Michael Fogler
Non-Ohmic
Hopping Transport
in 1D Conductors
Acknowledgements
ƒ Work in collaboration with: Randall Kelly, UCSD
ƒ Support: A. P. Sloan Foundation; C. & W. Hellman Fund
ƒ Reference: cond-mat/0504047
Systems of interest
1D
• carbon nanotubes
• quantum and nano-wires
Quasi-1D
• organic conductors
• chain-like compounds
K2NiF4
TMTSF
Why study 1D and quasi-1D systems?
1. Unusual physics:
ƒ dominance of quantum effects
ƒ strong correlations
ƒ strong effects of disorder
2. Intellectual challenge
ƒ non-perturbative theory
ƒ understanding real experiments
3. Promising applications
ƒ new materials: nanotubes, composites, organic-inorganic hybrids
ƒ new electronics: plastic & molecular
ƒ new devices: nano-sensors, flat-panel displays, etc.
Disorder-induced localized states
1D nanostructures are often have lots of defects
Energy
Wavefunction decays
exponentially:
e −| x| / a
Position
a = localization length
Materials with localized electrons states are insulators
Variable-Range Hopping = Transport Mechanism in Insulators
ε
ε2
phonon
ε2
tunneling decay
x
ε1
e −∆ x / a
ε1
∆x
a = localization length
 2∆ x ε 2 − ε 1 
−

a
T


Hopping rate ~ exp  −
Conductivity of the entire system is determined by the optimal
network of hopping sites. Typical hopping distance varies
(decreases) with temperature; hence, Variable Range Hopping.
Mott (1960’s)
Non-Ohmic hopping transport
ƒ Recent experiments: 1D VRH transport is easily made non-Ohmic
Polydiacetylene:
Aleshin et al. (2004)
Nanotube arrays:
Tang et al. (2000);
Tzolov et al. (2004)
Also individual nanotubes:
Cumings and Zettl (2004)
ƒ Surprisingly little theoretical work on the non-Ohmic VRH in 1D
• McInnes, Butcher, and Triberis (1990)
• Malinin, Nattermann, Rozenow (2004)
ƒ VRH in higher dimensions is essentially different
Basic ingredients of the model
ε
ε j, f j
Hopping rate from
Γij
x
ε i , fi
 2∆ x
 E j − Ei  
, 0  
Γ ji = exp  −
− max 
a

 T
 
Ei ≡ ε i − Fxi
I ji = Γ ji − Γij
fi =
Net current from
1
µi
Equation to solve:
i to j
Occupation factor
exp[(ε i − µi ) / T ] + 1
Local chemical potential
∑ I ji = 0
i
µi = ?
i to j
f i (1 − f j )
F is the electric field
Ohmic VRH. Equivalence to Resistor Network
I ji =
 ξi − ξ j
T
sinh 
R ji
 T




Net current from
i to j
ξ i = µi − Fxi is the local electro-chemical potential
| ε i − µi | + | ε j − µ j | + | ε i − ε j
2
R ji = exp  ∆ xij +
2T
a
I ji ≅
At small electric fields
|



R ji
i
ξi − ξ j
R ji
Resistor network:
K
i−2
i −1
i
i +1
i+2
K
x
j
Problem of finding the optimal current path
ε
xM
Conventional argument:
εM
x
−εM
xM
typical length of the links
εM
typical energy change across each link
ε M ≥ 1 / g xM
g
is the density of states
 2 xM ε M 
+

T 
 a
Minimize the typical resistance R ~ exp 
xM ~ a
T0
,
T
T0 ≡ 1 / g a
ε M ~ T0 T ,
ln R ~
T0
T
1D analog of Mott’s formula
energy level spacing within a localization volume
This argument and the result for the resistance are WRONG in 1D !
ε
au
“Breaks” on the current path
ε
T0
Tu
x
x
− Tu
− T0
Non-optimal break
Kurkijarvi (1973); Lee et al. (1984)
Resistance of the break:
Probability of the optimal break:
Optimal break
Raikh and Ruzin (1989)
R ~ exp (u )
 T 2
P ~ exp (− g A ) = exp  −
u 
 2T0 
A = (a u ) (T u ) / 2 area of the break
Breaks dominate over typical resistors:
T
T 
R ~ exp  0  >> exp 0
T
 2T 
1D Mott’s law is incorrect!
Statistical distribution of the conductance
The conductance of a finite-length wire is random and depends on what
configuration of breaks is realized
Bar plot:
Distribution function of conductance through
a 1.8 x 0.2 mu GaAs device
Hughes et al. (1996)
Solid curves:
The best fit to the theory
Raikh and Ruzin (1989)
New questions addressed in this work
ƒ What is the highest electric field F at which the VRH is still Ohmic?
ƒ Do “breaks” continue to play a role at larger fields?
ƒ How does the resistivity depends on F at such fields?
Early “break”-down of the Ohmic transport
L
L ~
Voltage drop per optimal break:
Current through the break:
Ohmic regime is valid if
I =
1
T 
~ xM exp  0 
P
 2T 
∆ξ ~ F L
T
 ∆ξ 
sinh 

Rbr
 T 
∆ξ << T
 T 
F << exp  − 0 
 2T 
Non-Ohmic transport. Intermediate fields
ƒ Breaks are still relevant
ƒ Electrochemical potential (voltage) drops mainly
on the breaks, in a cascade fashion
ƒ Hardest breaks are progressively circumnavigated
as the current (electric field) increases
ξ
Non-Ohmic break
x
Chemical and electrochemical potentials
ξ
Non-Ohmic break
x
µ = ξ + Fx
Chemical potential adjusts itself to
the existing distribution of breaks
x
Calculational strategy
ƒ Assume a given fixed current I
ƒ Each link of the optimal path is a “voltage generator”
I
V1 ( I )
Vi (I )
V2 ( I )
V3 ( I )
is determined by inverting
I=
K
T
V 
sinh  i  = const
Ri , i +1
T 
ƒ Most effective generators operate in the non-linear regime
ƒ Need to determine their geometry and distribution function
ƒ Averaging over this distribution we obtain the electric field needed to sustain
the given current
Similar approach was used by Shklovskii (1976) in 3D
Optimal non-Ohmic breaks
Intermediate current (field) regime is defined by:
T
T0
<< u I ≡ ln ( I 0 / I ) << 0
T
T
Breaks still control the transport.
Optimal breaks are shaped as hexagons.
They have average linear density
 T 2 Vi

u I 
P ~ exp (− g A ) = exp  −
uI −
2T0 
 2T0
Vi ≅ µi − µi +1 ~
T0
>> T
uI
Field needed to sustain this current is
F ~
 T 2
Vi
~ Vi P ~ exp  −
u I 
L
 2T0 
1D Mott’s law is recovered at F ~ k BT / a
ln ρ ~
2T0  k BT 
ln 

T
 Fa 
Non-Ohmic transport. Strong fields
ƒ Breaks are no longer relevant
ƒ Electron distribution function is driven far from equilibrium
ƒ Only forward hops are important
ε
x*
F >>
ε*
− ε*
x
k BT
a
Calculational strategy: directed percolation
ε
x*
x

 2
I = f i ( 1 − f i +1 ) exp  − xi − xi +1 

 a
xi − xi +1 ≤
a
uI
2
E = ε − Fx
1/ 2
 k T
ρ = exp  C B  ,
 Fa 
x
C =8
Like the 1D Mott’s law with an effective temperature
k BTeff ~ Fa
slope
F =
dE
dx
The real temperature plays no role
Previous work: the exponential differs by a large log
Malinin, Nattermann, Rozenow (2004)
Predictions for experiments
ƒ In an ensemble of finite-length wires, there are enormous resistance variations
ƒ The distribution (or, simply, the mean) of log-resistance should be studied instead
ƒ Temperature should be low enough; otherwise, one has Nearest-Neighbor-Hopping
instead of the Variable-Range-Hopping
ƒ Three types of behavior are predicted for the log-averaged resistivity:
ln ρ ~ − eV / 2k BT ,
k BT << eV << k B TT0 / ln L
ln ρ ~
2T0  k BT 
ln 
,
T
Fa


ln ρ =
8
k BT
,
Fa
F <<
k BT
a
F >> k BT / a
(“weak”)
(intermediate)
(strong)
Conclusions and future challenges
ƒ Constructed the theory of a 1D Variable-Range Hopping in finite electric
fields with a due account of rare events specific for the 1D geometry
ƒ The dependence of the resistivity on the field shows a rich structure with three different
functional laws in weak, intermediate, and strong electric fields
ƒ Need to compute the statistical distribution of the resistivity in finite-length wires
ƒ Need to account for small transverse hopping in quasi-1D systems
ƒ Need to include Coulomb interaction effects