Transcript P. Grigoriev

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?
The answer is yes,
and we consider what are these differences.
[1] P. D. Grigoriev, Phys. Rev. B 83, 245129 (2011).
[2] P. D. Grigoriev, JETP Lett. 94, 48 (2011) [arXiv:1104.5122].
Coherent and incoherent interlayer electron transport
2D electron gas
2D electron gas
2D electron gas
3
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.
“Weakly coherent” interlayer magnetotransport:
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?
Introduction
Generally accepted opinion
P. Moses and R. H. McKenzie, Phys. Rev. B 60, 7998 (1999).
This conclusion is incorrect and was obtained because the authors have
used oversimplified model for the electron interaction with impurities.
1
They have used the following G (n,  ) 
,
R
   2 D n, k y   i0
2D electron Green’s function
disorder
wrong
Motivation
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.
2
Motivation (monotonic growth)
2a
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)
Introduction
The coherent regime of interlayer
magnetotransport is well understood.
If the electron dispersion ε(p) is known, the background conductivity is
given by the Shockley tube integral (solution of transport equation):
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):
Yamaji angles
4
Introduction
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):
One can derive the relation between the first coefficients kmn and Amn !
[First order: C. Bergemann et al., PRL 84, 2662 (2000); Adv. Phys. 52, 639 (2003).
Second order relation between kmn and Amn : P.D. Grigoriev, PRB 81, 205122 (2010).]
Model
The model of weakly coherent regime
is the same as in the coherent regime, but the
parameters and approach to the solution differ.
The Hamiltonian contains 3 terms:
2D electron gas
1
3
2
2D electron gas
The 2D free electron Hamiltonian in
magnetic field summed over all layers:
2D electron gas
the coherent electron tunneling between any two adjacent layers:
and the point-like
impurity potential:
where
7
Approach to the solution of the problem
8
Calculation of interlayer conductivity
in the weakly incoherent regime
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:
2D electron gas
szz
2D electron gas
2D electron gas


d

d
r
d
r
'
A
(
r
,
r
'
,
j
,

)
A
(
r
'
,
r
,
j

1
,

)

n
F ( ) ,

 2
where the spectral function A(r , r' , j,  )  2 Im GR (r , r' , j,  )
e 2 t z2 d
s zz 
Lx L y
2
2
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, =>
e 2 t z2 d
d
2
2

s zz 
d
r
d
r
'

n
F ( ) A( r , r ' , j ,  ) A( r ' , r , j  1,  ) ,


Lx L y
2


The electron Green’s function in 2D layer with disorder in Bz
9
The point-like impurities are included in the
“non-crossing” approximation, which gives:
G ( r1 , r2 ,  )   n0, k y ( r2 )n0, k y ( r1 )G  , n ,
n, k y
E  E g 1  ci    E  E1  E  E 2 
GR  E , n 
,
2E Eg
where
E1  E g


2
ci  1 , E 2  E g

Tsunea Ando, J. Phys. Soc.
Jpn. 36, 1521 (1974).

2
2
 B, ci  2 l Hz N i  N i / N LL .
ci  1 , E g  V0 / 2 l Hz
2
The density of states on each Landau level has the dome-like shape:
D E   
Im GR E 


Landau level width
E  E1 E2  E 
2 E E g
D(E)
,
Density of states
ci  1
E
Bare LL
Broadened LL
B  E2  E1  / 2  2 E g ci  B .
4 c
In strong magnetic field the effective electron B

 1
level width is much larger than without field: 0
 0
10
Monotonic part of conductivity for B || z
The averaging over impurities on two adjacent layers is not correlated.
For B = BZ we get
In weak magnetic field this gives
In strong magnetic field we substitute the Green’s function from the noncrossing approx. and obtain the monotonic part of interlayer conductivity
2
where ci  2 l Hz N i  N i / N LL
0   ci E g2 / c .
and
 s0
In the SC Born approximation s zz  s 0
0
8
c 3 
s0
0
c
0
c
1 .5 .
1.13 .
The shape of LLs is not as important as their width!
20
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 ~Bz1/2:
D(E)
DoS
D(E)
DoS
ci  1
ci  1
E
E
bare LL
bare LL
broadened LL
broadened LL
The conductivity is not sensitive to the shape of LLs,
but strongly depends on their width.
B / 
Therefore, we can take the DoS:



where B  0 4c /  0  1
2
1/ 4
D E  
E 
2
2
B
.
and 0 is the electron level width
without magnetic field
The corresponding Green’s function is GR (n,  ) 
which gives s zz  s 0  0 / 4c  0.89s 0 0 / c .
1
,
n, k y  iB
   2D 

Result 1 Comparison with experiments on interlayer MR R (B)
zz
11
(magnetic field dependence: background and MQO)
Rzz
Theory on MR
6
New
5
4
Old
3
2
1
B
5
10
15
20
On experiments MR grows with Bz
even in the minima of MQO!
W. Kang et al., PRB 80, 155102 (2009)
Sometimes,
MR grows too
strongly with
increasing Bz
MR growth appears
also at large tz as in
F. Zuo et al., PRB 60, 6296 (1999).
-(BEDT-TTF)2SF5CH2CF2SO3
PRL 89, 126802 (2002);
B
Physical reason for the decrease of interlayer
conductivity in high magnetic field
1
2
BZ
12
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 ~ BZ1/2
Why W ~ BZ1/2 ? Because the area where e0, approximately, S ~1/BZ,
and the number of effectively interacting with the electron impurities
ci SNi ~1/BZ , fluctuates as ci1/2 ~ BZ-1/2, => the average shift of electron
energy due to impurities W=SNiV0 fluctuates as W/ci1/2 ~(SNi)1/2V0~ BZ1/2 .
The same physical conclusion comes from more
accurate averaging of electron Green’s function
BZ
1
2
13
The impurity potential shifts the energy
of each electron state, given by W=Re .
This shift is random with the distribution
The interlayer conductivity contains averaged electron Green’s functions
2s 00
d
2

s zz 
d
r

n
F ( ) Im G R ( r , j ,  ) Im G R ( r , j  1,  ) ,


 n2 D
2
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:


Magnetic quantum oscillations of
conductivity in the weakly incoherent regime
Result 2.
23
MQO of interlayer conductivity are given by

 2 ikm  k B  2k 2T / c
 2 k B 
k
s zz  s 0 B    1 exp

1 
.
2
c
c 
k  

 sinh 2k T / c  
Dingle temperature
background MR


e 2 t z2n F d
1
2


s
B


,
B  0 4c /  0   1
where 0
B
B
1/ 4
 Bz .
Comparison of the results on Rzz of standard theory (coherent regime)
and new theory (weakly incoherent) :
amplitude of MQO differs because
Rzz
6
5
Rzz the Dingle temperature increases
with field
background MR
grows with Bz
15
New result
4
10
3
Old result
2
5
1
B
5
10
15
20
5
10
15
20
B
21
Calculation of the angular dependence of MR
The impurity averaging on adjacent layers can be done independently:


e 2 t z2d 2 2
d

s zz 
d
r
d
r
'
A
(
r
,
r
'
,
j
,

)
A
(
r
'
,
r
,
j

1
,

)

n
F ( ) .


Lx L y
2
where the spectral function A(r , r' , j,  )  iGA (r , r' , j ,  )  GR (r , r' , j,  ).
In tilted magnetic field B  Bx , 0, Bz   B sin , 0, B cos  
the vector potential is A  0, xBz  zB x , 0 , the electron wave
functions on adjacent layers acquire the coordinate-dependent phase
difference  r    yBx d   yBd sin , and the Green’s functions
acquire the phase GR (r , r' , j  1,  )  GR (r , r' , j,  ) exp ier   r' ,


2
2e 2 t z2 d

d
 eByd

2

s

d
r

n
(

)
G
(
r
,

)
cos
sin



The expression
 R
zz
F


h
2
 h / 2


for conductivity
has the form:
GRGR

2
 ieByd
 
 Re G R ( r ,  ) exp
sin    .
 h / 2
 

GRGA
Result 3.
Angular dependence of magnetoresistance
in the weakly incoherent regime


Angular dependence of
2





J

2
n
interlayer conductivity is s zz  s 0 BZ   J 0    2
,
given by old expression:
n 1 1  nc B 






where   kF d tan , but  depends on Bz:  B   0 0 / B   1 / B cos
and the prefactor acquires s B  1 / B  1 / B cos .
0
Z
Z
the angular dependence:
zz
B=5T
0.35
szz()
0.35
0.30
0.30
0.25
0.25
New result
0.20
50
Old result *1/2
zz
0.20
0.15
0.15
0.10
0.10
0.05
0.05
50
50
B=10T
50
The difference comes from the high harmonic contributions and from the prefactor
22
Appendix 1.
Comparison with experiment on angular
oscillations of magnetoresistance (AMRO)
24
Experiment:
Theory (qualitative view):
“Clean” sample
Rzz
B:
100
R, Ohm
3 T
100
0.5 T
80
10
0.12 T
-100
-50
0
60
50
100
150
, deg.
“Dirty” sample
40
35
B:
30
20
Old result
50
new result
P. Moses and R.H. McKenzie,
Phys. Rev. B 60, 7998 (1999).
R, Ohm
50
25
3T
20
0.5 T
12
11
10
0.12 T
-100
-50
0
50
100
150
, deg.
M. Kartsovnik et al.,
PRB 79, 165120 (2009)
25
Further work
Above analysis is applicable to the high-field limit C>0, tz.
There is still much work to do:
1.
2.
3.
4.
5.
6.
The crossover 2D --> quasi-2D --> 3D (tz ~ 0)
The crossover weak --> strong magnetic field (C ~ 0).
Very high field, when the growth of Rzz(B) is faster than ~B1/2 .
Change in angular dependence of harmonic amplitudes of MQO
Influence of chemical potential oscillations and electron reservoir.
Quasi-1D anisotropic metals.
Summary
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:
1. The background interlayer MR grows ~B1/2 with increasing field B||s.
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].
Thank you for attention!
26
Strongly incoherent interlayer magnetotransport
is very model-dependent
6
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.