An empirical pseudopotential approach to surface and line-edge roughness scattering in...

An empirical pseudopotential approach to surface and line-edge roughness scatteringin nanostructures: Application to Si thin films and nanowires and to graphenenanoribbonsMassimo V. Fischetti and Sudarshan Narayanan

Citation: Journal of Applied Physics 110, 083713 (2011); doi: 10.1063/1.3650249 View online: http://dx.doi.org/10.1063/1.3650249 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/110/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Physical insight into reduced surface roughness scattering in strained silicon inversion layers Appl. Phys. Lett. 101, 073504 (2012); 10.1063/1.4742772 Surface roughness scattering model for arbitrarily oriented silicon nanowires J. Appl. Phys. 110, 084514 (2011); 10.1063/1.3656026 Empirical pseudopotential calculations of the band structure and ballistic conductance of strained [001], [110],and [111] silicon nanowires J. Appl. Phys. 110, 033716 (2011); 10.1063/1.3615942 Monte Carlo study of surface roughness scattering in Si inversion layer with improved matrix element J. Appl. Phys. 100, 044513 (2006); 10.1063/1.2218029 Theoretical investigation of surface roughness scattering in silicon nanowire transistors Appl. Phys. Lett. 87, 043101 (2005); 10.1063/1.2001158

[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

194.69.13.215 On: Mon, 12 May 2014 07:43:05

http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1349655507/x01/AIP-PT/JAP_ArticleDL_050714/Awareness_LibraryF.jpg/5532386d4f314a53757a6b4144615953?x

http://scitation.aip.org/search?value1=Massimo+V.+Fischetti&option1=author

http://scitation.aip.org/search?value1=Sudarshan+Narayanan&option1=author

http://scitation.aip.org/content/aip/journal/jap?ver=pdfcov

http://dx.doi.org/10.1063/1.3650249

http://scitation.aip.org/content/aip/journal/jap/110/8?ver=pdfcov

http://scitation.aip.org/content/aip?ver=pdfcov

http://scitation.aip.org/content/aip/journal/apl/101/7/10.1063/1.4742772?ver=pdfcov

http://scitation.aip.org/content/aip/journal/jap/110/8/10.1063/1.3656026?ver=pdfcov




http://scitation.aip.org/content/aip/journal/apl/87/4/10.1063/1.2001158?ver=pdfcov

An empirical pseudopotential approach to surface and line-edgeroughness scattering in nanostructures: Application to Si thinfilms and nanowires and to graphene nanoribbons

Massimo V. Fischettia) and Sudarshan NarayananDepartment of Materials Science and Engineering, University of Texas at Dallas,800 West Campbell Road RL 10, Richardson, Texas 75080, USA

(Received 4 May 2011; accepted 29 August 2011; published online 25 October 2011)

We present a method to treat scattering of electrons with atomic roughness at interfaces,

surfaces, and edges on nanometer-scale structures based on local empirical pseudopotentials.

This approach merges the computational advantages of macroscopic models based on the shift

of a phenomenological “barrier potential,” with the physical accuracy of models based on

modifications of the atomic configuration at the interface/surface/edge. We illustrate the

method by considering the dependence of the scattering matrix element on the confinement

(inversion) field in free-standing H-terminated Si inversion layers, on the thickness in similarly

H-terminated thin-Si bodies, on the diameter of free-standing [100] cylindrical Si nanowires, and on

the width of armchair-edge graphene nanoribbons. For these latter structures, we find extremely

large scattering rates, whose magnitude — ultimately due to the chirality dependence of the bandgap

— renders perturbation theory invalid and prevents us from drawing quantitative conclusions about

transport properties. Yet, they show clearly the dominant role played by line-edge roughness in

controlling electronic transport in these structures, in agreement with suggestions that transport in

narrow and rough ribbons does not occur via extended Bloch states. VC 2011 American Institute ofPhysics. [doi:10.1063/1.3650249]

I. INTRODUCTION

Scattering with roughness at the interfaces between two

semiconductors, a semiconductor and an insulator, or at the

free surface of a semiconductor has been treated at very dif-

ferent levels of approximations. Fully atomistic models

based on density functional theory (DFT) — describing

roughness as the presence of individual “protrusions” or

“vacancies” in the interfacial region — have been proposed1

and applied to the problem of the enhanced mobility in

tensely strained Si inversion layers,2 but these remain com-

putationally expensive methods, which have yet to find a

wider range of applications. Deferring to a later section the

discussion of line-edge roughness (LER or simply ER in the

following) in graphene nanoribbons (GNR), full-band

description of atomistically roughened interfaces and surfa-

ces (such as the tight-binding studies of Si nanowires

(NW)3–5) as well as phenomenological descriptions of the

roughness via “geometrically” roughened surfaces or interfa-

ces have been used to study the effect of roughness on elec-

tronic transport using three-dimensional (3D) classical6 or

quantum (based on the non-equilibrium Green’s function

(NEGF) approach) simulations of double-gate field-effect

transistors (DGFETs),7 NEGF simulations of Si NWs,3–5 or

two-dimensional (2D) Master-equation studies of DGFETs.8

These atomistic models or models which treat roughness

with geometrical modifications of the interfaces provide

some welcome kind of “ab initio” flavor. Unfortunately, this

gain must be weighed against the need to simulate ensembles

of many different geometrical configurations in order to

extract the desired transport properties of the “average” de-

vice. For this reason, more pervasive in the literature is the

use of the semi-phenomenological Ando’s model9 (see, for

example, recent work on thin bodies10–12), originally applied

to the case of Si inversion layers and recently extended by

Jin et al.13,14 to the case of thin bodies13 and circular cross-

section nanowires.14

With the exception of the DFT1,2 and tight-binding3–5

studies mentioned above, all of these approaches are based on

the effective-mass approximation and either assume as

“potential” the Hartree-like spatial variation of the conduction-

band minimum, including “barriers” due to band-edge discon-

tinuities, or introduce the concept of a “barrier potential”. For

interfaces between a semiconducting film of thickness ts and

oxide “barriers” located at z ¼ 0 and at z ¼ ts, the barrier-

potential Vb takes the form (see Eq. (9) of Ref. 13)

VbðzÞ ¼ Vox½hð�zÞ þ hðz� tsÞ�; (1)

where z is the coordinate along the direction perpendicular to

the interface, Vox is the barrier height, and h is the step func-

tion. The roughness is introduced by shifting rigidly along

the z-direction the Hartree-like or barrier potential by an

amount of �DR varying randomly over the plane of the inter-

faces. The power-spectrum hjDQj2i is usually taken to be ei-

ther a Gaussian real-space correlation function leading to9

hjDQj2i ¼ pD2K2e�Q2K2=4 (2)

or a correlation spectrum of the forma)Electronic mail: [email protected].

0021-8979/2011/110(8)/083713/20/$30.00 VC 2011 American Institute of Physics110, 083713-1

JOURNAL OF APPLIED PHYSICS 110, 083713 (2011)


194.69.13.215 On: Mon, 12 May 2014 07:43:05

http://dx.doi.org/10.1063/1.3650249

http://dx.doi.org/10.1063/1.3650249

http://dx.doi.org/10.1063/1.3650249

http://dx.doi.org/10.1063/1.3650249

hjDQj2i ¼ pD2K2ð1þ Q2K2=2Þ�n�1; (3)

where D and K are the root-mean-square (rms) roughness

and correlation length, respectively. For n¼ 1/2, this corre-

sponds to the exponential correlation proposed by Goodnick

et al.,15 while alternative choices correspond to Bessel-type

real-space correlation, such as the case n¼ 2, which has

been proposed by Leadley and coworkers.16

Here, we abandon model-concepts, such as effective

mass and barrier potentials, and rely, instead, on atomic

potentials to describe the change of the interfacial/surface

potential caused by the atomic roughness. As an example,

we employ here empirical pseudopotentials calibrated to

experimentally known band structure and the workfunctions,

which are responsible for the confinement and the shape of

the interfacial potential. The use of alternative atomic poten-

tials, such as self-consistent pseudopotentials or tight-

binding approaches, are clearly possible, but will not be

discussed here. Clearly, the choice of local empirical pseudo-

potentials limits our approach, since we cannot treat realistic

oxides (although local empirical pseudopotentials for oxygen

and oxides have been used in the past17–19) so that insulating

barriers must consist of either vacuum — albeit with proper

H terminations of dangling bonds to avoid states in the gap

— or fictitious “barrier materials” of the type employed by

Williamson and Zunger as insulators for InAs quantum

dots.20 In addition, we are unable to account for atomic

relaxation, which occurs, for example, in small diameter Si

NWs21 and narrow armchair-edge GNRs.22 Yet, the simplic-

ity afforded by empirical pseudopotentials allows us to

investigate the ability of the method presented here to handle

a variety of realistic structures. Also, one should not underes-

timate the surprising “transferability” of the empirical pseu-

dopotentials available in the literature,23,24 which we employ

here,25 their ability to capture the essential physics and pro-

vide a sufficiently accurate band structure — compared with

self-consistent ab-initio methods — and the flexibility they

exhibit by definition in matching experimental information

when properly calibrated, in some cases exceeding (at the

expense of predictive power) the quality of ab-initio results.

The basic idea consists in following Ando’s or Jin’s

approaches, but rather than defining the roughness as a shift

z! zþ DR of the barrier potential given by Eq. (1) above,

we consider a similar shift of the atomic coordinates entering

the pseudopotentials or consider the deformation of the inter-

face caused by the addition or deletion of atomic lines. We

shall limit our study to the case of roughness caused by dis-

placed atoms, excluding the equally interesting case of chemi-

cal changes of the interface or surface (such as the oxygen

protrusions considered in Ref. 1, for example). Similarly, we

shall ignore the all-important Coulomb contributions to the

roughness-induced scattering rates, as they can be treated ei-

ther in the “conventional way” (such as in Refs. 9, 13, and

14), or by using approaches in which the self-consistent Har-

tree potential provides the shift of the electron charge and the

polarization dipoles caused by the deformation of the inter-

face. Also, we shall employ first-order perturbation theory

(more precisely, the first Born approximation, as discussed

below), having in mind diffusive transport. The case of ballis-

tic (coherent) transport requires the study of transmission

amplitudes, which are the subject of the quantum-transport

approaches mentioned above. Therefore, we shall focus here

on an attempt to deal with diffusive transport, employing a

pseudopotential-based atomistic model and doing so while

accounting for an ensemble-average of the interfacial or sur-

face roughness (SR) without the need to perform calculations

using a statistically significant number of different micro-

scopic configurations. The cases of free-standing H-termi-

nated Si inversion layers, thin bodies, cylindrical nanowires

(NWs), and armchair-edge graphene nanoribbons (AGNRs)

will illustrate the method and will provide interesting insight

on the dependence of the roughness-induced scattering on

confinement, thickness, diameter, and width, respectively. The

extremely large scattering rates found in AGNR will be the

main unanticipated result we will present.

We should stress that, ultimately, there is nothing

intrinsically new in using atomic potentials to deal with

roughness and its effect on ballistic (coherent) transport.

What we believe is different, and hopefully useful in the con-

text of electronic transport, is the use we make of these

atomic potentials, the explicit use of local pseudopotentials,

and the fact that we employ them within a perturbation-

theory framework amenable to transport studies in the diffu-

sive (scattering-dominated) regime. In this context, surprising,

but in hindsight expected, is the large role played by line-

edge roughness in AGNRs.

The paper is organized as follows: In Sec. II, we present

the method in the case of two-dimensional transport in inver-

sion layers and thin films and discuss three alternative for-

mulations for the roughness-induced scattering potential in

order to account for different degrees of correlation of the

roughness at different interfaces. We also discuss the validity

of perturbation theory. In Sec. III, we consider one-

dimensional (1D) transport in AGNRs and cylindrical Si

NWs (Secs. III A and III B, respectively), stressing the role

of the chirality dependence of the bandgap of AGNRs in

causing the very large scattering rates induced by line-edge

roughness. Conclusions are finally drawn in Sec. IV.

II. TWO-DIMENSIONAL TRANSPORT

In this section, we consider the case of two-dimensional

transport in inversion layers, quantum wells, and thin films.

This is the class of structures in which surface roughness-

induced scattering has been studied originally9 and provides

the ideal systems to check the validity of the method pre-

sented here.

Ideally, the geometry considered here consists of a thin

semiconductor film of thickness ts with barrier layers at both

surfaces, such as silicon-on-insulator (SOI) systems, SiO2-

Si-SiO2, or III-V heterostructures, such as AlInAs-InGaAs-

AlInAs.

To fix the ideas, we consider here a Si film with (100)

surfaces of thickness given by 9 cells with vacuum padding

of thickness equivalent to 2 Si cells and dangling bonds satu-

rated by H. Results obtained using slightly larger supercells

(11 Si cells separated by vacuum padding of thickness

083713-2 M. V. Fischetti and S. Narayanan J. Appl. Phys. 110, 083713 (2011)


194.69.13.215 On: Mon, 12 May 2014 07:43:05

equivalent to 4 Si cells) will also be presented below and do

not yield significantly different results.

In order to account for the additional confining potential

of an inversion layer, a self-consistent solution of the associ-

ated Schrodinger/Poisson problem can be obtained (see, for

example, Ref. 25). For simplicity here, we mimic this effect

by adding a parabolic potential of the form

VðextÞðzÞ ¼ V0 1� 2z

Lcþ z2

L2c

� �; (4)

(where Lc is the extension of the supercell in the z direction,

Lc ¼ a0N, with N the total number of cells employed and a0

the Si lattice constant. Also, V0 ¼ FsLc=2 is the total voltage

drop in the cell expressed in terms of the surface field Fs).

This potential is added to the lattice pseudopotential when

expressed in terms of its Fourier components FsL=3 for

Gz ¼ 0 and

VðextÞGz¼ �Fs

2

2

LcG2z

� i

Gz

� �(5)

for Gz 6¼ 0.

Figure 1 shows the band structure of this film for a sur-

face field Fs ¼ 106 V/cm, obtained employing the local Si

and H empirical pseudopotentials proposed by Zhang, Yeh,

and Zunger23 and ignoring the spin-orbit interaction. Note

the “almost” doubly degenerate quantized bands at the zone-

center, corresponding to the “unprimed” subbands, the re-

moval of the double degeneracy (“valley splitting,”9 barely

visible in the figure) being caused by the breaking of the

inversion symmetry (“parity”) caused both by the external

confining field and by the finite extent of the film. Note also

the appearance of additional conduction bands at the �X point,

bands already obtained by Esseni and Palestri26 using a lin-

ear combination of bulk bands (LCBB) and denoted by them

as M3 and M4. Figure 2 shows the variation of the direct and

indirect bandgaps with film thickness. This smooth depend-

ence of the bandgaps with thickness will be in contrast with

a similar plot we shall show for AGNRs. This behavior has

strong consequences on the magnitude of the SR or LER

scattering rates.

We shall now consider three different approaches we

can use to describe atomic roughness: A uniform shift by an

amount D along the z direction of the entire structure (corre-

sponding to correlated roughness at both surfaces); an effec-

tive shift caused by the addition of an atomic line; or the

changes in thickness caused by the addition and removal of

atomic lines. The latter ones allow us to deal with anticorre-

lated or uncorrelated roughness at the interfaces.

A. Perturbation Hamiltonian and matrix element:Correlated roughness

We start, as usual, by assuming that the roughness is

described by a shift DR of the atomic coordinates along the zdirection and function of the coordinate R on the plane of

the interface or surface. The shift is defined in terms of Fou-

rier components DQ, so that

DR ¼X

Q

DQeiQ�R: (6)

(Here and in the following, upper-case bold symbols refer to

two-dimensional vectors on the plane of the surface or inter-

face, with the exception of the wavevectors of the reciprocal

lattice, G, for which we follow the conventional notation,

and we shall denote by Gk and Gz their in-plane and out-of-

plane components, respectively.) The rigid shift just defined

FIG. 1. Band structure of a (100) Si slab in vacuum terminated by H atoms.

The slab thickness is 9 Si cells, and a vacuum padding of thickness equiva-

lent to 2 Si cells separates the slabs. A parabolic potential with a surface

field of 106 V/cm has been employed to strengthen the confinement of the

conduction-band states. Note the quantized subbands in the conduction and

valence bands, the widening of the gap caused by the confinement, and,

barely visible, the lifting of the twofold degeneracy of the unprimed states

(“valley splitting”) caused by the symmetry breaking due to the external

potential. Note also, at the �X-point, the presence of two additional 2D

valleys.

FIG. 2. (Color online) Dependence on film thickness of the band-gap of a

free-standing, H-terminated (100) Si layer for the conduction-band minima

at the �C -point (associated with the “unprimed” ladder of subbands) and

along the �D symmetry line (“primed” ladder). The lines connecting the cal-

culated points are only a guide to the eye.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

describes correlated roughness at the two interfaces. Semi-

phenomenological approaches to extend the approach to

more general cases will be discussed below. As a conse-

quence of the atomic shift, the lattice (pseudo)potential will

be modified as follows:

VðlatÞðrÞ ¼X

a

VðaÞðr� saÞ

¼X

a

XG

VðaÞG eiG�ðr�saÞ

¼X

G

VðlatÞG eiG�r ! VðlatÞðr þ zDRÞ

¼X

G

eiGzDR VðlatÞG eiG�r �

XG

ð1þ iGzDRÞVðlatÞG eiG�r

¼X

G

VðlatÞG eiG�r þ i

XQG

DQVðlatÞG Gze

iðG�rþQ�RÞ (7)

to the first order in the atomic displacement. In this expres-

sion, VðlatÞðrÞ is the total lattice (pseudo)potential, VðaÞðrÞ is

the pseudopotential of atom of species a, VðlatÞG and V

ðaÞG are

their respective Fourier components, sa are the coordinates

of atom a in the supercell, and z is the unit vector along the zdirection.

From Eq. (7), we see that the perturbation Hamiltonian

caused by the roughness can be identified with the term

HðSRÞðrÞ ¼ iXQG

DQVðlatÞG Gze

iðG�rþQ�RÞ; (8)

which replaces the shift of the barrier potential employed in

models based on the effective-mass approximation. There-

fore, to the first order in DQ, the matrix element associated

with the process in which roughness causes a Bloch wave of

in-plane crystal momentum �hK in band n to scatter into a

Bloch wave of crystal momentum �hK0 in band n0 will take

the form

VðSRÞK0;K;n;n0

¼ iX

GkG0kG00kGz

DK�K0þGk�G0kþG

00kGzV

ðlatÞG00k ;Gz

�ð

dznðn0Þ�

G0k;K0 ðzÞeiGzznðnÞGk;K

ðzÞ; (9)

where we have introduced the functions

nðnÞGkKðzÞ ¼ 1

L1=2c

XGz

uðnÞG;KeiGzz; (10)

where the quantities uðnÞG;K are the Fourier coefficients of the

periodic component of the Bloch waves. Note that, at this

stage, the roughness fluctuations, DQ, are still “coupled” to

the in-plane variation of the wavefunctions (due to their Bloch

components eiGk�R), because of the presence of the vectors Gkand G

0

k. However, thanks to the fast-decaying fluctuations DQ

at large Q, this expression may be simplified, since the sum

over some of the Gk vectors is effectively truncated for a large

enough magnitude of K�K0 þGk �G0

k þG00

k (the sub-

scripts of D in the equation above). Physically, this means that

the roughness has negligible fluctuations at wavelengths short

enough to be felt by the in-plane Bloch functions. In order to

see this, it is convenient to rearrange the sums in a more con-

venient form. Defining g¼G0

k �Gk and, in turn, g0 ¼G0

k �gand renaming g0 !G

0

k and g!G00

k, this expression can be

rewritten as

VðSRÞK0;K;n0;n

¼ iX

GkG0kG00kGz

DK�K0þG0kGzV

ðlatÞG0kþ

00k ;Gz

ðdznðn

0Þ�kþG

00k ;K

0 ðzÞeiGzznðnÞGk;KðzÞ

¼ iX

GG00

G0k

DK�K0þG0kGzV

ðlatÞG u

ðn0Þ�GkþG

00k �G0k;GzþG00z ;K

0uðnÞG00;K

¼XG0k

DK�K0þG0kCðGPNÞ

K0;K;n0;n;G0k; (11)

where the last step defines the quantity CðGPNÞ, the

“generalized Prange-Nee” matrix element, which, as we

shall discuss below, plays the role of the matrix element

originally proposed by Prange and Nee27 in the effective-

mass, barrier potential approximation. In this equation, we

have also given an alternative form of the matrix element

expressed in terms of the “bulk” Bloch components uðnÞG;K.

This form may be more suitable for numerical evaluation,

especially in the case of small supercells, when the number

of G-vectors is relatively small, while in large cells, the real-

space integration can be more efficient. However, this k-

space form hides somewhat the physical picture, which is

more intuitively emphasized when employing the wavefunc-

tions nðnÞGk;KðzÞ or the wavefunctions fðnÞK ðzÞ introduced below.

Since the power spectrum of the roughness, hjDQj2idecays very quickly with increasing magnitude of the in-

plane transfer wavevector Q, we can retain only the term

G0

k ¼ 0 in the equation above or the term corresponding to

the minimum G0

k required to map the momentum transfer

K�K0 into the first two-dimensional Brillouin zone (2D

BZ). When only the term G0

k ¼ 0 is retained, this approxima-

tion amounts to considering only normal (N), ignoring

Umklapp processes. Swapping now G00

k with Gk and renam-

ing G00

k ! G0

k, we have

VðSRÞK0;K;n0;n

� iDK�K0X

G

GzVðlatÞG

XG0k

ðdznðn

0Þ�G0kþGk;K

0 ðzÞeiGzznðnÞG0k;KðzÞ

¼ iDK�K0XGG0

GzVðlatÞG u

ðn0Þ�GþG0;K0

uðnÞG0;K

¼DK�K0CðGPNÞK0;K;n0;n;0

:

(12)

A further simplification may be obtained by ignoring the

Bloch-function overlap effects by using the envelope

wavefunctions

fðnÞK ðzÞ ¼1

L1=2c

XGz

uðnÞGk¼0;Gz;K

eiGzz; (13)

which constitute the average of the wavefunctions nðnÞGk;KðzÞ

over the area of the cell on the plane of the interface/surface.

We have referred to these functions as “envelopes,” since



194.69.13.215 On: Mon, 12 May 2014 07:43:05

they represent the envelope approximation on the plane of

the interface, varying only over length scales larger than the

size of the in-plane unit cell. This approximation is justified

as long as the power spectrum hjDQj2i is negligible for Q of

the order of 1=a, where a is the lattice constant in the ðx; yÞplane. This means that there will be negligible roughness at

wavelengths of the order of the in-plane lattice constant,

assumption of which will not result in any significant loss of

information about the nature of the roughness, the only

“loss” being the mistreatment of the Bloch overlap effects.

Thus we can replace both the lattice potential VG as well as

the full wavefunctions nðnÞGk;Kwith their cell-averaged func-

tions, VGk¼0;Gzand fðnÞK , respectively. In so doing, we obtain

the following expression, which can be evaluated numeri-

cally in a relatively efficient way:

VðSRÞK0;K;n0;n

� iDK�K0XGz

GzVðlatÞ0;Gz

ðdzfðn

0Þ�K0ðzÞeiGzzfðnÞK ðzÞ

¼ iDK�K0XGz

GzVðlatÞ0;Gz

~Ið2DÞK;K0;n;n0 ðGzÞ; (14)

where the overlap factor ~Ið2DÞK;K0;n;n0 ðqzÞ is also employed in the

context of electron-phonon scattering.

B. Physical interpretation

The expressions we have just derived for the matrix ele-

ment VðSRÞK0;K;n0;n

are suitable to numerical evaluation, but hide

the physical meaning of the process. Therefore, it is worth

retracing, in part, our steps, rewriting the matrix element in

its “N-process form,” Eq. (12), as

VðSRÞK0;K;n0;n

¼ DK�K0XGkG

0k

ðdznðn

0Þ�G0kþGk;K

0 ðzÞd ~VðlatÞGkðzÞ

dznðnÞ

G0k;KðzÞ

� DK�K0

ðdzfðn

0Þ�K0ðzÞ

d ~VðlatÞGk¼0ðzÞdz

fðnÞK ðzÞ

¼ DK�K0

ðdzfðn

0Þ�K0ðzÞ dhV

ðlatÞðzÞidz

fðnÞK ðzÞ; (15)

where we have introduced the 2D Fourier transform of the

lattice pseudopotential

~VðlatÞGkðzÞ ¼

XGz

VðlatÞG eiGzz (16)

and have considered only the cell-averaged lattice potential

hVðlatÞðzÞi ¼P

GzV0;Gz

eiGzz when replacing the functions

nðnÞGk;Kwith their corresponding cell-average fðnÞK , since the

latter functions are insensitive to the position R within a cell.

Note that the equation above resembles very closely the

model given by Eq. (7) of Ref. 10, the main — and very sig-

nificant — difference being the presence of the atomic (pseu-

do)potential instead of the Hartree-like potential in Eq. (15).

More important is the fact that Eq. (15) has a very clear

physical interpretation: The matrix element for scattering

with roughness is mostly controlled by the location at which

the atomic (pseudo)potential exhibits the largest change as a

function of z. This occurs at the interfaces where

dhVðlatÞðzÞi=dz is largest, since it is related to the change of

workfunction as we move from one material (Si or InGaAs

in the examples above) to another (SiO2 or AlInAs, respec-

tively). Thus a large matrix element will be obtained when-

ever there will be a large workfunction variation (that is, at

interfaces with large band discontinuities) and whenever the

wavefunctions peak in proximity of these interfaces (such as

in the case of a large interfacial field or strong geometric

confinement in thin films). Note also that, for n0 ¼ n and

K ¼ K0, the integral in Eq. (15) represents the first-order

shift, DEnðKÞ, of the dispersion of band n under a change

�D of the layer thickness. This is equivalent to the “usual”

assumption CðGPNÞK;K;n;n;0 � �DEnðKÞ=D � �dEnðKÞ=dts (see

Eq. (39) of Ref. 13, model first proposed in Ref. 28), relating

surface-roughness scattering to the fluctuations of the energy

of quantum-confined states in a quantum well. Finally note

that our approach accounts for the “smearing” of the interfa-

cial ionic potential barrier (over the interfacial region in

which dhVðlatÞðzÞi=dz is significantly different from zero)

and for whatever penetration of the wavefunction across the

interface happens to occur.

Figure 3(a) helps to visualize the situation. The figure

refers to the H-terminated 9-cell-thin (100) Si layer described

FIG. 3. (Color online) (a) Derivative of the cell-averaged lattice potential,

dhVðlatÞðzÞi=dz, as a function of position for a (100) H-terminated free-stand-

ing Si film, with thickness given by 9 Si cells and vacuum padding with thick-

ness equivalent to 11 Si cells. (b) Perturbation potentials due to the deletion

(positive potentials, dashed line, black online) or insertion (negative poten-

tials, solid lines, red online) of a single Si atom from each interface. Also

shown are the real parts of the two ground-state quasi-doubly-degenerate

wavefunctions at flat-band conditions (thin lines, blue online). Note that, while

they both exhibit the same sine-like envelopes, their real parts show different

parity, as required by orthogonality.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

above. The thick line in (a) shows the derivative of the cell-

averaged lattice potential, dhVðlatÞðzÞi=dz, as a function of

position in the supercell. The variation of the confining poten-

tial at the Si/vacuum interface is clearly seen as the positive

(negative) peak at the bottom (top) interface. For wavefunc-

tions affected by an electric field which confines them at the

top (z¼ 0) interface, it is clear that the matrix element, con-

trolled by the overlap between the wavefunctions and the de-

rivative of the pseudopotential, will increase with increasing

confinement. This is illustrated by the black dots (labeled

“uniform shift”) in Fig. 4. This figure shows the squared ma-

trix element jVðSRÞK;K;n;n0=Drmsj2 (that is, normalized to the rms

average of the atomic displacement, Drms ¼ hjD0j2i1=2) for

intra- and inter-subband transitions at the �C-point, involving

the two-fold almost-degenerate ground state (n ¼ 16) calcu-

lated from Eq. (15). The labels “þ” and “–” identify the high-

energy and the low-energy state of each quasi-degenerate pair.

This lack of perfect degeneracy — known as “valley

splitting,” as mentioned previously — is due to the presence

of the confining potential and to the finite thickness of the

layer, effects of which both break inversion symmetry. Note

that one state of each pair is associated to a symmetric (even)

wavefunction, the other state to an odd wavefunction,

although the envelopes of their squared amplitudes are both

even and of almost identical form, and they both resemble the

envelope of the ground-state sine-like wavefunction obtained

using the effective-mass approximation (see Fig. 3(b)). In

order to follow the “conventional” effective-mass nomencla-

ture, which ignores valley splitting, we refer to the ðn�; nþÞtransitions as “intra-subband,” although we have the rather un-

usual feature of dealing with even-odd intra-subband proc-

esses. Similarly, we lump into the same subband n the pair of

quasi-two-fold degenerate states nþ and n� and average the

four squared matrix elements ðn6; n6Þ. Also note that we

plot inter-subband matrix elements at the �C symmetry point,

as they give additional information about the physical picture,

even though energy conservation obviously forbids many of

these transitions. Also shown is the field-dependence of the

Prange-Nee matrix element calculated according to the varia-

tional wavefunction approximation.9 For intrasubband transi-

tions within the ground state in the electric quantum limit, this

matrix element — known to be quantitatively incorrect when

employed using Eq. (17), but still providing the correct de-

pendence of Fs — yields

VðSRÞK;K;n;n0

Drms� �h2

2mL

dfð0Þðz ¼ 0Þdz

��2

� �h2b3

2mL� 33

32

e2

�sns ¼

33

32eFs;

(17)

where b3 ¼ 12mLe2½nd þ ð11=32Þns�=ð�s�h2Þ, e is the magni-

tude of the electron charge, mL is the longitudinal electron

effective mass, nd and ns are the depletion and electron den-

sities, respectively, and �s is the static Si dielectric constant.

In Eq. (17), we have neglected the depletion density nd com-

pared to the electron density ns at these large surface fields

and we have used the fact that Fs ¼ ens=�s.

C. Correlated, anticorrelated, and uncorrelatedroughness

As we have already mentioned, in the case of thin films,

the approach we have just described amounts to assuming

correlated roughness at both interfaces, since Eq. (7) repre-

sents a rigid shift on the z direction of the entire structure,

albeit of magnitude randomly varying on the plane of the

interface. Indeed, for perfectly z-symmetric structures, the

integral in Eq. (15) vanishes in the case of odd-to-odd, even-

to-even transitions and, of course, of intraband processes as

well, since, in this case, the energy shift of the dispersion

vanishes, dEnðKÞ=dts ¼ 0. Obviously, in addition to the case

of correlated roughness in a thin-body structure, the matrix

elements above can be employed to treat surface-roughness

scattering in thick films or, in thin films, at confining fields

large enough to localize the wavefunctions at one of the two

interfaces (the “top” interface, to fix the ideas) and so render-

ing the roughness of the bottom interface ineffective, since

the factor nðn0Þ�

Gk;K0 ðzÞnðnÞGk;K

ðzÞ is negligible around z � ts.

In order to deal with more general situations, we must

resort to a quasi-atomistic framework. The reason why the

model described above can handle only correlated roughness

stems from the fact that we have initially assumed a homoge-

neous shift of the entire structure. On the contrary, account-

ing for the effect of different degrees of correlation between

the two interfaces requires decoupled (uncorrelated rough-

ness) or opposite (anticorrelated roughness) shifts of the

atoms at the interfaces. To accomplish this, let us consider

the effect of removing a single atom at one interface only,

the “top” one to fix the ideas. Accounting also for the fact

FIG. 4. (Color online) Comparison of the (1,1) squared generalized Prange-

Nee matrix elements obtained using the uniform atomic shift, as in Eq. (15)

(labeled “uniform shift”), the displaced average interface potential obtained

by removing one atomic layer, as in Eq. (22) (labeled “average interfacial

potential”), and the interface potentials associated to the insertion or deletion

of one atomic layer, as in Eq. (25) (labeled “interfacial potentials”). Note that

the results differ only by 65%, despite the vastly different assumptions. The

dotted (blue online) line represents the value of the squared matrix element

obtained employing the variational wavefunction multiplied by a factor of 4.5

to align it with the pseudopotential results. While the variational wavefunction

used with Eq. (17) is known to yield inaccurate results, it still provides the

correct dependence of the squared matrix element on the surface field Fs. The

lines connecting the calculated points are only a guide to the eye.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

that, in H-terminated surfaces, this requires shifting two H

atoms to new positions, this will cause a change of the

lattice-averaged potential given by

hDVðtÞðzÞi ¼XGz

eiGzzX

a

e�iGzsaz VðaÞ0Gz�Xa0

e�iGzsa0z Vða0Þ0Gz

" #:

(18)

More generally, we can view this as the lattice average of

the perturbation Hamiltonian caused by this change of the

lattice potential, the Hamiltonian which takes the form

HðSRÞðrÞ ¼XGQ

eiðG�rþQ�RÞ

�X

a

e�iG�saVðaÞG �

Xa0

e�iG�s0aVG

ða0Þ" #

: (19)

In these equations, the index a runs over the atoms added

(that is, the two H atoms in the new positions) and the index

a0 runs over the atoms deleted from the structure (i.e., the Si

atom removed together with the two H atoms saturating its

dangling bonds). The change of the lattice potential

expressed by Eq. (18) can be viewed as caused by a down-

ward shift D ¼ a0=4 of the top interface. Of course, the pres-

ence of random roughness implies, in general, the interface

may shift “downwards,” as we have just assumed, or

“upwards” (which is equivalent to the addition on an atomic

layer). This general case can be handled in two different

ways, which, however, will yield very similar end results, as

we shall see shortly.

Following a first approach, we can simply shift the func-

tion hDVðtÞðzÞi by an amount �a0=8 (that is, upward) to repre-

sent a shift of rms amplitude < jD00j2 >¼ a0=4 of the top

interface. Proceeding similarly for the bottom interface, defin-

ing a function hDVðbÞðzÞi by removing a single Si atom from

the bottom interface (and re-arranging two H atoms as well),

we can define a matrix element, which accounts for the

roughness-induced potential at the two interfaces separately as

follows:

VðSRÞðacÞK0;K;n0;n

� DK�K0

ðdzfðn

0Þ�K0ðzÞ dhV

ðtÞðzÞidz

fðnÞK ðzÞ�

þð

dzfðn0Þ�

K0ðzÞ dhV

ðbÞðzÞidz

fðnÞK ðzÞ�

¼ DK�K0 IðSR;topÞK0;K;n0;n

þ IðSR;bottomÞK0;K;n0;n

h i; (20)

where dhVðtÞðzÞi=dz¼ð4=a0ÞDVðtÞðzþ a0=8Þ and dhVðbÞðzÞi=dz ¼ ð4=a0ÞDVðbÞðz� a0=8Þ are the “top” and “bottom” inter-

facial roughness potentials normalized to the displacement

amplitude a0=4, from which have been calculated and shifted

by amounts 6a0=8 to represent vanishing rms shifts at either

interface. These potentials are shown by the dashed lines,

black online (labeled “removal”) in Fig. 3(b). We see the

similarity with the derivative of the lattice potential shown in

frame (a) of that figure, but now the “peaks” at the two interfa-

ces are fully decoupled. Of course, Eq. (20) represents the

effect of anti-correlated roughness at the two interfaces, since

the interfacial potentials have been obtained by removing one

atom at each interface, so effectively shifting the two interfaces

by equal and opposite amounts. The effect of correlated or

uncorrelated roughness can be obtained defining the matrix

elements

VðSRÞðcÞK0;K;n0;n

¼ DK�K0 IðSR;topÞK0;K;n0;n

� IðSR;bottomÞK0;K;n0;n

h i; (21)

VðSRÞðucÞK0;K;n0;n

�� 2� hjDðtopÞ

K�K0j2ijI ðSR;topÞ

K0;K;n0;nj2 þ hjDðtopÞ

K�K0j2ijIðSR;bottomÞ

K0;K;n0;nj2; (22)

respectively. In the last expression, DðtopÞ (DðbottomÞ) are the

random atomic shifts for the top (bottom) interface. This

expression can be regarded as describing uncorrelated shifts

of spatially averaged amounts hDðtopÞ0 irms and hDðbottomÞ

0 irms at

the two interfaces.

In a second approach, we consider now both the deletion

of an atomic layer, as we have just discussed, as well as the

addition of a new atomic layer. Thus, the perturbation poten-

tial resulting from correlated, anticorrelated, and uncorre-

lated roughness at both interfaces can be expressed as

VðSRÞðcÞK0;K;n0;n

�� 2 ¼ 1

4hjDK�K0 j2i I

ðSR;top;þÞK0;K;n0;n

þ IðSR;bottom;�ÞK0;K;n0;n

�� 2�þ IðSR;top;�Þ

K0;K;n0;nþ IðSR;bottom;þÞ

K0;K;n0;n

�� 2� (23)

for correlated roughness, as


�� 2 ¼ 1

4hjDK�K0 j2i I

ðSR;top;þÞK0;K;n0;n

þ IðSR;bottom;þÞK0;K;n0;n

�� 2�þ IðSR;top;�Þ

K0;K;n0;nþ IðSR;bottom;�Þ

K0;K;n0;n

�� 2� (24)

for anticorrelated roughness, and as


�� 2¼ 1

4hjDðtopÞ

K�K0j2i IðSR;top;þÞ

K0;K;n0;n

�� 2þ IðSR;top;�ÞK0;K;n0;n

�� 2� ��þhjDðbottomÞ

K�K0j2i IðSR;bottom;þÞ

K0;K;n0;n

�� 2�þ IðSR;bottom;�Þ

K0;K;n0;n

�� 2�� (25)

for uncorrelated roughness. In these expressions, the multipli-

cative factor of 1=4 is due to the fact that each matrix element

must be normalized by twice the displacement of the atomic

lines at a single interface, and the form factors IðSR;top;6ÞK0;K;n0;n

and

IðSR;bottom;6ÞK0;K;n0;n

represent integrals of the form of Eq. (20) where

the perturbation potential is given by the difference pseudopo-

tential obtained by inserting (þ) or deleting (�) an atomic

layer at the top or bottom interface, respectively. Note that the

potential resulting from the insertion of an atomic layer is real

and negative, while that resulting from the deletion of an

atomic layer is real and positive. Thus, anti-correlated rough-

ness, for example, is expected to result in the largest matrix

element. The scattering (pseudo)potentials correspond to the



194.69.13.215 On: Mon, 12 May 2014 07:43:05

dashed lines, black online (“removal”) and solid lines, red

online (“addition”) shown in Fig. 3(b).

One major concern arises when abandoning the

perturbation-potential described by the “shifted” scattering

potential, Eq. (8), moving to the atomistic approach, which

employs the scattering potential due to the removal or addi-

tion of atomic line, Eq. (18): The former potential is “weak”

in the perturbation parameter DR, so that perturbation theory

applies for small shifts DR. On the contrary, the difference-

potential given by Eq. (18) is of the same magnitude as the

unperturbed ionic (pseudo)potential. Therefore, the use of

perturbation theory seems to be highly questionable. How-

ever, one should look at the problem with a perspective simi-

larly embraced when using the (first) Born approximation in

dealing with Coulomb scattering with ionized impurities or

fixed charges.

Two criteria must be considered: The strength of the scat-

tering potential and the average density of the scatterers.

Regarding the first criterion, we are at the very edge of the

range of validity of the Born approximation. Assuming the

potential to be localized within a scale-length a � hjD0j2i1=2

comparable to an inter-ionic separation, for low-energy

electrons, ka < 1, we must have jHðSRÞj ¼ jhjD0j2i1=2

ðdhVðlatÞi=dzÞj << Ea, where Ea is the energy of an electron

confined within the length-scale a. Assuming that is due to a

square-well potential, Ea � p2�h2=ð2ma2Þ, which is of the

order of one-to-a-few Rydbergs, depending on the exact shape

of the scattering potential. For high-energy electrons,

ka >> 1, we must have, instead, the jHðSRÞj << �hk=ðmaÞcondition met at sufficiently large energies. So, limiting our-

selves to the more worrisome low-energy range, from Fig. 3,

we see that we are within the range of validity, but not too

comfortably so, depending on the shape of the scattering

potential. While this may indeed worry us, we should note

that we satisfy the conditions of validity of the Born approxi-

mation much more comfortably than when dealing with scat-

tering with the Yukawa-type dielectrically screened Coulomb

potential of an ionized-impurity, VðimpÞ. Indeed, in this case,

a � 1=b, (where b is the Debye-Huckel or Thomas-Fermi

screening wavevector) the static dielectric screening length,

which is usually much larger than hjD0j2i1=2, so that Ea is cor-

respondingly smaller and the condition jVðimpÞj << Ea is met

with even less certainty. Note, finally, that, in principle, we

could bypass these limitations and go beyond the perturbative

approach by considering the full wavefunctions of the per-

turbed system — i.e., with atomic lines added or deleted —

when calculating the scattering matrix elements. However,

this would require a significant computational overhead, since

we should have to compute the full band structure of at least 2

additional systems, a tough computational proposition.

Regarding the density of the scatterers, additional prob-

lems arise at densities large enough to cause concerns about

multiple-centers scattering and interference between succes-

sive collisions. Drawing once more on the analogy with Cou-

lomb scattering with ionized impurities of electron-electron

single-particle collisions, we can consider the Born approxi-

mation valid as long as the interfacial steps are separated by

an average distance (the “correlation length” K) larger than

the electron wavelength or mean-free-path due to additional

collisions, so the coherence over successive SR scattering

events is excluded. Note also that, for large K, the quantity

hjDQj2i1=2becomes smaller, thus relaxing also the conditions

considered in the previous paragraph. However, for very

short correlation lengths, one should employ approaches

which account for the coherence of the electronic wavefunc-

tions over many steps, such as the approaches of Refs. 4 and

5 or Refs. 29–38, which give a full quantum treatment of the

transmission probability across surface or edge steps in the

case of nanowires or graphene nanoribbons, respectively.

The necessity of using these approaches will become even

more evident below, when we will find that the ER-induced

mean-free-path in narrow AGRNs predicted by perturbation

theory may even be much smaller than any reasonable value

of K itself, a clearly inconsistent result.

Finally, concerns may arise also regarding the issue of

unitariety: Considering a perturbation caused by a shift of

the interface resulting in widening the layer by an atomic

layer, we should have

jhfðwide;n0ÞK0

jHðSR;þÞjfðnarrow;nÞK ij ¼ jhfðnarrow;nÞ

K jHðSR;�Þjfðwide;n0ÞK0

ij;(26)

where fðnarrow=wide;nÞK are the wavefunctions (at the same

energy, although here we shall assume that they do not

depend on energy here for discussion’s sake) of the wide or

narrow layer and HðSR6Þ is the perturbation Hamiltonian due

to a step resulting in the widening (þÞ or narrowing (�) of

the layer. The formulation presented here (and its variant dis-

cussed below for nanowires and graphene nanoribbons) sat-

isfies this requirement — which amounts to unitariety —

only in the limit in which the perturbation to the

wavefunctions,

DfðnÞK ¼ fðwide;nÞK � fðnarrow;nÞ

K ; (27)

can be ignored, as it constitutes a second-order contribution

to the matrix element above, Eq. (26). Once more, we should

consider Df negligible for “dilute” roughness-steps after a

suitable spatial average over the interface, so that

hDfðnÞK i � hjDRj2i1=2. Therefore, the treatment followed here

will be valid as long as the correlation length K of the inter-

face or edge roughness is larger than the electronic wave-

length, while it will fail at small K (and, perhaps, at small

carrier energies).

Figure 4 shows that treating correlated roughness via

Eqs. (15), (21), or (23) yields essentially the same result, as

the matrix elements for intra-subband transition (1,1) com-

puted using these different models differ by no more than

65%. Of course, in this case, we do not expect the degree of

correlation at the interfaces to affect the result because of the

strong confinement at the top surface. This effect will be

most evident in thinner films and under flat-band conditions.

Before discussing this case as well as the thickness de-

pendence of the SR matrix elements, we should make two

observations, which we will emphasize again in the follow-

ing: First, even considering a fully “symmetric” quantum



194.69.13.215 On: Mon, 12 May 2014 07:43:05

well, at the atomistic level, no full inversion symmetry can

be obtained; regardless of how the two interfaces are termi-

nated, no symmetry under inversion around the axis of the

well can be obtained. As a consequence, the wavefunctions

— as well as the perturbing potentials — will not yield the

selection rules, which one would naıvely expect in an

effective-mass context. This is of course related to the

valley-splitting mentioned above. Second, as we analyze the

thickness dependence by varying the number of atomic

planes, we should expect some “noise” arising from the ab-

rupt variation of the atomic (pseudo)potential. In Si, the issue

is further complicated by the alternating symmetry exhibited

by the quasi-twofold degenerate subbands. Similarly, the

thickness ts of the film cannot be defined rigorously (this will

be an ever bigger issue when attempting to define the radius

of a cylindrical nanowire). Here, we have taken it as the

number of cells times the cell thickness, Nca0, but we should

keep in mind the presence of H atoms and of an “interfacial”

region, over which the atomic potential varies significantly.

In Fig. 5, we show the squared matrix element — calcu-

lated using Eqs. (23), (24), and (25) for the various corre-

lated, anticorrelated, and uncorrelated cases, respectively —

for SR-induced processes as a function of the thickness of an

H-terminated free-standing Si layer at flat-band conditions.

Only the four lowest-energy subbands at the �C-point are con-

sidered. The top (a), middle (b), and bottom (c) frames of

Fig. 5 are relative to the case of correlated, uncorrelated, and

anti-correlated roughness at the two interfaces. The top

frame shows that the ð1; 2Þ inter-subband transition yields a

non-negligible matrix element, as expected, since uncorre-

lated or anticorrelated roughness is required to trigger intra-

subband even-even or odd-odd processes. The surprising

presence of significant ð1; 1Þ and ð2; 2Þ matrix elements is

indeed exclusively due to ð1þ; 1�Þ and ð2þ; 2�Þ even-odd

processes, while the even-even and odd-odd ðn�; n�Þ and

ðnþ; nþÞ processes exhibit a vanishing matrix element. The

completely opposite situation is seen in the bottom frame:

Anticorrelated roughness allows only even-even and odd-

odd intra- and inter-subband processes, even-odd mode-

mixing (i.e., inter-subband transitions involving states of

opposite symmetry) being forbidden. Thus, the presence of

the ð1; 1Þ and ð2; 2Þ processes is due exclusively to

ð1�; 1�Þ, ð1þ; 1þÞ, ð2�; 2�Þ, and ð2þ; 2þÞ processes,

while the ð1�; 1þÞ and ð2�; 2þÞ transitions are forbidden.

Similarly, ð2; 1Þ processes arise exclusively from the even-

even or odd-odd ð2þ; 1þÞ and ð2�; 1�Þ transitions, since

the even-odd ð1þ; 2�Þ and ð1�; 2þÞ transitions are forbid-

den, in complete contrast to the case of correlated roughness.

In all frames, note how the well-known t�6s dependence

of the process is approximately reproduced, as expected also

from the expression28

VðSRÞK;K;n;n

Drms� dEn

dts� � �h2p2

mLt3s

; (28)

which is shown in Fig. 5 as dashed lines (obtained using

mL ¼ 0:91m) for the ð1; 1Þ and ð2; 2Þ intrasubband processes.

A dependence on layer-thickness slightly slower than t�6s is

actually to be expected from the strong nonparabolic effects

arising at high energies (so, especially in thinner films) and

from the fact that, at lower ts, the energy of the eigenstate

grows substantially, thus causing an increased spreading of

wavefunctions inside the vacuum padding region and, thus, a

reduced quantum confinement.

D. Numerical evaluation

The numerical evaluation of the surface-roughness scat-

tering rate can be performed in the standard way: The

Golden-Rule (or, better, Born-approximation) expression for

the scattering rate will be

1

sðSRÞn ðKÞ

¼ 2p�h

XK0n0

VðSRÞK0;K;n0;n

�� 2d EnðKÞ � En0 ðK0Þ½ �: (29)

FIG. 5. (Color online) Squared generalized Prange-Nee matrix element —

normalized to the rms atomic displacement — as a function of layer-

thickness, ts, calculated from Eqs. (23), (24), and (25) for correlated (a),

uncorrelated (b), and anti-correlated (c) roughness at the two interfaces of an

H-terminated, free-standing Si layer at flat-band conditions. A large number

of cells (20 total) have been employed for all values of the Si thickness ts. The

lines connecting symbols are only a guide to the eye, while the dashed (purple

online) lines correspond to the analytical results obtained using Eq. (28) for

the ð1; 1Þ and ð2; 2Þ intrasubband transitions, as indicated in the top frame.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

In order to perform the integration over the constant-energy

surface, we can follow the algorithm proposed by Gilat and

Raubenheimer39 adapted to a two-dimensional integration.

Thus, we first discretize the 2D BZ into squares of sides DKcentered at points Kj. We then calculate the energy Ejn and

the gradient r2DEjn at the center of the square for each band

n. Then, the density of states at energy E for band n in the jthsquare can be expressed as ½1=ð2pÞ2�LðwjnÞ=r2DEjn. Here,

w ¼ ðE� EjnÞ=jr2DEjnj is the distance in k-space between

the center of the square and the equienergy surface at energy

E along the direction of r2DEjn and LðwÞ is the length of the

segment perpendicular to r2DEjn intersecting the square at a

distance w away from the center. Only squares spanning the

required energy E (that is, �w1 w < w1 with terms defined

below) will contribute to the sum, and, for these squares, the

length LðwÞ is given by (assuming without loss of generality

axes rotation such that cos a> sin a)

LðwÞ ¼

DK

cos aðw w0Þ

w1 � w

cos a sin aðw0 w w1Þ

8>><>>: ; (30)

where w0¼ðDK=2Þðcosa�sinaÞ, w1¼ðDK=2ÞðcosaþsinaÞ,a is the angle between the Ky-axis and r2DEjn. Employing

this algorithm, Eq. (29) can be evaluated numerically as

1

sðSRÞn ðKÞ

¼ 2p�h

Xjn0

0VðSRÞKj;K;n0;n

�� 2 1

ð2pÞ2Lðwjn0 Þjr2DEjn0 j

; (31)

the “primed sum” indicates that only “energy-conserving”

squares contribute. The evaluation of matrix element VðSRÞK0;K;n0;n

presents only limited difficulties: The generalized Prange-Nee

term in the form given by Eq. (15), for example, involves only

the numerical evaluation of a one-dimensional overlap inte-

gral for each pair (K;K0).Ignoring Coulomb-related terms, Fig. 6 shows the gener-

alized Prange-Nee component of the SR scattering rate in an

11-cell-thick Si layer with the additional confinement of a par-

abolic potential: At small electron kinetic energies, the con-

finement effect of the field is significant, since the scattering

rates in the first subband increases significantly when the sur-

face field increases from 106 to 2:5� 106 V/cm. At larger

energies, the larger density of states at lower fields and geo-

metric confinement (since the wavefunctions “extend” more

and more toward the bottom interface as their energy

increases) dominate and the rate becomes weakly dependent

on the field.

Note that, here and in the following, we shall not discuss

the low-field carrier mobility, since this requires the calcula-

tion of the relaxation rates due to additional processes, such

as scattering with phonons (possibly confined), the effort of

which goes beyond the scope of the present work.

III. ONE-DIMENSIONAL TRANSPORT

In this section, we discuss scattering caused by rough-

ness at the free surfaces of nanowires or at the edges of gra-

phene nanoribbons, which can be treated following an

approach similar to what we have discussed in Sec. II.

AGNRs are of interest because of their potential technologi-

cal applications enabled by the chirality dependence of the

gap40 and also because they illustrate the outcome of the

method presented here in case in which roughness-induced

scattering takes is of a unique flavor. Si NWs, of course, are

of interest because gate-all-around Si NW field-effect tran-

sistors (FETs) provide excellent electrostatic control at the

10-nm gate length41 and roughness can present a significant

challenge if we want to maximize their performance.

A. Armchair-edge graphene nanoribbons

Let us consider first the case of armchair-edge graphene

nanoribbons with axis along the z direction, cross section on

the ðx; yÞ plane, and width W along the y axis fixed by the

number Na of atomic lines.

In Fig. 7, we show the band structure of a 9-AGNR

(Na ¼ 3p, where p is an integer), a 10-AGNR (Na ¼ 3pþ 1),

and an 11-AGNR (Na ¼ 3pþ 2) obtained using the C and H

local empirical pseudopotentials proposed by Kurokawa etal.42 This figure shows directly the energy gap for the three

chiral “ladders” considered in Fig. 8. These results are in

good quantitative agreement with DFT results, which ignore

many-body corrections,22 while GW corrections predict the

same qualitative trends, but yield even larger values for the

energy gap.43 Note that nearest-neighbor empirical tight-

binding models based on a single pz orbital predict qualita-

tively incorrect gaps and an incorrect ordering of the three

ladders.22 It should be remarked that experiments44 have not

FIG. 6. (Color online) Surface-roughness scattering rates (Prange-Nee com-

ponent only) for electron in the inversion layer of an 11-cell thick Si layer

with a 4-cell thick vacuum “insulator” and an external parabolic potential

with a surface field of 106 (dashed line, blue online) and 2:5� 106 V/cm

(solid line, black online). Energies are measured from the bottom of the

ground state subband, and 8 subbands have been considered. Note that, at

the low carrier energies which determine the low-field carrier mobility, the

scattering rates take values in the range of 1013/s, as expected. Note also the

effect of the field at low energy — a higher surface field inducing a larger

rate — and the major effect of the density of states and of the geometric

confinement at larger energies yielding a very high rate — casting doubts on

the validity of perturbation theory — which becomes almost independent of

the confining potential. Uncorrelated roughness at both interfaces has been

considered together with a quasi-exponential correlation spectrum of the

form given in Eq. (3) with n¼ 2, as suggested in Ref. 16.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

confirmed this chirality dependence of the bandgap. Querlioz

and co-workers have attributed this to disorder or, equiva-

lently, to line-edge roughness and disorder of the chirality.45

Tseng et al.29 have similarly argued that the clustering of the

gap around values corresponding to the largest 3pþ 1 gap is

the result of line edge roughness, which allows electron

transmission only at the largest “local” gap.

In complete analogy to Sec. II, we now describe the

roughness by a shift of the atoms along the y direction by an

amount �Dz varying along the length of the ribbon,

described by its Fourier decomposition,

Dz ¼X

q

Dqeiqz: (32)

We shall assume for its power spectrum hjDqj2i a Gaussian

or exponential form similar to Eqs. (2) or (3), respectively,

accounting, however, for the lower dimensionality. Thus, for

the exponential case, we have

hjDzj2i ¼ D2e�jzj=ðffiffi2p

KÞ ! hjDqj2i¼

ffiffiffi2p

D2Kð1þ q2K2=2Þ�1: (33)

The displacement of the atoms modifies the potential as

follows:

VðlatÞðrÞ ¼X

a

VðaÞðr� saÞ

¼X

a

XG

VðaÞG eiG�ðr�saÞ

¼X

G

VðlatÞG eiG�r ! VðlatÞðr þ yDzÞ

¼X

G

eiGyDz VðlatÞG eiG�r �

XG

ð1þ iGyDzÞVðlatÞG eiG�r

¼X

G

VðlatÞG eiG�r þ i

XqG

DqVðlatÞG GyeiðG�rþqzÞ (34)

to the first order in the atomic displacement, so that the per-

turbation Hamiltonian caused by the edge roughness can be

expressed as

HðERÞðrÞ ¼ iXqG

DqVðlatÞG GyeiðG�rþqzÞ: (35)

Defining the following wavefunctions analogous to Eq. (10),

nð2DÞðnÞGz;kz

ðRÞ ¼ 1

A1=2c

XGk

uðnÞG;kz

eiGk�R (36)

(where Ac is the area of the supercell on the cross-sectional

plane), the matrix element associated with a transition from

a state in band n and wavenumber kz to a state in band n0 and

wavenumber k0z can be written as

VðERÞkz;k0z;n;n

0 ¼ iX

GzG0zG00z

XGk

GyDkz�k0zþGz�G0zþG00Z

VðlatÞGk;G

00z

�ð

dR nð2DÞðn0Þ�G0z;k

0zðRÞ eiGk�R nð2DÞðnÞ

Gz;kzðRÞ: (37)

With the usual reshuffling of the dummy summation wave-

vectors, we can rewrite this expression in the more conven-

ient form

FIG. 7. Band structure of (a) an Na¼ 9, (b)

an Na ¼ 1 (b), and (c) an Na ¼ 11 graphene

nanoribbon with armchair edges with H ter-

mination of the edge C atoms. The exten-

sion of the supercell along the axial zdirection is az ¼ 3aC, where aC is the C–C

bond length � 0.142 nm. The variation of

the energy gap for the various cases

Na ¼ 3pþ 1 and Na ¼ 3pþ 2 compared to

the Na ¼ 3p can be easily seen.

FIG. 8. (Color online) Dependence of the bandgap of AGNRs on ribbon-

width. The lines connecting the calculated points are only a guide to the eye.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

VðERÞkz;k0z;n;n

0 ¼ iX

GzG0zG00z

XGk

Gy Dkz�k0zþG0z VðlatÞGk;G

00z

ðdRnð2DÞðn0Þ�

Gz�G0zþG00z ;k0z

�ðRÞeiGk�R nð2DÞðnÞGz;kz

ðRÞ

¼ iX

GzG0zG00z

XGkG

00k

GyDkz�k0zþG0z VðlatÞG u

ðn0Þ�GkþG

00k ;Gz�G0zþG00z ;k

0z

uðnÞG;kz

¼XG0z

Dkz�k0zþG0zCð2DÞðGPNÞkz;k0z;n;n

0;G0z: (38)

Recognizing the fast decay of Dq with increasing q, we can

ignore Umklapp terms with G0z 6¼ 0, so that the matrix ele-

ment simplifies to (renaming G00 ! G0)

VðERÞkz;k

0z ;n;n

0 � iDKz�k0z

XGzG

0z

XGk

GyVðlatÞGk;G

0z

�ð

dRnð2DÞðn0Þ�GzþG0z;k

0z

ðRÞeiGk�rnð2DÞðnÞGz;kz

ðRÞ

¼ iDKz�k0z

XGG0

GyVðlatÞG u

ðn0Þ�GþG0;k0z

uðnÞG0;kz

¼ DKz�k0zCð2DÞðGPNÞ

kz;k0z ;n;n

0;0: (39)

Defining wavefunctions fð2DðnÞkz

in analogy to Eq. (13) as

fð2DÞðnÞkz

ðRÞ ¼ 1

A1=2c

XGk

uðnÞGk;Gz¼0;kz

eiGk�r; (40)

we reach an approximated form convenient for numerical

evaluation,46

VðERÞkz;k

0z ;n;n

0 � iDKz�k0z

XGk

GyVðlatÞGk;Gz¼0

�ð

dRfð2DÞðn0Þ�k0z

ðRÞeiGk�rfð2DÞðnÞkz

ðRÞ

¼ iDkz�k0z

XGk

GyVðlatÞG

~Ið1DÞkz;k0z;n;n

0 ðGkÞ; (41)

having used in the last step the expression for the overlap

factor

~Ið1DÞkz;k0z;n;n

0 ðQÞ ¼ð


ðRÞeiQ�rfð2DÞðnÞkz

ðRÞ: (42)

Note that, in simpler models which ignore Bloch-functions

effects,47,48 the reciprocal-lattice vectors disappear and the

scattering rate for edge-roughness-induced processes van-

ishes for inter-subband transitions, since, whenever only one

final state k0z ¼ �kz is assumed to exist, ~Ið1DÞkz;�kz;n;n0

ð0Þ ¼ dn;n0 .

The expression above, Eq. (41), represents the case of

roughness correlated at both edges. As we have noticed above

discussing the case of thin films, such a correlated roughness

(representing a “snaking” ribbon with constant width), can

induce only a moderate mode-mixing, but no intra-subband

transitions. Therefore, we should consider the more interesting

case of anti-correlated and uncorrelated edge roughness.

These cases can be treated in a way similar to the approach,

which had led us to Eqs. (23)–(25), as demanded by the fact

that the addition or the deletion of an atomic line in an AGNR

can have a dramatically different effect in view of the

chirality-dependence of the AGNR band-gap (see Fig. 8).

This chirality dependence also makes us expect a “noisy”

width dependence, since, for each different ribbon, a change

of width will cause a dramatically different change of the sub-

band energy. More importantly, questions should be raised

about the validity of the perturbation theory (the Born and

independent-collisions approximation) we are employing,

since large matrix elements will imply scattering lengths

shorter than the electronic wavelength, even in the presence of

“dilute” roughness. As we shall see from the large magnitude

of the matrix elements, a correct approach — already men-

tioned above — would have to rely on the calculation of the

transmission probability across a ribbon of varying width, as

done by Tseng et al.,29 by Betti and co-workers,30 and

others,31–35,37,38 and even in the context of GNR-based FETs

by Luisier and Klimeck.36 Therefore, our results should be

interpreted as providing a qualitative trend and clearly indicat-

ing most definitely that LER scattering plays a huge role in

controlling electronic transport in AGNRs, but their quantita-

tive correctness — and especially their applicability to trans-

port calculations — should be questioned.

Here, we consider the change of the lattice (pseudo)po-

tential caused by removing (�) or adding (þ) only a single

line of C atom (and terminating H atoms) from the left and

right edges, obtaining scattering potentials hDðleft;6ÞðRÞi and

hDðright;6ÞðRÞi given by expressions similar to Eq. (18). We

ignore the case of abrupt variations of the AGNR-width by

more than a single atomic line at each edge, since these

would correspond to (hopefully) unrealistically large values

of hjDq¼0j2i1=2. Thus, the matrix element associated to the

roughness at the two edges can be written as

VðERÞðcÞk0z;kz;n0;n

�� 2¼ 1

4hjDKz�k0z j

2i IðER;left;þÞk0z;kz;n0;n

þ IðER;right;�Þk0z;kz;n0;n

�� 2�þ IðER;left;�Þ

k0z;kz;n0;nþ IðER;right;þÞ

kz;kz;n0;n

�� 2� (43)

for correlated roughness, by

VðERÞðacÞk0z;kz;n0;n

�� 2¼ 1

4hjDKz�k0z j

2i IðER;left;þÞk0z;kz;n0;n

þ IðER;right;þÞk0z;kz;n0;n

�� 2�þ IðER;left;�Þ

k0z;kz;n0;nþ IðER;right;�Þ

k0z;kz;n0;n

�� 2� (44)

for anti-correlated roughness, and by

VðERÞðacÞk0z;kz;n0;n

�� 2¼ 1

4hjDðleftÞ

Kz�k0zj2i IðER;left;þÞ

k0z;kz;n0;n

�� 2þ IðER;left;�Þk0z;kz;n0;n

�� 2� ��þhjDðrightÞ

Kz�k0zj2i IðER;right;þÞ

k0z;kz;n0;n

�� 2þ IðER;right;�Þk0z;kz;n0;n

�� 2� ��(45)

for uncorrelated roughness. In these expressions, the form fac-

tors IðER;left;6Þk0z;kz;n0;n

and IðER;right;6Þk0z;kz;n0;n

represent integrals of the form



194.69.13.215 On: Mon, 12 May 2014 07:43:05

IðER;lr;6Þk0z;kz;n0;n

¼ð


ðRÞ dhVðlr;6ÞðRÞi

dxfð2DÞðnÞ

kzðRÞ; (46)

where the perturbation potentials dhVðlr;6ÞðRÞi=dx are given

by the difference pseudopotentials obtained by inserting (þ)

or deleting (�) an atomic layer at the left (l) or right (r) edge,

respectively, where, exactly as discussed after Eq. (18),

dhVðleftÞðRÞi=dx¼ ½2=ffiffiffi3p

aCÞ�DVðleftÞðzÞ and dhVðrightÞðRÞi=dx¼ ½2=

ffiffiffi3p

aCÞ�DVðleftÞðzÞ are the “left” and “right” edge rough-

ness potentials normalized to the displacement amplitudeffiffiffi3p

aC=2 from which have been calculated. Here, aC is the

C–C bond length, � 0.142 nm. These potentials are shown in

Fig. 9. The contour plots of the ground-states conduction-band

wavefunctions are shown in the top (a) and third frames (c) to-

gether with the contour plots of the scattering potentials (only

the potential due to the deletion of an atomic line is shown for

clarity) for a 7-AGNR and an 11-AGNR, respectively. They

can be viewed as 2D “cross-sections” of the nanoribbons. The

second frame from the top (b) and the bottom frame (d) show

the same quantities, but averaged over the thickness of the rib-

bon. Both potentials (resulting from the insertion or deletion

of atomic line) are shown in this case. Comparing these

frames — in which the “thickness-averaged” wavefunctions

are plotted, retaining the relative normalization ratio — we

can see how the overlap between the ER-induced scattering

potential and the wavefunctions is reduced as the width of the

ribbon increases. Note also — comparing the 2D and the 1D

views of the ER potential and wavefunction — that two-

dimensional effects are quite significant.

Figure 10 shows the squared matrix element as a func-

tion of the width of the nanoribbon for armchair-edge rib-

bons for the case of uncorrelated roughness. For wide

AGNRs, we expect a W�4 dependence, weaker than the t�6s

dependence observed in thin films. This dependence is

expected from the fact that, in a simple analytical approxi-

mation, the dispersion of the ribbons can be expressed as

EnðkzÞ � �htF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

n þ k2z

q; (47)

where tF is the Fermi velocity in graphene and kn is the

quantized wavenumber along the width of the ribbon for

band n given by gnp=W, where g is a number of the order of

unity, which varies depending on the model chosen.37,47–49

If additional effects (to be discussed momentarily) are

ignored, in analogy with Sasaki’s expression, we expect for

the matrix element due to line-edge roughness

VðERÞðucÞ0;0 � dEnðkz ¼ 0Þ

dW� � g�htFpn

W2; (48)

hence the W�4 dependence for the squared matrix element.

This is indeed observed for the (2,2) transitions (Fig. 10(b))

for 3p-wide AGNRs at the largest W we have considered,

but it does not appear to be a general feature as a result of

the chirality-dependence of the AGNR bandgap not captured

by the simple model, Eq. (47).

Indeed, two features stand out in Fig. 10: First, the

“noise” observed, especially at small W, and, second, the

magnitude of the squared matrix element, some two orders

of magnitude larger than the magnitude (of the order of

1 eV/nm for W¼ 1 nm) expected from Eq. (48). Both fea-

tures can be understood, looking back at Fig. 8. Since, at

the atomic level, the edge roughness stems from a fluctuat-

ing edge affected by the removal or addition of an atomic

line, the gap, and so the energy of each kz state is not cor-

rectly described by Eq. (47), but “jumps” quite suddenly as

the number of atomic lines along the width of the ribbons

FIG. 9. (Color online) (a) Contour plot of the cell-averaged squared ground-

state conduction-band wavefunction (thin lines, black online) and LER scat-

tering potentials due to the insertion of an atomic line at the left (thick blue

lines) and right (thick contour lines, black online) edges for a 7-AGNR.

Open circles underneath the contour lines represent C atoms, solid circles

(partially hidden by the contours of the ER potential) H atoms. (b) The same

quantities plotted in the top frame, but now averaged over the thickness of

the ribbon. In this case, both scattering potentials due to the insertion (solid

lines) or deletion (dashed lines) of an atomic line have been plotted. (c), (d)

As in the top two frames, but for an 11-AGNR.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

jumps among 3p, 3pþ 1, or 3pþ 2 (although this effect may

be masked by disorder45), hence the large fluctuations —

actually, the chirality dependence — and the large magnitude

seen in Fig. 10. This effect has not been considered in

previous mobility studies based on a “smooth” analytic band-

structure and associated roughness models,47,48 but is obvi-

ously implicitly present in atomistic approaches,30,36,37,50–52

and it may explain the low electron mobility seen experimen-

tally in narrow ribbons,54,55 much lower than in large gra-

phene sheets.56 Thus, perfect control of the nanoribbon width

— with whatever process one may envision, such as bottom-

up synthesis57 or chemically “unzipping” carbon nanotubes

(CNTs)58,59 — may be necessary in order to obtain AGNRs

with good transport properties, at least in the diffusive (long

GNRs) regime. How crucial our observations really are in

practice is still hard to assess. Indeed, we should add a brief

“disclaimer” stressing once more the (perhaps) obvious fact

that our discussion is valid in the diffusive regime. On the

contrary, in the quasi-ballistic regime, the quantum-

transmission approaches of Refs. 29–35, 37, and 38 are to be

preferred also, because, as mentioned above, electronic trans-

port in rough (or disordered) GNRs may be affected by vari-

ous confinement effects, depending on the electron coherence

length,56 by Anderson localization,33,35 as discussed in the

review article by Cresti et al.,60 or by transport among quan-

tum dots formed by lithographically induced ER61 or induced

by nearby charged impurities.62,63 Finally, while we have con-

sidered free-standing ribbons, we should keep in mind possi-

ble interactions with the substrate, which may give rise to

strong effects related to remote coupling with polar substrate

phonons.64,65

Curiously, relatively smaller matrix elements are

observed for AGNRs of width given by a small odd integer

of the form 3pþ 1. This is definitely against the expectations

we could draw from Ref. 29. There, it is shown that, in

AGNRs of varying width, good transmission occurs only for

states at energy corresponding to the ground-state energy of

the nearest sector with 3pþ 1 width (which exhibits the larg-

est gap). Thus, we should expect ð3pþ 1Þ AGNRs to exhibit

the smallest matrix elements. On the contrary, we find that

the two narrowest odd-ð3pþ 2Þ AGNRs we have considered

(Na¼ 5 and 11) exhibit the smallest overlap between the per-

turbing potential and the electron wavefunction. These rib-

bons are characterized by the smallest gap, like all ð3pþ 2ÞAGNRs, and also by their full inversion symmetry at the

atomic level:60 AGNRs of even width do not exhibit inver-

sion symmetry about their axial direction and only odd-

ð3pþ 2Þ AGNRs exhibit this symmetry fully (at the A and Bsublattice level). For example, the edge C atoms and the cen-

ter C atom along a cross-section of the ribbon belong to the

same sublattice in this case. This allows the wavefunctions

of states in the first conduction band to be fully symmetric

with small amplitude at the edges, as in the naıve effective-

mass sine-like approximation. This is seen in Fig. 9(d). This

minimizes the overlap integral with the roughness-induced

difference-potential, leading to a small matrix element. We

are unable to provide any deeper physical explanation for

this surprising observation.

Finally, the ER scattering rate can be evaluated using a

1D discretization analogous to the 2D expression of Eq. (31),

1

sðERÞn ðkzÞ

¼ 2p�h

Xjn0

0 1

2pdEn0j

dkz

�� 1

VðERÞkzj;kz;n0;n

�� 2; (49)

and similarly for the velocity relaxation rate,

1

sðERÞnt ðkzÞ

¼ 2p�h

Xjn

0

12p

dEn0j

dkz

�� 1

VðERÞkzj;kz;n0;n

�� 2 1� tn0j

tnðkzÞ

� �: (50)

This requires a discretization of the 1D BZ into segments of

length Dkz labeled by an integer j centered around the wavenum-

ber kzj, with central energy Enj given by the dispersion EnðkzjÞin band n. We denote the central derivative dEnðkzjÞ=dkzj

by dEnj=dkz and group velocity by tnj ¼ dEn0j=dkz=�h.

The “primed” sum means that only energy-conserving

segments (i.e., such that En0j � ðdEn0j=dkzÞDkz=2 EnðkzÞ< En0j þ ðdEn0j=dkzÞDkz=2) should be considered. Note that, in

FIG. 10. (Color online) Squared magnitude of the generalized Prange-Nee

edge-roughness matrix element at the zone-center in the case of uncorrelated

toughness for the (1,1) (top frame) and (2,2) (bottom frame) matrix ele-

ments, emphasizing the chirality dependence. For the (1,1) matrix element, a

much different power-law dependence is observed for each different chiral-

ity of the ribbons. The two narrowest odd-(3pþ 2)-AGNRs exhibit the

smallest matrix element. For the (2,2) matrix element, roughly the same

power-law is observed. These differences can be attributed to the chirality

dependence of the band-gap of AGNRs. Indeed, in the bottom frame, one

can clearly see the quasi-periodic oscillation of the matrix element as we

cycle through 3p-, (3pþ 1)-, and (3pþ 2)-wide AGNRs.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

the sum appearing in Eq. (49), the initial state itself must be

excluded, since this process amounts to a first-order self-energy

renormalization of the electron dispersion. (The fact that this ob-

servation matters is specific to 1D transport, since 3D or 2D scat-

tering to the initial state itself constitutes a zero-measure volume

of k-space and gives no contribution to the total rate.) Also, the

“microscopic” velocity relaxation rate defined by Eq. (50) —

different from the relaxation rate used when performing mobility

calculations — may take negative values in the presence of

strong forward inter-subband scattering to states with larger

group velocity. This is a frequent occurrence for 1D processes

with the matrix element decreasing with increasing wavenumber

transfer, which is the case here because of the dependence of

hjDqj2i on q. While not shown explicitly below, this indeed hap-

pens in AGNRs at large energies and in Si NWs even at small

energies.

Figure 11 shows the uncomfortably large magnitude of

the generalized Prange-Nee component of the LER scatter-

ing rates (top) or velocity relaxation rates (bottom) in 7-, 13-,

and 19-AGNRs, the effect already discussed above. Since a

quantitative estimate of the scattering rates is now available,

we can rephrase the discussion — and present our concerns

— in more precise terms: From Fig. 2 (or even Fig. 14, to be

discussed below), we see that a change D � 0.1 nm of the

thickness (or diameter) of Si layers (or NWs) causes a

change of the bottom of the conduction band DEc � 0.1 eV.

On the contrary, for AGNRs, the chirality dependence of

their energy gap implies that a change of width of a similar

D due to the addition or deletion of a single atomic line

causes a change DEc one order of magnitude larger, hence in

turn, ER matrix elements and scattering rates two orders of

magnitude larger, as indeed shown in Fig. 11. From this per-

spective, zigzag-edge ribbons are not expected to suffer so

severely, as indeed found by Cresti and Roche.38 As already

mentioned, the large magnitude of these scattering rates

clearly presents difficulties in translating these results into

measurable properties of electron transport: Even assuming

the very high Fermi velocity tF � 108 cm/s of carriers near

the graphene Dirac point, electrons would not survive farther

than � 0.l nm for sðERÞ � 10�16 s, a distance even shorter

than any reasonable ER correlation-length K. While one may

improve the confidence in these perturbation theory results

by going beyond the first Born approximation (for example,

by employing as final “scattered” wavefunctions those calcu-

lated by considering ribbons of width different from those

used to compute the initial “incident” waves), the matrix ele-

ments are likely to remain extremely large. This is a clear

symptom of the fact that the extended Bloch waves of the

homogeneous system cannot be employed reliably to deal

with electronic transport in narrow and rough AGNRs — as

implied by the ridiculously short lifetimes shown in Fig. 11

— and quantum-transport simulations of the full inhomoge-

neous system become a necessity. Nevertheless, if we were

willing to take these quantitative results seriously, from the

magnitude of the rates shown in this figure, we could crudely

estimate an ER-limited electron mobility lERÞ in narrow

AGNRs of the order of 1 cm2/V s or even less, assuming

naıvely lðERÞ � esðERÞt =m�, (m� � 0:1m being the effective

mass in the lowest-energy band of AGNRs with width in the

range of 2 to 3 nm53), in rough qualitative (and probably ac-

cidental) agreement with recent experimental observations.54

So, while clearly the numerical values shown in Fig. 11 can-

not be taken seriously, nevertheless, they tell us a strong

message which confirms, from our different perturbative per-

spective, the dramatic importance of edge disorder in

AGNRs,37,61–63 with the likely implication that transport in

narrow and rough AGNRs does not occur via extended

states. Therefore, it appears that the optimistic statements

made in Ref. 47 regarding the “robustness” of GNRs with

respect to ER scattering — based on Eq. (47) and the ques-

tionable assumption that only intra-subband transitions are

allowed — are not supported by our approach, at least in the

FIG. 11. (Color online) (a) Line-edge-roughness scattering rate (Prange-Nee component only) as a function of electron energy (measured from the bottom of

the conduction band) in three H-terminated armchair-edge GNRs with width given by Na ¼ 3pþ 1 atomic lines (Na ¼ 7, 13, and 19). A total number of 6 sub-

bands have been employed together with an exponential autocorrelation. Note the very large scattering rates originating from the chirality-dependence of the

gap shown in Fig. 8. (b): As in the top frame, but showing the velocity relaxation time. Note that, for the 7-AGNR at large energies, the relaxation rate

increases as a result of inter-subband scattering to the lower-velocity, higher density-of-states second subband. Scattering and relaxation rates so large obvi-

ously bear implications on the electron mobility of narrow AGNRs, but their magnitude casts doubts on the suitability of perturbation theory (Born approxima-

tion) and, more appropriately, of employing Bloch states to deal with this issue.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

diffusive regime. On the contrary, what is most worrisome is

the fact that the “slower” width dependence shown in Fig. 10

for any chirality, if maintained at widths larger than what we

can handle here, implies that ER-scattering may remain very

strong — stronger than in NWs or thin films of similar dimen-

sions and roughness — also in wider ribbons.

B. Circular cross-section nanowires

Considering now cylindrical Si nanowires, Fig. 12

shows the band structure and density of states of an

H-terminated [100] Si NW with diameter D of 2 nm. The

same local empirical pseudopotentials from Ref. 23 have

been employed, and spin-orbit interaction has been

neglected. The squared amplitudes of the wavefunctions

of the lowest-energy conduction-band states for the 2-nm-

diameter NW are illustrated in Fig. 13. Note that, in these

small wires, the ground state wavefunctions have polar

symmetry (being characterized by an angular momentum

quantum number l ¼ 0), but the first excited state has quad-

rupole symmetry (l ¼ 2). This is due to the fact that the

states at C originate from a superposition of states in the 4

bulk ellipsoidal equienergy surfaces with transverse and

longitudinal masses along the principal x and y directions.

The smaller energy thus corresponds to those states whose

“lobes” probe the larger longitudinal mass by extending

along the positive and negative x and y directions, resulting

in the fourfold symmetry of the lattice and so in s-wave

envelopes. On the contrary, the dipole-like (l ¼ 1) states

result from states whose lobes probe the smaller transverse

mass, thus yielding a larger kinetic energy. Finally, the de-

pendence of the bandgap as a function of nanowire diame-

ter is shown in Fig. 14. The expected D�2-dependence is

seen, deviations at the smallest diameters being caused by

the finite confinement potential.

In order to account for roughness at the surface of the

wire, we consider an atomic shift along the radial directionbR by an amount �D/;z function of the position on the surface

of the wire characterized by the axial and angular coordi-

nates z and /, respectively, employing cylindrical coordi-

nates. We can decompose the displacement D/;z into its

Fourier components,

D/;z ¼Xl;q

Dl;qeiðl/þqzÞ; (51)

and assume an exponential power spectrum of the form14,66

hjDl;qj2i ¼D2K2

2Rs1þ K2

2

l2

r2s

þ q2

� �� 3=2

; (52)

where Rs is the radius of the nanowire and D and K are, as

usual, the roughness rms amplitude and correlation length.

Note that, while this expression is valid only for 2pRs >> K,

in the following, we shall use it indiscriminately in all situa-

tions, since our interest here is not so much on the micro-

scopic nature of the roughness itself, but on the effect of the

roughness on electronic transport. Then, under such a shift,

the change of the lattice potential — and so the perturbation

Hamiltonian — will be

FIG. 12. Band structure of a [100] cylindrical H-terminated Si nanowire

with diameter of 2 nm. The extension of the supercell along the axial z direc-

tion is az ¼ a0, where a0 is the bulk Si lattice constant.

FIG. 13. Contours of the square amplitude of the wavefunctions of the 6

lowest-energy conduction-band states for the 2-nm-diameter, H-terminated

[100] Si NWs of the previous figures. The angular momentum quantum

number l indicates the most significant lowest-l component of each

wavefunction.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

HðSRÞðRÞ ¼ dVðlatÞðRÞ � iX

G

Xl;q

G � bRDl;qeiðl/þqzÞeiG�r;

(53)

where bR ¼ R=R is the unit vector along the radial direction.

Employing the wavefunctions nð2DÞðnÞGz;kz

ðR;/Þ (expressed in

polar coordinates), the scattering matrix element can be writ-

ten as follows:

VðSRÞk0z;kz;n0;n

¼ iX

l

XGz;G0z;G

00Dl;kz�k0zþG0z V

ðlatÞG00 G00k

ð2p

0

d/cosð/� hG00 Þeil/

�ð

dRRnð2DÞðn0Þ�Gz�G0zþG00z ;k

0zðR;/ÞeiG00kRcosð/�hG00 Þnð2DÞðnÞ

Gz;kzðR;/Þ

¼X

l

XG0z

Dl;kz�k0zþG0zCðGPNÞðlÞk0z;kz;n0;n;G0z

;

(54)

where we have expressed Gk � bR as Gk cosð/� hGÞ and hG

is the polar angle of Gk, hG ¼ acosðGx=GkÞ. Considering, as

usual, only N processes (i.e., retaining only the term G0z ¼ 0

in the sum above),

VðSRÞk0z;kz;n0;n

� iX

l

Dl;kz�k0z

XGz;G0

VðlatÞG0 G0k

ð2p

0

d/ cosð/� hG0 Þeil/

�ð

dR Rnð2DÞðn0Þ�GzþG0z;k

0zðR;/ÞeiG0kR cosð/�hG0 Þnð2DÞðnÞ

Gz;kzðR;/Þ

¼X

l

Dl;kz�k0zCðGPNÞðlÞk0z;kz;n0;n;0: (55)

Our final “usual” approximation consists of ignoring the

Bloch oscillations within the wavefunctions nð2DÞðnÞGz;kz

, replac-

ing them with the envelopes fð2DÞðnÞkz

, while simultaneously

considering the lattice potential averaged over the unit cell

in the axial direction. Thus, the matrix element takes the sim-

pler form

VðSRÞk0z;kz;n0;n

� iX

l

Dl;kz�k0z

XGk

VðlatÞGk;Gz¼0Gk

�ð2p

0

d/ cosð/� hGÞeil/

�ð

dRRfð2DÞðn0Þ�k0z

ðR;/ÞeiGkR cosð/�hGÞfð2DÞðnÞkz

ðR;/Þ:

(56)

This expression is still too complicated for numerical evalua-

tion: In order to calculate the scattering rate, for each pair

ðkz; k0zÞ, we would have to compute a two-dimensional integral

for several values of angular momentum l and for several val-

ues of Gk. Therefore, we shall seek an alternative formulation,

while retaining unaltered the major physical properties of the

process. In order to do this, let us rewrite Eq. (56) as

VðSRÞk0z;kz;n;n0

�X

l

Dl;kz�k0z

ð2p

0

d/

�ð

dRRfð2DÞðn0Þ�k0z

ðR;/Þ @hVðlatÞðR;/Þi@R

eil/fð2DÞðnÞkz

ðR;/Þ;

(57)

where

hVðlatÞðR;/Þi ¼XGk

VðlatÞGk;Gz¼0eiGk�r (58)

is the lattice potential averaged over a unit cell in the z direc-

tion. This form of the matrix elements emphasizes two major

aspects of the problem. First, it illustrates the physical mean-

ing of the integration: The dominant contribution to the inte-

gral above will arise from the interfacial/surface region,

where the confining supercell pseudopotential changes

abruptly. Indeed, as we had noticed in the case of thin layers,

for l ¼ 0, kz ¼ k0z, and n ¼ n0, Eq. (57) expresses the change

of the energy of the state kz; n under a change �D of the NW

diameter. Therefore, we recover the customary physical

interpretation of the process, which applies to models

employing “barrier potentials,” having, however, replaced

such potentials with the atomic pseudopotential of the super-

cell. The terms with l 6¼ 0, instead, express the energy

change due to angular variations of the wire boundaries. Due

to the “faceting” necessarily resulting from the quasi-circular

nature of the cross section, these effects are quite important,

especially in small-diameter wires.

The second virtue of Eq. (57) stems from the following

observation: In circular NWs, the potential which confines

the wavefunctions within the wire exhibits “approximately”

cylindrical symmetry, deviations from this symmetry being

due to the individual ionic potentials at the ion locations.

Therefore, we can make use of this “quasi-circular” geome-

try by considering the angular Fourier components of the

wavefunctions and of the lattice potential defining

qðnÞkz;lðRÞ ¼ 1

ð2pÞ1=2

ðd/e�il/fð2DÞðnÞ

kzðR;/Þ (59)

FIG. 14. (Color online) Diameter-dependence of the band-gap of

H-terminated, circular, cross-section Si nanowires. The expected depend-

ence on inverse of the square of the diameter is shown by the dashed line

(blue online). Deviations at the smallest values of the diameter are due to the

finite height of the Si-vacuum confining potential barrier. The lines connect-

ing the calculated points are only a guide to the eye.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

so that

fð2DÞðnÞkz

ðR;/Þ ¼ 1

ð2pÞ1=2

Xl

qðnÞkz;lðRÞeil/: (60)

Similarly, we can define the angular Fourier components of

the lattice potential averaged over a unit cell along the axial

direction, hVðlatÞðR;/Þi,

tðlatÞl ðRÞ ¼ 1

ð2pÞ1=2

XGk

VðlatÞGk;Gz¼0

ð2p

0

d/eiGkR cosð/�hGÞe�il/

¼ 1

ð2pÞ1=2

ð2p

0

d/hVðlatÞðR;/Þie�il/: (61)

Thus,

hVðlatÞðR;/Þi ¼ 1

ð2pÞ1=2

Xl

tðlatÞl ðRÞeil/: (62)

A similar Fourier decomposition can be obtained for

dhVðlatÞðR;/Þi=dR. So, inserting these expansions into Eq.

(57), the matrix element takes the form

VðSRÞk0z;kz;n;n0

� 1

ð2pÞ1=2

Xll0l00

Dl�l0þl00;kz�k0z

�ð

dR Rqðn0Þ�

k0z;l0 ðRÞ

dtðlatÞl00 ðRÞdR

qðnÞkz;lðRÞ: (63)

The squared matrix element must be obtained by adding

“incoherently” the various q and l contributions of the SR

spectrum, consistently with Eq. (52),

jVðSRÞk0z;kz;n;n0

j2 ¼ hjVðSRÞk0z;kz;n;n0

j2i

� 1

2p

Xll0l00hjDl�l0þl00;kz�k0z j

2ij

�ð

dR Rqðn0Þ�

k0z;l0 ðRÞ

dtðlatÞl00 ðRÞdR

qðnÞkz;lðRÞ

��2

: (64)

In order to illustrate the behavior of the matrix elements in

wires of different diameters independent of the particular

form chosen for the power spectrum D/;z, let us consider the

case of circularly symmetric change of the wire diameter and

consider the matrix element

VðSRÞkz;kz;n;n0

� 1

ð2pÞ1=2D0;0

Xll0

ðdR Rqðn

0Þ�k0z;l0 ðRÞ

dtðlatÞl�l0 ðRÞdR

qðnÞkz;lðRÞ;

(65)

(i.e., the circular-symmetric term of Eq. (63) for kz ¼ k0z).The square of this matrix element, normalized to the squared

rms displacement D2rms ¼ hjD0;0j2i, is plotted in Fig. 15. This

figure shows the dependence of the SR matrix element on

the radius Rs of circular cross-section (100) Si NWs at flat

bands. As done previously in the case of thin Si films, quasi-

twofold degenerate pairs are observed, the degeneracy being

lifted by the presence of different facets at the wire boundary

(effects equivalent to the valley splitting seen in thin films).

As we have already remarked in the case of thin Si films, rec-

ognizing how each element of the pair evolves as the diameter

of the wire is changed is a hard task which can be avoided by

lumping together the pair of quasi-degenerate states and aver-

aging the squared matrix element over the transitions among

“equivalent” states. Note that the R�6s dependence expected

for the generalized Prange-Nee term is seen at large values of

the radius and for low-energy states, but as the radius

decreases and/or the energy of the states increases, “leakage”

into the vacuum reduces the strength of the confinement and

so of the matrix elements. Only transitions of significant am-

plitude are shown, processes like (2,1) or (3,2) transitions

being forbidden (so, no mixing of these modes) for the partic-

ular circular-symmetric atomic shift considered in the figure

via Eq. (65), especially at the low energies at which quantum

confinement induces strong angular-momentum selection

rules. Note that the expected R�6s -dependence is observed in

larger wires, while nonparabolic and “leakage into vacuum”

effects take over in smaller-diameter wires.

Finally, the scattering rate caused by interfacial roughness

can be calculated from Eq. (49) just replacing VðERÞkzj;kz;n0;n

with

VðSRÞkzj;kz;n0;n

. Figure 16 shows the generalized Prange-Nee com-

ponent of the SR scattering rates as a function of electron

energy (measured from the bottom of the ground-state sub-

band) in [100] Si NWs with a diameter of 1, 2, and 3 nm.

Note, at low energy, the much lower scattering rates within

the ground-state subband for the larger wires. At higher ener-

gies, instead, the larger DOS in the 2- and 3-nm-diameter

wires dominates.

FIG. 15. (Color online) Squared generalized Prange-Nee matrix element for

the ðn; n0Þ transitions as a function of the radius of circular, cross-section

(100) Si NWs under flat-band conditions. The matrix element has been cal-

culated for the kz ¼ k0z, l¼ l0 transitions and have been consequently normal-

ized to the rms atomic displacement hD0;0i. Because of the quasi-two-fold

degeneracy of the states and the fact that the ordering of same-symmetry

states changes with varying NW diameter, the states have been lumped into

doublets and the label ðn; n0Þ in the figure indicates the average of the

squared matrix elements for the transitions ðn�; n�Þ, ðn�; nþÞ, and

ðnþ; nþÞ, where n6 denotes the low/high energy member of the pair. The

dotted line shows the “usual” R�6s dependence expected for the generalized

Prange-Nee term (Ref. 14). The lines connecting the calculated points are

only a guide to the eye.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

IV. CONCLUSIONS

We have presented a method to study scattering with

surface/interface and line-edge roughness in nanometer-scale

structures based on atomic potentials, considering explicitly

the case of local empirical pseudopotentials. The method

bridges the gap between, on the one hand, physically accu-

rate atomistic and geometric approaches, which require com-

putationally expensive ensemble averages to extract the

behavior of the “average” device, and on the other hand,

numerically efficient macroscopic models, which lack the ac-

curacy of atomistic models. We have considered thin Si films

and cylindrical nanowires as well as armchair-edge graphene

nanoribbons and shown that the method provides the

“expected” confinement and width dependence in known

cases, but provides extremely large scattering rates in the

case of AGNRs as a result of the chirality dependence of

their band-gap. While these scattering rates are too large to

be employed correctly in transport simulations, they show

clearly the dominant role LER plays in controlling electronic

transport in AGNRs and confirms that transport in narrow

and rough AGNRs may not occur via extended Bloch states.

ACKNOWLEDGMENTS

We gratefully acknowledge the support provided by the

Semiconductor Research Corporation (SRC), by the Micro-

electronics Advanced Research Corporation (MARCO)

Focus Center Research Project (FCRP) for Materials, Struc-

tures and Devices (MSD), and by Samsung Electronics Ltd.

1M. H. Evans, X.-G. Zhang, J. D. Joannopoulos, and S. T. Pantelides, Phys.

Rev. Lett. 95, 106802 (2005).2G. Hadjisavvas, L. Tsetseris, and S. T. Pantelides, IEEE Electron. Device

Lett. 28, 1018 (2007).3J. Wang, E. Polizzi, A. Ghosh, S. Datta, and M. Lundstrom, Appl. Phys.

Lett. 87, 043101 (2005).4M. Luisier, A. Schenk, and W. Fichtner, Appl. Phys. Lett. 90, 102103

(2007).5S. G. Kim, A. Paul, M. Luisier, T. B. Boykin and G. Klimeck, IEEE Trans.

Electron Dev. 58, 1371 (2011).

6C. Riddet, A. R. Brown, C. L. Alexander, J. R. Watling, S. Roy, and A.

Asenov, IEEE Trans. Nanotechnol. 6, 48 (2007).7A. Martinez, S. Svizhenko, M. P. Anantram, J. R. Barker, A. R. Broen,

and A. Asenov, 2005 IEEE International Electron Devices Meeting,

Washington, DC, 5 December 2005, p. 616.8S. E. Laux, A. Kumar, and M. V. Fischetti, J. Appl. Phys. 95, 5545 (2004).9T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).

10F. Gamiz, J. B. Roldan, P. Cartujo-Cassinello, J. A. Lopez-Villanueva, and

P. Cartujo, J. Appl. Phys. 89, 1764 (2001).11D. Esseni, M. Mastrapasqua, G. K. Celler, C. Fiegna, L. Selmi, and E. San-

giorgi, IEEE Trans. Electron Devices 48, 2842 (2001); D. Esseni, A.

Abramo, L. Selmi, and E. Sangiorgi, IEEE Trans. Electron Devices 50,

2445 (2003); D. Esseni, IEEE Trans. Electron Devices 51, 394 (2004).12T. Ishihara, K. Uchida, J. Koga, and S. Takagi, Jpn. J. Appl. Phys. 45,

3125 (2006).13S. Jin, M. V. Fischetti, and T.-W. Tang, IEEE Trans. Electron Devices 54,

2191 (2007).14S. Jin, M. V. Fischetti, and T.-W. Tang, J. Appl. Phys. 102, 083715

(2007).15S. M. Goodnick, D. K. Ferry, C. M. Wilmsen, Z. Liliental, D. Fathy, and

O. L. Krivanek, Phys. Rev. B 32, 8171 (1985).16D. R. Leadley, M. J. Kearney, A. I. Horrell, H. Fisher, L. Risch, E. H. C.

Parker, and T. E. Whall, Semicond. Sci. Technol. 17, 708 (2002).17M. L. Cohen, P. J. Lin, D. M. Roessler, and W. C. Walker, Phys. Rev. 155,

992 (1967).18K. Hubner, Phys. Status Solidi A 48, 147 (1978).19S. K. Ghoshal, M. R. Sakar, and M. Rohani, J. Korean Phys. Soc. 58, 256

(2011).20A. J. Williamson and A. Zunger, Phys. Rev. B 61, 1978 (2000).21A. Palaria, G. Klimeck, and A. Strachan, Phys. Rev. B 78, 205315 (2008).22Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803

(2006).23S. B. Zhang, C.-Y. Yeh, and A. Zunger, Phys. Rev. B 48, 11204 (1993).24L.-W. Wang and A. Zunger, Phys. Rev. B 51, 17398 (1995).25M. V. Fischetti, B. Fu, S. Narayanan, and J. Kim, “Semiclassical and quan-

tum electronic transport in nanometer-scale structures: Band structure,

Monte Carlo simulations and Pauli master equation,” in Selected Topics inSemiclassical and Quantum Transport Modeling, edited by D. Vasileska

(Springer, New York, 2011, in press).26D. Esseni and P. Palestri, Phys. Rev. B 72, 165342 (2005).27R. E. Prange and T.-W. Nee, Phys. Rev. 168, 779 (1968).28H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, Appl.

Phys. Lett. 51, 1934 (1987).29F. Tseng, D. Unluer, K. Holcomb, M. R. Stan, and A. W. Ghosh, Appl.

Phys. Lett. 94, 223112 (2009).30A. Betti, G. Fiori, G. Iannaccone, and Y. Mao, 2009 IEEE International

Electron Devices Meeting, Baltimore, MD, 7-9 December 2009.31D. A. Areshkin, D. Gunlycke, and C. T. White, Nano Lett. 7, 204 (2007).

FIG. 16. (Color online) (a) Generalized Prange-Nee component of the surface-roughness scattering rate as a function of electron energy in [100] circular

cross-section Si nanowires with diameter of 1, 2, and 3 nm. The energy is measured from the bottom of the ground-state subband; a roughness correlation spec-

trum of the form given by Eq. (52) has been used. The lowest-energy 16 conduction (sub)bands have been considered and have been decomposed into 32 angu-

lar momentum components. At the lowest energy of importance for mobility calculations, the intrasubband scattering rate within the ground-state subband is

reduced significantly as the diameter increases. On the contrary, at larger energies, the largest density of final states in the larger wires controls the scattering

rate. (b) As in (a), but showing the velocity relaxation rate. The “noise” is due to forward inter-subband transitions to states with higher group velocity, giving

negative contribution to the total relaxation rate.



194.69.13.215 On: Mon, 12 May 2014 07:43:05

http://dx.doi.org/10.1103/PhysRevLett.95.106802


http://dx.doi.org/10.1109/LED.2007.906471


http://dx.doi.org/10.1063/1.2001158

http://dx.doi.org/10.1063/1.2001158

http://dx.doi.org/10.1063/1.2711275

http://dx.doi.org/10.1109/TED.2011.2118213


http://dx.doi.org/10.1109/TNANO.2006.886739

http://dx.doi.org/10.1063/1.1695597

http://dx.doi.org/10.1103/RevModPhys.54.437

http://dx.doi.org/10.1063/1.1331076

http://dx.doi.org/10.1109/16.974714

http://dx.doi.org/10.1143/JJAP.45.3125


http://dx.doi.org/10.1063/1.2802586

http://dx.doi.org/10.1103/PhysRevB.32.8171

http://dx.doi.org/10.1088/0268-1242/17/7/313

http://dx.doi.org/10.1103/PhysRev.155.992

http://dx.doi.org/10.1002/pssa.v48:1

http://dx.doi.org/10.3938/jkps.58.256








http://dx.doi.org/10.1063/1.98305

http://dx.doi.org/10.1063/1.98305

http://dx.doi.org/10.1063/1.3147187

http://dx.doi.org/10.1063/1.3147187

http://dx.doi.org/10.1021/nl062132h

32D. Gunlycke, D. A. Areshkin, and C. T. White, Appl. Phys. Lett. 90,

142104 (2007).33E. R. Mucciolo, A. H. Castro Neto, and C. H. Lewenkopf, Phys. Rev B 79,

075407 (2009).34I. Martin and Y. M. Blanter, Phys. Rev B 79, 235132 (2009).35M. Evaldsson, I. V. Zozoulenko, H. Xu, and T. Heinzel, Phys. Rev B 78,

161407 (2008).36M. Luisier and G. Klimeck, Appl. Phys. Lett. 94, 223505 (2009).37M. Bresciani, P. Palestri, D. Esseni, and L. Selmi, in Proceedings of 38th

European Solid State Device Research Conference (ESSDERC), Athens,

Greece, 14-18 September 2009, pp. 480–483.38A. Cresti and S. Roche, New J. Phys. 11, 095004 (2009).39G. Gilat and L. J. Raubenheimer, Phys. Rev. 144, 390 (1966).40E. Ezawa, Physica E 40, 1421 (2008).41C. P. Auth and J. D. Plummer, IEEE Trans. Electron Devices 18, 74

(1997).42Y. Kurokawa, S. Nomura, T. Takemori, and Y. Aoyagi, Phys. Rev. B 61,

12616 (2000).43L. Yang, C.-H. Park, Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys.

Rev. Lett. 99, 186801 (2007).44X. Li, X. Wang, L. Zhang, S. W. Lee, and H. J. Dai, Science 319, 1229

(2008).45D. Querlioz, Y. Apertet, A. Valenkin, K. Huet, A. Bournel, S. Galdin-

Retailleau, and P. Dollfus, Appl. Phys. Lett. 92, 042108 (2008).46In the context of AGNRs, the idea of replacing the full wavefunction

nð2DÞðnÞGz ;kz

ðRÞ with its cell-average fð2DÞðnÞkz

ðzÞ — thus ignoring Bloch-

functions and symmetry-related overlap effects — may appear question-

able. Indeed, recently it has been shown by K. Wakabayashi, Y. Tanake,

M. Yamamoto, and M. Sigrist, New J. Phys. 11, 095016 (2009) (see also

K. Wakabayashi, Y. Takane, and M. Sigrist, Phys. Rev. Lett. 99, 036601

(2007); M. Yamamamoto, Y. Takane, and K. Wakabayashi, Carbon 47,

124 (2009); and T. Ando, J. Phys. Soc. Jpn. 74, 095016 (2005) for the case

of graphene and nanotubes) that backscattering may be suppressed in both

zigzag-edge GNRs — because of “missing” backscattering channels —

and also in AGNRs because of the phase structure of the wavefunctions,

despite the mixing of the A and B-sublattices-related chiralities at the Kþand K� graphene symmetry points. This would give rise to a perfectly con-

ducting channel (PCC). However, this happens only in the presence of

long-range scattering potentials (impurities in the case considered by

Wakabayashi et al.) which suppress intervalley scattering and only when

employing a nearest-neighbor, single-orbital (pz) empirical tight-binding

model. In our case, the presence of all orbitals and of beyond-nearest-

neighbor interactions implicitly included by the pseudopotential approach

alters the band structure (see, for example, Ref. 60) and also the phase

structure of the wavefunctions, thus removing the selection rules, which

prevent backscattering. In addition, the ER scattering potential we con-

sider here exhibits short-range components which assist intervalley

scattering.47T. Fang, A. Konar, H. Xing, and D. Jena, Phys. Rev. B 78, 205403 (2008).

48L. Zheng, X. Y. Liu, G. Du, J. F. Kang, and R. Q. Han, in Proceedings ofthe 2009 Simulation of Semiconductor Processes and Devices (SISPAD),San Diego, CA, 9-11 September 2009.

49A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K.

Geim, Rev. Mod. Phys. 81, 109 (2009).50Y. Yoon and J. Guo, Appl. Phys. Lett. 91, 073103 (2007).51Y. Ouyang, X. Wang, H. Dai, and J. Guo, Appl. Phys. Lett. 92, 243124

(2008).52P. Zhao and J. Guo, J. Appl. Phys. 105, 034503 (2009).53While this value for the effective mass m* has been obtained by empirical-

pseudopotential band-structure calculations, the emergence of a nonzero

mass and an estimate of its magnitude can be obtained, considering the

broken translational symmetry when moving from graphene to ribbons. In

graphene, the energy dispersion near the Dirac point, EðKÞ � �htFK (where

tF � 108 cm/s is the Fermi velocity) implies that the Green’s function of

the free-electron Hamiltonian is ð�htFKÞ�1. This “propagator” of a mass-

less, two-dimensional particle can be viewed as the Fourier transform of a

force with infinite range. In a nanoribbon of width W, the range of this

force will be finite, � W. Thus, p/W can be viewed as a “screening param-

eter” (with “screening” due to the edges, which play the role of Higgs par-

ticles) yielding the Fourier transform ½�htFðK þ p=WÞ��1. This can now be

viewed as the “relativistic” propagator (with tF playing the role of the

speed of light C) of a massive two-dimensional particle, ½�htFK þ m�t2F��1

,

with mass m� ¼ �hp=ðWtFÞ � 0.3 m0=W, with W measured in nm.54A. Sinitskii, A. A. Fursina, D. V. Kosynkin, A. L. Higginbotham, D. Natel-

son, and J. M. Tour, Appl. Phys. Lett. 95, 253198 (2009).55Y. Yang and R. Murali, Electron. Device Lett. 31, 237 (2010).56B. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J.

Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer,

Science 312, 1191 (2006).57J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M.

Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Muellen, and R. Fasel, Na-

ture 466, 470 (2010).58L. Jiao, L. Zhang, X. Wang, G. Diankov, and H. Dai, Nature 458, 877

(2009).59D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A.

Dimiev, B. K. Price, and J. M. Tour, Nature 458, 872 (2009).60A. Cresti, N. Nemec, B. Biel, G. Niebler, F. Triozon, G. Cuniberti, and S.

Roche, Nano Res. 1, 361 (2008).61F. Sols, F. Guinea, and A. H. Castro Neto, Phys. Rev. Lett. 99, 166803

(2007).62K. Todd, H.-T. Cou, S. Amasha, and D. Goldhaber-Gordon, Nano Lett. 9,

416 (2009).63C. Stampfer, J. Guttinger, S. Hellmuller, F. Molitor, K. Ensslin, and T.

Ihn, Phys. Rev. Lett. 102, 056403 (2009).64S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008).65M. Bresciani, P. Palestri, D. Esseni, and L. Selmi, Solid State Electron. 54,

1015 (2010).66S. DasSarma and X. C. Xie, Phys. Rev. B 35, 9875 (1987).



194.69.13.215 On: Mon, 12 May 2014 07:43:05

http://dx.doi.org/10.1063/1.2718515




http://dx.doi.org/10.1063/1.3140505

http://dx.doi.org/10.1088/1367-2630/11/9/095004


http://dx.doi.org/10.1016/j.physe.2007.09.031

http://dx.doi.org/10.1109/55.553049




http://dx.doi.org/10.1126/science.1150878

http://dx.doi.org/10.1063/1.2838354

http://dx.doi.org/10.1088/1367-2630/11/9/095016

http://dx.doi.org/10.1088/1367-2630/11/9/095016

http://dx.doi.org/10.1088/1367-2630/11/9/095016

http://dx.doi.org/10.1088/1367-2630/11/9/095016


http://dx.doi.org/10.1103/RevModPhys.81.109

http://dx.doi.org/10.1063/1.2769764

http://dx.doi.org/10.1063/1.2949749

http://dx.doi.org/10.1063/1.3073875

http://dx.doi.org/10.1063/1.3276912


http://dx.doi.org/10.1126/science.1125925

http://dx.doi.org/10.1038/nature09211




http://dx.doi.org/10.1007/s12274-008-8043-2


http://dx.doi.org/10.1021/nl803291b



http://dx.doi.org/10.1016/j.sse.2010.04.038


Date post:	25-Dec-2016
Category:	Documents
Upload:	sudarshan
View:	214 times
Download:	2 times

An empirical pseudopotential approach to surface and line-edge roughness scattering in...

Documents