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Acknowledgements
Work in collaboration with: Randall Kelly, UCSD
Support: A. P. Sloan Foundation; C. & W. Hellman Fund
Reference: cond-mat/0504047
Systems of interest
• carbon nanotubes
• quantum and nano-wires
1D Quasi-1D
• organic conductors
• chain-like compounds
TMTSFK2NiF4
Why study 1D and quasi-1D systems?
1. Unusual physics:dominance of quantum effectsstrong correlationsstrong effects of disorder
2. Intellectual challengenon-perturbative theoryunderstanding real experiments
3. Promising applicationsnew materials: nanotubes, composites, organic-inorganic hybrids new electronics: plastic & molecularnew devices: nano-sensors, flat-panel displays, etc.
Disorder-induced localized states
Materials with localized electrons states are insulators
Wavefunction decaysexponentially:
a = localization length
1D nanostructures are often have lots of defects
axe /||−
Energy
Position
Variable-Range Hopping = Transport Mechanism in Insulators
ε
x1ε
2ε
x∆
Hopping rate
−
−∆
−Ta
x 122exp~ εε
axe /∆−
tunneling decay
1ε
2ε
Conductivity of the entire system is determined by the optimalnetwork of hopping sites. Typical hopping distance varies(decreases) with temperature; hence, Variable Range Hopping.
phonon
a = localization length
Mott (1960’s)
Non-Ohmic hopping transport
Recent experiments: 1D VRH transport is easily made non-Ohmic
• McInnes, Butcher, and Triberis (1990)• Malinin, Nattermann, Rozenow (2004)
Surprisingly little theoretical work on the non-Ohmic VRH in 1D
Also individual nanotubes:Cumings and Zettl (2004)
VRH in higher dimensions is essentially different
Polydiacetylene:Aleshin et al. (2004)
Nanotube arrays:Tang et al. (2000); Tzolov et al. (2004)
Basic ingredients of the model
ε
x
)1(0,max2exp jiij
ji ffTEE
ax
−
−−
∆−=Γ
ii f,ε
jj f,ε
ijΓHopping rate from toi j
ijjijiI Γ−Γ= Net current from toi j
1]/)exp[(1
+−=
Tf
iii µε
iµ
Occupation factor
Local chemical potential
0=∑i
jiIEquation to solve:
iii FxE −≡ ε F is the electric field
?=iµ
Ohmic VRH. Equivalence to Resistor Network
iii Fx−= µξ is the local electro-chemical potential
−=
TRTI ji
jiji
ξξsinh
−+−+−+∆=
Tx
aR jijjii
ijji 2||||||2exp
εεµεµε
Net current from toi j
ji
jiji RI
ξξ −≅At small electric fields
i jjiR
Resistor network:
xi 1+i 2+i1−i2−i KK
Problem of finding the optimal current path
ε
x
Conventional argument:
typical length of the linksMx
Mε typical energy change across each link
MM xg/1≥ε
+
TaxR MM ε2exp~Minimize the typical resistance
TTRTT
TTax MM
00
0 ~ln,~,~ ε 1D analog of Mott’s formula
g
agT /10 ≡
is the density of states
energy level spacing within a localization volume
Mx
Mε
Mε−
This argument and the result for the resistance are WRONG in 1D !
“Breaks” on the current path
Kurkijarvi (1973); Lee et al. (1984)
ε
x
Tu
ε
x
Non-optimal break Optimal breakRaikh and Ruzin (1989)
Probability of the optimal break:
−=− 2
02exp)(exp~ u
TTAgP
area of the break
Resistance of the break: )(exp~ uR
Tu−
au
Breaks dominate over typical resistors:
2/)()( uTuaA =
TT
TTR 00 exp2
exp~ >>
1D Mott’s law is incorrect!
0T
0T−
Statistical distribution of the conductance
Bar plot:
Distribution function of conductance through a 1.8 x 0.2 mu GaAs deviceHughes et al. (1996)
Solid curves:
The best fit to the theoryRaikh and Ruzin (1989)
The conductance of a finite-length wire is random and depends on whatconfiguration of breaks is realized
New questions addressed in this work
What is the highest electric field F at which the VRH is still Ohmic?
Do “breaks” continue to play a role at larger fields?
How does the resistivity depends on F at such fields?
Early “break”-down of the Ohmic transport
Voltage drop per optimal break:
L
TTx
PL M 2
exp~1~ 0
LF~ξ∆
Ohmic regime is valid if T<<∆ξ
−<<
TTF2
exp 0
∆=TR
TIbr
ξsinhCurrent through the break:
ξ
x
Non-Ohmic transport. Intermediate fields
Non-Ohmic break
Breaks are still relevant
Electrochemical potential (voltage) drops mainlyon the breaks, in a cascade fashion
Hardest breaks are progressively circumnavigatedas the current (electric field) increases
Chemical and electrochemical potentials
ξ
x
Non-Ohmic break
Fx+= ξµ
x
Chemical potential adjusts itself tothe existing distribution of breaks
Calculational strategy
Assume a given fixed current
Each link of the optimal path is a “voltage generator”
Most effective generators operate in the non-linear regime
Need to determine their geometry and distribution function
Averaging over this distribution we obtain the electric field needed to sustainthe given current
I
)(1 IV )(2 IV )(3 IV K
I
constsinh1,
=
=
+ TV
RTI i
ii
)(IVi is determined by inverting
Similar approach was used by Shklovskii (1976) in 3D
Optimal non-Ohmic breaks
TTIIu
TT
I0
00 )/(ln <<≡<<
TuTVI
iii >>−≅ +0
1 ~µµ
−−=− I
iI u
TVu
TTAgP
0
2
0 22exp)(exp~
Field needed to sustain this current is
Intermediate current (field) regime is defined by:
Breaks still control the transport.Optimal breaks are shaped as hexagons.They have average linear density
− 2
02exp~~~ Ii
i uTTPV
LVF
FaTk
TT Bln2~ln 0ρ
1D Mott’s law is recovered at aTkF B /~
Non-Ohmic transport. Strong fields
Breaks are no longer relevant
Electron distribution function is driven far from equilibrium
Only forward hops are important
ε
x
*x
*ε
*ε−
aTkF B>>
Calculational strategy: directed percolationε
x
*x
dxdEF =
−−−= ++ 112exp)1( iiii xxa
ffI
Iii uaxx21 ≤− +
FxE −=ε
x
slope
8,exp2/1
=
= C
FaTkC Bρ
Previous work: the exponential differs by a large logMalinin, Nattermann, Rozenow (2004)
Like the 1D Mott’s law with an effective temperature
FaTkB ~eff
The real temperature plays no role
Predictions for experiments
In an ensemble of finite-length wires, there are enormous resistance variations
The distribution (or, simply, the mean) of log-resistance should be studied instead
Temperature should be low enough; otherwise, one has Nearest-Neighbor-Hoppinginstead of the Variable-Range-Hopping
Three types of behavior are predicted for the log-averaged resistivity:
aTkF
FaTk
TT BB <<
,ln2~ln 0ρ
LTTkeVTkTkeV BBB ln/,2/~ln 0<<<<−ρ
aTkFFaTk
BB /,8ln >>=ρ
(“weak”)
(intermediate)
(strong)
Conclusions and future challenges
Constructed the theory of a 1D Variable-Range Hopping in finite electricfields with a due account of rare events specific for the 1D geometry
The dependence of the resistivity on the field shows a rich structure with three differentfunctional laws in weak, intermediate, and strong electric fields
Need to compute the statistical distribution of the resistivity in finite-length wires
Need to account for small transverse hopping in quasi-1D systems
Need to include Coulomb interaction effects