Download Electrodynamic Properties of Metals and Superconductors

Electrodynamic
Properties of
Metals and
Superconductors
Metals
Summary:
ü  Drude model
ü  Conduction current in metals
ü  EM Wave Propagation in Metals
ü  Skin depth
ü  Plasma Frequency
ü  Dielectric constant of Metals
ü  Dispersion relation for EM waves
The free electron theory of metals
Paul Drude (1900): theory of electrical and thermal conduction
in a metal .
The theory is based on the application of the kinetic theory of
gases to a metal, which is considered as a gas of electrons.
Mobile negatively charged electrons are confined in a metal by
attraction to immobile positively charged ions
Definitions:
eZa= nucleus charge
Z = valence electrons are weakly bound to the nucleus
(participate in chemical reactions)
Za – Z = core electrons are tightly bound to the nucleus (play
much less of a role in chemical reactions)
In a metal :
-  the core electrons remain bound to the nucleus to form the
metallic ion
-  the valence electrons wander far away from their parent
atoms. This electrons are called conduction electrons or
electrons
Density of conduction electrons in metals ~1022 – 1023 cm-3
rs – measure of electronic density
rs is radius of a sphere whose volume is equal to the volume per
electron
4π rs3 V 1
= =
3
N n
1/ 3
⎛ 3 ⎞
rs = ⎜
⎟
⎝ 4πn ⎠
~
in metals
rs ~ 1 – 3 Å rs/a0 ~ 2 – 6 2
a = ! = 0.529 Å – Bohr radius 0
me2
1
n1 / 3
mean inter-electron spacing
ü electron densities are thousands times greater than those of a
gas at normal conditions
ü t here are strong electron-electron and electron-ion
electromagnetic interactions
in spite of this the Drude theory treats the electron gas by the
methods of the kinetic theory of a neutral dilute gas
The basic assumptions of the Drude model
1. between collisions, the interaction of a given electron
- with the other electrons is neglected (independent electron
approximation)
- with the ions is neglected (free electron approximation)
2. collisions are instantaneous events: Drude considered electron
scattering off the impenetrable ion cores
Notes:
-  electron-electron collisions (analogue of the collision
mechanism in an ordinary gas) are not considered
-  “naïve” picture: we can only say there is some scattering
mechanism
3. an electron experiences a collision with a probability per unit
time 1/τ
dt/τ – probability to undergo a collision within small time dt
randomly picked electron travels for a time τ before the next
collision
τ is known as the relaxation time, the collision time, or the mean
free time
τ is independent of an electron position and velocity
4. after each collision an electron emerges with a velocity that is
randomly directed and with a speed appropriate to the local
temperature Drude model
Drude Model : Lorenz model (Harmonic oscillator model) without restoration force
(that is, free electrons which are not bound to a particular nucleus)
Linear Dielectric Response of Ma2er Lorentz Model (harmonic oscillator model) Charges in a material are treated as harmonic oscillators
ma = F
+F
+F
( one oscillator )
d r
dr
m
+ mγ
+ Cr = −eE exp ( −iωt )
dt
dt
el
E , Local
Damping
Spring
2
2
L
C = elastic constant
γ = damping coefficient
DC electrical conductivity of a metal
V = RI Ohm’s law
the Drude model provides an estimate for the resistance
introduce characteristics of the metal which are independent on the
shape of the wire
E = ρj
j=I/A – the current density
ρ – the resistivity
R=ρL/A – the resistance
s = 1/ρ - the conductivity
j = σE
L A j = −env
v is the average electron velocity (drift velocity) due to collisions
eE
v=− τ
m
⎛ ne 2τ ⎞
⎟⎟E
j = ⎜⎜
⎝ m ⎠
j = σE
ne 2τ
σ=
m
τ=
m
ρne 2
At room temperatures resis2vi2es of metals are typically of the order of microohm cen2meters (µohm-­‐cm) and τ is typically 10-­‐14 – 10-­‐15 s Mean free path
l = v0τ
v0 : the instantaneous electron speed (between collisions) is esBmated from classical equiparBBon of energy: 1 2 3
mv0 = K
BT
→ v0~107 cm/s (@RT) 2
2
I : measures the average distance an electron travels between collisions esBmate for v0 at Drude’s Bme → l ~ 1 – 10 Å consistent with Drude’s view that collisions are due to electron bumping into ions However: v0 is actually temperature-­‐independent (and one order of magnitude larger) at low temperatures very long mean free path can be achieved l > 1 cm ~ 108 interatomic spacings! The electrons do not simply bump off the ions! The Drude model can be applied only where a precise understanding of the sca2ering mechanism is not required Sommerfeld theory of metals
Shortly after the discovery of the Pauli exclusion principle, Sommerfeld
applied the same principle to the free electron gas model.
Basically, in most applications Sommerfeld’s model is nothing more
than Drude’s model, with the single modification that the electronic
velocity distribution is quantistic and not classical.
The classical Maxwell-Boltzmann distribution must be replaced by the
Fermi-Dirac distribution.
Quantum theory of an electron gas:
In the free and independent approximation the electron ground state is a
sphere in the k-space, called the Fermi sphere, having radius kF (Fermi
wavevector)
Fermi momentum
p = k
F
F
vF = pF / m
1
EF = mvF2 =  2 kF2 / 2m
2
Fermi velocity
Fermi energy
Electrical conductivity and Ohm’s law
Equation of motion: Newton’s law
dv
dk
=h
= − eE
dt
dt
eE
k (t ) − k (0) = −
t
h
eE
k avg = − τ
h
hk
eE
v avg = avg = − τ
m
m
m
In the absence of collisions the Fermi sphere in k-­‐space is displaced as a whole at a uniform rate by a constant applied electric field. Because of collisions the displaced Fermi sphere is maintained in a steady state in an electric field. ky j = − nev avg
F kx Fermi sphere kavg dp(t )
p(t )
=−
+ f (t ) = 0
dt
τ
p = fτ = −eEτ
Ohm’s law ⎛ ne 2τ ⎞
⎟⎟E
j = ⎜⎜
⎝ m ⎠
ne 2τ
σ=
m
1
m
ρ= = 2
σ ne τ
Electrical conductivity and Ohm’s law
the mean free path l = vFτ
because all collisions involve only electrons near the Fermi surface vF ~ 108 cm/s for pure Cu: at T=300 K
at T=4 K τ ~ 10-­‐14 s l ~ 10-­‐6 cm = 100 Å τ ~ 10-­‐9 s l ~ 0.1 cm kavg << kF : 22
-­‐3
2
8
for n = 10 cm and j = 1 A/mm vavg = j/ne ~ 0.1 cm/s << vF ~ 10 cm/s Conduction current in metals
The equation of motion of a free electron (not bound to a particular ion: C=0)
m dr
d 2r
dv
me 2 = Cr − e
− eE ⇒ me
+ meγv = −eE
dt
τ dt
dt
Lorentz model (Harmonic oscillator model) Drude model (C=0) The current density is defined as :
→
→
! C $
J = −Ne v with units of #
" s-m 2 &%
Substituting in the equation of motion we obtain :

→
" Ne 2 % 
dJ
+γ J = $
'E
dt
# me &
τ=
1
γ
≈ 10 −14 s
Conduction current in metals
Assume that applied electric field and the conduction current density are given by:
" "
E = E0 exp(−iωt )
 
 
J (r, t) = σ (ω )E(r, t)
" !
J = J 0 exp(−iωt )
Local approximaBon to the current-­‐
field relaBon (λ >> l) Assumption valid for:
Ø  High temperature
Ø  Low frequency
Substituting into the equation of motion we obtain :
"
"
"
"
⎛ Ne 2 ⎞ !
J 0 exp(−iωt )
⎟⎟ E0 exp(−iωt )
+ γ J 0 exp(−iωt ) = −i ω J 0 exp(−iωt ) + γ J 0 exp(−iωt ) = ⎜⎜
dt
⎝ me ⎠
[
]
Multiplying through by exp( +iωt ) :
! ⎛ Ne 2 ⎞ !
(- iω + γ )J 0 = ⎜⎜ ⎟⎟ E0
⎝ me ⎠
Conduction current in metals
For static fields ( ω = 0) we obtain :
 ! Ne 2 $ 

Ne 2
J = ##
= static conductivity
&& E = σ E ⇒ σ =
meγ " meγ %
For the general case of an oscillating applied field :
$
 !

σ
#
& E = σ E σ = dynamic conductivity
J=
ω ω # 1− iω / γ &
"
%
(
)
Low frequencies ( ω / γ ) << 1 : the dynamic conductivity is purely real and the electrons follow the electric
field.
As the frequency of the applied field increases the inertia of electrons introduces a phase lag
in the electron response to the field, and the dynamic conductivity becomes complex.
Very high frequencies ( ω / γ ) >> 1 : the dynamic conductivity is purely imaginary and the electron oscillations
are 90° out of phase with the applied field.
Propagation of EM waves in metals

Maxwell's equations in the presence of a specified current density J :
 
∇⋅ E = 0
 
∇⋅ H = 0

 
∂B
∇× E = −
∂t

 
 1 ∂E
∇ × B = µ0 J + 2
c ∂t
 
  
2 
∂(∇ × B)
∇ × ∇ × E = −∇ E = −
∂t
µ0 =
1
ε0c 2
Propagation of EM waves in metals
Maxwell's relations give us the following wave equation for metals:


2
!
!

1∂ E
1 ∂J
2
P
=
0
,
J
≠0
∇ E=
+ 2
2
c ∂t
ε0c ∂t
 $
'
σ
But J = &
)E
%1− (iω / γ ) (
Substituting in the wave equation we obtain :


2

$
'
1
∂
E
1
σ
∂
E
∇2 E =
+ 2&
)
2
c ∂t
ε0c %1− (iω / γ ) ( ∂t
The wave equation is satisfied by electric fields of the form :
 
 
$
E = E 0 exp %i( k ⋅ r − ω t)'(
where :
ω 2 $ σωµ0 '
k = 2 + i&
)
1−
(i
ω
/
γ
)
c
%
(
2
c2 =
1
ε0µ0
Skin depth
ω << γ
2
k =
⇒
ω2
⎡ σωµ0 ⎤
⎛ π ⎞
+
i
≅
i
σωµ
=
exp
⎜ i ⎟σωµ0
0
⎢
⎥
(
)
c2
1
−
i
ω
/
γ
⎝ 2 ⎠
⎣
⎦
⎡
σωµ0
⎛ π ⎞
⎛ π ⎞
⎛ π ⎞
⎛ π ⎞⎤
k ≅ exp⎜ i ⎟σωµ0 = exp⎜ i ⎟ σωµ0 = ⎢cos ⎜ ⎟ + i sen⎜ ⎟⎥ σωµ0 = (1 + i )
= k R + ik I
2
⎝ 2 ⎠
⎝ 4 ⎠
⎝ 4 ⎠
⎝ 4 ⎠⎦
⎣
!c$
σ c 2µ0
σ
n R = # & kR =
=
= nI
"ω %
2ω
2ωε 0
σωµ 0
k R = kI =
2
In the metal for a wave propagating in the z - direction :
 

& z)
"
$
E = E0 exp −k I z exp #i k R z − ω t % = E0 exp ( − + exp k R z − ω t
' δ*
(
)
(
2ε 0 c 2
1
2
δ= =
=
kI
σωµ 0
σω
)
skin depth
(
Real and imaginary parts coincide )
For copper the static conductivity at 10 GHz:
σ = 5.76 ×10 7 Ω−1m −1
→ δ=
2
= 0.66µ m
σ ⋅ 2πν ⋅ 4π 10 −7
Skin effect
Skin effect is the tendency of an alternaBng electric current (AC) to become distributed within a conductor such that the current density is largest near the surface of the conductor, and decrease with greater dephts in the conductor. From a theoreBcal point of view the current density J (the current flowing through the unit area) in a conductor decreases exponenBally as it penetrates from the outer surface in its interior. At the depth d the current density J is approximately: where J0 is the current density on the surface of the conductor Classical skin effect: δ>>l
ω2
ω2 !
4π $ ω 2 4π
k = 2 ε = 2 #1+ i
σ&≈i
σ
c
c "
ω % c2 ω
2
12
(2πσω )
k=
c
(1+ i)
! ( 2πσω )1 2 $
E ∝ exp(−iω t + ikr) ∝ exp ## −
r &&
c
"
%
E ∝ e− r δ = e
−Kr
Ø  At low frequency, the electric field is distributed in the enBre secBon of the conductor, the overall resistance is the lowest. Ø  Increasing frequency, the electric field is concentrated on the surface and the overall resistance increases. Skin depth for some common metals (@RT)
Anomalous skin effect δ<<l (pure metals at low temperatures) The usual (local) theory of electrical conducBvity is no longer valid: the electric field varies rapidly over l. Further, not all electrons are parBcipaBng in the wave absorpBon and reflecBon. δ’
l
Only electrons moving inside the skin depth for most of the mean free path ℓ are capable of picking up much energy from the electric field. Only a fracBon of the electrons δ’/ℓ contribute to the conducBvity. c
c
δ'=
≈
(2πσ ' ω )1 2 ⎛ δ ' ⎞1 2
⎜ 2π σω ⎟
l
⎝
⎠
13
⎛ lc2 ⎞
⎟⎟
δ ' = ⎜⎜
2
πσω
⎝
⎠
The threshold of angular frequency that separates the normal from the abnormal skin effect is typically 1013 rad/s. (THz regime) At these frequencies the depth of penetraBon is worth a few tens of nanometers Plasma frequency
Let's come back and consider again the general case:
ω 2 " σωµ 0 %
2
k = 2 + i$
'
c
$#1− (iω / γ ) '&
(* σ c 2 µ
,*
(* σ c 2 µ
,*
c2 2
i
γ
2
0
0
n = 2 k = 1+ i )
- = 1+ i )
"
%
"
%
ω
iγ *+ ω #1− (iω / γ )& *.
*+ ω #1− (iω / γ )& *.
2
2
2
⎛
⎞
γ
σ
c
µ
Ne
Ne
0
⎟⎟c 2 µ0 =
n2 = 1 − 2
→ ω2p = γ σ c 2 µ0 = γ ⎜⎜
ω + iωγ
meε 0
⎝ meγ ⎠
The refractive index of the medium is given by :
n2 = 1 −
ω2
p
ω 2 + iωγ
Plasma
frequency
Plasma frequency
At frequencies high enough to satisfy ω / γ >> 1
the refractive index of the medium is given by :
2
ω
p
n 2 = 1− 2 = ε
ω
ε (ω ) Is the dielectric constant of the metal For ω < ωp (ε real and negaBve) no radiaBon can propagate For ω > ωp (ε real and posiBve) metal becomes transparent to the em radiaBon Alkali metals have been observed to become transparent in the UV The high frequency condiBon is well saBsfied in the neighborhood of the plasma frequency Dielectric constant in metals (general case)
If we model a metal as a free-­‐electron gas, then: 1) Infinite relaxaBon Bme 2) No interband transiBon γ = 0 Plasma frequency
λ  > λp : n is complex and radiaBon is aqenuated λ < λp : n is real and radiaBon propagates CriRcal wavelength Metal
Lithium Sodium Potassium Rubidium Caesium λ observed [nm]
205 210 315 360 440 λ calculated [nm] Neff/N 150 0,54 210 1 290 0,85 320 0,79 360 0,67 •  Below the criBcal wavelength, metal become transparent •  Above the criBcal wavelength, metal is opaque and highly reflecBng For “cold” electrons (metals and semiconductors) For best metals [Au, Ag, Pt, Al] Plasma frequency
The Nature of Plasma Oscillations: Corresponds to a displacement of the entire electron gas at a distance d with
respect to the positive ion background. This creates surface charges σ = nde and
thus an electric field E = 4πnde (cgs units)
If the electrons in a plasma are displaced from a
uniform background of ions, electric fields will
be built up in such a direction as to restore the
neutrality of the plasma by pulling the electrons
back to their original positions.
Because of their inertia, the electrons will
overshoot and oscillate around their equilibrium
positions with a characteristic frequency known
as the plasma frequency.
2
EquaRon of Nm ∂ d = − NeE = − Ne(4πnde) OscillaRons at the ∂t 2
Plasma Frequency MoRon 2
∂d
2
+
ω
d =0
p
2
∂t
ωp
2
4πne 2
=
m
Bulk vs Surface Plasmons
ElectromagneBc waves propagaBon through metals happens above plasma frequency Surface plasmons are coherent electron oscillaBons exisBng at the metal/dielectric interface, where real part of the dielectric funcBons changes sign across the interface SPs have lower energy than bulk plasmons which quanBze the longitudinal electron oscillaBons about posiBve ion cores within the bulk of an electron gas. Plasma waves
Plasma oscillation = density fluctuation of free electrons
+
-
+ +
-
-
k
+
-
+
Plasmons in the bulk oscillate at p determined by
the free electron density and effective mass
Ne 2
drude
p
m 0
Plasmons confined to surfaces that can interact
with light to form propagating “surface plasmon
polaritons (SPP)”
1 Ne 2
particle
3 m 0 SPP modes
Confinement effects result in resonant
in nanoparticles
drude
Dispersion relation for EM waves [bulk plasmons]
Wave equaBon: For ω > ωp → K2 > 0, K is real, waves with ω > ωp propagate in the media with the dispersion relaBon. The electron gas is transparent. For ω < ωp → K2 < 0, K is imaginary, waves with ω
< ωp incident on the medium do not propagate. The electron gas is totally reflecBve. Dispersion relation for EM waves [surface plasmons]
Dispersion relation for EM waves [surface plasmons]
CondiBon of SP existence From the conBnuity relaBons on the boundaries, we can obtain the dispersion relaBon. Dispersion relation for EM waves [surface plasmons]
x direcBon: z direcBon: --> εm < -­‐ εd kx must be real: εm < 0 Dispersion relation for EM waves [surface plasmons]
Nanophotonics with plasmonics: A logical next step?
Why not electronics?
Why not photonics?
Why plasmonics?
Why nanophotonics needs plasmons?
Superconductors
(in a microwave field:
basic principles)
BASIC MICROWAVE PROPERTIES Surface impedance E x ( 0)
Z≡
∫
∞
0
=
j x ( z )dz
iωµ
σ 1 − iσ 2
=
4π E x (0)
⋅
= Rs + iX s
c H y (0)
x vacuum δ, λ
for a superconductor Two-­‐fluid analogy: !
! ! !
j = jn + js = (σ 1TF − iσ 2TF ) E
σ 1TF
nn e 2τ
=
m(1 + ω 2τ 2 )
σ 2TF
ns e 2
ωnn e 2τ 2
=
+
ωm m(1 + ω 2τ 2 )
(super)conductor z y !
jn
!
j
!
js
In the microwave regime ωτ<<1 and relaxaBon effects can be ignored BASIC MICROWAVE PROPERTIES Comparison of superconductors with normal conductors: N.C. S.C. Skin depth δ = 2 ∝ ω -­‐1/ 2 ωµσ n ⎯ T⎯ ⎯ 0 → const . →
PenetraBon depth (anomalous skin effect) Surface resistance Rs ⎯T⎯
⎯→ X s / 3 ∝ ω
→0
2/3
independent of ω
(no dispersion) strongly T dependent Surface resistance ωµ ωµδ
Rs = X s =
=
2σ n
2
∝ ω 1/ 2
1
1
λ∝
∝
ns
1− t4
R s << X s = ωµλ (crossover frequency) ∝ ω 2 (anomalous regime) ⎯ T⎯ ⎯ 0 → 0 →
(ideally) BASIC MICROWAVE PROPERTIES Comparison of superconductors with normal conductors: N.C. δ = Nb-­‐Ti Skin depth 2 ∝ ω -­‐1/ 2 ωµσ n ⎯ T⎯ ⎯ 0 → const . →
Copper PenetraBon depth (anomalous skin effect) Surface resistance Rs ⎯T⎯
⎯→ X s / 3 ∝ ω
→0
2/3
independent of ω
(no dispersion) strongly T dependent Surface resistance ωµ ωµδ
Rs = X s =
=
2σ n
2
∝ ω 1/ 2
1
1
λ∝
∝
ns
1− t4
S.C. R s << X s = ωµλ (crossover frequency) ∝ ω 2 (anomalous regime) (from A. Andreone) ⎯ T⎯ ⎯ 0 → 0 →
(ideally) (from S. Inagaki) BASIC MICROWAVE PROPERTIES BCS theory: Surface resistance: (for T < Tc/2) ⎡ (!ω ) 2 ⎤ ⎛ 4k BT ⎞
R (T , ω ) ≈ ⎢
⎟ exp(−Δ / k BT )
⎥ ⋅ ln⎜
⎣ k BT ⎦ ⎝ !ω ⎠
2
Rs (T , ω ) = RsBCS (T , ω ) + Rres (ω ); Rres ∝ ω
BCS
s
In LTS Rres describes residual losses due to extrinsic mechanisms: defects, dust, chemical residues, insulaBng or normal phases, … (see s.c. caviBes) In HTS, even taking into account unconvenBonal paring mechanisms (q.-­‐p. excitaBons at zero temperature), surface losses are sBll higher than predicted (from M. Hein) BASIC MICROWAVE PROPERTIES 4 K 77 K (From M. Lancaster) NONLINEAR MICROWAVE PROPERTIES Superconductors are non linear materials A nonlinear response is detrimental for microwave devices NONLINEAR MICROWAVE PROPERTIES Sources of nonlinearity: Extrinsic: •  weak links
(bulk materials, granular films, …) •  thermal effects (defects, dust, substrate heaBng, ...) •  geometrical effects
(in microstrips, striplines, coplanar lines) • inhomogeneiBes (changes in the currents distribuBon) •  nucleaBon and growth of vorBces in normal phase Intrinsic: •  nonequilibrium excitaBons •  unconvenBonal pairing
(close to Tc) (excess of q.-­‐p.) Consequences in microwave devices: •  reduced power handling capability •  harmonic and intermodulaBon products generaBon NONLINEAR MICROWAVE PROPERTIES Power handling capability: experimental response of a HTS filter 0
-10
LaAlO3 +12 dbm
+22 dbm
+27 dbm
+32 dbm
+37 dbm
+38 dbm
+39 dbm
+40 dbm
@77 K -20
S12 [dB]
-30
1
2
5 cm An interdigital YBCO filter for PCS applicaBons -40
-50
-60
-70
-80
1,60
1,65
1,70
1,75
1,80
1,85
1,90
f [GHz]
Breakdown @ 40 dBm = 10 W input power (from A. Andreone) NONLINEAR MICROWAVE PROPERTIES IntermodulaBon distorBon: NONLINEAR MICROWAVE PROPERTIES IMD scenarios in the wireless environment Antenna 1. In-­‐Band Carriers, In-­‐Band IMD Carriers IMD Spurs RX Filter Antenna 2. Out-­‐of-­‐Band Carriers, In-­‐Band IMD RX Filter TX Tones (from S. K. Remillard) Superconductors
(applications)
I. 
HTS PASSIVE MICROWAVE DEVICES
DELAY LINES A delay line is a simple analog passive device which adds a Bme delay to a signal applied to the input. SomeBmes used for more sophisBcated devices such as chirp filters and transversal filters. Real delay lines always introduce some aqenuaBon and deviaBon from the linear phase shix. Advantages of superconducBng technology: •  low propagaBon losses •  compact geometry •  wide bandwidth •  no frequency dispersion (λ independent of frequency) Spiral stripline pattern for the bottom substrate of a HTS delay line
Frequency response
(from S. Talisa) FILTER BASICS Ideal resonator: Q = QUnloaded λ/2
Substrate InserBon loss (dB) Isolated resonator L C R Frequency Coupling λ/2
Real resonator: QLoaded < QUnloaded (there is a world outside …) QU Coupling Substrate IN Frequency pulling OUT ~ RS VS Source Coupling L C R Coupling RL Load InserBon loss (dB) Resonator QL QU Frequency FILTER BASICS From resonator to filter: C R . . . Resonator n L C R Coupling R L Coupling C Coupling L Resonator 2 InserBon loss (dB) Coupling Resonator 1 The coupling networks “pull” the frequency of the resonators and broaden their bandwidths, to form the filter passband Frequency IN OUT Substrate For forward coupling, the number n of resonators gives the order of the filter THE ADVANTAGES OF HTS TECHNOLOGY: MiniaturisaBon: Filter Technologies at PCS Frequency: Reduced volume ——> reduced mass Superior electrical performance: 5 M Hz W ide Chebyshev Filters
Insertion Loss (dB)
0
Q=3,000
Q=20,000
Q=60,000
-5
Simulated transmission response of a 10-­‐pole 5 MHz bandwidth Chebyschev filter as a funcBon of the quality factor of each resonator -10
-15
Increased sensiBvity and selecBvity -20
1.91 10
9
9
1.92 10
Frequency (Hz)
1.93 10
9
(From S. Remillard) RF filter technologies: a pictorial comparison Rectangular single mode waveguide Circular dual-­‐mode waveguide Dielectric resonator Hybrid dielectric resonator/HTS HTS thin film (All filters have the same order and operate at 4 GHz) (from R. R. Mansour) INTERDIGITAL FILTERS 1% fractional bandwidth 5-pole interdigital filter at 2 GHz
2 1 3 4 5 Forward (resonators 2,3 and 4)and backward
resonators 1, 2 and 4, 5) coupling. Resonators have 10-Ω characteristic impedance.
Frequency response at 60 K for
different input powers
Input and output feed lines have a 50-Ω characteristic
impedance
Major advantages: •  simple design •  moderate power handling capability Drawbacks: •  relaBvely large volume •  low wafer “efficiency” DUAL MODE PLANAR FILTERS The basic element is a dual-­‐mode planar microstrip resonator, excited in correspondence of a degenerate mode. Patch resonators ——> high Q Circular Square
Triangular
The introducBon of a small perturbaBon splits the degeneraBon, separaBng the modes in the frequency domain Dual-­‐mode patch resonator filters ——> high power handling Planar Filter Typology (size vs power handling) Lumped
Miniaturiza2on InterDigital
Dual Mode
Power Handling Superconductors
(applications)
II.
CAVITIES FOR PARTICLE ACCELERATORS
Metamorphosis of a LC circuit into a high frequency pill -­‐ box cavity a)
b)
c)
d)
d R Ea
MagneBc Field e)
Electric Field The geometrical factor Γ Ea
V
c π dR 2
Γ ≅ µ0ω0 ≈ µ0 2.4
S
R 4π R 2
if d = R ⇒ Γ = 0.6µ 0 c = 230Ω
Π -­‐ mode Bunch
Ea
Beam axis
Bunch
Beam axis
350 MHz Nb cavity measured at CERN S.C. CAVITIES FOR PARTICLE ACCELERATORS 4 K S.C. CAVITIES FOR PARTICLE ACCELERATORS Comparison with normal conducBng caviBes Copper @ 500 MHz, 300 K: σ ≈ 6 .107 (Ωm)-­‐1 δ=
2
ωµσ n
≈ 3 µm Rs =
1
σδ
∝ 1/ σ
≈ 6 mΩ For a given acceleraBng field Eacc = 5 MV/m the r.f. power necessary to maintain it per unit length is 1 MW/m For a niobium cavity at T = 4.2 K Rs ≈ 70 nΩ, the dissipated power is only 10 W/m ! (+ cooling power ~ 5 KW/m ...) There is no benefit to cool a normal conducBng cavity to low temperatures. Anomalous skin effect : ℓ >> δ
83
Rs =
9
3µ 0ω 2 ℓ
16πσ
independent of T Need for Superconductors Superconductors (BCS !) Normal Metals Cu : RS = 6mΩ (77 K ,500MHz)
Q = 4 ⋅10 4
Pd
∝ Ea2Γ Q
L
Pd
≅ 100 kW m
L
Eamax ≅ 2 MV m
RealisBc Field limit ! (Pulse ?) Nb :
9
RS = 50nΩ (4.2 K ,500MHz ) Q = 5⋅10
RS < 5nΩ
{
(1.8 K )
Q = 5⋅1010
RS = RS BCS (T ) + Rres
⎡ Δ ⎤
1
RBCS ∝ exp⎢−
⎥
T
⎣ k BT ⎦
Intrinsic Q limit : ∞ ! ( Rres ?)
H a ≤ H c 2 = 240mT
Eamax ≅ 50 MV m
Intrinsic Field Limit ! Quality factor vs AcceleraBng field