phys. stat. sol (a) 195, No. 1, 3 – 10 (2003)/DOI theDOIprefix/theDOIsuffix
3
Non-equilibrium Green function implementation of boundary
conditions for full band simulations of substrate-nanowire
structures
4
Cristian Rivas*1,2 and Roger Lake**1
5
6
1
7
8
Received zzz, revised zzz, accepted zzz
Published online zzz
9
PACS 71.20Mq, 72. , 73. , 73.20.At, 73.21.Hb, 73.63.Nm
1
2
2
1
Department of Electrical Engineering, University of California, Riverside, CA 92521-0204
Eric Jonsson School of Engineering, University of Texas at Dallas, Richardson, TX 75083-0688
10
11
12
13
14
15
16
17
18
19
20
The density of states of a nanowire exhibits peaks at energies in which the individual transverse modes
begin to propagate. Under ideal conditions these modes are eigenstates solely determined by the cross
sectional size and shape of the nanowire. However, for realistic nanowires, which are grown on semiconductor substrates, the density of states of the subsrate-nanowire structure is dependent on the distance
from the nanowire endpoints. Near the substrate, the density of states is is nonzero far below the energy
corresponding to the first eigenstate of the ideal nanowire. This initial increase in the density of states occurs at energies near the conduction and valence band edges of the semiconductor substrate on which the
nanowire is grown. Away from the substrate, the density of states begins to acquire ideal nanowire characteristics. In the present work this effect is captured by imposing boundary conditions, with properties of
bulk material, at the nanowire base. The calculations utilize a first neighbor sp3s*d5 orbital basis within
the non-equilibrium Green function formalism.
21
Introduction
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
The atomic dimensions of the crystal lattice, the random statistical fluctuations of dopants and alloys,
and the finite, countable number of electrons are factors which will ultimately limit the level of
miniaturization of future Si-based devices. However, a better understanding of the electrical behavior of
nano-devices is needed at the design stages of the process. The utilization of device physics concepts
beyond tranditional CMOS will require the aid of computational tools capable of modelling the quantum
transport of a finite number of electrons. A set of devices falling under this category are self-assembled
Si/SiGe nanopillars, which utilize Si nanowires oxidized to a certain radius [1] and contain Ge rich cores
[2], possibly displaying quantum dot behavior. Prediction of the electrical characteristics of such devices
is cleary necessary but requires the development of a three-dimentional (3D), atomistic, full-band
simulator. In this work, the starting point for such a tool is the non-equilibrium Green function
formalism, which is described in detail in [3] for a 1D planar system. Utilizing an sp3s*d5 localized
orbital model [4], we study the effects of incorporating more realistic boundary conditions at the
nanowire base, which is composed of bulk semiconductor and illustrated in Figure 1. We present a more
efficient algorithm than iterative methods [5] for computing the surface green functions of a nanowire.
Comparisons are made using results obtained with the application of ideal nanowire boundary
conditions at both ends. The density of states (DOS) is analyzed along the the axis of the nanowire in
oder to study surface effects emanating from the substrate base.
Generalized eigenvalue problem for substrate base
*
**
Corresponding author: e-mail: [email protected]
[email protected]
© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0031-8965/03/0101-0003 $ 17.50+.50/0
000000 physica (a) 193/3
WordXP
Art.: W0000/Autor
H:\cristian\work_2001_backup_b\NSF_4_year_contract\FY_2003_work\San_Francisco_Feb_2003\motorola\word\FIN
AL_COPY\email_alex_demkov_02_05_03\CR_RL_2002\CR_RL_3D_bcs.doc
&R31; insgesamt 7 Seiten
Diskette
17.01.2003
Bearb.:
2
Author, Author, and Author: Short title
1
2
3
4
Our approach uses two Hamiltonians in the sp3s*d5 orbital basis [4] for a zincblende lattice. The first
Hamiltonian (H1D) is formulated in planar form as in [6], but the second Hamiltonian (H3D) is formulated
in non planar form as in [5]. We begin by formulating the planar problem and write two Schrodinger
equations,
5
Dn , B ;n , A χ n , A + Dn , B ;n , B χ n , B + tn , B ;n +1, A χ n +1, A = I ⋅ E χ n , B ,
tn, A;n −1, B χ n −1, B + Dn , A;n, A χ n, A + Dn , A;n, B χ n, B = I ⋅ E χ n , A
(1.1)
6
7
8
where the index n indicates the location of the zincblende primitive basis, which is composed of atoms
A and B at positions 0,0,0 and a/4,a/4,a/4 relative to each other, as illustrated in Figure 2. Introducing a
propagation factor z with properties,
9
χ n +1, A = z χ n, A ,
10
11
12
χ n , B = z χ n −1, B
allows Equations (1.1) to be rewritten as a generalized eigenvalue problem,
 − Dn, B ;n , A

0

I ⋅ E − Dn , B;n, B 
 tn , B;( n +1), A
χ = z

tn, A;( n −1), B 
 I ⋅ E − Dn , A;n, A

χ,
− Dn, A;n , B 
0
(1.3)
which produces a matrix of eigenvectors
13
14
(1.2)
 χ AL
 L
 χB
χ AR 

χ BR 
(1.4)
 zL

 0
0 
.
zR 
(1.5)
and a matrix of eigenvalues
15
Dn , B ;n , A , appearing in equation (1.3) is of size 10, the number of sp3s*d5
16
Each matrix block, such as
17
18
orbitals, but this size increases to 20 if spin orbit coupling is included in the calculation.
19
20
21
In order to obtain an expression for the surface green function of the substrate we perform a matrix operation of the type
(1.6)
( I ⋅ E − h) g − I = 0
22
where the matrix
Green function for substrate surface
 Dn, A;n , A
t
h =  n , B ;n , A
 0

 0
23
24
0
tn , B;( n +1), A
r
r
0
0 
r

r
(1.7)
is block tridiagonal and
 g n , A; n , A
g
g =  n , B ;n , A
 
 25
26
t n , A; n , B
Dn, B ;n , B
r
0
g n , A; n , B
g n , B ;n , B





(1.8)
is a full matrix. Utilizing the upper left corner elements of (1.6) yields the equations
000000 physica (a) 193/3
WordXP
Art.: W0000/Autor
H:\cristian\work_2001_backup_b\NSF_4_year_contract\FY_2003_work\San_Francisco_Feb_2003\motorola\word\FIN
AL_COPY\email_alex_demkov_02_05_03\CR_RL_2002\CR_RL_3D_bcs.doc
&R31; insgesamt 7 Seiten
Diskette
17.01.2003
Bearb.:
pssa header will be provided by the publisher
−tn, A;n, B g n, B ;n , A + ( I ⋅ E − Dn, A;n , A ) g n, A;n , A − I = 0
(1.9)
−tn , B ;( n +1), A g ( n +1), A;n , A + ( I ⋅ E − Dn, B ;n , B ) g n, B ;n , A − tn , B ;n , A g n , A;n , A = 0.
(1.10)
1
2
3
4
and
3
The remaining equation needed is obtained by expanding the green function,
g ( n +1), A, β ;n , A, µ = ∑ (n + 1), A, β (n + 1) χ RA, j ×
j ,φ
5
(n + 1) χ
(1.11)
n, A, φ n, A, φ g n, A, µ ,
R
A, j
6
in the last expression in terms of dummy sets of complete states. Utilizing the second propagation in
7
Equation set (1.2) and the orthogonality relations between the various
8
tion (1.11) in matrix form,
11
12
13
allows one to rewrite Equa-
−1
g( n +1), A;n, A = χ RA zR  χ RA  g n , A;n, A .
9
10
χ RA, j
Combining Equations (1.9), (1.10) , and (1.12) yields
g n , A; n , A
(
(1.12)
)
−1
−1
−1
 −t
tn , B ;( n+1), A χ RA zR  χ RA  + tn , B ;n , A + 
n , A; n , B ( I ⋅ E − Dn , B ; n , B )
 , (1.13)
=
I ⋅ E − D

n , A; n , A


which is wavevector dependent. The surface green function of the substrate is finally transformed into
the nanowire basis via
g n , A;n, A (r, r', E ) =
14
1
VBZ
d 2k ik ⋅( r −r')
g n, A;n , A (k , E ),
∫ 4π e
BZ
(1.14)
15
16
17
18
where VBZ is a two dimensional integral with an integrand of unity. The position vectors r and r' span
atomic positions in the first atomic plane of the nanowire primitive basis, which is discussed in the following section.
19
20
21
22
For the boundary at the non substrate end of the nanowire in Figure 1 the formulation of the eigenvalue
problem involves layers, which are composed of atoms belonging to the nanowire primitive basis, as
shown in the Figure 3. We begin by writing the analog of the equations in (1.1) as
t0;−1 χ −1 + D0;0 χ 0 + t0;1 χ1 = I ⋅ E χ 0 ,
(1.15)
23
24
where the subscripts are layer indices. Utilizing the properties of
the generalized eigenvalue problem
Generalized eigenvalue problem for ideal wire
 t0;−1 0 
 I ⋅ E − D0;0
 0 I χ = z
I



25
26
−t0;1 
χ,
0 
(1.16)
which produces the matrix of eigenvectors
27
28
z in (1.2) leads to the formulation of
 χ −L1
 L
 χ0
χ −R1 

χ 0R 
(1.17)
 zL

 0
0 
.
zR 
(1.18)
and the matrix of eigenvalues
29
000000 physica (a) 193/3
&R31; insgesamt 4 Seiten
WordXP
Diskette
Art.: W0000/Autor
19.08.2002
V:\Lohnbelichtung\pha\STEHSATZ\vorlage_WordXP\pss_2000.dot
Bearb.:
4
1
2
3
Author, Author, and Author: Short title
Each of the four matrix blocks appearing on either side of (1.16) is of size 10N due to the 10 sp3s*d5
orbitals and N atomic positions inside the nanowire primitive basis. This size is doubled if spin orbit
coupling is included in the calculation. Utilizing Equation (1.6) and the block matrices,
  h=
 0 t−1;−2

0
0
4
5


g =



and
10
11
g −1;−1
g0;−1


,
g −1;0 

g 0;0 
(1.20)
−t0;−1 g −1;0 + ( I ⋅ E − D0;0 ) g 0;0 − I = 0
(1.21)
−t−1;−2 g −2;0 + ( I ⋅ E − D−1;−1 ) g −1;0 − t−1;0 g 0;0 = 0.
(1.22)
As was done with Equation (1.11), one can perform the expansion of the green function,
g −2, β ;0, µ = ∑ −2, β −2 χ Lj
12
j ,φ
13
(1.19)
one can extract the lower right elements,
8
9
0 
0 
t−1;0 

D0;0 
and
6
7
0
D−1;−1
t0;−1
−2 χ Lj 0, φ 0, φ g 0, µ ,
(1.23)
appearing in the last expression. The matrix equation,
−1
−1
−1
g −2;0 = χ L  z L   z L   χ L  g0;0 ,
14
(1.24)
15
can then be written with the aid of the equations in (1.2) and orthogonality between the various
16
Combining equations (1.21), (1.22), and (1.24) leads to the surface green function
17
g 0;0
(
)
χ Lj .
−1
 −t ( I ⋅ E − D )−1 t χ L  z L  −1  z L  −1  χ L  −1 + t

0; −1
−1; −1
−1;−2 0 
−1;0 +
    0 
 .
=
I ⋅ E − D

0;0


(1.25)
18
19
20
21
22
This method of obtaining the ideal nanowire surface green function is more efficient than iterative methods [5] since it does not involve repetitive calculations, which may depend on the energy E . However,
as a backup option [6] we have available an iteration routine, which is based on the recursive Green
function algorithm discussed in the following section.
23
24
25
26
27
For the computation of transport properties we utilize the recursive Green function algorithm discussed
in [3]. We begin by separating the nanowire primitive basis into 4 layers labelled A-D as shown in the
Figure 4. For the nanowire segment, which is making contact with the substrate, the recursive Green
function algorithm begins with the equation (1.14) and creates right connected green functions by travelling along the length of the nanowire via the repetitive use of the relations
Recursive Green function algorithm
000000 physica (a) 193/3
WordXP
Art.: W0000/Autor
H:\cristian\work_2001_backup_b\NSF_4_year_contract\FY_2003_work\San_Francisco_Feb_2003\motorola\word\FIN
AL_COPY\email_alex_demkov_02_05_03\CR_RL_2002\CR_RL_3D_bcs.doc
&R31; insgesamt 7 Seiten
Diskette
17.01.2003
Bearb.:
pssa header will be provided by the publisher
5
g(Rn −1), D;( n −1), D =  I ⋅ E − D( n −1), D;( n −1), D − t( n−1), D;n , A gR
n , A;n , At n , A;( n −1), D 

−1
g(Rn −1),C ;( n −1),C =  I ⋅ E − D( n −1),C ;( n −1),C − t( n −1),C ;( n−1), D g R
( n −1), D ;( n −1), D t( n −1), D ;( n −1),C 

1
g(Rn −1), B ;( n −1), B =  I ⋅ E − D( n −1), B ;( n −1), B − t( n −1), B ;( n −1),C gR
( n −1),C ;( n −1),C t( n −1),C ;( n −1), B 

−1
(1.26)
−1
−1
R
gR
( n −1), A;( n −1), A = 
 I ⋅ E − D( n −1), A;( n −1), A − t( n−1), A;( n −1), B g( n −1), B;( n −1), B t( n −1), B;( n −1), A  .
2
3
4
At the opposite end of the nanowire the algorithm begins with the lower right corner block of (1.25) and
creates left connected green functions by travelling along the length of the nanowire via the use of the
relations
g1,LA;1, A =  I ⋅ E − D1, A;1, A − t1, A;0, D g 0,LD;0, Dt0, D;1, A 
g1,LB;1, B =  I ⋅ E − D1, B;1, B − t1, B ;1, A g1,LA;1, At1, A;1, B 
5
L
1,C ;1,C
g
−1
−1
=  I ⋅ E − D1,C ;1,C − t1,C ;1, B g1,LB;1, B t1, B ;1,C 
(1.27)
−1
−1
g1,LD;1, D =  I ⋅ E − D1, D;1, D − t1, D;1,C g1,LC ;1,C t1,C ;1, D  .
6
At the observation plane, the density of states may be computed by
DOS =
7
∑
j = A , B ,C , D
8
where the various
−2 Im tr { g L , j ;L , j },
(1.28)
g L , j ; L , j are obtained from
g L , A;L , A =  I ⋅ E − DL , A; L , A − t L , A; L −1, D g LL−1, D; L −1, Dt L −1, D; L , A − t L , A; L , B gRL , B; L , B t L , B; L , A 
g L , B ;L , B =  I ⋅ E − DL , B ; L , B − t L , B ; L , A g LL, A; L , At L , A; L , B − t L , B ;L ,C gRL ,C ; L ,C t L ,C ; L , B 
−1
g L ,C ;L ,C =  I ⋅ E − DL ,C ; L ,C − t L ,C ;L , B g LL, B ; L , B t L , B ; L ,C − t L ,C ; L , D g R
L , D ; L , D t L , D ; L ,C 

9
−1
−1
(1.29)
−1
 I ⋅ E − DL , D; L , D − t L , D; L ,C g LL,C ; L ,C t L ,C ; L , D − 
g L, D;L, D = 
 ,
R
t L , D; L +1, A g L +1, A; L +1, At L +1, A; L , D

where the green functions marked with a L belong to the set of equations (1.27) which are connected to
the ideal wire surface green function. The label R indicates the green functions from the group in
10
11
12
13
(1.26) which are connected to the surface green function of the substrate. The transmission through the
complete substrate-nanowire structure can then be obtained from
14
T = tr {Γ( A − g L , D ;L , D Γg L† , D ; L , D )}
(1.30)
Γ = i ( Σ − Σ† )
(1.31)
A = i ( g L , D; L , D − g L† , D ; L , D )
(1.32)
15
[3], where
16
17
and
18
19
20
are spectral functions. The self energy,
Σ = t L , D; L ,C g LL,C ;L ,C t L ,C ; L , D ,
21
000000 physica (a) 193/3
&R31; insgesamt 4 Seiten
WordXP
Diskette
Art.: W0000/Autor
19.08.2002
(1.33)
V:\Lohnbelichtung\pha\STEHSATZ\vorlage_WordXP\pss_2000.dot
Bearb.:
6
Author, Author, and Author: Short title
1
2
3
4
is obtained from the third term inside the last expression in (1.29). Each trace appearing in equation
(1.28) is over the atoms which compose each atomic plane of the nanowire primitive basis, and the trace
in (1.30) is over the atoms composing the last atomic plane in the nanowire primitive basis.
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
The transport properties of a nanowire depend on the shape and size of the wire cross section, but these
properties also depend on the contacting surfaces at the wire endpoints. The DOS and transmission
through the structure depicted in Figure 1 will be analyzed numerically utilizing the theory previously
discussed. The silicon atoms along the circular perimeter of the nanowire will be terminated with hydrogen atoms in order to eliminate surface effects associated with such boundary [5]. Figure 5 displays the
results of computing Equations (1.28) and (1.30) for two different cases. The first case does not incorporate a substrate-nanowire contact and amounts to employing Equation (1.25) and its analog for both ends
of the nanowire. The transmission values extending from unity to a value of six are due to the quantization of the six X valleys of silicon. The DOS peaks sharply at each energy corresponding to the onset
of propagation of a transverse mode as mentioned in Ref. [7], which analyzes a 2D quantization simulation. The second case investigated corresponds to again utilizing equation (1.25) for one end of the
nanowire, but equation (1.14) is incorporated for the opposite end. This condition is investigated for two
different values of an imaginary potential η , which is added to the total energy E in order to simulate
diffusive scattering events at each end of the nanowire. For an imaginary potential of 10meV the DOS
and transmission through the structure clearly become diffuse in comparison to the ideal wire characteristics. In addition, the DOS develops a long tail towards lower energies due to the lower band edge of
bulk silicon, which is located at the base of the nanowire. For an imaginary potential of 1meV the DOS
and transmission become oscillatory but follow closely the 10meV characteristics. The nature of these
oscillations is under investigation, but it is believed that they are due to the spurious reflections caused
by the energy difference between the band edges of the nanowire and its bulk silicon base. Figures 6
and 7 display a comparison between the ideal wire characteristics of Figure 5 and the 10meV substrate
characteristics evaluated at distances of 20 and 40 atomic planes from the base of the nanowire. Away
from the substrate, the transmission and DOS clearly begin to acquire properties associated with the
nanowire.
30
31
32
33
34
35
36
37
38
A step has been taken toward the 3D simulation of nanopillars, which are grown on semiconductor
substrates. An atomistic full-band simulator has been developed in order to gain an improved
understanding of the electrical characteristics of such structures. The effect that the substrate has on the
DOS and transmission through the overall structure has been studied by incorporating a non-ideal
nanowire boundary condition, which exhibits bulk material properties, at the base of the nanowire. The
DOS and transmission characteristics are found to depend on the distance from the nanowire base. Close
to the substrate the DOS possesses a long energy tail consistent with the bandedge of bulk silicon at the
base of the nanowire. As the distance from the substrate increases the DOS acquires properties
associated with an ideal wire, which is not grown on a semiconductor substrate.
Results
Conclusion
39
Fig. 1
Silicon nanopillar grown on semiconductor substrate.
40
Fig. 2
Zincblende lattice projected on the y-z plane.
41
Fig. 3
Two sets of atoms composing two repetitions of the primitive basis of the nanowire in Figure 1.
42
Fig. 4
Atomic planes composing the primitive basis of the nanowire in Figure 1.
43
44
Fig. 5 Transmission and DOS characteristics for ideal nanowire and nanowire on substrate and evaluated at a distance of one atomic plane from the substrate.
45
46
Fig. 6 . Transmission and DOS characteristics for ideal nanowire and nanowire on substrate and evaluated
at a distance of 20 atomic planes from the substrate.
000000 physica (a) 193/3
WordXP
Art.: W0000/Autor
H:\cristian\work_2001_backup_b\NSF_4_year_contract\FY_2003_work\San_Francisco_Feb_2003\motorola\word\FIN
AL_COPY\email_alex_demkov_02_05_03\CR_RL_2002\CR_RL_3D_bcs.doc
&R31; insgesamt 7 Seiten
Diskette
17.01.2003
Bearb.:
pssa header will be provided by the publisher
7
1
2
Fig. 7 . Transmission and DOS characteristics for ideal nanowire and nanowire on substrate and evaluated
at a distance of 40 atomic planes from the substrate.
3
4
5
6
7
8
Acknowledgements This work was supported by the NSF NIRT grant DMR-0103248 and DOD/DARPA/DMEA
under grant number DMEA90-02-2-0216. The computer simulations were performed on the UCR Institute of Geophysics and Planetary Physics (IGPP) Beowulf computer Lupin. The authors are grateful to Young J. Ko for providing a set of sp3s*d5 parameters for silicon termination. These parameters permitted the verification of the operation of the 3D simulator and allowed the authors to draw comparisons with genetically optimized parameters for Si
and Ge termination.
References
9
10
11
12
13
14
15
16
17
18
[1]
[2]
[3]
[4]
[5]
[6]
H. I. Liu, D. K. Biegelsen, F. A. Ponce, N. M. Johnson, and R. F. W. Pease, Appl. Phys. Lett., 64, 1383 (1994).
D. Nayak, K. Kamjoo, J. C. S. Woo, J. S. Park, and K. L. Wang, Appl. Phys. Lett., 56, 66 (1989).
R. Lake, G. Klimeck, R. C. Bowen and D. Jovanovic, J. of Appl. Phys., 81, 7845 (1997).
J. M. Jancu, R. Scholz, F. Beltram, and F. Bassani, Phys. Rev. B, 57, 6493 (1998).
Y-J. Ko, M. Shin, S. Lee, and K. W. Park, J. Appl. Phys., 89, 374 (2001).
C. Rivas, R. Lake, G. Klimeck, W. R. Frensley, M. V. Fischetti, P. E. Thompson, S. L. Rommel, and P. R.
Berger, Appl. Phys. Lett., 78, 814 (2001).
[7] A. Svizhenko, M. P. Anantram, T. R. Govindan, B. Biegel, And R. Venugopal, J. Appl. Phys., 91, 2343 (2002).
000000 physica (a) 193/3
&R31; insgesamt 4 Seiten
WordXP
Diskette
Art.: W0000/Autor
19.08.2002
V:\Lohnbelichtung\pha\STEHSATZ\vorlage_WordXP\pss_2000.dot
Bearb.:
Layer n+1
3.6
B
A
3.2
B
Layer n
z (a)
3.4
3
A
2.8
0.4
0.6
0.8
y (a)
1
1.2
Layer 1
4
3
Layer 0
z (a)
3.5
2.5
2
3
y (a)
2
1
2
x (a)
3
A
3
2.8
D
z(a)
2.6
C
2.4
2.2
B
2
A
1.8
3
y(a)
2
1
1
2
3
x (a)
160
6
2
140
120
100
80
0
60
-2
40
-4
-6
1.8
20
1.9
2
2.1 2.2 2.3
Energy (eV)
2.4
0
2.5
DOS (1/eV)
Transmission
4
ideal wire
wire on substrate
h=1meV
wire on substrate
h=10meV
160
6
4
ideal wire
wire on substrate
140
100
80
0
60
-2
40
-4
-6
1.8
20
1.9
2
2.1 2.2 2.3
Energy (eV)
2.4
0
2.5
DOS (1/eV)
Transmission
120
2
160
6
ideal wire
wire on substrate
4
140
100
80
0
60
-2
40
-4
-6
1.8
20
1.9
2
2.1 2.2 2.3
Energy (eV)
2.4
0
2.5
DOS (1/eV)
Transmission
120
2