Michael FoglerNon-Ohmic

Hopping Transportin 1D Conductors

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