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arXiv:cond-mat/0611493v1 [cond-mat.mtrl-sci] 18 Nov 2006

Linear scaling calculation of band edge states

and doped semiconductors

H. J. Xiang, Jinlong Yang,

∗

J. G. Hou, and Qingshi Zhu

Hefei National Laboratory for Physical Sciences at Microscale,

University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China

(Dated: February 6, 2008)

Linear scaling methods provide total energy, but no energy levels and canonical wavefuctions.

From the density matrix computed through the density matrix puriﬁcation methods, we propose

an order-N (O(N)) method for calculating both the energies and wavefuctions of band edge states,

which are important for optical properties and chemical reactions. In addition, we also develop an

O(N) algorithm to deal with doped semiconductors based on the O(N) method for band edge states

calculation. We illustrate the O(N) behavior of the new method by applying it to boron nitride

(BN) nanotubes and BN nanotubes with an adsorbed hydrogen atom. The band gap of various BN

nanotubes are investigated systematicly and the acceptor levels of BN nanotubes with an isolated

adsorbed H atom are computed . Our methods are simple, robust, and especially suited for the

application in self-consistent ﬁeld electronic structure theory.

I. INTRODUCTION

Traditional electronic structure algorithms calculate

all eigenstates associated with discrete energy levels. The

disadvantage of this approach is that it leads to a diago-

nalization problem that has an unfavo rable cubic scaling

in the computational eﬀort. Linear scaling density func-

tional or Ha rtree-Fock methods are an essential tool for

the calculation of the electronic structure of large systems

containing many atoms.

1

The key point of the success

of most linear scaling methods is that only the density

matrix or localized Wa nnier functions which span the oc-

cupied manifold is calculated. In these O(N) methods,

no canonical wavefunctions or eigenvalues are available.

However, in many cases, one may be interested in some

eigenstates, especia lly the states near the Fer mi level, i.e.,

band edge states. For instance, from the theory of fron-

tier orbitals, many mo le c ular pro perties are determined

by the highest occ upied molecular orbital (HOMO) and

lowest unoccupied molecular orbital (LUMO), and fron-

tier orbitals play an important role in chemical reactions.

On the other hand, there are some linear scaling algo-

rithms such as the Kim-Mauri-Galli (KMG)

2

method

need the Fermi level which can be estimated from the

HOMO and LUMO energy.

There are some methods which can be us e d to o bta in

band edge states. The most popular method for cal-

culating states near a reference energy ǫ

ref

is the folded

sp e c trum method.

3

However, in this method, by squaring

the Hamiltonian, the condition number is also squared

and thus the diﬃculty of solving the equation is also

increased. To so lve this problem, Tacke tt et al.

4

pre-

sented the Jacobi-Davidson method in which the con-

dition number and diﬃculty in solving for the s e lected

eigensolutions is the s ame as the original eigenvalue equa-

tion. Unfortunately, the implementation of the Jacobi-

Davidson method is rather involved and its application

is not widespread. In the ﬁeld of computational mathe-

matics, the shift-and-invert Lanczos algorithm is a well-

known method for calculating a pair of eigenvalue and

eigenvector near a reference energ y. This method was

used by Liang et al.

5

to obtain the Fermi level in the con-

text of linear s caling Fermi operator expansion method.

In this method, the Lanczos method is applied to the so-

called shift-and-inver t matrix, (H − ǫ

ref

I)

−1

, where H

and I are the Hamiltonian and identity matrices, respec-

tively, and ǫ

ref

is the reference energy. These matrices

are not, of course, formed explicitly. Instead, e ach time

the Lanczos metho d requir e s a multiplication of a vector

v by matrix (H − ǫ

ref

I)

−1

, a linear solver subroutine is

called to solve the corres po nding linea r systems. If these

linear sys tems are solved suﬃciently accura tely, the con-

vergence of the Lanczos method is typically much faster

compared to that when the matrix H is used in the Lanc-

zos method. The diﬃculty now is that accurate numer-

ical solution of linear systems, needed on each iteration

of the Lanczos method, can be costly. Besides the diﬃ-

culties of these methods mentioned a bove, when they are

applied to get the frontier orbitals, another inconvenience

is that a reference energy ǫ

ref

must be selected.

Here in this work, we present an alternative simple

method to get states near gap base d on linear scaling

density matrix methods. In our method, we do not need

the reference energy. The new O(N) method is particu-

larly useful for calculating frontier orbitals in the frame-

work of self-consistent ﬁeld (SCF) electronic structure

theory. Using this method, we also pr opose a promising

linear scaling method which can be utilized to explore

the energetics, defective levels, and gemoetry of doped

semiconductors.

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

present o ur new O(N) methods for calculating band

edge states and dealing with doped semico nductors. In

Sec. III, we des c rib e the details of the implementation

and perfo rm some test calculations to illustrate the rig ht-

ness, robustness, and linear-scaling b e havior of our meth-

ods. In Sec. IV, we use our new methods to calculate the

band gap of b oron nitride (BN) nanotubes and the accep-

tor level of a single H adsorbed BN nanotubes. Finally,

our concluding r e marks are given in Sec. V.

2

II. THEORY

A. Calculation of band edge states

Within our method, we must ﬁrst obtain density ma-

trix ρ corresponding a given Hamiltonian H befor e we

proceed to calculate band edge states. However, it is

not an inconvenience in the framework o f linear scaling

SCF elec tronic structure theory. In principles, any lin-

ear scaling density matrix metho ds can be used to obtain

the density ma trix.

1,6,7

Moreover, O(N) localized orbital

based methods can also be used to c onstruct the density

matrix.

1,8,9

In the representation of molecular canonical orbitals,

density matrix ρ and Hamiltonian H are diagonal matri-

ces of the following forms:

ρ = diag(1, 1, . . . , 1, 0, 0, . . . , 0),

H = diag(ǫ

1

, ǫ

2

, . . . , ǫ

N

e

/2

, ǫ

N

e

/2+1

, . . . , ǫ

N

b

),

(1)

where N

e

is the number of electrons of a closed-shell sys-

tem, and N

b

is the number of basis functions. Without

loss of generality, we assume that

ǫ

1

≤ ǫ

2

≤ . . . ≤ ǫ

N

e

/2

≤ ǫ

N

e

/2+1

≤ . . . ≤ ǫ

N

b

, (2)

then ǫ

N

e

/2

and ǫ

N

e

/2+1

will be the HOMO and LUMO

energies respectively. It can be easily seen that:

ρH = Hρ = diag(ǫ

1

, ǫ

2

, . . . , ǫ

N

e

/2

, 0, . . . , 0). (3)

If ǫ

N

e

/2

> 0, then ǫ

N

e

/2

will be the largest eigenvalue

of ρH. Otherwise, we can shift the Hamiltonian H to

H + λI (λ > 0) so that λ + ǫ

N

e

/2

> 0. Clearly, λ +

ǫ

N

e

/2

is the lar gest eigenvalue of ρ(H + λI). Using the

similar argument, we can prove that if −λ + ǫ

N

e

/2+1

<

0, −λ + ǫ

N

e

/2+1

will be the smallest eigenvalue of (I −

ρ)(H − λI). We should note that the parameter λ can

be set to be a large positive value without degrading the

eﬃciency of the method. In practice, we ﬁnd that it is

usually reliable by setting λ to be 1 Ry. The largest

(smallest) eigenvalue and its corresponding eigenvector

of ρ(H + λI) ((I − ρ)(H − λI)) can be computed easily

using the well-known O(N) Lanczo s method. Up to now,

we discuss the problem in the representation of molecular

canonical orbitals ψ. In the representation of orthogonal

basis orbitals φ, molecular canonical orbitals ψ can be

expressed as

ψ

i

=

X

µ

φ

µ

C

µi

, (4)

where the coeﬃcient matrix C is a unitary matrix . Thus

in the representation o f orthogonal basis orbitals, density

matrix ρ

or

and Hamiltonian H

or

can be calculated as:

ρ

or

= CρC

+

,

H

or

= CHC

+

.

(5)

Moreover, ρ

or

H

or

can also be obtained through a unitary

transformation of ρH. Since the unitary transformation

of a matrix does not change its eigenvalues, we can see

that the above results deduced using the repre sentation

of molecular canonical orbitals also hold in the represen-

tation of orthogonal basis orbitals. The procedure for

obtaining HOMO and LUMO states are illustra ted in

Fig. 1(a).

Since many ﬁrst principles codes use non-orthogonal

atomic orbitals, here we discuss the case of non-

orthogonal basis. A general method is transforming the

non-orthog onal basis to or tho gonal basis. We achieve

this by tr ansforming the atomic orbital (AO) Hamilto-

nian matrix H

ao

to an orthonormal basis using H

or

=

ZH

ao

Z

T

and obtaining the AO density matrix ρ

ao

using

ρ

ao

= Z

T

ρ

or

Z, where the inverse factor Z = L

−1

, and L

is the Cholesky factor for which S = LL

T

. The Cholesky

transformation has been used in severval linear scaling

densit matrix programs.We next show how to get wave-

function in the non-orthogona l basis. In non-orthogonal

basis, a generalize d eigenvalue problem should be solved:

H

ao

ψ

ao

= ǫSψ

ao

, (6)

where ψ

ao

is the wavefunction in the non-orthogonal ba-

sis. Given the wavefunction in the orthogona l basis ψ

or

,

which satisﬁes

H

or

ψ

or

= ZH

ao

Z

T

ψ

or

= ǫψ

or

, (7)

we have

H

ao

Z

T

ψ

or

= ǫZ

−1

Z

−T

Z

T

ψ

or

= ǫSZ

T

ψ

or

,

ψ

ao

= Z

T

ψ

or

.

(8)

We also present another method to calculate band

edge states in non-orthogona l basis without tr ansform-

ing to orthogo nal basis. This method is particularly

useful when localized orbitals based O(N) algorithms

are employed. From ρ

ao

H

ao

= Z

T

ρ

or

H

or

Z

−T

, one

can easily see that ρ

ao

H

ao

ψ

ao

= ǫψ

ao

is equivalent to

ρ

or

H

or

Z

−T

ψ

ao

= ǫZ

−T

ψ

ao

. Thus ρ

ao

H

ao

has the same

eigenvalues as ρ

or

H

or

. We can also prove that ρ

ao

(H

ao

+

λS) has the same eigenvalues as ρ

or

(H

or

+ λ). Thus the

largest e igenvalue of ρ

ao

(H

ao

+λS) will be ǫ(HOMO)+λ.

We should point out that ρ

or

(H

or

+ λ) is hermitian, but

ρ

ao

(H

ao

+ λS) is not. However, this doesn’t pose any

problem since the Lanczos algo rithm can also be used to

get the extreme eigenvalues of a non- hermitian matrix.

We can see that the calculation of HOMO state is simple

since only ρ

ao

, H

ao

, and S are needed. However, the cal-

culation of the LUMO state is a diﬀerent story. We can

easily prove that (I − ρ

ao

S)(S

−1

H

ao

− λI) has the same

eigenvalues as (I − ρ

or

)(H

or

− λI) and −λ + ǫ(LUMO)

is the smallest eigenvalue of (I − ρ

ao

S)(S

−1

H

ao

− λI).

As can be seen, to ca lculate the LUMO state, besides

ρ

ao

, H

ao

, and S, we must also have S

−1

or S

−1

H

ao

. The

inverse of S is usually a formidable task. Fortunately,

Gibson et al. introduced an O(N) metho d to calculate

S

−1

H

ao

.

10

3

B. Treatment of doped semiconductors

To our best knowledge, most linear scaling methods

are mainly applied to semiconductors or insulators with

an energy gap. When the system is metallic or gapless,

these O(N) methods fail or lose of eﬀectiveness since these

methods rely on the sparsity of the density matrix and

the convergence of many of these methods is determined

by the magnitude of band gap. Partial occupation is an-

other obstacle fo r many popular linear sca ling methods

due to the non-idempotence of the density matrix. Here

we propose an O(N) method to deal with doped semi-

conductors where dopa nts or defects exist. Our method

has the similar s pirit as that proposed by Raczkowski and

Fong in tha t a subspace larger than the occupied space is

used.

11

In their seminal work, the subspace optimization

method formulated in terms o f localized nonorthogonal

orbitals was employed. However, besides two O(N

3

) steps

in the Grassmann conjugate gradient (GCG) algorithm,

an additional O(N

3

) step of diagonalization is needed.

Another problem is that when the orbital loc alization is

used to acheive linear scaling, loca l minima might oc-

cur in the subspace optimization method, resulting in a

stalling of GCG algorithm during the last several SCF

steps.

11

In our method, we treat the valence bands using the

density matrix method, and other defective bands a re

calculated using our O(N) method for band edge states.

For simplicity, we consider the cases where only an elec-

tron or hole is present in a semiconductor, as shown in

Fig. 2. In case of n-type doping (Fig. 2(a)), the total

density matrix ρ can be calculated as

ρ = ρ

val

+ 0.5|ψ

N+1

ihψ

N+1

|, (9)

where ρ

val

is density matrix corresponding to the valence

band. In case of p-type do ping (Fig. 2(b)), the total

density matrix ρ can be calculated as

ρ = ρ

val

− 0.5|ψ

N

ihψ

N

|. (10)

Both ψ

N+1

and ψ

N

are computed throug h the newly de-

veloped O(N) method for ba nd edge states. Using the

block La ncz os algorithm, our method can also be used

when several doped levels are present. In this case, the

Fermi distribution can be used to get the occupation of

doped levels. Since the valence band a re well separated

from the conduction band, ρ

val

is sparse, and the calcu-

lation of ρ

val

can be carried out using traditional O(N)

methods , such as the tr ace-correcting density matrix pu-

riﬁcation (TC2) method.

6

Since canonical orbitals ψ

N+1

and ψ

N

are usually delocalized, the total density matrix

ρ is much denser than ρ

val

. It is diﬃcult to deal with

the full density matrix. However, we notice that in fac t

only a small par t of the full density matrix is used in

the construction of the new Hamiltonia n. Thus in prac-

tice, we only construct a small part of the full density

matrix. To make our O(N) method for the treatment

of doped semiconductors more clear, we show the ﬂow-

chart of a typical calculation in Fig. 1(b). Our metho d

is very simple and applicable to many doped systems.

We should mention that our method is not a black-box

method since some knowledge of the studied system must

be known prior. For instance, we should know the doping

type and number of doping levels. Typically, we can get

this information from intuition or deduction from other

smaller systems with similar characters.

III. IMPLEMENTATION AND RESULTS

A. Implementation

Our newly developed method has been implemented

in SIESTA,

12

a standard Kohn-Sham density-functional

program using norm-conserving pseudopotentials and nu-

merical atomic orbitals as basis sets. In SIESTA, periodic

boundary conditions are employe d to simulate both iso-

lated and periodic systems. Here we use the O(N) TC2

method

6

to get the density matrix since it is very simple,

robust, and eﬃcient. The details about the implementa-

tion of the TC2 method can be found in Ref. 13.

In our O(N) method for doped s emiconductors, to

obtain atomic forces, it is necessary to get the energ y

weighted density matr ix E when using atomic basis sets.

Take the case as shown in Fig. 2(a) as an example,

E = E

val

+ 0.5ǫ

N+1

|ψ

N+1

ihψ

N+1

|, (11)

where E

val

is calculated from ρ

val

. For energy weighted

density matrix E, we also compute and save only a part of

the full matrix. To speed up the calculation, we adopt the

block Lanczos method to calculate the defect levels, since

the vectors produced by the previous SCF step c an be

reused in the s ubse quent step. Usually, in the last several

SCF steps, we don’t need any matrix-vector multiplica-

tions in the calculation of band edge states. Thus, when

a geometry o ptimization is performed, the extra amount

for computing defect levels using our O(N) method is al-

most negligible. This contrasts to the method proposed

by Raczkowski and Fong.

8,9,11

B. Validity and performance of the O(N) method

for band edge states calculation

All our calculations rep orted in this work are done in

the local density approximation (LDA).

14

Unless oth-

erwise stated, the double-ζ plus polarization functions

(DZP) basis set is used in the calculations.

We ﬁrst validate our method by computing the HOMO

and LUMO of H

2

O molecule. The energ ies of HOMO and

LUMO are −7.532 (−7.53257) and −1.375 (−1 .37292)

eV, respectively (values in parenthesis are results fr om

the diagonalization method). We also compare the

HOMO and LUMO wavefunctions with those from the

diagonaliza tion method, and ﬁnd that the agreement is

remarkable.

4

To check the performance of our method, we calcu-

late the HOMO and LUMO of BN(5,5) na notubes with

diﬀerent number of atoms in the supercells. The CPU

time used is shown in Fig. 3. We can clearly see the lin-

ear scaling behavior of our new method for both single-ζ

(SZ) and DZP basis sets.

For the purpose of comparison, we also calculate the

LUMO of BN(5,5) nanotube with 400 atoms using the

folded spectrum method. The SZ basis is adopted. Since

the performance of the folded spectrum method is very

sensitive to the choice of the reference energy, several dif-

ferent r e ference energies varying from the midgap posi-

tion to the LUMO energy are chosen. The precision of the

calculation is within 3 meV with respect to the value from

the diagonalizatio n. As shown in Fig. 4, the CPU time

used is very large, especially when the reference energy

is close to the LUMO energy (the HOMO and LUMO

energies are -7.075 and -2.577 eV respec tively in the cur-

rent computing parameters setting). Even when the the

reference e nergy is chosen to be optimal, the folded sp e c -

trum method is still slower by seventeen times than our

new O(N) method (387 s v.s. 22 s).

C. Validity and performance of the O(N) method

for doped semiconductors

We will take BN(8,0) zig-zag nanotubes with an ad-

sorbed H atom as an e xample to illustrate the correct-

ness and eﬃciency of our new method. As shown by Wu

et al., a H atom prefers to ads orb on a B atom, and the

system is a p-type semiconductor.

15

For a BN(8,0) nan-

otube (128 atoms in the supercell) with an adsorbed H

atom, the energy diﬀerence between o ur result and that

from the diagonalization method is only 6 meV. And the

force diﬀerences between our result and that from the

diagonaliza tion method do not exceed 0.6 meV/

˚

A. Both

the energy and forces agreement validates our new O(N)

method for doped semiconductors. We also deal suc-

cessfully with a BN(8,0) nanotub e with two adsorbed H

atoms on two B sites, indicating that our method also

works in c ase of systems with multi defect levels.

Here we show in Fig. 5 the CPU time us e d in an

ion step for supe rcells with diﬀerent number of atoms.

Clearly, our new method for doped semiconductors dis-

plays a linear scaling behavior.

IV. APPLICATIONS

A. Band gap of BN chiral nanotubes

Previous study showed that for small zigzag (chiral an-

gle α = 0

◦

) nanotubes the energy gap decreas e s rapidly

with the decrease of radius, while armchair nanotubes

(chiral a ngle α = 30

◦

) almost have a constant energy

gap.

16

Although previous experiments

17

indicated a pref-

erence for zig-zag and near zig-zag BN tubes and a plau-

sible explanation

16

was propo sed, a very recent high-

resolution electron diﬀraction study on BN nanotubes

grown in a ca rbon-free chemical vapor deposition pro-

cess revealed a disp e rsion of the chiral angles.

18

Thus a

thorough knowledge of the dependence of the band g ap

upon the chirality of BN nanotubes is des irable. Chiral

BN nanotubes usually contain large number of atoms in

a unit cell, e.g., a BN(14,1 ) nanotube has 844 atoms in

the unit cell. These nanotubes are diﬃcult to be treated

using traditional methods. Here we calculate systemat-

icly the band g ap of BN nanotubes including chiral BN

nanotubes. Whenever the system is large enough to be

sampled using Γ-point, we use the new O(N) method

for calculating band edge states. The results are shown

in Fig. 6. Two general trends are observed: ﬁrst for

BN nanotube s with similar radius, BN nanotubes with

larger chiral ang le s have larger band gaps, secondly, for

BN nanotube s with chiral angles clo se to zero, BN nan-

otubes with larger radius have la rger band ga ps . In

addition, we can see that for BN(n,m) nanotubes with

n +m = k, the band gap of BN(n,k − n) doe s not depend

monotonously on the n value due to the competition of

the two trends mentioned above, however, the band gap

of BN(k − [

k

2

],[

k

2

]) (Here [

k

2

] denotes the maximal integer

no larger than

k

2

) nanotub e is the largest, and BN(k,0)

nanotube usually has the smallest band gap except that

the band gap of BN(8,2) nano tube is small than that of

BN(10,0) nanotube. The band gaps of some BN na n-

otubes were reported previously and the results are in

accord with ours,

16,19

and a more complete picture for

the trend of the band g ap of BN nanotubes is presented

here.

B. Acceptor level of H adsorbed BN nanotubes

Wu et al.

15

investigated the adsorption of a hydrogen

atom on zigzag BN(8,0) nanotube using a supercell con-

taining 32 boro n and 32 nitrogen atoms and found H

prefers to adsorb on the boron atom which introduces

an acceptor state in the gap. They also showed that the

dispersion of the defect band is as large as 0.2 eV. Our

test calculations in the Γ-only approximation also show

that the acc e pto r levels of a single H adsorbed BN(8,0)

nanotube using a 64-a toms or 128-atoms supercell are

1.064 eV and 1.180 eV, respectively (Here, the accep-

tor level is deﬁned as the energy diﬀerence between the

acceptor state and the top of the valence band). In addi-

tion, the adsorption energy of the H atom also dep e nds

on the chosen supercell: For instance, the adsorption en-

ergy is − 0.353 (−0.246) eV when using a 64-atoms (320-

atoms) BN(8,0) supercell and the diagonalization (our

linear scaling) method. All these facts suggest that larger

supe rcells should be used to predict the properties of BN

nanotubes with an isolated ads orbed H atom. Here with

the O(N) method for do ped semiconductors developed

in this paper, we can trea t much larger radius BN nan-

otubes with truely isolated adsorbed H atom throug h us-

5

ing huge supercells. Three B N nanotubes are considered:

BN(8,0) nanotube simulated using a supercell with 320

atoms, BN(15,0 ) nanotube simulated using a supercell

with 720 atoms, and BN(13,2) nanotube with 796 atoms

in the unit cell. Here we show the distribution of the ac-

ceptor state and the highest orbital of the valence band in

Fig. 7. Clearly, the acceptor state is a relatively localized

state around the adsorbed H atom, which agrees with

the result reported by Wu et al.

15

. However, the highest

orbital of the valence band is delocalized and mainly c on-

tributed by N 2p

z

orbitals. As can be seen from Fig. 6,

BN(15,0) nano tube and BN(13,2) nanotube have similar

radius but diﬀerent chirality, and the radius of BN(8,0)

nanotube is smaller. The calculated acceptor levels in-

troduced by an isolated H atom are 1.184 eV, 1.55 7 eV

and 1.563 eV for BN(8,0), BN(15,0) and B N(13,2) nan-

otubes, respectively. Thus the position of the defect level

is closer to the to p of valence bands for smaller radius BN

nanotubes, but does not depend signiﬁcantly on the chi-

rality.

V. CONCLUSIONS

We present a simple O(N) method for calculating band

edge states using the density matrix obtained from O(N)

electronic structure methods. Based on the O(N) method

for calculating band edge states, we further develop an

O(N) algorithm to deal with doped semiconductors. In

our methods, no reference energy is needed to obtain

the band edge states, and they are especially suited for

the applica tion in SCF electronic str ucture theory. The

O(N) behavior of the new methods is demonstrated by

applying it to bare and H adsorbed BN nanotubes. The

band gap of various BN nanotube s are investigated sys-

tematicly and the acceptor levels of BN nanotubes with

an isolated a ds orbed H atom a re calculated. Our al-

gorithms could be generalized straightforwardly to spin-

unrestricted systems,

13

such as magnetic semiconductors

and diluted ma gnetic semiconductors.

20

This work is partially supported by the Nationa l Nat-

ural Science Foundation of China (50121202, 20533030,

10474087), by the USTC-HP HPC project, and by the

SCCAS and Shanghai Superc omputer Center.

∗

Corresponding author. E-mail: jlyang@ustc.edu.cn

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6

FIG. 1: Schematic illustration of (a) the O(N) method for

calculating the HOMO (the program ﬂow for the LUMO cal-

culation is similar except for some modiﬁcations as described

in the text) and (b) the O(N) method for dealing with doped

semiconductors. Here “density matrix” is abbreviated to

“DM”.

FIG. 2: (Color online)Schematic illustration of the electronic

structure of doped semiconductors: (a) n-typ doping and (b)

p-type doping.

FIG. 3: Total CPU time for calculating HOMO and LUMO

of BN(5,5) nanotubes using the linear scaling method. Here,

both SZ and DZP basis sets are used. All calculations were

carried out on a 1.5 GHz Itanium 2 CPU workstation running

RedHat Linux Advanced Server V2.1.

FIG. 4: Total CPU time for calculating the LUMO of BN(5,5)

nanotube with 400 atoms using the folded spectrum method

with diﬀerent reference en ergy. Here, SZ basis set is used.

FIG. 5: Total CPU time for calculating of BN(8,0) nanotube

with a H atom adsorbed on a boron atom using the O(N)

method for doped semiconductors. Here, double-ζ (DZ) basis

set is used.

FIG. 6: Band gap for various BN(n,m) nanotubes. BN(n,m)

nanotubes with n + m = k are connected with a line to guide

the eyes.

FIG. 7: (Color online) (a) The acceptor state and (b) the

highest orbital of the valence band of a BN(13,2) nanotube

with an isolated adsorbed H atom. The insets show the en-

larged plots around the adsorbed H atom.

7

trial DM

ρ

0

trial DM

TC2: new DM

mixing of DM

set up Hamiltonion H

(a)

ρ

0

Converged?

No

ρ

Yes

K

new

max

: Max eigenvalue of K

max

Κ −λε(ΗΟΜΟ) =

ρ(Η+λ)

Calculate K=

Energy, force etc

val

new

ρ

tot

new

Yes

Converged?

No

new total DM

O(N) method: defect level

ρ

TC2: new valence DM

mixing of DM

set up Hamiltonion H

(b)

Fig. 1 of Xiang et al.

8

(b)

ConductionConduction

Band Band

Valence Valence

BandBand

(a)

n−type p−type

N

N+1

Fig. 2 of Xiang et al.

9

0

1000

2000

3000

4000

CPU Time (s)

0 200 400

600

800

0

10

20

30

40

50

60

70

CPU Time (s)

Number of Atoms

DZP

SZ

Fig. 3 of Xiang et al.

10

-4.5

-4

-3.5

-3

-2.5

Reference Energy (eV)

0

1000

2000

3000

4000

5000

6000

CPU Time (s)

Fig. 4 of Xiang et al.

11

0 100 200 300 400

500 600

700 800 900 1000

Number of Atoms

0

5000

10000

15000

CPU Time (s)

Fig. 5 of Xiang et al.

12

2.5

3

3.5

4

4.5

2 2.5 3 3.5 4 4.5 5 5.5 6

Energy Band Gap (eV)

Radius of BN nanotubes (Å)

(3,3)

(4,2)

(5,1)

(6,0)

(7,0)

(5,2)

(6,1)

(4,4)

(4,3)

(6,2)

(5,3)

(7,1)

(8,0)

(5,4)

(8,1)

(9,0)

(6,4)

(10,0)

(7,2) (8,2)

(9,1)

(7,3)

(6,3)

(5,5)

(8,7)

(9,6)

(10,5)

(11,4)

(12,3)

(13,2)

(14,1)

(15,0)

Fig. 6 of Xiang et al.

13

(a)

(b)

B

N

H

Fig. 7 of Xiang et al.