AMRO in Q2D - UCSBonline.kitp.ucsb.edu/online/lowdim09/yakovenko/pdf/...Victor Yakovenko Angular...

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


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.


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


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

−=


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


AMRO in intercalated graphiteIye, Baxendale, Mordkovich, J. Phys. Soc. Jpn. 63, 1643 (1994)

( )1/ 4tan −=NF

Nk d

πθ

kFd


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


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⊥ →%


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||


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


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)


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


SdH and AMRO in bilayer with parabolic dispersion

Contourplot of thecalculatedσzz vs.Bx and Bz


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


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


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


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


Nonperturbative treatment of t⊥ in graphene bilayer

Electron dispersion is parabolic; Dirac points are lost.


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||


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


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


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


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:


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)


The two-parameter pattern of AMRO in σc

Calculated for (ωcτ)2=50

Circles –maxima

Squares –minima

Diagonal line –the third angular effect


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.


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


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

AMRO in Q2D - UCSBonline.kitp.ucsb.edu/online/lowdim09/yakovenko/pdf/...Victor Yakovenko Angular...

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