Pavel D. Grigoriev

L. D. Landau Institute for Theoretical Physics, Russia

Consider very anisotropic metal, such that the interlayer tunneling time of electrons is longer than the in-layer mean scattering time and than the cyclotron period. Does the interlayer magnetoresistance has new qualitative features? Do we need new theory to describe this regime?

[1] P. D. Grigoriev, Phys. Rev. B 83, 245129 (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv:1104.5122].

The answer is yes, and we consider what are these differences.

2D electron gas

2D electron gas

2D electron gas

The coherent interlayer tunneling conserves the in-plane electron momentum p|| . This gives the well-defined 3D electron dispersion ε(p)=ε (p) +2tz cos(kzd) and Fermi

surface as a warped cylinder. This also assumes tz>>h,

where is the in-plane mean free time. Examples: most anisotropic metals.

The incoherent interlayer electron tunneling does not conserve the in-plane momentum. The 3D electron dispersion and FS do not exist (only in-layer 2D). Examples: compounds with extremely small interlayer coupling, where the interlayer electron transport goes via local crystal defects or by the absorption of bosons.

p is conserved in interlayer tunneling, but the tunneling time is longer than the cyclotron and/or mean free times. The 3D FS and electron dispersion are smeared. Examples: all layered metals with small tz in strong magnetic field. Candidates: some organic metals, heterostructures, high-Tc cuprates. Are the standard formulas for magnetoresistance applicable in this case? Does this regime contains new physics?

3

Generally accepted opinion

This conclusion is incorrect and was obtained because the authors have used oversimplified model for the electron interaction with impurities.

P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998 (1999).

,,1

),(02

ikn

nGyD

R They have used the following

2D electron Green’s function disorderwrong

Introduction

Motivation2

This question is rather general. The weakly coherent regime appears in very many layered compounds: high-Tc cuprates, pnictides, organic metals, heterostructures, etc. Magnetoresistance (MQO and AMRO) are used to measure the quasi-particle dispersion, FS, scattering, ..

Experimentally observed transitions coherent – weakly coherent – strongly incoherent interlayer coupling show many new qualitative feature: monotonic growth of interlayer magnetoresistance, different amplitudes of MQO and angular dependence of magnetoresistance, etc.

Motivation (monotonic growth)

Monotonic growth of interlayer magnetoresistance, observed in many layered compounds when magnetic field is layers (parallel to electric current)

-(BEDT-TTF)2SF5CH2CF2SO3

F. Zuo et al., PRB 60, 6296 (1999).W. Kang et al., PRB 80, 155102 (2009)

2a

The coherent regime of interlayer magnetotransport is well understood.

For axially symmetric dispersion and

in the first order in tz it simplifies to:[R. Yagi et al., J. Phys. Soc. Jap. 59, 3069 (1990)]

This gives angular magnetoresistance oscillations (AMRO):

If the electron dispersion ε(p) is known, the background conductivity is given by the Shockley tube integral (solution of transport equation):

Yamaji angles

4Introduction

5

Harmonic expansion of Fermi momentum

Harmonic expansion of the angular dependence of FS cross-section area (measured as the frequency of magnetic quantum oscillations):

[First order: C. Bergemann et al., PRL 84, 2662 (2000); Adv. Phys. 52, 639 (2003). Second order relation between k and A: P.D. Grigoriev, PRB 81, 205122 (2010).]

One can derive the relation between the first coefficients k and A !

Introduction

The model of weakly coherent regime

The Hamiltonian contains 3 terms:

The 2D free electron Hamiltonian in magnetic field summed over all layers:

the coherent electron tunneling between any two adjacent layers:

and the point-like impurity potential:

where

2D electron gas

2D electron gas

2D electron gas

is the same as in the coherent regime, but the parameters and approach to the solution differ.

1 3 2

7Model

Calculation of interlayer conductivity in the weakly incoherent regime

2D electron gas

2D electron gas

2D electron gaszz

The interlayer transfer integral tz<<0 is the smallest parameter. We take it into account in the lowest order (after the magnetic field and impurity potential are included as accurately as possible). Interlayer conductivity is calculated as the tunneling between two adjacent layers using the Kubo formula:

,)(),1,,'(),,',(2

'2222

F

yx

zzz njrrAjrrA

drdrd

LL

dte

),,',(Im2),,',( jrrGjrrA Rwhere the spectral function

Approach to the solution of the problem 8

includes magnetic field and impurity scattering.

The impurity distributions on two adjacent layers are uncorrelated, and the vertex corrections are small by the parameter tZ/EF, =>

,),1,,'(),,',()(2

'2222

jrrAjrrAn

drdrd

LL

dteF

yx

zzz

The electron Green’s function in 2D layer with disorder in Bz

,

2

1, 21

g

igR EE

EEEEcEEnEG

,,)()(),,( 1,2,,

2100 nGrrrrG

yy

y

knknkn

The point-like impurities are included in the “non-crossing” approximation, which gives:

whereTsunea Ando, J. Phys. Soc. Jpn. 36, 1521 (1974).

,1,12

2

2

1 igig cEEcEE ,2/ 20 BlVE Hzg ./2 2

LLiiHzi NNNlc

The density of states on each Landau level has the dome-like shape:

,

2Im 21

g

R

EE

EEEEEGED

Density of states

E

D(E)

Bare LLBroadened LL

1ic

.22/12 BcEEE igB Landau level width

14

00

cBIn strong magnetic field the effective electron

level width is much larger than without field:

9

Monotonic part of conductivity for B || z

The averaging over impurities on two adjacent layers is not correlated. For B = BZ we get

In strong magnetic field we substitute the Green’s function from the non-crossing approx. and obtain the monotonic part of interlayer conductivity

LLiiHzi NNNlc /2 2 ./20 cgi Ec where and

In weak magnetic field this gives

10

In the SC Born approximation .5.13

8 00

00

cc

zz

.13.100c

The shape of LLs is not as important as their width!The inclusion of diagrams with intersection of impurity lines in 2D electron layer with disorder only gives the tails of the DoS dome. The width of this dome remains unchanged and ~Bz

1/2:

DoS

E

D(E)

bare LL broadened LL

1ic

DoS

E

D(E)

bare LL broadened LL

1ic

Therefore, we can take the DoS: .

/22B

B

EED

The corresponding Green’s function is ,,1

),(2 ByD

R iknnG

4/1200 1/4 cB

20

The conductivity is not sensitive to the shape of LLs, but strongly depends on their width.

where and 0 is the electron level width without magnetic field

which gives ./89.04/ 0000 cczz

Comparison with experiments on interlayer MR Rzz(B)(magnetic field dependence: background and MQO)

Sometimes, MR grows too strongly with increasing Bz

On experiments MR grows with Bz even in the minima of MQO!

11

5 10 15 20B

1

2

3

4

5

6

R zz Theory on MR

W. Kang et al., PRB 80, 155102 (2009)

New

Old

F. Zuo et al., PRB 60, 6296 (1999).-(BEDT-TTF)2SF5CH2CF2SO3

MR growth appears also at large tz as in

BPRL 89, 126802 (2002);

Result 1

Physical reason for the decrease of interlayer conductivity in high magnetic field

12

BZ1

2

The impurity distributions on adjacent layers are different. When an electron tunnels between two layers, its in-plane wave function does not change, but the energy shift due to impurities differs by the LL width W (0 C)1/2 ~ BZ

1/2

Why W ~ BZ1/2 ? Because the area where e0, approximately, S ~1/BZ,

and the number of effectively interacting with the electron impurities ci SNi ~1/BZ , fluctuates as ci

1/2 ~ BZ-1/2, => the average shift of electron

energy due to impurities W=SNiV0 fluctuates as W/ci1/2 ~(SNi)1/2V0~ BZ

1/2 .

The same physical conclusion comes from more accurate averaging of electron Green’s function

BZ1

2

The impurity potential shifts the energy of each electron state, given by W=Re . This shift is random with the distribution

,),1,(Im),,(Im)(2

2 2

2

00

jrGjrGn

drd RRF

Dzz

The averaging of electron Green’s function over impurities must include this averaging over the energy shift, which increases the effective imaginary part of the electron self energy:

The interlayer conductivity contains averaged electron Green’s functions

13

5 10 15 20 B

5

10

15

Rzz

amplitude of MQO differs because the Dingle temperature increases with field

5 10 15 20B

1

2

3

4

5

6

R zz

Magnetic quantum oscillations of conductivity in the weakly incoherent regime

23

.

21

/2sinh/22

exp1 2

2

0

c

B

c

c

c

Bk

kzz

k

TkTkkik

B

,122

0 B

dteB

B

Fz

.1/44/12

00 zcB B where

New result

Old result

Comparison of the results on Rzz of standard theory (coherent regime)

and new theory (weakly incoherent) :

Result 2.

MQO of interlayer conductivity are given by

Dingle temperaturebackground MR

background MRgrows with Bz

Calculation of the angular dependence of MR

.)(),1,,'(),,',(2

'2222

F

yx

zzz njrrAjrrA

drdrd

LL

dte

cos,0,sin,0, BBBBB zx In tilted magnetic field the vector potential is , the electron wave functions on adjacent layers acquire the coordinate-dependent phase difference and the Green’s functions acquire the phase

0,,0 xz zBxBA

,sinyBddyBr x ,exp),,,(),1,,( r'riejr'rGjr'rG RR

.),,',(),,',(),,',( jrrGjrrGijrrA RA where the spectral function

.sin2/

exp),(Re

sin2/

cos),()(2

2

2

2222

hieByd

rG

heByd

rGnd

rddte

R

RFz

zz hThe expression for conductivity has the form:

GRGAGRGR

The impurity averaging on adjacent layers can be done independently:

21

50 500 .05

0 .10

0 .15

0 .20

0 .25

0 .30

0 .35

zz

Angular dependence of magnetoresistance in the weakly incoherent regime

22Result 3.

,1

22

1

200

BcZzz

JJB

cos/1/00 BBB

Angular dependence of interlayer conductivity is given by old expression:

50 500 .05

0 .10

0 .15

0 .20

0 .25

0 .30

0 .35

zz

B=5T B=10T

New result

Old result *1/2

The difference comes from the high harmonic contributions and from the prefactor

,tan dkFwhere but depends on Bz:

.cos/1/10 BBB ZZ and the prefactor acquires the angular dependence:

zz()

Comparison with experiment on angular oscillations of magnetoresistance (AMRO)

“Clean” sample

-100 -50 0 50 100 150

10

100

R,

Oh

m

, deg.

B :

3 T

0.5 T

0.12 T

“Dirty” sample

-100 -50 0 50 100 150

10

11

12

20

25

30

35

R,

Oh

m

, deg.

0.5 T

0.12 T

B :

3 T

Theory (qualitative view):

new resultOld result

Experiment:

Appendix 1.

50 50

20

40

60

80

100

Rzz

M. Kartsovnik et al.,PRB 79, 165120 (2009)

24

P. Moses and R.H. McKenzie, Phys. Rev. B 60, 7998 (1999).

Further work

1. The crossover 2D --> quasi-2D --> 3D (tz ~ 0)2. The crossover weak --> strong magnetic field (C ~ 0). 3. Very high field, when the growth of Rzz(B) is faster than ~B1/2 . 4. Change in angular dependence of harmonic amplitudes of MQO5. Influence of chemical potential oscillations and electron reservoir.6. Quasi-1D anisotropic metals.

Above analysis is applicable to the high-field limit C>0, tz.

There is still much work to do:

25

Summary26

In the “weakly coherent” regime of interlayer conductivity, i.e. when the interlayer tunneling time is longer than the electron mean free time in the layers, the effect of impurities is much stronger and the Landau level width is much larger than in the standard 3D theory. This strongly changes the angular and field dependence of magnetoresustance:

Thank you for attention!

1. The background interlayer MR grows ~B1/2 with increasing field B||.2. The Dingle temperature grows ~B1/2 , which leads to the weaker

increase of the amplitude of MQO with increasing B.3. The angular dependence of MR changes: additional (cos)-1/2 factor

appears and the maxima of AMRO are weaker.

[1] P. D. Grigoriev, Phys. Rev. B 83, 245129 (2011). [2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv:1104.5122].

Strongly incoherent interlayer magnetotransport is very model-dependent

Usually, the conductivity in this regime has non-metallic exponential temperature dependence (thermal activation or Mott-type). It has weak angular dependence of background magnetoresistance (contrary to the coherent case) [A. A. Abrikosov, Physica C 317-318, 154 (1999); U. Lundin and R. H. McKenzie, PRB 68, 081101(R) (2003); A. F. Ho and A. J. Schofield, PRB 71, 045101(2005); V. M. Gvozdikov, PRB 76, 235125 (2007); D. B. Gutman and D. L. Maslov, PRL 99, 196602 (2007) ; PRB 77, 035115 (2008); etc.]

Exception gives the following model [PRB 79, 165120 (2009)]:

E0

1 2

The interlayer transport goes via local hopping centers (resonance impurities). Resistance contains 2 in-series elements:

The hopping-center resistance Rhc is almost independent of magnetic

field and has nonmetallic temperature dependence. The in-plane

resistance R|| between nearest hopping centers depends on magnetic

field and has the metallic temperature dependence.

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