Time Operator

INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 16 (2004) S5289 S5296 PII: S0953-8984(04)85017-3

Thermally driven hopping and electron transport in

amorphous materials from density functional

calculations

Tesfaye A Abtew and D A Drabold

Department of Physics and Astronomy, Ohio University, Athens, OH 45701-2979, USA

E-mail: *****@******.***.*****.*** and *******@****.***

Received 17 August 2004

Published 22 October 2004

Online at stacks.iop.org/JPhysCM/16/S5289

doi:10.1088/0953-8984/16/44/025

Abstract

In this paper we study electron dynamics and transport in models of amorphous

silicon and amorphous silicon hydride. By integrating the time-dependent

Kohn Sham equation, we compute the time evolution of electron states near

the gap, and study the spatial and spectral diffusion of these states due to lattice

motion. We perform these calculations with a view to developing ab initio

hopping transport methods. The techniques are implemented with the ab initio

local basis code SIESTA, and may be applicable to molecular, biomolecular

and other condensed matter systems.

(Some gures in this article are in colour only in the electronic version)

1. Introduction

Mike Thorpe has worked intensely on practically every aspect of the physics of disordered

systems, always with an eye toward general insights. To many attending this meeting, his work

on oppy modes and the entire view of disordered systems it has engendered is the best known

single contribution. Yet Mike has also contributed deeply to issues of structure and of theory

of spin systems, and with Denis Weaire has offered another cornerstone of understanding

of amorphous insulators: the Weaire Thorpe theorem, which explains the existence of an

optical gap in terms of the connectivity of an amorphous covalently bonded network. With

characteristic elegance, the problem is formulated in a one-band tight-binding (essentially

bond-orbital) picture, and the analysis concludes with an appeal to a theorem of linear algebra.

It is a delight in re-reading these papers to see Mike s lifelong friend, scienti c foil and

colleague Jim Phillips referenced and his ideas discussed in the context of the work of Thorpe

and Weaire [1]. This result has become such a basic part of our thinking that it is worth reading

the comments of Ziman [2] on the non-trivial nature of the work.

0953-8984/04/445289+08$30.00 2004 IOP Publishing Ltd Printed in the UK S5289

S5290 T A Abtew and D A Drabold

In this paper, we shall be concerned with mixing two of Michael s favourite things: electron

states and vibrational dynamics. The essential underlying physical processes were deduced

by Sir Nevill Mott [3], Miller and Abrahams [4] and Conwell [5] in the 1950s; the methods

have been nicely updated by Shklovskii and Efros [6]. In this paper we take these ideas

seriously, and use current density functional descriptions of electron states, and computational

schemes for integrating the time-dependent Schr dinger (Kohn Sham) equation to work out

o

the consequences of a dynamic lattice on electron states (thermally driven hopping). Phonon-

induced delocalization is clearly seen. An accessible explanation of the idea, and a review of

the eld up to the early 1980s, is given by Zallen [7].

Because of nanoscience and microelectronics, there is currently a renaissance of interest

in transport calculations, and creation of predictive theories of transport. While the present

theory lacks the rigour of the best current techniques, it is easy to implement and easy to

understand. Our scheme also has the strength that it is based upon a dynamic lattice, unlike

the most sophisticated methods (where one obtains conductivities due to external elds with a

frozen lattice). We expect that this additional realism will give improved results for localized

states with their large electron phonon coupling [8], and this gives us cause to hope that the

method will have broad utility [9].

2. Methodology

2.1. Electronic structure

In our calculations we used the code SIESTA [10 12] within the local density approximation

and the parametrization of Perdew and Zunger [13] for the exchange correlation functional.

To remove core electrons from the calculations, norm conserving Troullier Martins [14]

pseudopotentials factorized in the Kleinman Bylander [15] form were used. We employed

single- (SZ) basis sets [16] where one s orbital for the H valence electron and one s and three

p orbitals for the Si valence electrons were used. The point was used to sample the Brillouin

zone. To track the time evolution of a given state an additional subroutine has been added to

the SIESTA code. In the simulation, 1000 molecular dynamics steps for a time step of 0.50 fs

were performed. (We have checked a time step of 0.05 fs and the result does not change.)

For this rst trial of the approach, we have used a non-self-consistent version of SIESTA, an

investigation we shall explore in subsequent work.

Elsewhere [8], it has been shown that the electron phonon coupling is always large for

localized states, which suggests the need for a non-perturbative solution for the time evolution

of localized electrons in a dynamic network. In particular, it was shown there that Kohn

Sham eigenvalues conjugate to localized eigenstates may thermally uctuate by tenths of an

electronvolt, larger than kT . The spectral and spatial electronic diffusion can be obtained by

tracking the time evolution of the state by solving the time-dependent Kohn Sham equation:

(t ) = H (t ) (t )

i

h (1)

t

where H is the Hamiltonian operator and = i Ci i is the single-electron wavefunction

written in the basis of non-orthogonal orbitals { i }. The L wdin transformation is used

o

1/2

to convert to an orthonormal basis { i } which is de ned by i = j (S )i j j where

Si j = i j d r is the overlap matrix [17]. With these transformations (1) becomes in an

3

explicit matrix representation

C =HC

i

h (2)

t

Electron dynamics and transport in amorphous materials S5291

where H = S 1/2 H S 1/2 and C = S 1/2 C . The time development can be obtained by using

the Crank Nicholson scheme with the approximate evolution operator linking time t to t +,

where is a small time interval between two consecutive time steps:

1

C (t + ) = 1 + i H (t )/2 1 i H (t )/2 C (t ).

h h (3)

This evolution operator is unitary for any (of course it only tracks the correct solution for

small enough).

The participation ratio, P, is a quantity that tells us how many sites participate in a given

eigenstate. For an ideally localized state P = 1, and for extended states P could reach the

number of atoms N . The inverse of P is de ned as the inverse participation ratio I . Given the

eigenvector C j ( Ri ) for state j which is de ned at each site Ri, I is de ned as

C j ( R i ) 4

i

I j = (1/P ) j = . (4)

2

C j ( R i ) 2

i

Since we are able to get the time evolution of the eigenvector C (t ) from equation (3), the

spatial and spectral diffusion of the localized states can be obtained from equation (4). For I

large we have a well localized state; I small means that the state is extended.

2.2. Structural models

Perhaps both the strength and weakness of our calculation is that we employ fairly realistic

structural models of a-Si [18] and a-Si:H [19] (we name these asi-64 and asi:H-138

respectively). The virtue of the approach is that the true topological disorder is encoded into

the coordinates of the supercells we employ. Furthermore, the lattice motion is determined

from conventional ab initio thermal simulation, and thus should also be fairly realistic. The

a-Si:H model has no defects; the a-Si model has one pair of coordination (under-coordinated

dangling bond and over-coordinated oating bond) defects.

The shortcomings are not to be minimized, and arise from the small model sizes, and

for low temperatures classical lattice dynamics, rather than a phonon picture, which strictly is

needed for T

as only an extremely sparse sampling of the gap and tail states is possible for small models as

we discuss here. This limitation appears to be somewhat ameliorated by the large thermally

driven uctuations of the localized states that we do have, but this point needs to be carefully

studied in future work.

2.3. Transport theory

At low temperature where the density of states at the Fermi level is nite but states near the

Fermi energy are localized, Mott and Davis [20] described the phenomenon of variable range

hopping with the temperature dependent DC conductivity

(T ) exp[ (T0 / T )1/4 ] (5)

where T is the temperature of the system and T0 is a constant that depends on the decay length of

the exponentially localized states and density of states at the Fermi level. Charge transport

by thermally assisted hopping of electrons between localized states is the basic assumption in

the derivation of the T 1/4 law.

Starting from Mott s model, Ambegaokar et al [21] derived the same temperature

dependence of the conductivity [22]. The basic step in their evaluation of the conductivity

is the reduction of the hopping model to an equivalent random resistance network as Miller

S5292 T A Abtew and D A Drabold

0.1

asi:H 138

asi 64

0.08

I (t = 0)

0.06

0.04

0.02

40-120-***-**-*** 280 380

Eigenstates Eigenstates

Figure 1. I (localization) plotted against eigenvalue index for most of the states for asi-64 and

asi:H-138 at time t = 0. The peak in I shows strong localization. The Fermi level is near the

eigenvector with maximum localization in both curves.

and Abrahams [4] reported earlier. In their model Miller and Abrahams showed that the

hopping regime conductivity problem is equivalent to nding an effective resistance of a

random impedance network in which each pair of centres i and j (the centres of localized

electronic states) is connected by a resistance Z i j whose inverse is given by

e2

Z i 1 = n i (1 n j ) i j, (6)

j

kB T

where i j is the transition rate for an electron hopping from centre i to j and may be

approximated for a localized state with a single well de ned centre and spherical symmetry as

0 exp( 2 i Ri j ) exp[( E i E j )/ kB T ] if E i E j

where 0 is a constant which depends on phonon density of states, electron phonon coupling

strength and other properties of the material, Ri j is the distance between the centres i and

j whose respective electronic energy levels are E i and E j (the resonance term), i 1 is the

localization length that describes the spatial extent of the wavefunctions localized at centre i .

n i is the equilibrium occupation of centre i which is given by

n i = (1 + exp[( E i E f )/ kB T ]) 1 (8)

and E f is the energy of the Fermi level.

3. Results

3.1. Spectral and spatial diffusion

The localization I for the two models at t = 0 is depicted in gure 1. Highly localized states

are clearly seen near the Fermi level. The values of I for asi:H-138 are normalized with respect

to the value of I of asi-64.

To study the delocalization process in these models we chose two states near the Fermi

level: the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular

orbital (LUMO). For asi-64, the HOMO is a state which is localized around the dangling bond

Electron dynamics and transport in amorphous materials S5293

20

I (t = 0.5ps)

10

0

20

I (t=0)

10

0

0 120

**-**-**-*-**-**

asi 64 atoms asi:H 138 atoms

Figure 2. Spatial diffusion: contribution of each atom on the highest occupied molecular orbital

(HOMO) in the case of asi-64, and lowest unoccupied molecular orbital (LUMO) in the case of

asi:H-138, as a percentage at t = 0 and 0.5 ps, plotted against atom number.

with an energy splitting of 0.34 eV from the next occupied orbital and 0.88 eV below the

LUMO. In the case of asi:H-138, the HOMO is an extended state with only 0.06 eV to the

next occupied orbital and 1.53 eV below the bandtail localized state LUMO which is 0.11 eV

below the next unoccupied state. These localized edge states are characterized by their higher

I value and concentration on a few atoms for both models at initial time t = 0, and diffuse as

time goes on. To illustrate this, we calculated the contribution of every individual atom to I of

both states at t = 0 and 0.50 ps and in gure 2 we show the contribution of individual atoms

for the localized states HOMO (for asi-64) and LUMO (for asi:H138). For asi-64, at t = 0 the

HOMO state is localized mainly on the dangling bond atom with 23% contribution to the I .

This picture changes at a later time t = 0.50 ps, where the contribution of the dangling bond

atom to I drops to 6%. Similarly, for asi:H-138 the higher contribution to I for the LUMO

state changes from 10% at t = 0 to 2.60% at t = 0.5 ps.

Understanding how diffusion is driven by thermal disorder is the key to understanding the

electronic dynamics and the hopping mechanism. Having this in mind we computed the time

evolution of the two localized states, HOMO and LUMO, for three different temperatures 100,

300 and 500 K for both models we used and present the result in gure 3.

In both models we observed that increasing temperature increases the diffusion. In the

case of asi-64 the HOMO takes about 0.2 ps and in case of asi:H-138 the LUMO takes about

0.15 ps to completely diffuse through the tiny cell at a temperature of 300 K. This diffusion

is explained to be due to quantum mechanical mixing (resonant tunnelling) when other states

get close in energy to the state that we are tracking. This is a manifestation of the resonance

condition [4] from direct calculation in our work.

S5294 T A Abtew and D A Drabold

Homo (temp = 100) Lumo (temp = 100)

0.09

Homo (temp = 300) Lumo (temp = 300)

Homo (temp = 500) Lumo (temp = 500)

I (asi 64)

0.06

0.03

0

I (asi:H 138)

0.03

0.01

0-400-***-*-*** 800

time (*0.5fs) time (*0.5fs)

Figure 3. Thermally induced delocalization: the time evolution of the localization I for highest

occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of asi-64

and asi:H-138 respectively, at different temperatures.

The spectral diffusion of an electron state to a energy-adjacent state can be characterized

by calculating the transition amplitude between the two states. The transition probability to

the state is C (0) C (t ) 2 . Figure 4 shows the hopping probabilities from the

initial LUMO state to the nearby states HOMO and LUMO + 1 in asi:H-138. It is evident from

the energy difference of 1.53 eV between the HOMO and LUMO states in asi:H-138 that the

hopping probability from one state to the other is very small and hence their spectral diffusion

is very limited. On the other hand, the spectral diffusion from LUMO to LUMO + 1 (0.06 eV

energy splitting) is larger. Elsewhere [23, 24], we have shown that the mixing proceeds via

thermally driven resonant cluster proliferation [24]. In a subsequent calculation on a much

larger system, the island structure of localized states [24, 25] will be important. That is,

the idealization of localized states having well de ned single centres is incorrect for states

suf ciently far into the band tails, but still on the localized side of the mobility edge. We

note that the present calculation is not designed to properly model carrier trapping in localized

states. To accomplish this, the trap must be unoccupied, and an electron in the conduction tail

should be added and its time dependence studied.

Figure 4 may also be interpreted providing examples of the phonon-driven transitions

(essentially equation (7)) for LUMO HOMO and LUMO LUMO + 1, the latter transition

being relevant to a weakly n-doped material. The interesting feature of these calculation is that

there are no assumptions made as in Miller Abrahams theory (such as simple radial exponential

decay of wavefunctions, that the existence and character of the defect states arises from real

topological disorder and, perhaps most importantly, our localized states are not static, but

rather are time dependent according to the detailed dynamics of the network). In subsequent

work, we will report the use of these ab initio transition probabilities for computing hopping

Electron dynamics and transport in amorphous materials S5295

0.8

Lumo+1 Lumo (T = 300k)

Lumo Lumo+1 (T = 300k) b) Lumo+1 Lumo (T = 500k)

a)

Lumo Lumo+1 (T = 500k)

0.6

Spectral projection

0.4

0.2

0

Lumo Homo (T = 300k) Homo Lumo (T = 300k)

c) d)

Lumo Homo (T = 500k) Homo Lumo (T = 500k)

0.2

Spectral projection

0.1

0

0-200-***-***-*** 200-***-***-***

time(*0.5fs) time(*0.5fs)

Figure 4. The spectral leakage of the selected states to the nearby states (and back) for asi:H-138

at different temperatures, giving examples of the hopping process between states.

conductivity. We have not yet fully implemented the Miller Abrahams type picture, but we

note that, in principle, all the ingredients are present in this work. It is also possible to

imagine a very direct approach to transport in which we subject the supercell to an electric

eld (for simplicity in a slab geometry with the eld transverse to the slab) and track the time

development of electron packets.

4. Conclusion

We have presented a simulation of the dynamics of the localized states in the presence of

thermal disorder by integrating the time-dependent Kohn Sham equation. We have found that

all localized states diffuse for suf ciently high temperatures and long timescales. The present

work should be developed in various ways.

(1) The case of a spin-polarized system should be considered. Such a calculation in the

local spin density approximation should be directly comparable to electron paramagnetic

resonance experiments, and should provide an interesting study in the differences between

spin and charge diffusion [26].

(2) Self-consistent eld methods should be employed.

(3) Larger model systems should be studied (to reduce admittedly serious nite-size effects).

(4) The scheme should be applied to interesting molecular and biomolecular systems.

Acknowledgments

We thank Professor Pablo Ordej n for help with SIESTA, and many discussions. We also thank

o

Professor Eric Schiff for helpful insights and advice. Finally, we thank the National Science

S5296 T A Abtew and D A Drabold

Foundation for support under grants DMR 0310933 and DMR 0205858 and P A Fedders for

providing us with his unpublished a-Si:H model.

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a o

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