1Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Angular Oscillations in a Tilted Magnetic Field in Layered Metals and Bilayers: Q2D, Q1D, Graphite/Graphene, etc.
Angular magnetoresistance oscillations (AMRO) in Q2D metals: 20 years since discovery in 1988AMRO in semiconducting bilayers: theory and experimentPhysica E 34, 128 (2006); PRB 78, 155320 (2008)Proposal for AMRO in graphene bilayers and multi-layersTwo-parameter pattern of AMRO in Q1D materials, arrays of quantum wires, driven superconducting qubits, ... PRL 96, 037001 (2006); PRB 78, 125404 (2008)
Victor M. Yakovenko with Benjamin K. Cooper and Anand Banerjee
Department of Physics, University of Maryland, College Park, USA
2Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
The (BEDT-TTF)2X organic conductors are layered, quasi-2D metals.
They consist of BEDT-TTFlayers intercalated with layers of anions X=I3, IBr2,etc.
3Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
β-(BEDT-TTF)2IBr2, B = 14 T, T=1.4 KKartsovnik et al., JETP Lett. 48, 541 (1988)
Angular magnetoresistance oscillations
θ-(BEDT-TTF)2I3, Kajita et al.,Sol. St. Comm. 70, 1079 (1989)
B ⊥ cB || c
Rc
B || c B ⊥ c B || c
Peaks of Rzz (and Rxx) at the “magic angles” Θn
4Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
β-(BEDT-TTF)2IBr2, Kartsovnik et al., J. Phys. I (France) 2, 89 (1992)
Angular magnetoresistance oscillations
Resistance maxima:n is an integer, d is the distance between layers, kF is the in-plane Fermi momentum.
Reconstruction of the Fermi surface from AMRO
( )1/ 4tan n
F
nk d
πθ
−=
5Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
AMRO in other layered materialsIntercalated graphite, Iye et al., J. Phys. Soc. Jpn. 63, 1643 (1994)
Sr2RuO4, Ohmichi et al., Phys. Rev. B 59, 7263 (1999)
High-Tc superconductor Tl2Ba2CuO6, Hussey et al., Nature 425, 814 (2003)
GaAs/AlGaAs superlattices, Kawamura et al., Physica B 249-251, 882 (1998) Notice high temperature 25 K
6Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
AMRO in intercalated graphiteIye, Baxendale, Mordkovich, J. Phys. Soc. Jpn. 63, 1643 (1994)
( )1/ 4tan −=NF
Nk d
πθ
kFd
7Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
β-(BEDT-TTF)2IBr2, T=0.4 K, Wosnitza et al., J. Phys. I (France) 6, 1597 (1996)
Shubnikov-de Haas and de Hass-van Alphen oscillations beatings
B
B
v
///
2
zz
z
z
kS PSS P
m
ε
ε
π
∂=
∂
∂ ∂= −
∂ ∂∂ ∂
= −
Yamaji JPSJ 1989: Smax=Smin at the magic angles ⇒ no beatings.The average velocity ‹vz›→0,so σz→0, Rz→∞.
magic angle
nonmagic angle
8Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Theory of AMRO in bilayers
a
b
B⊥z
Rc
d
B|| x
t⊥
Interlayer tunneling, gauge Az=B||x( ) /†
aˆ ˆ ˆ( ) ( ) h.c.zieA d cbH t eψ ψ⊥ ⊥= +
rr r h
For quasiclassical cyclotron motion in the plane, the effective amplitude of inter-plane tunneling is obtained by averaging of the phase over time t:
( ) /, ( ) cos( ), /c c c F
ieB x t d c
tt t e x t R t R k c B eω⊥ ⊥ ⊥= = =
h% h
0 cos 4F FB B
t t J k d t k dB B
π⊥ ⊥ ⊥
⊥ ⊥
⎛ ⎞ ⎛ ⎞= ∝⎜ ⎟ ⎜
⎝ ⎠ ⎝ ⎠% − ⎟
( 1/ 4)tan nF
B nB k d
πθ⊥
−= =
The layers decouple, so σz→0 (Rz→∞), and beatings disappear.
The effective amplitude when
– Bessel function
0t⊥ →%
9Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
AMRO as Aharonov-Bohm interference
( ) ( )tann
nF
B n CB k d
πθ⊥
+= =
Real-space picture: Periodicity in the magnetic flux of B||( ) 0
02 ( ), 2n F
c ckB R d n C R
Bφφ
π ⊥Φ = = + =
Momentum-space picture: Interlayer tunneling /ixeB d c
t e⊥h
injects the in-plane momentum Δk||=eB||d/c.
Interference between trajectories a and b:( ) ( )
2( ) 2 ( )Fn n
Fce
B k dn C B k k B
n Cπ π⊥ ⊥+ = Δ ⇒ =
+h
Interlayer matrix element between Landau wave functions:Hu & MacDonald, PRB 46, 12554 (1992) ' ' 0ˆ', ,a b
n nn n nn H n L J J
−⊥ −∝ → →
Kurihara, J. Phys. Soc. Jpn. 61, 975 (1992): Laguerre → Bessel for n>>1
d B||
2Rc
B⊥z xa
b
a bkF
B⊥
B|| Δk||
10Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Interlayer conductance σz with finite τShockley tube integral: Yagi et al., JPSJ 59, 3069 (1990)
[ ]( ' ) /' v ( )v ( ') , v ( ) 2 sin ( )t tz z z z ztt
dt t t e t t k t dτσ∞
− −⊥∝ =∫
Bkz
Interlayer tunneling: McKenzie & Moses, PRL 81, 4492 (1998), PRB 60, 7998 (1999)
2'( ) ( ')
'zt
ieB d t tx t x tc
t
t d t e e τσ∞
⊥
−−⎡ ⎤⎣ ⎦ −∝ ∝∫ h ⎨
⎩
⎧ 2t τ⊥% for ωcτ>>12t τ⊥ for ωcτ1 and no AMRO for ωcτ
11Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Beating period in B⊥ increases with B||Boebinger et al., PRB 43, 12673 (1991)
Experiments in bilayers
Interlayer conductance vs. B||Eisenstein et al., PRB 44, 6511 (1991)
12Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Harff, Simmons et al., PRB 55, 13405 (1997)Experiments in bilayers
B⊥(T)
( )
( )Fn B k dB
n Cπ⊥≈
+AMRO in bilayers?Interferenceoscillations? Rzz is needed
13Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
SdH and AMRO in bilayer with parabolic dispersion
Contourplot of thecalculatedσzz vs.Bx and Bz
14Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Gusev et al., PRB 78, 155320 (2008)
Experimental observation of AMRO in bilayers
Landau level N=1
Spin Zeeman splitting
Bilayer splitting ΔSAS=2t⊥
Bilayer splitting vanishesfor certain values of B||
Observation of AMRO for high Landau levels, described by semiclassicalcyclotron orbits
15Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Graphene – the single-layer graphite
Real-space structure:hexagonal lattice,two sublattices
Energy dispersion: twobands touching at points
Linear dispersionnear the Dirac point
Graphics courtesy of Philip Kim
16Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Parabolic dispersion Linear dispersion(semiconductors) (graphene)
ε( p) = p
2
2m ε( p) = vF p
ohn nyx .|),( =ψ ⎟⎟⎠
⎞⎜⎜⎝
⎛−
=oh
ohn n
nyx
.
.
1||
),(ψ
( )2/1+= nmeBE zn
h 22 Fzn venBE h±=
In a perpendicular magnetic field
H( p) =
0 px + ipypx − ipy 0
⎛
⎝ ⎜
⎞
⎠ ⎟
ψ( p) − scalar ψ( p) − spinor
17Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Shubnikov-de Hass oscillations and AMRO in graphene bilayers with ABAB stacking
Contour plot of σzz vs. Bx and Bz
Sublattice A of one layer couples to sublattice B’of another layer
18Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Nonperturbative treatment of t⊥ in graphene bilayer
Electron dispersion is parabolic; Dirac points are lost.
19Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Dispersion in a parallel magnetic field
The parallel magnetic field shifts the in-plane momentum of electrons by q when they tunnel between layers.
In a parallel magnetic field,the Dirac cones reappear.
Electronic properties of a graphene bilayer can be controlled by the orbital effect of a parallel magnetic field.
B||
20Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Graphene bilayer with a twist
The van Hove singularity in DOS from the saddle point in the electron dispersion due to ΔKrot for a twisted bilayer was observed with STM by Eva Andrei.
Theory: Lopes dos Santos, Peres, Castro Neto, PRL 99, 256802 (2007)
φ
ΔKrot=Kφ=(4π/3a)φ=6×108 m-1 for φ=2o
ΔKmagΔKrot
ΔKrot
The magnetic shift of the in-plane momentum is much smaller that the rotational shift:ΔKmag/ΔKrot=0.7×10-2
In general,ΔKmagΔK rot
=3a
4πϕeB||d
h=
34π
ΦadΦ0
⎛
⎝ ⎜
⎞
⎠ ⎟
1ϕ
,
where Φad = B||ad and Φ0 = h/e.
Brillouinzone
21Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
c
a
The TMTSF molecule A stack of TMTSF molecules,with anions X=PF6, ClO4
c
b
Q1D conductors (TMTSF)2X (Bechgaard salts)
View along the TMTSF chains
22Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Different types of AMRO in Q1D conductors
Lebed magic angles:(z,y) rotation, magnetic field points from one chain to another
Chashechkina, Chaikin, PRL 80, 2181 (1998)
DKC effect: (x,z) rotation
Danner, Kang, Chaikin, PRL 72, 3714 (1994)
Lee, Naughton, PRB57, 7423 (1998)
Third angular effect:(x,y) rotation.Generic rotation –all effects together.B
23Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Tunneling between two Q1D layersInterlayer tunneling:
2 † ( )1 2
ˆ ˆ ˆ( ) ( ) H.c.
,
φψ ψ
φ⊥ = +
= = −∫ rr r
h
ic
edz z x yc
H t d r e
A A B y B x
Quasiclassical in-plane motion:( )0 0
2( ) , ( ) sinbF cF z
t cx t x v t y t y tev B
ω= ± = m
The condition n=By’ corresponds to the Lebed magic anglesOscillations of Jn(Bx’) represent the Danner-Kang-Chaikin effect
( )( ) forc c c n x yi t tt t e t J B B nφ = ′ ′= =%2, , yF z x bc x y
z F z
Bebv B B t d dB Bc B v B b
ω ′ ′= = =h h
Effective amplitude of inter-plane tunneling is obtained by phase averaging:
24Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Interlayer conductivity σcShockley tube integral (solution of the Boltzmann equation): Yagi et al. (1990), Danner, Kang, Chaikin (1994)
[ ]( ' ) /' v ( )v ( ') , v ( ) 2 sin ( )t tc z z z c ztt
dt t t e t t k t dτσ∞
− −∝ =∫
Interlayer tunneling conductance: McKenzie, Moses, Lundin (1998-2004), Osada (2002-2003)
2 ( ) ( ') ( ' ) /'c ct
i t i t t tt
t d t e φ φ τσ∞
− − −∝ ∝∫ ⎨⎩
⎧ 2ct τ% for ωcτ>>12ct τ for ωcτ>>1
2
2 2
( ) ( )(0) 1 ( ) ( )
c n x
nc c y
J Bn B
σσ ω τ
∞
=−∞
′=
′+ −∑B tctc 1 2 tt'
Kubo formula: Osada, Kagoshima, Miura (1992), Lebed’, Naughton (2003)
25Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
The two-parameter pattern of AMRO in σc
Calculated for (ωcτ)2=50
Circles –maxima
Squares –minima
Diagonal line –the third angular effect
26Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Interlayer interference in momentum space
Interlayer tunneling changes the in-plane momentum by q:
( ), -ed y xc B B=qThe Fermi surfaces of the two layers are shifted by the vector q. Tunneling is possible only at k1, k2, k3, etc.
Constructive interference between k1 and k3 due to Bz requires(S1+S2)c/heBz=(2π)2n, equivalent to the Lebed condition By’=n.
Constructive (destructive) interference between k1 and k2 requiresS1c/heBz=(2π)2(j+1/4) with j integer (half-integer) – the DKC scillations.
If the displaced Fermi surfaces do not cross, there are no interference and no AMRO – the third angular effect.
27Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
Coherent quantum interference in a driven superconducting qubit and Q1D organic conductors
Cooper, Yakovenko, PRL 96, 037001 (2006) – theory of AMRO in Q1D conductors
Oliver, Yu, Lee, Berggren, Levitov, Orlando, Science 310, 1653 (2005) –experiment and theory for a driven superconducting qubit
28Angular magnetoresistance oscillations (AMRO) in Q2D and Q1D conductorsVictor Yakovenko
ConclusionsAMRO result from periodic modulation of the effective interlayer tunneling amplitude due to the quantum Aharonov-Bohm interference between electron trajectories in the real or momentum space.
AMRO have been observed in many layered metals and recently in semiconducting bilayers.
An observation of AMRO in graphene bilayers and multilayers is desirable.
In Q1D conductors, there are two Aharonov-Bohm interference areas, soAMRO are two-parameter oscillations described by σc(Bx’,By’). These results can be also applied to arrays of quantum wires.
The formula σc(Bx’,By’) derived for Q1D conductors can be applied to a driven superconducting qubit and to the Shapiro steps in Josephson junctions with a finite decoherence time.
The quantum interference effects are limited by decoherence time τ due to loss of phase memory.
Nonperturbative treatment of t in graphene bilayerDispersion in a parallel magnetic fieldGraphene bilayer with a twist