V/I
Rd= dV/dI
1 ⎛ 2e ⎞ Rd2
∆f = ⎜ ⎟
kBT
π⎝ h ⎠ R
Line shrinking near cavity resonance
Lin and AEK, Phys. Rev. B, 2013
0.20
|E(ω,0)|
Normalized magnitude
IB=0.14
2
1 ⎛ 2e ⎞ Rd2
∆f = ⎜ ⎟
kBT
π⎝ h ⎠ R
0.25
0.15
0.10
0.05
0.00
1.111
Frequency ω/2π
1.112
0.25
ν = 0.02, L = 0.4
2νT = 7.1 10-7
Normalized magnitude
IB=0.16
IB
0.20
0.20
0.15
0.10
0.05
0.00
0.15
1.2195
1.2200
1.2205
Frequency ω/2π
1.2210
0.10
6.0
6.5
7.0
V
7.5
8.0
Normalized magnitude
IB=0.18
0.4
0.2
0.0
1.2355
38
1.2360
1.2365
Frequency ω/2π
1.2370
Line width of synchronized stack
Dominating quasiparticle current dissipation
Effective noise current
1
J% =
N
N
∑ %j
n =1
n
2
J% =
%j
n
N
2
→ 1/N narrowing
2
T
4 R ⎛e⎞
Line width ∆f 0 =
→
⎜ ⎟ k BT
2
π NLx Lyν c l
π N ⎝h⎠
R=
sρ c
Lx Ly
Estimate
ρc = 50 ohm cm, Lx=80 µm, Ly=200 µm, N=600, T=60K → ∆f0 =0.2 MHz
Much smaller than best experimental observation (∆f0 ~ 20 MHz)
Mesa arrays
Synchronized mesa arrays → Route to enhance power
Synchronization via base crystal
Radiation field:
Lin and AEK
Physica C 491 24 (2013)
Two resonances
Uniform cavity mode in mesa → uniform standing wave in crystal
2L
a=
1−
ω2p
(c π / L )2
Strongest interaction between the mesas
Ndecay(ω)=1/Im[qk(ω)]
ω2 = c 2 ( π / L )2 = ω2p + c 2 (2π / a )2
k=2π/a
propagating
waves
decaying
waves
ω2 = 1 + l 2 k 2
ω
Hy
L
a
Mesa arrays, experiment
Tim Benseman et al., Appl. Phys. Lett. 103, 022602 (2013): 3 mesas synchronized
Total power up to 610 µW
Summary
• Artificial and intrinsic Jos. junction arrays
• Resonant emission from BSCCO mesas
– Frequency and polarization → Josephson origin
– Frequency ∝ 1/width, from 0.4 to 0.85 THz, power up to 50 µW
– Line width ~ 20 MHz
• Scientific issues and properties
– radiation losses, radiation power in ideal case
– external and self-generated modulation of Josephson current
• alternating kink state
– synchronization in inhomogeneous mesas
• partial synchronization
– intrinsic line width
• Mesa arrays
Potential for powerful, efficient and compact THz source!