Lecture Notes for Nanolab spring school 19 - 23 May 2003, Toulouse, France
Single Particle Transport in Nanostructures : Theory, Implementation and Examples
M.P. Anantram
NASA Ames Research Center, Mail Stop: N229-1,
Moffett Field, CA 94035-1000, USA
E-Mail: [email protected]
(Dated: May 1, 2003)
This course introduces elements of the Landauer-Buttiker and the Non-Equilibrium Green’s Function methods, in modeling transport through nanostructures. The equations and algorithms used
in modeling these structures are discussed. Examples relevant to silicon nanoelectronics (nanotransistors/MOSFET), molecular structures (carbon nanotubes) and bio-inspired systems (DNA) are
discussed. An example of the use of molecular dynamics in determining the transport properties of
nanostructures (deformed nanotubes) is also discussed.
Lectures 1 and 2: Theory and Algorithm
Lectures 3 and 4: Applications: Nanotransistors, Carbon Nanotubes, DNA
Copyright: Lecture notes for most part are based on papers published in Physical Review, Journal
of Applied Physics and Applied Physics Letters. These journals hold the copyright.
2
I
Non rigorous introduction to Green’s functions
I A Uniform Tight binding Hamiltonian
I B Tight binding Hamiltonian for a one dimensional device
I C Eliminating the Left and Right semi-infinite leads
I D Landauer-Buttiker Approach
I E Relationship to Green’s functions and self-energies
I F Self-energy due to scattering
I G Examples
II
Summary of the Non-equilibrium Green’s Function Equations Solved
II A Crib Sheet
III Dyson’s equation for layered structures
III A Dyson’s equation for Gr
III B Dyson’s equation for G<
IV Algorithm to calculate Gr and G<
IV A Recursive algorithm for Gr
IV B Recursive algorithm for G<
V
Two Dimensional Solution of Ballistic MOSFETs: A computational experiment
V A The governing equations
V B Gr and G< : Discretized matrix equations
V C Expressions for Contact Self-energies (ΣrS , ΣrD and ΣrP )
V D Results and Discussion
V E Id versus Vg - Effect of polysilicon depletion region
V F Id versus Vg - Comparison to Medici
V G Id versus Vd
V H Isotropic versus anisotropic effective mass
V I Gate leakage current
V J 2D Ballistic MOSFET Summary
Lectures 2 and 3 (supplemenatry material):
See attached papers:
- Role of scattering in nanotransistors
- Conductance in carbon nanotubes with defects: A numerical study
- Electronic Transport through Carbon Nanotubes: Effects of Structural Deformation and Tube Chirality
- Environment and structure influence on DNA conduction
3
References
Some useful journal references on the non-equilibrium Green’s function method:
1) C. Caroli, R. Combescot, P. Nozieres and D. Saint-James, Direct calculation of the tunneling current, J. Phys. C:
Solid St. Phys. 4, 916 (1971);
2) G. D. Mahan, Quantum transport equation for electric and magnetic fields, Physics Reports 145, 251 (1987); A
number of useful relationships involving the non equilibrium Green’s functions can be found in this paper; Discussion
of electron-phonon interaction in the non-equilibrium Green’s function method; Relationship to Boltzmann equation
and much more); Relevant to sections I and II.
3) Y. Meir and N.S. Wingreen, Landauer Formula for the Current through an Interacting Electron Region, Phys. Rev.
Lett. 68, 2512 (1992). Relevant to section II.
4) R. Lake, G. Klimeck, R. C. Bowen and D. Jovanovic, Single and multiband modeling of quantum electron transport
through layered semiconductor devices, J. Appl. Phys. 81, 7845 (1997). Relevant to section II.
5) A. Svizhenko et al., Two Dimensional Quantum Mechanical Modeling of Nanotransistors, J. of Appl. Phys., 91,
2343 (2002). Relevant to sections III, IV and V.
Some useful books:
1) G. D. Mahan, Many Particle Physics, Second Edition, Plenum Publishing Corporation, New York, 1990. Some
discussion on non-equilibrium Green’s function but has extensive discussions on equilibrium Green’s function; Relevant
to electron-phonon scattering.
2) S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, UK, 1997. Very
nice discussions of the non-equilibrium Green’s function method and covers a lot of nanoscale physics; Relevant to
sections I and II.
For the theoretically inclined (Extensive discussion of non-equilibrium Green’s function theory.):
1) L. V. Keldysh, Diagram Technique for nonequilibrium processes Sov. Phys. JETP, 20, 1018 (1965)
2) R. A. Craig, Perturbation Expansion of Real-Time Green’s Function, J. of Math. Phys., 9, 605 (1968)
3) P. Danielewicz, Quantum Theory of Nonequilibrium Processes, 1, Annals of Physics, 152, 239 (1984)
4
I.
NON RIGOROUS INTRODUCTION TO GREEN’S FUNCTIONS
The purpose of this section is to start from Schrodinger’s equation for a ”Device” and transform it to a typical
Green’s function equation for the device, using familiar concepts from college quantum mechanics. This section will
help the reader get acquainted with the concepts of self-energy, and retarded and less-than Green’s function, starting
from Schrodinger’s equation. While useful, this section is not a substitute to more rigorous field theoretical treatments
that are valid in the presence of interactions such as the electron-phonon and electron-electron interactions.
A.
Uniform Tight binding Hamiltonian
Consider a system described by a set of one dimensional grid / lattice points with uniform spacing a. The diagonal
elements of the Hamiltonian (potential / on-site potential) and the first off-diagonal element of the Hamiltonian t
representing the interaction between nearest neighbor grid points i and i + 1, do not vary with position. Further and t are assumed to be real. The Hamiltonian of such an uniform system is,

• • •
 • •

•


−t E − −t




−t E − −t
(E − H)Ψ = 0 → 



−t E − −t



•
• • 
• • •

or
•
•
Ψ−1
Ψ0
Ψ+1
•
•





=0



−tΨq−1 + (E − )Ψq − tΨq+1 = 0,
(1)
(2)
where, E is the energy and Ψq is the wave function at grid point q. The solution of Eq. (2) can be verified to be,
E = + 2tcos(ka)
Ψq = eikqa .
(3)
(4)
2at
1 ∂E
=−
sin(ka) .
~ ∂k
~
(5)
The velocity is given by,
v=
Semi-infinite Uniform Tight binding Hamiltonian: Consider a system described by grid points placed uniformly with spacing a, in half-space, z ≤ 0. The region z ≥ 0 is vacuum. and t do not vary in this half space. The
Hamiltonian of such a semi-infinit semi-infinit uniform system is (for z ≤ 0),
−tΨq−1 + (E − )Ψq − tΨq+1 = 0 for q ≤ -1 and
−tΨ−1 + (E − )Ψ0 = 0.
B.
(6)
(7)
Tight binding Hamiltonian for a one dimensional device
The nearest neighbor tight binding Hamiltonian with variable potential in the Device region (variable q and tq ) is
defined by,
tq,q−1 Ψq−1 + (E − q )Ψq + tq,q+1 Ψq+1 = 0.
(8)
q is the on-site potential at grid point q. tq,q+1 is the Hamiltonian element connecting grid points q and q + 1 and
tq+1,q = t†q,q+1 .
5
FIG. 1:
We divide our device into three regions (Fig. 1): semi-infinite Left lead (L), Device (D) and semi-infinite Right lead
(R). The potential (on-site potential) and hopping parameter of the Left (Right) lead are held fixed at l (r ) and tl
(tr ) respectively. Schrodinger’s equation for this system can be written as,



EI − Hl
−Hld
0
Ψl
 −Hdl EI − HD −Hdr   ΨD  = 0 ,
(9)
0
−Hrd EI − Hr
Ψr
where,
• • •
•
 • •

−tl E − l −tl
EI − Hl = 

−tl E − l −tl
−tl E − l






E − r −tr
 −tr E − r −tr

−tr E − r −tr
EI − Hr = 

•
• •
• • •

E − 1 −t1,2
 −t2,1 E − 2 −t2,3

•
• •


• •
•
=

•
•
•


−tn−1,n−2 E − n−1 −tn−1,n
−tn,n−1 E − n

EI − HD
Hl and Hr are semi-infinite matrices as shown in Eq. (10).
C.



 (10)






 .



(11)
Eliminating the Left and Right semi-infinite leads
Schrodinger’s equation for the device with open boundaries / leads [Fig. 1, Eqs. (8) and (9)] is:
•
•
−tl Ψ−l1 + (E − l )Ψ−l0 − tl,d Ψ1
−td,l Ψ−l0 + (E − 1 )Ψ1 − t1,2 Ψ2
−t1,2 Ψ1 + (E − 2 )Ψ2 − t2,3 Ψ3
−tn−1,n Ψn−1 + (E − n )Ψn − tdr Ψrn+1
−tr,d Ψn + (E − r )Ψrn+1 − tr Ψrn+2
•
•
=
=
=
=
=
0
0
0
0
0
(12)
(13)
(14)
(15)
(16)
6
where, the bullets represent the semi-infinite Left and Right leads. The subscript lm (rm) of the wave function in the
Left (Right) lead represents grid point m of the Left (Right) lead. The numbering scheme for grid points is shown in
(L)
(R)
(loc)
Fig. 1. The wave function in region D (ΨD in Eq. (9) → {ΨD , ΨD ΨD }) can be thought to arise from:
(L)
(i) waves incident from the Left lead (ΨD ),
(R)
(ii) waves incident from the Right lead (ΨD ), and
(loc)
(iii) localized states in Device region (ΨD ).
Waves incident from the Left lead: Terminating the semi-infinite Left and Right leads
The wave function due to a wave with wave number kl that is incident from the Left lead, upon scattering from
the device is,
Ψln = (e+ikl n + sll e−ikl n )uln
Ψrn = srl eikr n urn
in region L
in region R,
(17)
(18)
and in the left lead [from Eq. (3)]:
E − l = 2tl cos(kl a) = tl (eikl a + e−ikl a ) .
(19)
sll and srl are the reflection and transmission amplitudes. Substituting Eqs. (17) and (19) in Eq. (12) helps terminate
the left semi infinite region as shown below. This substitution yields,
sll ul = t−1
l (−tl ul + tld Ψ1 ) .
(20)
Substituting Eqs. (17) and (20) in Eq. (13), we get,
(E − 1 − td,l e+ikl a t−1
l tl,d )Ψ1 + t1,2 Ψ2 = −2itd,l sin(kl a)ul
(21)
Eq. (21) is a modification of Schrodinger’s equation centered at grid point 1 of the Device to include the influence of
the entire semi-infinite Left lead.
Similarly, substituting Eqs. (18) and E − r = 2tr cos(kr a) in Eq. (16) we get,
srl ur = t−1
r td,r Ψn .
(22)
Substituting Eqs. (18) and (22) in Eq. (15) we can terminate the right semi infinite region:
−tn−1,n Ψn−1 + (E − n − td,r eikr t−1
r tr,d )Ψn = 0
(23)
Eq. (23) is a modification of Schrodinger’s equation centered at grid point n of the Device to include the influence of
the entire semi-infinite Right lead.
Rewriting equation (9) as a n x n matrix: The influence of the semi-infinite Left and Right leads of Eqs. (9)
have been folded into grid points 1 and n of the device, for the waves incident from the Left lead [Eqs. (21) and (23)].
(L)
The wave function in the device (ΨD ) due to waves incident from the left lead can now be obtained by solving:
(L)
AΨD = iL ,
(24)
(L)
where, A is a square matrix of dimension n, ΨD and iL are n by 1 vectors. iL is the source function at (k, E) due
to the Left lead. Matrix A is,
A = EI − HD − ΣrLead .
(25)
ΣrLead 1,1 = ΣrL = td,l e+ikl a t−1
l tl,d
(26)
The only non zero elements of ΣrLead are,
ΣrLead n,n
=
ΣrR
=
td,r e+ikr a t−1
r tr,d
.
(27)
Matrix A is not Hermitian due to the ΣrL and ΣrR terms. ΣrL and ΣrR are called the self-energies due to the Left and
Right leads. They represent the influence of the semi-infinite leads on the Device and are typically complex numbers.
7
The real part of the self-energy shifts the on-site potential at grid point 1 by 1 + Re(ΣrL ), and −2Im[ΣrL ] represents
the scattering rate of electrons from the Left lead to the Device.
For clarity, the only non zero elements of A and iL are:
A(1, 1) = E − 1 − ΣrL and A(n, n) = E − n − ΣrR
(28)
A(i, i) = E − i , A(i, i + 1) = −ti,i+1 and A(i + 1, i) =
−t†i,i+1
(29)
iL (1) = −2itd,l sin(kl a)ul ,
(30)
and the expanded form of Eqs. (24) is:

E − 1 − ΣrL −t1,2
0
•
•
•
†

−t
E
−
−t
0
•
•
2
2,3
1,2

†

0
−t2,3 E − 3 −t3,4
0
•


•
•
•
•
•
•


•
•
•
•
•
•
•
•
•
0 −t†n−1,n E − n − ΣrR
  (L)
Ψ1
(L)

Ψ2

(L)

Ψ3


•


 •
(L)
Ψn


−2itd,l sin(kl a)ul

 
0
 
0
 
=
•
 
 
•

0




 .


(31)
Waves incident from the Right lead: Terminating the semi-infinite Left and Right leads
The wave function due to a wave with wave number kr that is incident from the Right lead, upon scattering from
the device is,
Ψrn = (e−ikr n + srr e−ikr n )urn
Ψln = slr eikl n uln
in region R
in region L,
(32)
(33)
and in the right lead [from Eq. (3)]
E − r = 2tr cos(kr a) = tr (eikr a + e−ikr a ) .
(34)
srr and slr are the reflection and transmission amplitudes for waves incident from the Right lead. Substituting Eqs.
(32) and (34) in Eq. (16) helps terminate the right semi infinite region as shown below. This substitution yields,
srr ur = t−1
r (−tr ur + trd Ψn) .
(35)
Using Eq. (35), Eq. (15) can be written as,
−tn−1,n Ψn−1 + (E − n − td,r e+ikr a t−1
r tr,d )Ψn = −2itd,r sin(kr a)ur .
(36)
(R)
The wave function in the device (ΨD ) due to waves incident from the right lead, can now be obtained by solving:
(R)
AΨD = iR ,
(37)
where, the definition of A is identical to that given above in Eqs. (25), (28) and (29), and i R , the source function at
(k, E) due to the Right lead is given by,
ir (n) = −2itd,r sin(kr a)ur
and
ir (i 6= n) = 0 .
The equivalent of Eq. (31) for waves incident from the right is,

E − d1 − ΣrL td1,d2
0
•
•
•
†

td1,d2
E − d2 td2,d3
0
•
•

†

0
t
E
−
t
0
•
d3 d3,d4
d2,d3


•
•
•
•
•
•


•
•
•
•
•
•
•
•
•
0 t†dn−1,dn E − dn − ΣrR
  (R)
Ψ1
(R)


Ψ
 2
  Ψ(R)
 3
 •

 •
(R)
Ψn



 
 
 
=
 
 

(38)
0
0
0
•
•
−2itd,r sin(kr a)ur







(39)
Synopsis: Given a potential profile in region D and that the potentials in leads L and R do not vary with position,
the wave function at all energies in the region D, transmission amplitude srl and reflection amplitude sll for waves
incident from the Left lead can be obtained by solving Eqs. (24) and (37), which are matrix equations of dimension
n.
8
D.
Landauer-Buttiker Approach
The Landauer-Buttiker approach is valid for phase coherent structures. It relates Eqs. (24) and (37) to an observable
Q in the following manner:
(L)
(R)
1) Find the contribution to Q due to all ΨD and ΨD at all energies E.
2) The expectation value of an observable Q, is then composed of two terms. The first term corresponds to the
(L)
contribution from all waves incident from the left (ΨD ) weighted by the Fermi factors of the left contact. The second
term is identical expect that it corresponds to waves incident from the right. The expectation value of Q is,
X
(L)
(L)
(R)
(R)
< Q >=
< ΨD |Q|ΨD > fL (E)+ < ΨD |Q|ΨD > fR (E) ,
(40)
where, fL (E) and fR (E) are Fermi factors of electrons incident from the Left and Right leads respectively.
3) If region D is connected to three contacts L, R, and G, then,
X
(L)
(L)
(R)
(R)
(G)
(G)
< Q >=
< ΨD |Q|ΨD > fL (E)+ < ΨD |Q|ΨD > fR (E)+ < ΨD |Q|ΨD > fG (E) .
(41)
E
4) The existence of localized states in region D is neglected above and should be separately accounted for.
In device modeling, we are typically interested in electron density (n) and current (J) at grid point q. Using Eq.
(40),
2
(R) 2
ρq (E) = |Ψ(L)
q | fL (E) + |Ψq | fR (E) and
"
#
(L)
(R)
e~
(L) † dΨD
(R) † dΨD
ΨD
fL (E) + ΨD
fR (E) − c.c
< Jq→q+1 (E) > =
2mi
dx
dx
E.
(42)
.
(43)
q, q + 1
Relationship to Green’s functions and self-energies
In this section, we discuss how the electron density and current [Eqs. (42) and (43)] can be expressed in terms of
the Green’s functions Gr and G< in the phase coherent limit corresponding to Eqs. (24) and (37).
The definition of the retarded Green’s function corresponding to an equation ÔX = 0 is,
ÔGr = I,
(44)
The Green’s function corresponding to Schrodinger’s equation ([E − H]Ψ = 0, Eq. (9)) for the system composed of
the device and leads [Fig 1] is [E − H]Gr = I.
We have seen in the discussions leading to Eqs. (24) and (37) that the concept of self-energies help us fold in the
influence of the semi-infinite Left and Right leads into the left-most and right-most grid points of the Device region.
The retarded Green’s function of the Device with Left and Right leads is (see also the next section),
AGr = I,
(45)
where A is the matrix defined by Eqs. (28) and (29). Using the definition for Gr in Eqs. (24) and (37), the wave
function in the Device due to waves incident from the Left and Right leads can be written as,
(L)
ΨD
(R)
ΨD
= G r iL
and
(46)
r
= G ir .
(47)
(L)
(R)
Note that only two columns of Gr , Gr:,1 and Gr:,n are necessary to calculate ΨD and ΨD .
Electron Density: Using Eqs. (46) and (47) in Eq. (42), the electron density is,
X
< nq > =
Grq,1 iL i†L Ga1,q fL + Grq,n iR i†R Gan,q fR ,
X
=
Gr (q, 1)[4td,l sin2 (kl a)tl,d fL ]Ga (1, q) + Gr (q, n)[4td,r sin2 (kr a)tr,d fR ]Ga (n, q),
(48)
(49)
9
where Ga = Gr † . The summation over k can be converted to an integral over E by,
Z
X
dE dk
→
|
|.
2π dE
(50)
k
Using Eq. (50) and | dE
dk | = 2a|t||sin(kl a)|, Eq. (49) becomes,
Z
dE r
a
r
<
a
< nq > = −i
Gq,1 (E)Σ<
L (E)G1,q (E) + Gq,n (E)ΣR (E)Gn,q (E)
2π
1
· ,
a
(51)
where,
1
|sin(kl a)|tl,d fL (E)
and
|t|
1
Σ<
|sin(kr a)|tr,d fR (E) .
R (E) = 2itd,r
|t|
Σ<
L (E) = 2itd,l
(52)
(53)
kl and kr at every energy E are determined by Eqs. (19) and (34) respectively. It can be seen from Eqs. (26), (27),
(52), (53):
r
Σ<
L (E) = −2iIm[ΣL (E)]fL (E)
r
Σ<
R (E) = −2iIm[ΣR (E)]fR (E) .
(54)
(55)
Eq. (51) can be written as,
< nq >= −i
Z
dE r
a
G (E)Σ<
Lead (E)G (E)|q,q ,
2π
(56)
where, the only non zero elements of Σ<
Lead are
<
Σ<
Lead 1,1 (E) = ΣL (E)
and
<
Σ<
Lead n,n (E) = ΣR (E).
(57)
(58)
Current: Eq. (43) can be written as,
< Jq→q+1 (E) > =
† (L)
† (R)
e~
(L) †
(R) †
[(Ψ(L)
Ψq+1 − Ψq+1 Ψq(L) )fL (E) + (Ψq(R) Ψq+1 − Ψq+1 Ψq(R) )fR (E)] .
q
2mai
Following the derivation for electron density above [Eq. 51], it is quite straight forward see that,
Z
e~
dE r
a
r
<
a
< Jq → q + 1 > =
[G (E)Σ<
L (E)G1,q+1 (E) + Gq,n (E)ΣR (E)Gn,q+1 (E)
2ma
2π q,1
a
r
<
a
−Grq+1,1 (E)Σ<
L (E)G1,q (E) − Gq+1,n (E)ΣR (E)Gn,q (E)]
Z
e~
dE
a
r
<
a
=
[ Gr (E)Σ<
Lead (E)G (E)|q,q+1 − G (E)ΣLead (E)G (E)|q+1,q ]
2ma
2π
(59)
(60)
(61)
a
Eqs. (56) and (61) are the diagonal and first off-diagonal components of Gr Σ<
Lead G .
Less-than Green’s function: More generally, we define the less-than Green’s function G < , which is an n by n
matrix obtained by solving,
a
AG< = Σ<
Lead G .
(62)
It is easy to verify from Eqs. (56), (61) and (62) [which is equivalent to G < = Gr Σ< Ga ] that the electron density and
current are related to G< by,
nq = −iG<
q,q ,
e~
Jq→q+1 =
[G<
− G<
q+1,q ].
2ma q,q+1
(63)
10
FIG. 2:
That is, the diagonal elements and first off-diagonal elements of G< are related to the electron density and current
respectively. Similarly, another function h(q) can be defined to represent the absence of electrons (”like holes”) at
grid point q:
Z
dE r
a
r
>
a
G (E)Σ>
(64)
< hq > = i
L (E)G1,q (E) + Gq,n (E)ΣR (E)Gn,q (E),
2π q,1
Z
dE r
a
< hq > = i
G (E)Σ>
(65)
Lead G (E)|q,q
2π
where, the only non zero elements of Σ>
Lead are,
More generally, we define G>
1
>
Σ>
Lead 1,1 (E) = ΣL (E) = −i2td,l sin(kl a)tl,d (1 − fL )
t
1
>
>
ΣLead n,n (E) = ΣR (E) = −i2td,r sin(kr a)tr,d (1 − fR ).
t
which is an n by n matrix obtained by solving,
(67)
AG> = Σ> Ga .
(68)
(66)
Now, it is easy to verify from Eqs. (64) and (68) that the hole density is simply,
h(q) = iG> (q, q) .
F.
(69)
Self-energy due to scattering
Eqs. (26) and (27) represent the self-energy due to leads. Again, ΣrL and ΣrR are the self-energy due to the Left and
Right leads. They represent the influence of the semi-infinite leads on the Device and are typically complex numbers.
The real part of the self-energy shifts the on-site potential at grid point n by n + Re(ΣrR ), and −2Im[ΣrR ] represents
the scattering rate of electrons from the Device to the Right lead. A similar statement is valid for Σ rL . It is well known
that electron-phonon interaction also causes scattering of electrons in and out of energy E at grid point i (Fig. 2).
The imaginary part of the self-energy due to phonon scattering represents the scattering rate due to electron-phonon
interaction.
Phonon scattering can scatter electrons into energy E from all other energies E 0 by either phonon emission or
absorption, at every grid point. For simplicity, if we take a phonon energy of ~ωη to be the only one active in mode
η, then the self-energy in the self-consistent Born approximation is [G. Mahan, Physics Reports 145, 251 (1987)],
h
i
X
η
<
<
D
n
(~ω
)G
(E
−
~ω
)
+
(n
(~ω
)
+
1)G
(E
+
~ω
)
,
(70)
Σ<
(E)
=
0
0
B
η
η
B
η
η
q
n q
n q
inel q
η
Dqη
represents the electron-phonon scattering strength at grid point q, and will be discussed in a little more detail in
the context of silicon later. Similarly, the greater than self-energy, Σ> is given by,
h
i
X
η
>
>
Σ>
(E)
=
D
n
(~ω
)G
(E
+
~ω
)
+
(n
(~ω
)
+
1)G
(E
−
~ω
)
.
(71)
0
0
B
η
η
B
η
η
q
n q
n q
inel q
η
11
FIG. 3:
The reader can easily verify the reason for nB being associated with G< (E − ~ωη ) and G> (E + ~ωη ) in Eqs. (70) and
(71) respectively. The above equations assume that phonon scattering is local for simplicity of discussion.
Fig. 2 illustrates the two types of self-energies that have been discussed: (i) ΣLead q (E) is the self-energy due to
leads, which is non-zero only at the device grid points where the leads (empty circles of Fig. 2) couple the device
(solid circle of Fig. 2 ) and (ii) ΣP honon q (E) is the self-energy due to phonon scattering, which is non-zero at all grid
points.
G.
Examples
1) Discretized Schrodinger’s Equation: The discretized form of the one dimensional Schrodinger’s equation
in a uniform grid with spacing a is,
EΨq +
~2
(Ψq+1 − 2Ψq + Ψq−1 ) − Vq Ψq = 0.
2ma2
(72)
tΨq−1 + (E − q )Ψq + tΨq+1 = 0,
(73)
This equation can be written as,
where,
t=−
~2
2ma2
and
q = Vq +
~2
.
ma2
(74)
When V (x) does not depend on x (Vq = V0 , a constant), the solution of equation 73 is,
E = 0 + 2tcos(ka)
Ψq = eikqa ,
(75)
(76)
2
~
where, 0 = V0 − ma
2 and n is the grid point number.
2) Carbon Nanotubes: A (6,0) zigzag nanotube will be discussed in the lecture.
II.
SUMMARY OF THE NON-EQUILIBRIUM GREEN’S FUNCTION EQUATIONS SOLVED
We will be concerned with layered structures such as those shown in Fig. 3. Each layer (between the two dashed
line) of these structures consists of rings of carbon atoms, DNA base pairs and a column of grid points in the case of
12
MOSFETs. It is a good approximation to assume in many cases such as these that each layer q interacts only with
itself and its nearest neighbor layers q − 1 and q + 1. Then, the single particle Hamiltonian matrix for the layered
structure is a block tridiagonal matrix.
Retarded Green’s Function: Here, we take a slightly different approach to fold the influence of the Left and
Right leads onto layers 1 and n, when compared to section I. The Schrodinger’s equation and the retarded Green’s
function equation for this system in the absence of scattering mechanisms (electron-phonon) are,
[EI − H]Ψ = 0 → [EI − H]Gr = I .
(77)
In the presence of scattering with phonons, electrons and so on, the Green’s function method is well established, and
the influence of these scattering mechanisms is represented by a self-energy, Σr [G. D. Mahan, Many Particle Physics,
Plenum, 1990]. The retarded Green’s function equation in the presence of scattering is,
[EI − H − ΣrP ]Gr = I ,
(78)
where a subscript P is added to the self-energy Σr to represent phonon scattering, which is the only type of scattering
considered here. Further, phonon scattering is treated only in the local approximation. That is, Σ rP i,j = ΣrP i δi,j , the
self-energy between layers i and j is non zero only if i = j.
Our layered structure consists of a Device region connected to Left and Right leads (Fig. 3). The motivation for this
partitioning has both a physical and mathematical reason. The Left and Right leads typically have a constant potential
and are in equilibrium / quasi-equilibrium (boundary conditions). The constant potential and quasi-equilibrium nature
of the leads results in a smaller size of the matrix equations for obtaining the electron density. The influence of the
Left and Right leads can be folded into the n Device layers as discussed below. Eq. (78) can be written as,
Define: A0 = [EI − H − ΣrP ] ,
(79)

 r
 
GL,L GrL,D GrL,R
I O O
A0LL A0LD O
 A0DL A0DD A0DR   GrD,L GrD,D GrD,R  =  O I O  ,
O O I
O A0RD A0RR
GrR,L GrR,D GrR,R

where,
A0LL
• • •
 • •
•

†
0
−T
A
=
l3 −Tl2
l3

†

Tl1
A0l2 −Tl1
†
−Tl1
A0l1




A0LD = 

O
O
O
O
−TLD
O
O
O
O
O
•
•
•
•
O
•
•
•
•
•
O
O
O
O
•
O
O
O
O
O







A0RR

A0r1 −Tr1

 −T † A0
r2 −Tr2
r1


†

0
=
−Tr2 Ar3 −Tr3



•
• • 
• • •







A0RD = 

†
A0DL = A0LD , A0DR = A0RD
A01 −T12
†
 −T12
A02 −T2,3


•
•
•

•
•
•
=


•
•

†

−Tn−2,n−1
(80)

−TRD
O
O
O
O
O
O
O
O
O
•
•
•
•
O
•
•
•
•
•
O
O
O
O
O
†





(82)
(83)

A0DD
O
O
O
O
•
(81)

•
A0n−1 −Tn−1,n
†
−Tn−1,n
A0n




 ,




(84)
where, the empty element and O both represent zero matrices. Note that A0LD and A0DL (A0RD , and A0DR ) are very
sparse matrices. Their only non-zero entry represents the coupling between the Left (Right) lead and Device region.
13
Eq. (80) can be simplified by eliminating the Left and Right leads. We note from Eq. (80),
0
r
GrLD = −A0−1
LL ALD GDD
0
r
GrRD = −A0−1
RR ARD GDD
0
r
0
r
ADL GLD + ADD GDD +
(85)
A0DR GrRD = I .
(86)
(87)
Substituting Eqs. (85) and (86) in Eq. (87), we have,
0−1 0
0
0
r
[A0DD − A0DL A0−1
LL ALD − ADR ARR ARD ]GDD = I .
(88)
Note that the Green’s function in the device can now be obtained by solving a system of equations defined only over
n layers. The additional complexity is however that the inverses of the semi-infinite matrices, A 0LL and A0RR are
0−1
required. From Eq. (78), A0−1
LL and ARR are simply the Green’s function of the semi-infinite Left and Right leads,
r
r
A0LL gL
= I and A0RR gR
=I .
(89)
Noting the sparsity of A0LD and A0RD , it follows that only the surface Green’s functions of the Left and Right
leads,
0−1
r
r
gL
= A0−1
LL 1,1 and gR n+1,n+1 = ARR 1,1
0,0
(90)
are required in Eq. (88). Then, we can rewrite Eq. (88) in terms of self-energies due to the Left and Right leads as,
ADD GrDD = I ,
(91)
where, the elements of the n by n ADD matrix are given by,
ADD 1,1 = A0DD 1,1 − ΣrL = [EI − H − ΣrP ]1,1 − ΣrL
(92)
ADD n,n = A0DD n,n − ΣrR = [EI − H − ΣrP ]n,n − ΣrR
(93)
ADD q,q =
A0DD q,q
= [EI − H −
ΣrP ]q,q
, where, q=2, 3, 4, ... n-1.
(94)
ΣrL and ΣrR are called the self-energies due to the Left and Right leads respectively. From Eqs. (88) and (90), the
self-energies can be easily identified to be
r
ΣrL = TDL gL
TLD
0,0
ΣrR
=
r
TDR gR
TRD
n+1,n+1
and
(95)
,
(96)
†
†
where, TDL = TLD
and TDR = TRD
.
Surface Green’s Function: For a system where the potential does not vary in the Left and Right leads, the
matrices A0LL and A0RR are semi-infinite periodic matrices. For example, the elements of A0LL [Eq. (81)] are:
Al1 = Al2 = Al3 = ... = Al
Tl1 = Tl2 = Tl3 = ... = Tl .
(97)
(98)
r
Noting that ALL is semi-infinite and using Eqs. (88) and (89), gL
can be obtained by solving the following matrix
0,0
quadratic equation:
r
r
[Al − Tl† gL
Tl ]gL
=I .
0,0
0,0
(99)
This equation can be solved iteratively by,
r <m−1>
r <m>
[Al − Tl† gL
Tl ]gL
=I ,
0,0
0,0
(100)
r
where, the superscript of gL
represents the iteration number. Note that the solution to Eq. (99) is analytic when the
dimension of Al is one [see discussion around Eq. (208)].
14
r
Caveat: Verify that for the one dimensional tight-binding model shown in Fig. 1 and discussed in section: g L
=
1,1
e+ikl a t−1
l . Thus Eqs. (26) and (27) of section I C satisfy Eq. (96).
r
gL
can often be solved more simply by transforming to an eigen mode basis using an unitary transformation (S),
such that
S −1 Al S = Ald and S −1 Tl S = Tld ,
(101)
where, both Ald and Tld are diagonal matrices. The surface Green’s function in this new basis is simple a diagonal
matrix, whose elements are obtained by solving the scalar quadratic version of Eq. (99). The Green’s function in the
original basis (in which Al is not diagonal) can be obtained using the inverse unitary transformation. A third method
to solve for the surface Green’s function of quite general Hamiltonians which is explicitly based on the Bloch states
of the Hamiltonian and does not involve an iterative solution is presented in R. Lake et. al, J. Appl. Phys. 81, 7845
(1997).
Less-than (G< ) and Greater-than (G> ) Green’s Function: At equilibrium, the electron density (ρ) at energy
E is,
ρ(~r, E) = N (~r, E)f (E) ,
(102)
where N (r, E) is the density of states at location ~r and f (E) is the Fermi factor. Away from equilibrium (with bias
and scattering), G< (E) is the distribution function, whose diagonal elements correspond to the electron density [See
the discussion of electron density in section I E.]. The off-diagonal elements of G< correspond to the off-diagonal
elements of the density matrix. The equation for G< is
a
[EI − H − ΣrP ]G< = Σ<
PG ,
(103)
<
in the
where Ga = Gr † and Σ<
P is the less-than self-energy due to phonon scattering. It can be derived that G
Device region can be obtained by solving,
<
a
ADD G<
DD = ΣDD GDD ,
(104)
where, GaDD = Gr†
DD , and ADD has been defined in Eqs. (92) - (94). The self-energy matrix has two contributions,
one due to phonon scattering and the other due to the Left and Right leads,
<
<
Σ<
DD 1,1 = ΣP 1,1 + ΣL
(105)
<
<
Σ<
DD n,n = ΣP n,n + ΣR
<
Σ<
DD q,q = ΣP q,q , where,
(106)
q=2, 3, 4, ... n-1.
(107)
For a concrete calculation of Σ<
P in the self-consistent Born approximation, see attached paper Role of scattering in
<
nanotransistors. The less-than self-energies due to the contacts, Σ<
L and ΣR are given by,
r
Σ<
L (E) = −2i Im[ΣL (E)]fL (E) = iΓL (E)fL (E)
<
ΣR (E) = −2i Im[ΣrR (E)]fR (E) = iΓR (E)fR (E) ,
(108)
(109)
ΓL (E) = −2 Im[ΣrL (E)]
ΓR (E) = −2 Im[ΣrR (E)] .
(110)
(111)
where,
fL and fR are the distribution functions in the Left and Right leads respectively (Fermi factors at equilibrium). The
self-energies ΣrL (E) and ΣrR (E) have been defined in Eq. (96).
The greater-than Green’s functions, G> (E), represents the absence of electrons or the presence of holes,
h(~r, E) = iG> (~r, ~r, E).
(112)
15
The equation for G> (E) is,
a
[EI − H − ΣrP ]G> = Σ>
PG ,
>
>
a
ADD GDD = ΣDD GDD
>
>
Σ>
DD 11 = ΣP 11 + ΣL
(113)
(114)
(115)
>
>
Σ>
DD nn = ΣP nn + ΣR
(116)
>
Σ>
DD ii = ΣP ii , where, i=2, 3, 4, ... n-1
r
Σ>
L (E) = 2i Im[ΣL (E)](1 − fL (E)) = −iΓL (1 − fL (E))
r
Σ>
R (E) = 2i Im[ΣR (E)](1 − fR (E)) = −iΓR (1 − fR (E))
(117)
,
(118)
(119)
where Σ>
P is the less-than self-energy due to phonon scattering. (1 − f L (E)) and (1 − fR (E)) are the probabilities for
the absence of electrons in the Left and Right leads respectively, at energy E.
Current Density: The current flowing between layers q and q + 1 is:
Z
e
dE <
Jq→q+1 =
Tr Tq,q+1 G<
q+1,q (E) − Tq+1,q Gq,q+1 (E) .
~
2π
(120)
This expression for current is often written in other forms, which are derived below. The derivation uses Dyson’s
equations [Eqs. (167), (169), (170)], which are derived in section III. Expanding both terms of Eq. (120) using Eq.
(181), of section III B we get,
Z
e
dE
<
a
JL =
Tr([TLD Gr1,1 (E)TDL gL
(E) + TLD G<
1,1 (E)TDL gL 0,0 (E)]
0,0
~
2π
<
a
−[TDL gL
(E)TLD Ga1,1 (E) + TDL gL
(E)TLD G<
(121)
1,1 (E)])
0,0
0,0
Z
e
dE
<
r
a
=
(122)
Tr [Gr1,1 (E) − Ga1,1 (E)]TDL gL
(E)TLD − G<
1,1 (E)TDL [gL 0,0 (E) − gL 0,0 (E)]TLD
0,0
~
2π
Using the relationships,
<
Σ<
L = TDL gL 0,0 TLD
−iΓL =
r
TDL [gL
0,0
−
(123)
a
gL
]TLD
0,0
,
(124)
Eq. (122) can be written as,
JL
Z
e
=
~
Z
e
=
~
dE
<
Tr([Gr1,1 (E) − Ga1,1 (E)]Σ<
L (E) + G1,1 (E)iΓL (E))
2π
dE
<
Tr([Gr (E) − Ga (E)]Σ̃<
Lead (E) + G (E)iΓ̃Lead (E))1,1 ,
2π
(125)
(126)
where, the only non-zero elements of Σ̃ and Γ̃ are
α
Σ̃α
Lead |1,1 = ΣL
and
α
Σ̃α
Lead |n,n = ΣR
Γ̃Lead |1,1 = ΓL
and
Γ̃Lead |n,n = ΓR .
(α ∈ r, >, <)
(127)
(128)
For phase coherent structures note that Σα i6=1,n = 0 and Γi6=1,n = 0. Then using
Gr − Ga |1,1 = −iGa Γ̃Lead Gr |1,1
<
G |1,1 =
=
a
G Σ̃<
Lead G |1,1
r
G1,1 [iΓL fL ]Ga1,1
and
(129)
r
+ Gr1,n [iΓR fR ]Gan,1
the current [Eq. (126)] in phase coherent structures can be written as,
Z
e
dE
JL =
Tr[Γ̃L Gr Γ̃R Ga ] [fL (E) − fR (E)] .
~
2π
,
(130)
(131)
(132)
16
A.
Crib Sheet
Equation Solved:
Retarded
Advanced
Less-than
Greater-than
Green’s
Green’s
Green’s
Green’s
Function:
Function:
Function:
Function:
[EI
[EI
[EI
[EI
− H − Σr ]Gr (E) = I → AGr = I
− H − Σa ]Ga (E) = I
− H − Σr ]G< (E) = Σ< (E)Ga (E) → AG< = Σ< Ga
− H − Σr ]G> (E) = Σ> (E)Ga (E) → AG> = Σ> Ga
(133)
(134)
(135)
(136)
α
Σα (E) = Σα
where, α ∈ r, <, >
Lead (E) + ΣP (E),
α
α
α
T
ΣLead 1,1 = ΣL (E) = TDL gL
LD
0,0
(137)
(138)
α
α
Σα
Lead n,n = ΣR (E) = TDR gR n+1,n+1 TRD
(139)
Σα
Lead i,i (i
(140)
6= 1, n) = 0
ΓL (E) = −2 Im[ΣrL (E)] , ΓR (E) = −2 Im[ΣrR (E)] and Γ(E) = −2 Im[Σr (E)]
Σ<
L (E) = iΓL (E)fL (E)
Σ<
R (E) = iΓR (E)fR (E) ,
Σ>
L (E) = −iΓL [1 − fL (E)]
Σ>
R (E) = −iΓR [1 − fR (E)] .
(141)
(142)
(143)
(144)
(145)
The diagonal and nearest neighbor off-diagonal elements of Gr and G< are computed repeatedly as they correspond
to physical quantities such as the density of states, electron density and current and (120)). Non local scattering
mechanisms would require calculation of further off-diagonal elements.
Useful Relationships:
Ga
Gr − G a
Σr − Σ a
Gr − G a
G<
†
=
=
=
=
=
Gr †
G> − G<
Σ> − Σ<
Gr [Σa − Σr ]Ga = Ga [Σr − Σa ]Gr
−iGr ΓGa = −iGa ΓGr
= −G<
transpose
(146)
(147)
(148)
(149)
(150)
(151)
(152)
Physical Quantities:
Scattering Rate:
~
= −2 Im[Σr (E)] = Γ(E).
τ (E)
(153)
(154)
Density of States at (~r, E):
1
N (~r, E) = − ImGr (~r, ~r, E).
π
(155)
Use recursive algorithm to calculate DOS (Do not invert A).
(156)
Electron Density at location ~r:
ρ(~r, E) =
Z
dE
[−iG< (~r, ~r, E)].
2π
Use recursive algorithm to calculate ρ (Do not use G< = Gr Σ< Ga ).
(157)
(158)
17
FIG. 4:
Current density flowing between layers q and q + 1:
Z
e
dE <
Tr Tq,q+1 G<
Jq→q+1 =
q+1,q (E) − Tq+1,q Gq,q+1 (E) .
~
2π
(159)
Current density flowing from the Left lead into layer 1 of Device:
Z
e
dE
<
JL =
Tr([Gr1,1 (E) − Ga1,1 (E)]Σ<
L (E) + G1,1 (E)iΓL (E))
~
2π
(160)
Current density flowing from the Left lead into layer 1 of Device (valid only for phase coherent structures):
Z
e
dE
JL =
Tr[Γ̃L Gr Γ̃R Ga ] [fL (E) − fR (E)].
~
2π
III.
(161)
DYSON’S EQUATION FOR LAYERED STRUCTURES
Note: Matrix A of this section is equivalent to Matrix ADD of section II
The version of Dyson’s equation presented here relates the Green’s function of the Device+Leads in terms of the
Green’s functions of two subsystems that comprise the Device+Leads. For example, in Fig. 4, the Device+Leads
consists of two subsystems Z and Z 0 . We are free to choose the partition. The Dyson’s equations, is very useful in
many particle physics but here, we use them in a more restrictive sense, which help us in solving the Green’s function
equations and in deriving the algorithm presented in section III.
Dyson’s equation for Gr
A.
The solution to (Fig. 4)
AZ,Z AZ,Z 0
AZ 0 ,Z AZ 0 ,Z 0
GrZ,Z GrZ,Z 0
GrZ 0 ,Z GrZ 0 ,Z 0
=
I O
O I
,
(162)
is
Gr = Gr0 + Gr0 U Gr
= Gr0 + Gr U Gr0 ,
(163)
(164)
where,
r
G =
GrZ,Z GrZ,Z 0
GrZ 0 ,Z GrZ 0 ,Z 0
,G
r0
=
Gr0
O
Z,Z
O Gr0
Z 0 ,Z 0
=
A−1
O
Z,Z
O A−1
Z 0 ,Z 0
and U =
O
−AZ,Z 0
−AZ 0 ,Z
O
.
(165)
18
The advanced Green’s function (Ga ) is by definition related to Gr by
Ga = Gr † = Ga0 + Ga0 U † Ga
= Ga0 + Ga U † Ga0 .
(166)
(167)
Eq. (163) is Dyson’s equation for the retarded Green’s function.
B.
Dyson’s equation for G<
The solution to (Fig. 4)
<
<
a
GZ,Z G<
ΣZ,Z Σ<
GZ,Z GaZ,Z 0
AZ,Z AZ,Z 0
Z,Z 0
Z,Z 0
=
<
<
AZ 0 ,Z AZ 0 ,Z 0
G<
Σ<
GaZ 0 ,Z GaZ 0 ,Z 0
Z 0 ,Z GZ 0 ,Z 0
Z 0 ,Z ΣZ 0 ,Z 0
(168)
can be written as
G< = Gr0 U G< + Gr0 Σ< Ga ,
(169)
where Gr0 and U have been defined in Eqs. (165), and G< and Ga are readily identifiable from Eq. (168). Using
Ga = Ga0 + Ga0 U † Ga , Eq. (169) can be written as
G< = G<0 + G<0 U † Ga + Gr0 U G<
= G<0 + Gr U G<0 + G< U † Ga0 ,
<0
where G
= Gr0 Σ< Ga0 .
IV.
(170)
(171)
(172)
ALGORITHM TO CALCULATE Gr AND G<
Note: Matrix A of this section is equivalent to Matrix ADD of section II
Why algorithm: A typical solution consists of solving Poisson’s equation self-consistently with the Green’s function equations. The input to Poisson’s equation is the charge density, which are the diagonal elements of G < . That
is, we do not require the entire G< matrix in most situations. The computational cost of obtaining the diagonal
elements of the G< matrix at each energy is approximately Nx3 Ny3 operations if G< = Gr Σ< Ga is used. Nx is the
dimension of the Hamiltonian of each layer and Ny (= n) is the total number of layers. The cubic dependence is the
operation count for matrix inversion [Gr = A−1 , Eq. (133)]. As the input to Poisson’s equation are only the diagonal
elements of G< (electron density), it is highly desirable to find methods that avoid inversion of the A matrix. One
such algorithm which is valid for the block tridiagonal form of matrix A is presented in this section. The operation
count of this algorithm scales as approximately Nx3 Ny . The dependence on Nx3 arises because we invert matrices
corresponding to the size of the sub Hamiltonian of each layer, and the dependence on N y corresponds to one such
inversion for each of the Ny layers.
Challenging problem: The algorithm presented here solves for Ny diagonal blocks of G< , each corresponding to
a layer. It is highly desirable to find a more efficient algorithm that solves only for the diagonal elements rather than
the diagonal blocks of G< .
The recursive algorithm to calculate the diagonal blocks of Gr and G< in Eqs. (133) and (135) are discussed,
using Dyson’s equation and the left-connected Green’s function. The concept of left-connected Green’s function is
introduced below for the cases of Gr and G< .
A.
Recursive algorithm for Gr
(i) Left-connected retarded Green’s function (Fig. 5): The left-connected (superscript L) retarded (superscript r)
Green’s function g rLq is defined by the first q blocks of Eq. (91) or (133) (includes the open boundaries attached to
layers 1 and n via the self-energy) by
A1:q,1:q g rLq = Iq,q , where, Iq = I1:q,1:q .
(173)
19
FIG. 5:
g rLq+1 is defined in a manner identical to g rLq except that the left-connected system is comprised of the first q + 1
blocks of Eq. (91). In terms of Eq. (162), the equation governing g rLq+1 follows by setting Z = 1 : q and Z 0 = q + 1.
Using Dyson’s equation [Eq. (163)], we obtain
rLq+1
rLq
gq+1,q+1
= Aq+1,q+1 + Aq+1,q gq,q
Aq,q+1
−1
.
(174)
rLN
Note that the last element gN,N
is equal to the fully connected Green’s function GrN,N , which is the solution to Eq.
(91).
(ii) Full Green’s function in terms of the left-connected Green’s function: Consider Eq. (162) such that A Z,Z =
A1:q,1:q , AZ 0 ,Z 0 = Aq+1:N,q+1:N and AZ,Z 0 = A1:q,q+1:N . Noting that the only nonzero element of A1:q,q+1:N is Aq,q+1
and using Eq. (163), we obtain
rLq
rLq
rLq
Grq,q = gq,q
+ gq,q
Aq,q+1 Grq+1,q+1 Aq+1,q gq,q
(175)
rLq
rLq
= gq,q
+ gq,q
Aq,q+1 Grq+1,q .
(176)
Both Grq,q and Grq+1,q are used in the algorithm for electron density, and so storing both sets of matrices will be useful.
In view of the above equations, the algorithm to compute the diagonal blocks G rq,q is given by the following steps:
rL1
• g11
= A−1
1 .
• For q = 1, 2, ..., N − 1, compute Eq. (174).
• For q = N − 1, N − 2, ..., 1, compute Eq. (176). Store Grq+1,q if memory permits for use in the algorithm for
electron density.
B.
Recursive algorithm for G<
(i) Left-connected g < (Fig. 5): g <Lq is the counter part of g rLq , and is defined by the first q blocks of Eq. (104):
aLq
A1:q,1:q g <Lq = Σ<
1:q,1:q g1:q,1:q .
(177)
g <Lq+1 is defined in a manner identical to g <Lq except that the left-connected system is comprised of the first q + 1
blocks of Eq. (104). In terms of Eq. (168), the equation governing g <Lq+1 follows by setting Z = 1 : q and Z 0 = q + 1.
<Lq+1
Using the Dyson’s equations for Gr [Eq. (163)] and G< [Eq. (170)], gq+1,q+1
can be recursively obtained as
aLq+1
<Lq+1
rLq+1 <
rLq+1
aLq+1
rLq+1 <
aLq+1
<
gq+1,q+1
= gq+1,q+1
Σq+1,q+1 + σq+1
gq+1,q+1 +gq+1,q+1
Σ<
q+1,q gq,q+1 + gq+1,q Σq,q+1 gq+1,q+1 ,
(178)
20
which can be written in a more intuitive form as
h
i
<Lq+1
rLq+1
†
aLq+1
<
<
aLq
rLq
<
gq+1,q+1
= gq+1,q+1
Σ<
+
σ
+
Σ
g
A
+
A
g
Σ
q+1,q
q+1,q+1
q+1
q+1,q q,q
q,q
q,q+1 gq+1,q+1 ,
q,q+1
(179)
<Lq+1
<
<Lq †
where σq+1
= Aq+1,q gq,q
Aq,q+1 . Eq. (179) has the physical meaning that gq+1,q+1
has contributions due to four
injection functions: (i) an effective self-energy due to the left-connected structure that ends at q, which is represented
<
by σq+1
, (ii) the diagonal self-energy component at grid point q +1 that enters Eq. (104), and (iii) the two off-diagonal
self-energy components involving grid points q and q + 1.
(iii) Full less-than Green’s function in terms of left-connected Green’s function: Consider Eq. (168) such that A Z =
A1:q,1:q , A0Z = Aq+1:N,q+1:N and AZ,Z 0 = A1:q,q+1:N . Noting that the only nonzero element of A1:q,q+1:N is Aq,q+1
and using Eq. (170), we obtain
†
<Lq
<Lq †
a
rLq
<
<0
a
G<
q,q = gq,q + gq,q Aq,q+1 Gq+1,q +gq,q+1 Aq+1,q Gq,q + gq,q Aq,q+1 Gq+1,q .
(180)
<
Using Eq. (171), G<
q+1,q can be written in terms of Gq+1,q+1 and other known Green’s functions as
†
r
<Lq
<
aLq
<0
<0
r
G<
q+1,q = gq+1,q + Gq+1,q Aq,q+1 gq+1,q +Gq+1,q+1 Aq+1,q gq,q + Gq+1,q+1 Aq,q+1 gq,q .
Substituting Eq. (181) in Eq. (180) and using Eqs. (163) and (164), we obtain
h
i
†
<Lq
rLq
aLq
<Lq †
a
r
<Lq
G<
Aq,q+1 G<
q,q = gq,q + gq,q
q+1,q+1 Aq+1,q gq,q + gq,q Aq,q+1 Gq+1,q + Gq,q+1 Aq+1,q gq,q
h
i
<0
<0
+ gq,q+1
A†q+1,q Gaq,q + Grq,q Aq,q+1 gq+1,q
,
(181)
(182)
where
<0
r0 <
a0
gq,q+1
= gq,q
Σq,q+1 gq+1,q+1
(183)
<0
gq+1,q
(184)
=
r0
a0
gq+1,q+1
Σ<
q+1,q gq,q
.
(185)
The terms inside the square brackets of Eq. (182) are Hermitian conjugates of each other.
In view of the above equations, the algorithm to compute the diagonal blocks of G < is given by the following steps:
<L1
r0 < a0
• g11
= g11
ΣL g11 .
• For q = 1, 2, ..., N − 1, compute Eq. (179).
• For q = N − 1, N − 2, ..., 1, compute Eqs. (182) - (185).
V.
TWO DIMENSIONAL SOLUTION OF BALLISTIC MOSFETS: A COMPUTATIONAL
EXPERIMENT
Same discussion as Two Dimensional Quantum Mechanical Modeling of Nanotransistors, J. of Appl. Phys., 91, 2343 (2002).
MOSFETs with channel lengths in the tens of nanometer regime have recently been demonstrated by various research labs1–3 . Design considerations to yield devices with desirable characteristics have been explored in references 4–8 .
Device physics of these MOSFETs were analyzed using simple quasi one dimensional models in 9–13 . The best modeling
approach for design and analysis of nanoscale MOSFETs is presently unclear, though a straightforward application of
semiclassical methods that disregards quantum mechanical effects is generally accepted to be inadequate. Quantum
mechanical modeling of MOSFETs with channel lengths in the tens of nanometer is important for many reasons:
(i) MOSFETs with ultra-thin oxide require an accurate treatment of current injection from source, drain and gate.
Gate leakage is important because it places a lower limit in determining the off current.
(ii) Ballistic flow of electrons across the channel becomes increasingly important as the channel length decreases.
(iii) The location of the inversion layer changes from the source to the drain end, and its role in determining the C-V
and I-V characteristics is most accurately included by a self-consistent solution of Poisson’s equation and a quantum
mechanical description to compute the charge density.
(iv) Approximate theories of quantum effects included in semi-classical MOSFET modeling tools are desirable from
practical considerations because semi-classical methods are numerically less expensive, and much of the empirical
21
and semi-classical MOSFET physics developed over the last few decades continues to hold true in many regions of
a nanoscale MOSFET. Examples of semiclassical methods that consider some quantum mechanical aspects are the
density gradient14,15 , effective potential16,17 approaches and quantum mechanical approximations used in the Medici
package18 . Fully quantum mechanical simulations can play an important role in benchmarking such simulators.
Central to quantum mechanical approaches to charge transport modeling is self-consistent solution of a wave
equation to describe the quantum mechanical transport, Poisson’s equation, and equations for statistics of the particle
ensemble. In the absence of electron-electron and electron-phonon interactions (state of the scatterer does not change),
the Landauer-Buttiker formalism19,20 is applicable. In this formalism, the wave equation is Schrodinger’s equation
and the statistics is represented throughout the device by the Fermi-Dirac distribution of particles incident from the
contacts (source, drain and gate). In the presence of electron-phonon interaction, the Wigner function (WF) and
non equilibrium Green’s function (NEGF) formalisms are used. The NEGF approach has been quite successful in
modeling steady state transport in a wide variety of one dimensional (1D) semiconductor structures 21 .
A number of groups have started developing theory and simulation for fully quantum mechanical two dimensional simulation of MOSFETs: references22,23 and24 use a real space approach, reference25 uses a k-space approach,
reference26 uses a Wigner function approach, and references13,27 and28 use the non equilibrium Green’s function approach. Others groups have taken a hybrid approach using the Monte Carlo method. The Monte Carlo approach, has
been quite successful in describing scattering mechanisms in MOSFETs, in comparison to fully quantum mechanical
approaches, and can also include ballistic effects and the role of quantized energy levels in the MOSFET inversion
layer in an approximate manner29–31 . Discussing the relative merits of various approaches and quantum-corrected
drift-diffusion approaches is important. In fact, such a comparison of methods using standard device structures has
been initiated32 but much work remains to be done in comparing and studying the suitability of different methods.
Here, we describe development of the NEGF approach, for numerical simulation of MOSFETs with two dimensional
(2D) doping profiles and charge injection from the source, drain and gate contacts. 2D simulation significantly increases
computational effort over the 1D case. Non-uniform spatial grids are essential to limit the total number of grid points
while at the same time resolving physical features. A new algorithm for efficient computation of electron density
without complete solution of the system of equations is presented. The computer code developed is used to calculate
the drain and gate tunneling current in ultra short channel MOSFETs. Results from the NEGF approach and Medici
are compared. Some specific results discussed are: role of polysilicon gate depletion V E, slopes of I d versus the gate
(Vg ) and drain (Vd ) voltages (sections V E - V G), role of gate tunneling current as a function of gate oxide thickness
and gate length in determining the off-current (section V I). It is emphasized that the calculations presented include
a self-consistent treatment of two dimensional gate oxide tunneling.
A.
The governing equations
We consider Nb independent valleys for the electrons within the effective mass approximation. The Hamiltonian of
valley b is
~2 d
1 d
d
1 d
d
1 d
Hb (~r) = − [
+
+
] + V (~r),
(186)
2 dx mbx dx
dy mby dy
dz mbz dz
where (mbx , mby , mbz ) are the components of the effective mass in valley b. The equation of motion for the retarded
(Gr ) and less-than (G< ) Green’s functions are20,33,34
Z
r
[E − Hb1 (~r1 )]Gb1 ,b2 (~r1 , ~r2 , E) − d~r Σrb1 ,b0 (~r1 , ~r, E)Grb0 ,b2 (~r, ~r2 , E) = δb1 ,b2 δ(~r1 − ~r2 )
(187)
and
r1 , ~r2 , E) −
[E − Hb1 (~r1 )]G<
b1 ,b2 (~
Z
d~r Σrb1 ,b0 (~r1 , ~r, E)G<
r , ~r2 , E) =
b0 ,b2 (~
Z
d~r Σ<
r1 , ~r, E)Gab0 ,b2 (~r, ~r2 , E),
b1 ,b0 (~
(188)
where Ga is the advanced Green’s function. In the above equations, the coordinate spans only the device (see Fig.
6). The influence of the semi-infinite regions of the source (S), drain (D) and polysilicon gate (P), and scattering
mechanisms (electron-phonon) are included via the self-energy terms Σrb1 ,b0 and Σ<
b1 ,b0 . We assume that charge is
α
injected independently from the contact into each valley. Then, Σα
b1 ,b2 ,C = Σb1 ,C δb1 ,b2 , where C represents the selfenergy due to contacts. Finally, the hole bands are treated using the drift-diffusion model, which is expected to be a
good approximation for n-MOSFETs.
22
-Lg/2
Lg/2
y
P
-Ly/2
S
-(LP + tox)
oxide
-tox
0
+Ly/2
D
-1 0 1
q q+1
semi-infinite
boundary
Ny
Ny+1
Ny+2
x
semi-infinite
boundary
+LB
semi-infinite
boundary
FIG. 6: The equations are solved in a 2D non uniform spatial grid, with semi-infinite boundaries as shown. Each column
q comprises the diagonal blocks of Eqs. (91) and (205). The electrostatic potential is held fixed at the beginning of the
semi-infinite regions closest to the device.
The electrostatic potential varies in the (x, y) plane, and the system is translationaly invariant along the z-axis. So,
all quantities A(~r1 , ~r2 , E) depend only on the difference coordinate z1 − z2 . Using the relation
Z
dkz ikz (z1 −z2 )
A(~r1 , ~r2 , E) =
e
A(x1 , y1 , x2 , y2 , kz , E) ,
(189)
2π
the equations of motion for Gr and G< simplify to
~2 kz2
− Hb (~r1 )]Grb (~r1 , ~r2 , kz , E) −
2mz
Z
~2 kz2
[E −
− Hb (~r1 )]Grb (~r1 , ~r2 , kz , E) −
2mz
Z
[E −
and
d~r Σrb (~r1 , ~r, kz , E)Grb (~r, ~r2 , kz , E) = δ(~r1 − ~r2 )
d~r Σrb (~r1 , ~r, kz , E)G<
r , ~r2 , kz , E) =
b (~
Z
d~r Σ<
r1 , ~r, kz , E)Gab (~r, ~r2 , kz , E),
b (~
(190)
(191)
where Zb = Zb,b , and for the remainder of this section ~r → (x, y).
The density of states [N (~r, kz , E)] and charge density [ρ(~r, kz , E)] are the sum of the contributions from the individual valleys:
X
1
N (~r, kz , E) =
Nb (~r, kz , E) = − Im[Grb (~r, ~r, kz , E)]
(192)
π
b
X
ρ(~r, kz , E) =
ρb (~r, kz , E) = −iG<
r , ~r, kz , E) .
(193)
b (~
b
B.
Gr and G< : Discretized matrix equations
Self-consistent solution of the Green’s function and Poisson’s equations requires repeated computation of the nonequilibrium charge density. This computation is often the most time consuming part in modeling the electronic
properties of devices. The electron density is given by,
ρb (~r, kz , E) = −iG<
r , ~r, kz , E)
b (~
(194)
23
In matrix form, Eqs. (190) and (191) are written as
A0 Gr = λ
and
0 <
< a
AG = Σ G .
(195)
(196)
The self-energies due to the S, D and P are non zero only along the lines y = yS = y1 , y = yD = yNy and x = xP
respectively (see Fig. 6). The A0 matrix has a dimension of Nx Ny and is ordered such that all grid points located
at a particular y-coordinate correspond to its diagonal blocks. The notation adopted is that A 0j1 ,j2 (i, i0 ) refers to the
off-diagonal entry corresponding to grid points (xi , yj1 ) and (x0i , yj2 ). The non zero elements of the diagonal blocks of
the A0 matrix are given by
A0j,j (i, i) = E 0 − Vi,j − Tj,j (i + 1, i) − Tj,j (i − 1, i) − Tj+1,j (i, i) − Tj−1,j (i, i)
−ΣrS (xi , xi )δj,1 − ΣrD (xi , xi )δj,Ny − ΣrP (yj , yj )δi,1 − Σr (xi , yj ; xi , yj )
A0j,j (i ± 1, i) = Tj,j (i ± 1, i) − ΣrS (xi±1 , xi )δj,1 − ΣrD (xi±1 , xi )δj,Ny
−Σr (xi±1 , yj ; xi , yj )
A0j,j (i, i0 ) = −ΣrS (xi , xi0 )δj,1 − ΣrD (xi , xi0 )δj,Ny , for i0 6= i, i ± 1 ,
(197)
(198)
(199)
where E 0 = E − ~2 kz2 /2mz and Vi,j = V (xi , yj ). The off-diagonal blocks are
A0j±1,j (i, i) = Tj±1,j (i, i) − ΣrP (yj , yj±1 )δi,1
A0j,j 0 (i, i0 ) = 0, for j 0 6= j, j ± 1.
(200)
The non zero elements of the T matrix are defined by
~2
2
1
2m±x xi+1 − xi−1 |xi±1 − xi |
~2
2
1
Tj±1,j (i, i) =
,
2m±y yj+1 − yj−1 |yj±1 − yj |
Tj,j (i ± 1, i) =
(201)
(202)
where m±x = mi±1,j2+mi,j and m±y = mi,j±12+mi,j . Non zero elements of ΣrP (yj , yj0 ), where j 0 6= j are neglected to
ensure that A0 is block tridiagonal (the algorithm to calculate Gr and G< relies on the block tridiagonal form of A0 ).
The λ appearing in Eq. (195) corresponds to the delta function in Eq. (190). λ is a diagonal matrix whose elements
are given by
λi,j;i,j =
4
.
(xi+1 − xi−1 )(yi+1 − yi−1 )
(203)
• • ••
Important Caveat: Matrix ADD of section II and matrix A of section IV are based on Hamiltonian matrices
that are Hermitian. Discretization over a non uniform grid gives rise to a non Hermitian Hamiltonian matrix. The
Hamiltonian can however be converted to a Hermitian form as discussed below.
The discretized version of Eq. (187) is,
A0 Gr =
I
, which is A0 Gr = λ [Eq. (195)].
Grid Spacing
Matrix A0 corresponds to a Hamiltonian matrix that is not Hermitian in a non uniform grid. Premultiplying Eq.
(195) by λ−1 gives
AGr = I ,
−1
(204)
0
where, matrix A = λ A corresponds to a Hermitian Hamiltonian.
Similarly, the discretized version of Eq. (188) is,
A0 G< = Σ < Ga
Premultiplying Eq. (196) by λ
−1
[Eq. (196).]
,
AG< = Σ< Ga ,
(205)
where Σ< in Eq. (205) is equal to λ−1 times the Σ< that appears in Eq. (196). Now, the equation in sections III and
IV can be applied.
• • ••
24
C.
Expressions for Contact Self-energies (ΣrS , ΣrD and ΣrP )
Potential and doping profiles in the semi-infinite regions to the (a) left of ’S’ and right of ’D’ are equal to the
value at q = 1 and Ny respectively (Fig. 6). That is, they do not vary as a function of the y-coordinate, and (b)
top of the ’P’ is equal to the value of the top most grid line of ’P’ (Fig. 6). That is, they are not a function of the
x-coordinate. The retarded surface Green’s functions of these semi-infinite regions are calculated from Eq. (204),
when the matrices involved are semi-infinite. All diagonal sub-matrices of the A matrix are equal to A 1,1 , ANy ,Ny and
AP , and all first upper off-diagonal matrices of the A matrix are equal to A1,2 , ANy −1,Ny and AP −1,P , in the source,
drain and polysilicon regions respectively. We spell out the entire matrix for the source semi-infinite regions below:
• • 0
 • • •
 • 0 0 0 0 0 0
•
•
•
•
0
0
0
0


•
0
0
0
0
0
•
•
0
0
0
0
•
•
A2,1
0
0
0
0
•
A1,1
A2,1
0
0
0
0
A1,2
A1,1
A2,1
0
0
0
0
A1,2
A1,1
A2,1
0
0
0
0
A1,2
A1,1
•
 ••
 •
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
g−3,−3
g−2,−3
g−1,−3
g0,−3
•
•
g−3,−2
g−2,−2
g−1,−2
g0,−2
•
•
g−3,−1
g−2,−1
g−1,−1
g0,−1
•
•
g−3,0
g−2,0
g−1,0
g0,0
 
=
0
0
0
0
0
0
•
0
0
0
0
0
0
•
0
0
0
0
0
0
I
0
0
0
0
0
0
I
0
0
0
0
0
0
I
0
0
0
0
0
0
I

.
(206)
The surface Green’s function of these regions can be obtained by using standard methods, two of which are given
below:
(i) After some algebra g0,0 is obtained by solving the following matrix equation:
[A1,1 − A2,1 g0,0 A1,2 ] g0,0 = I,
(207)
where A1,1 = λ−1 A01,1 , A1,2 = λ−1 A01,2 and A2,1 = λ−1 A02,1 .
(ii) The simplest procedure from a numerical viewpoint, which is applicable when the y-grid is uniform, is discussed
now. Transform the two dimensional wire representing the semi-infinite contacts with Nx grid points to Nx one
dimensional wires. That is, transform the problem from the real coordinate basis along x to an eigen basis. This
is achieved by defining a matrix S, whose columns are composed of the eigen vectors of A 1,1 . D = S −1 A1,1 S is a
diagonal matrix, and B = S −1 A1,2 S and C = S −1 A2,1 S are diagonal because A1,2 and A2,1 are equal to a constant
times the identity matrix, for a uniform grid only. The surface Green’s functions of the Nx one dimensional wires are
q
2
gi = Di ± Di − 4Bi Ci /(2Bi Ci ),
(208)
where the subscript i refers to the ith diagonal element. In most cases (absence of magnetic field) A 1,2 = A2,1 and
Bi = Ci . The surface Green’s function in the real space basis is now obtained by g00 = SgD S −1 , where gD is a matrix
whose only non zero elements correspond to gi along the diagonal. The various gi must be arranged in a consistent
manner with the construction of S.
The self-energy due to the contacts is:
ΣrS (kz , E)
ΣrD (kz , E)
ΣrP (kz , E)
Σ<
S (kz , E)
Σ<
D (kz , E)
=
=
=
=
=
A1,0 g0,0 (kz , E)A0,1
ANy ,Ny +1 gNy +1,Ny +1 (kz , E)ANy +1,Ny
AP gP (kz , E)AP
−2iA1,0 Im [g0,0 (kz , E)] A0,1 fS (E)
−2iANy ,Ny +1 Im gNy +1,Ny +1 (kz , E) ANy +1,Ny fD (E)
Σ<
P (kz , E) = −2iAP Im [gP (kz , E)] AP fP (E),
where fi (E) is the Fermi factor in contact i ∈ S, D, P .
~2 k 2
When Σα
r1 , ~r2 , kz , E) depends only on Exy = E − 2mzz , then Eqs. (190) and (191) simplify to
b (~
Z
[Exy − Hb (~r1 )]Grb (~r1 , ~r2 , Exy ) − d~r Σrb (~r1 , ~r, Exy )Grb (~r, ~r2 , Exy ) = δ(~r1 − ~r2 )
(209)
(210)
(211)
(212)
(213)
(214)
(215)
(216)
and
[Exy − Hb (~r1 )]G< (~r1 , ~r2 , Exy ) −
Z
Z
d~r Σrb (~r1 , ~r, Exy )G<
r , ~r2 , kz , E) =
b (~
d~r Σ<
r1 , ~r, Exy )Gab (~r, ~r2 , Exy ).
b (~
(217)
While solving the equations, to keep the problem two dimensional, mz has to be independent of (x,y). So, we assume
mz (SiO2 ) = mz (Si).
25
D.
Results and Discussion
The steady state characteristics of MOSFETS that are of practical interest are the drive current, off-current, slope
of drain current versus drain voltage, and threshold voltage. In this section, we show that quantum mechanical
simulations yield significantly different results from drift-diffusion based methods. These differences arise because of
the following quantum mechanical features:
(i) polysilicon gate depletion in a manner opposite to the classical case,
(ii) dependence of the resonant levels in the channel on the gate voltage,
(iii) tunneling of charge across the gate oxide and from source to drain,
(iv) quasi-ballistic flow of electrons.
The MIT well-tempered 25 nm device structure35 is chosen for the purpose of discussion (MIT 25 nm device
structure35 is hereafter referred to as MIT25). The method and computer code developed can however handle a wide
variety of two dimensional structures with many terminals. We first compare the potential profiles from a constant
mobility drift-diffusion solution and our quantum calculations at equilibrium. The motivation for this comparison
results from the observation that the classical and quantum potential profiles should be in reasonable agreement, if
the doping density is significantly higher than the electron and hole densities and the boundary conditions are the
same. The doping profile of MIT25 meets this requirement in the channel region at small V g , and we verify that the
potential profiles are in reasonable agreement at y = 0 (see ’Q1 flat band’ and ’DD flat band’ of Fig. 7). The legend
’flat band’ refers to the potential at x = −tox being fixed at the applied gate potential.
Conduction Band (meV)
500
DD flat band
Q1 flat band
Q1 q−poly
DD c−poly
400
300
200
Vg=Vd=0
y=0nm
100
0
−100
−200
−235
−5
−4
−3
−2
x (nm)
−1
0
1
FIG. 7: Potential profile at the y=0 slice of MIT25, calculated by four different methods. Note the qualitative difference of the
’Q1 q-poly’ case due to electron depletion in the gate.
An index of abbreviations used follows:
Length Scales:
tox − oxide thickness
LP − polysilicon gate thickness in x-direction
LB − boundary of substrate region in x-direction
Ly − Poisson’s and NEGF equations are solved from −Ly /2 to +Ly /2
Lg − length of polysilicon gate region in y-direction
26
Models:
Q1 − quantum mechanical calculations using an isotropic effective mass
Q3 − quantum mechanical calculations using an anisotropic effective mass
DD − drift diffusion
F lat band − potential in the polysilicon gate region is held fixed from x = −(tox + LP ) to −tox at bulk value
q − poly − potential in the gate polysilicon region is held fixed at x = −(tox + LP ) at the bulk value, and the
potential is computed quantum mechanically (self-consistently) for x > −(tox + LP )
c − poly − classical treatment of gate polysilicon region, as in DD
Current and voltage:
Id − drain current Ig − gate current
Vd − drain voltage Vg − gate voltage
Other constraints:
Electron effective mass of silicon: 0.3283 (isotropic), 0.19 and 0.98 (anisotropic)
Electron effective mass of SiO2 : mx = my = 0.5 and same as silicon in mz direction
Hole effective mass of silicon: 0.49
Band gap of silicon (SiO2 ): 1.12 eV (8.8 eV)
Energy barrier between the silicon and the oxide: ∆EC =3.1 eV
Dielectric constant of Si (SiO2 ): Si =11.9 (3.9)
kT = 0.02585 eV
E.
Id versus Vg - Effect of polysilicon depletion region
The quantum mechanically calculated electron density near the SiO2 barrier in the polysilicon region is smaller than
the uniform background doping density. This is because the electron wavefunction is small close to the barrier. As
a result, the conduction band in the polysilicon gate bends in a direction opposite to that computed semi-classically
(compare x and triangle in Fig. 7). The band bending in the polysilicon gate plays a significant role in determining
the threshold voltage and off-current. To emphasize the importance of band bending, we plot the drain current versus
gate voltage calculated with the gate polysilicon region treated as (i) ’flat band’ and (ii) ’q-poly’. We find that the
computed current is larger in (ii) because quantum mechanical depletion of electrons in the polysilicon gate region
close to the oxide causes lowering of the potential in the channel. The Id versus Vg curve shifts by approximately the
an amount equal to the band bending in the polysilicon gate, in comparison to the flat band case. This band bending,
which is measured from −(LP + tox ) to −tox at equilibrium, is about 130 meV at the given doping density (Fig. 7).
Computationally, a 2D treatment of the polysilicon gate region is expensive because of the additional grid points
required. Note that matrix inversion depends on the cube of the matrix dimension. We point out that for highly
doped polysilicon gate (in the absence of gate tunneling), a shift in the Id (Vg ) curve from (i) by the equilibrium 1D
built-in potential does a reasonable job of reproducing the quantum mechanical result (see triangles in Fig. 8). This
approximation becomes progressively poorer with increase in gate voltage, as can be seen from the figure.
F.
Id versus Vg - Comparison to Medici
In the absence of gate tunneling and inelastic tunneling, the quantum mechanical current is
Z
2e
Id =
dE TSD (E) [fS (E) − fD (E)] ,
h
(218)
where TSD is the transmission probability from source to drain, and fS and fD are the Fermi-Dirac factors in the
source and drain respectively. The total transmission (Fig. 9) is step-like with integer values at the plateaus inspite of the complicated two dimensional electrostatics. In visual terms, the energies at which the steps turn on are
determined by an effective ’subband dependent’ source injection barrier, in contrast to the source injection barrier in
27
−2
10
−3
10
−4
Id (A/µm)
10
−5
10
−6
10
−7
10
Vd=1V
−8
10
Q1 flat band
Q1 q−poly
−9
10
−10
10
0
0.2
0.4
0.6
Vg (V)
0.8
1
FIG. 8: Drain current versus gate voltage for Vd = 1 V. Quantum mechanical treatment of the polysilicon gate (Q1 q-poly)
results in much higher current.
drift-diffusion calculations10 . This subband dependent source injection barrier is simply the maximum energy of the
subband between source and drain due to quantization in the direction perpendicular to the gate plane (x-direction of
Fig. 1). From a practical view point, the following two issues are important in ballistic MOSFETs: (a) typically, the
total transmission assumes integer value at an energy slightly above the maximum in 2D density of states as shown
in the inset of Fig. 9, and (b) the steps develop over 50 meV (twice the room temperature thermal energy). So, the
shape of the steps is important in determining the value of current. Assuming a sharp step in total transmission with
integer values in a calculation of current as in reference9 is not quite accurate.
3
2.5 3
2
y=−7, 0,−4 nm
y=−4 nm
2
1
1.5 0
400
450 500 550
Energy (meV)
1
Vg=0V & Vd=1V
0.5
Transmission
DOS
0
200
300
400
500
600
700
800
Energy (meV)
FIG. 9: Transmission (+) and density of states (DOS) versus energy at a spatial location close to the source injection barrier,
at Vg = 0V and Vd = 1V. The peaks in the density of states represent the resonant levels in the channel. Inset: DOS at three
different y-locations and the total transmission. The points y = -7 and 0 nm are to the left and right of the location where the
source injection barrier is largest (close to y = -4 nm).
We compare the results from our quantum simulations with published results from quantum-corrected Medici 35 .
To compare the quantum and classical results, an estimate of the energy of the first subband minima (E r1 ) from
Fig. 9, and the location of the classical barrier height (Eb (classical)) (Fig. 10) are useful. The main features of this
comparison are:
(a) Subthreshold region: The slope d[log(Id )]/dVg is smaller in the quantum case when compared to Medici (Fig.
28
500
Eb(classical)
Er1
Er1−207meV
400
Energy (meV)
300
200
100
E
Fermi
0
−100
−200
0
0.2
0.4
0.6
0.8
1
V (V)
g
FIG. 10: Location of the first resonant level (Er1 ) versus gate voltage and the classical source injection barrier (Eb (classical)).
Note that Er1 decreases slower than Eb (classical) with gate voltage due to narrowing of channel potential well.
11). Further, the current resulting from the simple intuitive expression
I = Iq0 e
−Er1
kT
(219)
matches the quantum result quite accurately. Iq0 is a prefactor chosen to reproduce the current at Vg = 0 in Fig. 11.
This match is rationalized by noting that for the values of gate biases considered, E r1 is well above the source Fermi
energy and Er2 is many kT (thermal energy) above Er1 . The difference in slope between the classical and quantum
results can be understood from the slower variation of Er1 in comparison to Eb (classical) as a function of Vg (Fig.
10). We also find that the decrease of Er1 with increases in gate voltage is slower than the barrier height determined
from the quantum potential profiles. This arises because (neglecting 2D effects) E r1 is determined by a triangular
well (whose apex is the conduction band) that becomes progressively narrower with increase in gate voltage.
−2
10
−3
10
−4
−5
10
−6
10
d
I (A/µm)
10
−7
10
Vd=1V
−8
Medici
Q1 flat band
Q1 q−poly
10
−9
10
−10
10
0
0.2
0.4
0.6
V (V)
0.8
1
g
FIG. 11: Plot of drain current versus gate voltage from the quantum mechanical calculations and Medici, at V d = 1V. At small
gate voltages, the drain current from Medici35 are comparable to the ’Q1 flat band’ results. The drain current from ’Q1 q-poly’
is however significantly different at large gate voltages.
(b) Large gate biases: The drain current and slope d[log(Id )]/dVg are larger in the quantum case. The higher dId /dVg
at large gate voltages in the quantum case can be understood from the fact that E r1 is above the Fermi level while
Eb (classical) is below, at Vg = 1V (the quantum current is proportional to exp(−(Er1 − EF )/kT )). The mobility
model assumed in the classical case also plays a role in determining the slope.
29
G.
Id versus Vd
−3
2
x 10
1.75
Id (A/µm)
1.5
1.25
Vg=1V
1
Medici
Q1, poly depletion
Q1, flat poly
0.75
0.5
0.25
0
0
0.2
0.4
0.6
Vd (V)
0.8
1
FIG. 12: Plot of drain current versus drain voltage (Vd ) from the quantum mechanical calculations and Medici, at Vg = 1V.
Note the large difference in drive current and dId /dVd between Medici35 , ’Q1 flat band’ and ’Q1 q-poly’.
The values of dId /dVd and drive current are important in MOSFET applications because they determine switching
speeds7 . Figure 12 compares the drain current versus drain voltage for Vg = 0 and Vg = 1V . The drive current
(Vg = 1V ) calculated using Q1 with the polysilicon region treated in the flat band and q-poly approximations is more
than 100% and 200% larger than the results in35 . dId /dVd in the linear region is up to three times larger in Q1.
The subthreshold drain current is smaller in Q1. We however expect that with decreasing channel length, the sub
threshold Id will become larger than the Medici results due to quantum mechanical tunneling12 .
H.
Isotropic versus anisotropic effective mass
3
Transmission
2.5
2
(0.19,0.19)
(0.19,0.98)
(0.98,0.19)
isotropic
Vg=0V and Vd=1V
1.5
1
0.5
0
200
300
400
500
600
700
Energy (meV)
FIG. 13: Same as Fig. 9 but the anisotropic effective mass case is included. Note that the valley with the largest mass in the
x-direction has subband energies that are about 50 meV smaller than the isotropic effective mass case even at V g = 0.
The primary influence of anisotropic effective mass is to influence the energy of the subbands in the inversion layer.
Valleys with the largest effective mass perpendicular to the oxide (0.98m∗o ) have subband energies that are smaller
than the isotropic effective mass case. We see from the plot of transmission versus energy (Fig. 13) that the valleys
with (mx = 0.98m∗o , my = mz = 0.19m∗o ) have resonance levels that are more than 50meV lower in energy than
the isotropic effective mass case. The corresponding subthreshold current (Fig. 14) is a few hundred percent larger
than the value obtained from the isotropic effective mass case. This follows by noting that the subthreshold current
30
−2
10
−3
10
−4
10
−5
Id (A/µm)
10
−6
10
Vd=1V
−7
10
Q1 flat band
Q1 q−poly
Q3 flat band
Q3 q−poly
−8
10
−9
10
−10
10
0
0.2
0.4
0.6
V (V)
0.8
1
g
FIG. 14: Plot of drain current versus gate voltage for the isotropic and anisotropic effective mass cases, at V d = 1V. The much
higher current in the anisotropic effective mass case (Q3) is due to the lower suband energy shown in Fig. 13.
depends on exp(−Er1 /kT ). The drive current (Fig. 14) from the anisotropic effective mass case is more than twenty
five percent larger than the isotropic effective mass case. Note that for large gate voltages the dependence of current
on Er1 is sub exponential. We are not aware of any calculations that compare the relative importance of the current
carrying capacity of electrons in the three inequivalent valleys. We find that the valley with the largest m x (=0.98m∗o )
carries 89.22 % and 79.77 % of the current at Vg equal to 0 and 1V respectively (Vd = 1V). Thus all three valleys are
necessary for an accurate calculation of the ballistic current.
I.
Gate leakage current
A major problem in MOSFETs with ultra thin oxides is that tunneling from gate to drain will determine the
off-current. The gate leakage current versus y is plotted for the MIT25 device in Fig. 15. Note that while we use a
value of 3.0 for the dielectric constant of SiO2 , a value of 3.9 does not change the qualitative conclusions. At Vg = 0V
and Vd = 1V, the main path for leakage current is from the polysilicon gate contact on top of the oxide to the highly
doped (n+ ) regions associated with the drain (Source Drain Extension, SDE) as shown in Fig. 15 (a). At non zero
Vg , there is also an appreciable tunneling from the highly doped n+ regions near the source to the polysilicon region
on top of the gate (Fig. 15 (a)). For tox = 1.5 nm, gate tunneling increases the off-current by about two orders of
magnitude, and for smaller oxide thicknesses, the gate leakage current is significantly larger.
We propose that the gate leakage current can be reduced by a factor of 10-100 without significantly compromising
the drive current. The drive current in these ultra small MOSFETs is primarily determined by the source injection
barrier10,11 , or more correctly as discussed earlier by the resonant level at the source injection barrier. So any changes
that result in a reduction of the gate leakage current should not significantly alter the location of the resonant level at
the source injection barrier (and hence the drive current). Two methods (without regard to fabrication issues) that
help in this direction are discussed below:
(i) Shorter or asymmetric polysilicon gate region: We propose that the gate leakage current can be significantly
reduced by using shorter gate lengths. The main feature of the shorter gate lengths is a small overlap between the
polysilicon gate and the n+ region near the drain. This is pictorially represented in Figs. 16 (a) and (b) with ’long’
and ’short’ gate lengths. To simulate the long and short gate lengths, we consider the doping profile of MIT25 with
Lg = 25 nm and 50 nm (gate length in35 ). The off-current and gate leakage current are plotted in Fig. 17. We see
that the gate leakage current reduces by more than an order of magnitude, and the drive current is within two percent
of the Lg = 50 nm case, as desired (see inset of Fig. 17). The spatial profile of gate leakage current for L g = 25
nm is shown in Fig. 15 (b). Though the gate leakage current reduces significantly, a drawback of this scheme is the
requirement for very short (approximately equal to the distance between highly doped region near source and drain)
polysilicon gate lengths. A polysilicon gate placed asymmetrically with respect to y=0 such that its overlap with the
n+ regions near the drain is small, will also serve to reduce the off-current without compromising the drive current.
(ii) Graded oxide: The second proposal is to use a graded oxide, which is thinner close to the source end and thicker
31
−7
8
x 10
q−poly, V =0.1V, L =50nm
d
g
(a)
Gate Current (A / µm 2 )
4
0
Vg=−0.2V
Vg=0V
Vg=0.2V
−4
−8
−25 −20
−7
3
x 10
−10
0
10
20
25
20
25
q−poly, Vd=0.1V, Lg=25nm
(b)
2
1
0
−1
−25 −20
−10
0
10
y (nm)
FIG. 15: Plot of gate leakage current when the device is off (Vg = 0V) as a function of the y-direction, from the source to drain,
for Lg equal to (a) 50 nm and (b) 25 nm. Note the significant gate leakage current in the regions where the high doping in the
y ( nm)
source and drain overlap the gate in (a). A shorter gate eliminates
a large fraction of the gate leakage current as shown in (b).
P
P
D S
S
(b)
(a)
P
D S
D
(c)
FIG. 16: Polysilicon gate and oxide configurations that could reduce the off-current (Vg = 0V) significantly without drastically
reducing the drive current (Vg = 1V).
close to the drain end (Fig. 16 (c)). The thinner oxide near the source is not expected to alter the source injection
barrier significantly, while the tunneling rate from gate to drain will be significantly smaller because of the thicker
oxide in the drain-gate overlap region. We consider an oxide that is 1.5 nm thick for y < +10 nm and 2.5 nm for
y > 11 nm, with the thickness varying linearly in between. The polysilicon gate lengths is 50 nm. Comparison of
this device to the original MIT25 with an uniform oxide and Lg = 50 nm show that while the gate leakage current
decreases by one order of magnitude, the drive current decreases by only 30 %. Further optimization of this device
structure could yield a larger drive current, while keeping the gate leakage current small.
J.
2D Ballistic MOSFET Summary
This section consisted of a modeling framework to calculate properties of ballistic MOSFETs with open boundaries
at the source, drain and gate contacts have been developed. The algorithm used to calculate G r and G< was presented
in section IV. The algorithm avoids solving for the entire G< matrix even in the presence of non zero self energies
throughout the device. Note that the simulations presented are 2D in nature and also involve self-consistency. As
a result, they were numerically intensive and were typically performed on sixteen to sixty four processors of an SGI
Origin machine.
32
−4
Id, Lg=50nm
Ig, Lg=50nm
Id, Lg=25nm
I , L =25nm
−5
−6
g
g
off current
V =0V, V =1V
g
d
−8
−9
tox (nm)
0.8 1.2 1.6 2.0 2.4
−10
−4
d
−13
g
−14
d
on current
d
−12
V =0V, V =1V
−3
g
−11
I (V =V =1V)
Current (A/µm)
−7
Lg=50nm
Lg=25nm
−5
−15
0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
FIG. 17: Plot of drain and gate currents when the device is off (Vg = 0V) versus oxide thickness for Lg equal to 50 and 25
nm. Inset: Drain current for the the gate lengths when the device is on (Vg = 1V). At the larger values of tox , the gate current
(Ig ) is significantly smaller than the drain current (Id ), meaning that the drain current is determined by electron injected
from the source to drain. At smaller values of tox , the drain current is dominated by the gate leakage current as can be seen
by comparing Id and Ig in this figure. More importantly, note that the shorter gate length (Lg = 25 nm) gives an order of
magnitude smaller drain current when the device is off for the smaller values of tox . The inset shows that the drive current
(Vd = Vg = 1 V) is however not affected much by the shorter gate length.
The main results are:
(a) Polysilicon gate depletion causes the conduction band close to the oxide interface to bend in a manner opposite
to the semi-classical case (Fig. 2). This causes a substantial shift in the location of the conduction band bottom in
the channel, which gives rise to drain currents that are different from the semiclassical case by one to two orders of
magnitude. Performing quantum mechanical calculations with a flat polysilicon region, and then shifting the gate
voltage axis (in Id versus Vg ) by the quantum mechanical built-in voltage shown in Fig. 2 results in an order of
magnitude better agreement with results from a quantum mechanical treatment of the polysilicon region. This builtin voltage can simply be determined by 1D simulations or an analytical expression. In reality, treatment of discrete
dopants in the polysilicon region will give rise to results that are in between the ’flat band’ and ’q-poly’ cases presented
in this section.
A quantum mechanical treatment of the polysilicon gate region results in an off-current (V g = 0 V and Vd = 1 V)
that is more than 35 times larger than the off-current from a flat band treatment of polysilicon region and published
results35 based on a sophisticated semiclassical simulator.
(b) Resonant levels in the channel the from source to drain increase the effective source injection barrier for ballistic
electrons. Further, even in the ballistic limit the transmission versus energy reaches integer values over an energy
range that could be many times the thermal energy. Knowledge of the detailed shape of transmission versus energy is
important to accurately determine the ballistic current. The precise shape of these transmission steps depends on the
details of the channel to source and drain overlap regions and the resulting 2D potential profile. Assuming a sharp
step-like increases in the total transmission is incorrect.
The slope dId /dVd , whose importance was emphasized in7 , and the drive current (at Vg = 1V) are about 300%
larger than reported in35 . Further, inclusion of anisotropic effective mass in our calculation makes the quantum results
deviate further from the semiclassical results as shown in Fig. 14.
(c) Tunneling of charge across the gate oxide can put a limit on the off-current. Models of the tunnel current for
thin oxide MOSFETs are important. We model the gate leakage current in two dimensions and show that significant
reduction in the off-current is possible without altering the drive current significantly. This is accomplished by
changing either the gate length (Fig. 16 b) or by introducing a graded oxide (Figs. 16 (c)).
(d) Quasi-ballistic flow of electrons causes the slope of d[log(Id )/dVg to be larger than the values obtained from
33
drift-diffusion methods using field dependent mobility models.
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Abstracts. IWCE Glasgow 2000 (2000), p. 112.
M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 (1993).
U. Ravaioli, B. Winstead, C. Wordelman, and A. Kepkep, Superlattices and Microstructures 27, 137 (2000).
D. Vasileska, W. J. Gross, and D. K. Ferry, Superlattices and Microstructures 27, 147 (2000).
in Proceeding of The Third NASA Workshop on Device Modeling: Superlattices and Microstructures (2000).
G. D. Mahan, Many Particle Physics (Second Edition, Plenum Publishing Corporation, New York, 1990).
G. D. Mahan, Physics Reports 145, 251 (1987).
See 25 nm device in http://www-mtl.mit.edu:80/Well (.).
IEEE TRANSACTIONS IN ELECTRON DEVICES
1
Role of scattering in nanotransistors
Alexei Svizhenko and M. P. Anantram
Abstract
We model the influence of scattering along the channel and extension regions of dual gate nanotransistor. It is found that the reduction
in drain current due to scattering in the right half of the channel is comparable to the reduction in drain current due to scattering in the left
half of the channel, when the channel length is comparable to the scattering length. This is in contrast to a popular belief that scattering
in the source end of a nanotransistor is significantly more detrimental to the drive current than scattering elsewhere. As the channel length
becomes much larger than the scattering length, scattering in the drain-end is less detrimental to the drive current than scattering near the
source-end of the channel. Finally, we show that for nanotransistors, the classical picture of modeling the extension regions as simple series
resistances is not valid.
To appear in IEEE Transaction in Electron Devices on Electron Devices
I. Introduction
Experimental and theoretical work on nanotransistors has been a hot area of research because of significant advance
in lithography. The significant advances in lithography have led to the construction of nanotransistors with channel
lengths smaller than 25 nanometers (nm) [1], [2], [3]. It is believed that devices with channel lengths equal to 10 nm may
become possible in research laboratories [4]. In these nanotransistors, the length scales of the channel, gate, screening
and scattering lengths, begin to become comparable to one another. This is not the case for long channel MOSFETs,
where the channel and gate lengths are much larger than the scattering lengths. As a result of the comparable length
scales, it is expected that the physics of nanotransistors will begin deviating from that of long channel transistors.
lCh
lEx-s
GATE
lEx-d
n+ source
n+ drain
X
TCh
Y
GATE
Fig. 1. Schematic of a Dual Gate MOSFET (DG MOSFET). Ex-s and Ex-d are the extension regions and the hatched region is the channel. The white region between the
source / drain / channel and gate is the oxide. The device dimension normal to the page is infinite in extent.
The resistance of a MOSFET (Fig. 1) with a long channel length can be qualitatively thought of as arising in four
regions, Extension regions near the source (Ex-s) and drain (Ex-d), Channel (Ch), and Contacts. It is believed that
the resistance of the contacts and extension regions are extrinsic series resistances [5], while the channel resistance is
intrinsic to the MOSFET. For a given doping distribution, both electrostatics and scattering of the current carriers
play an important role in determining the drive current. Electrostatics dictates that the total carrier density in a long
channel MOSFET is approximately Cox (VG − VS ), where VG and VS are the gate and source voltages. The role of
scattering of current carriers in long channel transistors is modeled using the mobility. For nanotransistors with ultra
short channel lengths, there are some deviations in the electrostatics from the long channel case [5]. The role of scattering
is however not well understood in nanotransistors. Most work on nanotransistors use the drift diffusion equations which
are applicable to long channel MOSFETs or fully ballistic calculations based on the Schroedinger equation. A detailed
understanding of the influence of scattering is important as it is crucial in determining the on-current of nanotransistors.
The role of scattering is however not straight forward to determine without a calculation because scattering tends to
change the carrier and current densities in the channel and extension regions, both spatially and energetically. Further,
the physics of this redistribution depends sensitively on the channel and scattering lengths as demonstrated in this
paper.
The aim of this paper is to model the exact influence of scattering at different spatial locations along the channel and
extension regions of silicon n-MOSFETs. We consider only electron-phonon scattering, which is an important scattering
NASA Ames Research Center, Mail Stop: 229-1, Moffett Field, CA 94035-1000.
IEEE TRANSACTIONS IN ELECTRON DEVICES
2
mechanism in devices with undoped channels. References [6] and [7] have recently pointed out that electron-electron and
plasmon scattering may play an important role in degrading nanotransistor characteristics. Electron-electron scattering
in the drain side will lead to carriers having an energy larger than the source injection barrier. The resulting small tail
of hot carriers [8] will be reflected back into the source-end, there by causing an increase in the source injection barrier
and a corresponding decrease in drain current. The modeling of these effects and interface roughness is beyond the scope
of our current work.
In our calculations, we consider the dual gate MOSFET [9], [10], which is considered to be a promising candidate for
nanotransistors. The reason for this is the large on-current and better scaling properties it offers, when compared to
bulk-type MOSFETs [11], [12], [13], [14], [15], [16].
The outline of the paper is as follows. In section II, we present our simulation results on the role of scattering in
nanotransistors, where we show that scattering is important throughout the device, and not just in the source-end. This
is followed by a discussion explaining why drain-end scattering is important in nanotransistors (section III). In section
IV, we show that scattering in the extension regions cannot be modeled as simple series resistances. We conclude in
section V. All details of our method and approximations are given in the appendix.
II. Where is scattering important?: Simulation Results
Two devices (Fig. 1) are simulated with the following parameters:
Device A (similar to the Purdue dual gate MOSFET [17].): Channel length (LCh ) = 10 nm, channel extends from -5
nm to 5 nm, channel thickness (TCh ) = 1.5 nm, oxide thickness = 1.5 nm, gate work function = 4.25 eV, doping in the
extension regions = 1 E+20 cm−3 , no doping in the channel, drain voltage (VD ) = gate voltage (VG ) = 0.6 V, and the
dielectric constant of the oxide (ox )=3.9.
Device B: Same as Device A, except that LCh = 25 nm, channel extends from - 12.5 nm to 12.5 nm and VG = 0.56
V. In all simulations involving this device, scattering is included only in the channel. The gate length is equal to the
channel length for both devices A and B. The temperature assumed in all calculations in this paper is 300K.
We first discuss device A. To elucidate the role of scattering in different spatial regions, we calculate the drain current
(ID ) as a function of the right boundary of scattering, YR−Scatt . Scattering is included from the edge of the source
extension region (-20 nm) to YR−Scatt in Fig. 2. The ballistic current is 1.92 mA/µm, the value at YR−Scatt = −20 nm.
The channel extends from -5 nm to +5 nm. The main points of this figure are:
(i) The decrease in current from the ballistic value due to scattering in the source extension, channel and drain
extension regions are 11.5%, 15.5% and 4% respectively. These values point to the well appreciated result that either
reducing the length or flaring the source extension region will make a nanotransistor significantly more ballistic.
(ii) The decrease in drain current due to scattering over the entire channel is important. That is, scattering in the
right half of the channel (0 nm to 5nm) is almost as important as scattering in the left half of the channel (-5 nm to 0
nm).
(iii) The drain current continues to decrease significantly due to scattering in the drain extension region. An important
question is if this decrease is simply a series resistance effect (see section IV).
We now present results for device B, whose channel length is two and a half times larger than device A. The scattering
times are nearly the same for the two devices. As a result of the larger channel length, the probability for a carrier to
energetically relax is larger. Here, we find that scattering in the left (-12.5 nm to 0 nm) and right (0 nm to 12.5 nm)
halves of the channel reduces the drain current by 32% and 15% respectively from the ballistic value, and the over all
ballisticity (ratio of Current with scattering to Ballistic current) is 53% (dashed line of Fig. 3). Again, this points to
the importance of scattering in the drain-end.
In lieu of simulating devices with longer channel lengths, we increase the scattering rate of device B. The scattering
rate is increased by√a factor of five by artificially increasing the values of the deformation potential quoted in reference
[18] by a factor of 5. Note that device B has almost no DIBL and that we self-consistently solve the Green’s function
and Poisson’s equations with the larger deformation potentials. The ballisticity of device B with the larger scattering
rate is 38%, and the current decreases by 60% and 12% of the ballistic value due to scattering in the left and right halves
of the channel respectively (solid line of Fig. 3). It is also apparent from Fig. 3 that the effect of scattering on drain
current becomes relatively smaller as YR−Scatt approaches the drain-end (12.5 nm).
III. Discussion
The results of section II show that scattering at all locations in the channel is important in determining
the drain
current of nanoscale MOSFETs. We first discuss device A. For device A, the scattering time (h̄/2|Im Σrphonon |)
at an energy of Eb + 26 meV is 50 fs and 24 fs in the source and drain-ends respectively. The scattering times are
comparable to the semiclassical transit time of 26 fs (Table I). The scattering (11 nm) and channel lengths (10 nm) are
hence comparable (Table I). It is interesting to note that for this device, the argument that the energetic redistribution
IEEE TRANSACTIONS IN ELECTRON DEVICES
3
Device A, LCh = 10 nm, Lscatt = 11 nm
I (mA/µm)
2
c
ballisti
1.5
ring
scatte
−10
D
1
0.5
V (V)
1.7
0.2 0.4 0.6
1.6
−15
−20
b
D
0
0
E (meV)
1.8
D
I ( mA /µm )
1.9
−5
2
1.5
−25
1.4
1.3
−20
−10
0
10
YR−Scatt (nm)
−30
20
Fig. 2.
Plot of drain current (ID ) versus the right boundary of scattering (YR−Scatt ) for device A. The scattering time is comparable to the transit time through the
channel. Scattering is included from -20 nm to YR−Scatt . Note that scattering in the right half of the channel (0 nm to 5 nm), which is to the right of the ’k B T layer’, is
almost as deleterious to current flow as scattering in the left half of the channel (-5 nm to 0nm). The black crosses represent E b as a function of YR−Scatt . Inset: Ballistic
ID versus VD for VG = 0.6 V, showing substantial DIBL. Scattering is included both in the channel and extension regions.
Device B, L
Ch
= 25 nm
D
I ( mA / µm )
1.2
1
Lscatt = 11 nm
0.8
0.6
Lscatt = 2.2 nm
0.4
−10
−5
0
5
YR−Scatt (nm)
10
Fig. 3. Plot of drain current versus YR−Scatt for device B. Scattering is included from -12.5 nm to YR−Scatt . For Lscatt = 11 nm (dashed line) and 2.2 nm (solid
line), the effect of scattering in the right half of the channel (0 nm to 12.5 nm) corresponds to nearly a third and sixth respectively of the total reduction in drain current.
This figure points to the relatively smaller role of drain-end scattering in comparison to source-end scattering, when L Ch becomes much larger than Lscatt . Scattering is
included only in the channel for both cases.
of electrons in the channel to states with kinetic energy in the transport direction well below E b will make drain-end
scattering ineffective fails.
To understand why drain-end scattering is important for the parameters in device A, it is useful to plot the change
in barrier height (Eb ) with YR−Scatt . Fig. 2 shows Eb as a function of YR−Scatt . It is noted that Eb first decreases and
then increases, with increase in YR−Scatt . The decrease of Eb for −20 nm < YR−Scatt < −4 nm is due to the potential
drop in the source extension region arising from the increasing series resistance. Note that the location of the source
injection barrier (Yb ) is -4 nm (Fig. 4). For YR−Scatt > Yb , Eb increases with YR−Scatt . The reason for the increase in
Eb are the electrons reflected towards the source from the right of Yb . Electrostatics, more or less demands that the
charge in the gate should be approximately Cox (VG − VS ) [19], [20], like in long channel MOSFETs [5]. So, Eb floats
to higher energies to compensate for the increase in electron density from the reflected electrons. This increase in E b
contributes significantly to the decrease in the drain current even due to scattering in the right half of channel (0 nm
to 5 nm). The increase in Eb with increase in YR−Scatt becomes smaller in the right end of Fig. 2 because the electrons
IEEE TRANSACTIONS IN ELECTRON DEVICES
4
Device A
τscatt at source-end (s)
5.0 E-14
Devic B
5.0E-14 (1.0 E-14)
τscatt at drain-end (s)
2.5 E-14
2.4 E-14 (4.8 E-15)
τtransit at Eb+26 meV (s)
2.6 E-14
6.4 E-14
τtransit at 60 meV (s)
2.0 E-14
5.6 E-14
v at Yb, Eb+26 meV (m/s)
2.2 E+5
2.2 E+5
v at Yb, 60 meV (m/s)
3.5 E+5
2.8 E+5
LCh (nm)
10
25
v*(τscatt at Yb)
E=Eb+26meV (nm)
11
11 (2.2)
•
•
τscatt − scattering time (hbar/2Im(Σr))
τtransit - shortest semiclassical transit time for electron with a given total energy = integeral
•
•
∫dy / [2(E-V(y))/m]
v - semiclassical velocity at y = [2(E-V(y))/m]1/2
For Device B, quantities in brackets are for the case of five times larger scattering rate
1/2
TABLE I
Estimates of scattering time, transit time, velocity and scattering length.
scattered here contribute less significantly to the channel charge, as will be apparent from the discussion below.
Device A
0
Eb
Y
b
1
E (eV)
−0.2
−0.4
−0.6
−10
Fig. 4.
−5
0
Y (nm)
5
10
Energy of the lowest subband (E1 ) versus Y for device A in the ballistic limit. Eb and Yb are the energy and position of the source injection barrier respectively.
E1
.
e
Potential = −
We now discuss device B. Device B is different from device A in that its channel length is two and a half times longer
than that of device A. The importance of scattering in the right half of the channel is obvious for device B from the
dashed line of Fig. 3. Here, scattering in the left (-12.5 nm to 0 nm) and right (0 nm to 12.5 nm) halves of the channel
reduce the drain current by 32% and 15% respectively from the ballistic value. To complement the discussion of device
A in terms of Eb , we will discuss device B in terms of another useful quantity: J(Y, E), which is the current distribution
as a function of total energy E at Y . J(Y, E) gives us partial information about the energetic redistribution of current
due to scattering (see end of current section). When the channel length is comparable to the scattering length, J(Y, E)
is peaked in energy above Eb , in the right half of the channel (Fig. 5 (a)). Scattering causes reflection of this current
towards the source. This is the first reason for the reduction in drain current. The second reason is that the reflected
electrons lead to an increase in the channel electron density (classical MOSFET electrostatics). As the charge in the
channel should be approximately Cox (VG − VS ), the source injection barrier Eb floats to higher energies to compensate
IEEE TRANSACTIONS IN ELECTRON DEVICES
5
for the reflected electrons. The increase in Eb leads to a further decrease in drain current due to scattering in the right
half of the channel.
200
Device B, L
Ch
= 25 nm, L
= 11 nm
scatt
200
Device B, L
= 25 nm, L
Ch
= 2.2 nm
scatt
100
50
50
0
−50
−50
−100
−100
−150
−150
−200
−17.5 −12.5 −7.5
−2.5
2.5
7.5
12.5
17.5
−200
Energy Relaxed
Carriers
0
E1 (meV)
100
Hot
Carriers
150
1
E (meV)
(b)
150
−17.5 −12.5 −7.5
−2.5
2.5
7.5
12.5
17.5
Y (nm)
Y (nm)
Fig. 5. The solid lines represent J(Y, E) for Y equal to -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5 and 17.5 nm, from left to right respectively. The dashed lines represent the
first resonant level (E1 ) along the channel. The dotted lines represent the first moment of energy (mean) with respect to the current distribution function J(Y, E), which is
R
dEEJ(Y,E)
R
.
(a) and (b) correspond to Lscatt = 11 and 2.2 nm respectively in device B. Scattering is included every where in the channel. (a) and (b) correspond to the
dEJ(Y,E)
YR−Scatt = 12.5 nm data points of the dashed and solid lines of Fig. 3. Scattering is included everywhere in the channel but not in the extension regions.
To gain further insight into the role of carrier relaxation, we now discuss device B when the scattering length is
five times smaller. The scattering length Lscatt is defined in Table I. Scattering in the right half of the channel for
Lscatt = 2.2 nm is significantly less detrimental to the drain current relative to scattering in the left half of the channel,
when compared to the device with Lscatt = 11 nm. As LCh (25 nm) is much larger than Lscatt (2.2 nm), multiple
scattering events now lead to an energy distribution of current that is peaked well below the source injection barrier in
the right half of the channel as shown in Fig.
5 (b). The first moment of energy (mean) with respect to the current
R
distribution function, which is defined by R
dEEJ(Y,E)
dEJ(Y,E)
, is also shown in Fig. 5. This mean also shows that the carriers
relax in a manner akin to bulk MOSFETs as a function of Y in Fig. 5 (b). Carriers reflected in the right half of the
channel can no longer reach Yb due to the large barrier to the left, and so contribute less significantly to the charge
density. Thus, explaining the diminished influence of scattering in the right half of the channel relative to the left half
of the channel, for devices with the channel length much larger than the scattering length.
The above discussion would be incomplete without discussing the electrostatic potential profiles, with and without
scattering. The solid line in Fig. 6 is the electrostatic potential in the ballistic limit. Increasing Y R−Scatt from - 2.5
nm to 2.5 nm causes Eb to increase because of carriers reflected towards the source. Further increase in Y R−Scatt to 7.5
nm causes very little increase in Eb because scattering in the right half of the channel is less effective in changing the
channel electron density. The electrostatic potential changes appreciably to the right of Y b due to scattering. It is also
interesting to note that the electrostatic potential drop for YR−Scatt = 7.5 nm is linear to the right of Yb compared to
the ballistic case because of scattering in the channel.
We now comment briefly on two issues:
- The quantity Eb − 2kT that has been discussed before in references [17] and [21].
- The influence of elastic scattering without any inelastic scattering.
For devices A and B, the potential profile in the right half of the channel is well below Eb − 2kT . Yet, scattering in the
right half of the channel is detrimental to drain current, relative to scattering in the left half of the channel. The reason
for this are the hot electrons in the right half of the channel that are reflected to the source-end / Y b . However, if the
scattering rate in the left half of the channel is large enough to energetically relax the electrons to energies comparable
to Eb − 2kT , then the scattering of these electrons in the drain-end are relatively less detrimental to the reduction in
drain current because the carriers cannot easily gain an energy of few times the thermal energy. This phenomenon of the
diminished role of scattering in the channel at the drain-end relative to the source-end because of thermalized carriers
is seen in Fig. 3 (solid line).
In the presence of elastic scattering processes such as interface roughness scattering, the electron does not loose total
energy. However, the kinetic energy in the transport direction can diminish at the expense of a corresponding gain in
IEEE TRANSACTIONS IN ELECTRON DEVICES
6
50
Device B, L
Ch
= 10 nm, L
scatt
= 2.2 nm
E1 (meV)
0
−50
−100
−150
ballistic
YR−Scatt = − 2.5 nm
YR−Scatt = 2.5 nm
YR−Scatt = 7.5 nm
−15 −10
−5
0
Y (nm)
5
10
Fig. 6. Electrostatic potential versus Y for device B. Scattering from -12.5 nm to 2.5 nm causes a large change in the source injection barrier (E b ). Scattering to the right
of 2.5nm causes a much smaller change in Eb . In the absence of scattering, the potential profile in the channel tends to flatten. The potential drop (or E 1 ) along the channel
is more ohmic / linear in the presence of scattering.
h̄2 kz2
.
2mn
z
The additional density of states for scattering that is available in the drain-end in comparison to the source-end
will also make drain-end scattering less effective than source-end scattering. While we included such process in our
calculations, the quantity J(Y, E) captures only the effect of change in total energy. A physically motivated study
quantifying the relative roles of elastic and inelastic scattering will be a useful future study.
IV. Failure of the classic series resistance picture for nanotransistors
We ask the question if scattering in the extension regions is a simple series resistance. The classic series resistance
picture [5] relates the current in a device with long extension regions to the current in the same device without (or with
much smaller) extension regions. The relationship is particularly simple for the case where the series resistance in the
source extension region is negligible [5],
scatt
noscatt
ID
(VD ) ∼ ID
(VD − δVD ) ,
(1)
scatt
noscatt
where, ID
(VD ) and ID
(VD − δVD )RD ) are the drain currents with and without scattering in the drain extension
scatt
region, at drain biases of VD and VD − δVD respectively. δVD = ID
(VD )RD , is the electrostatic potential drop in the
drain extension region, which has a series resistance of RD . To answer the question on the appropriateness of the classic
series resistance picture, we consider a case where the channel and source extension region
are ballistic. Scattering
√
is introduced only in the drain extension region with deformation potentials that are 5 times larger than in silicon
(scattering time is 5 times smaller).
Fig. 7 shows the decrease in drain current with YR−Scatt . The striking point of Fig. 7 is the super-linear decrease of
drain current. The ID (VD ) curves (inset of Fig. 7) predict a significantly smaller decrease in drain current with increase
in YR−Scatt when Eq. 1 is used. It is helpful to estimate the drain current from Eq. 1 and compare it to the calculated
value. For Device A in Fig. 7, the voltage drop in the drain extension region with scattering is approximately 100 mV
(plot not shown). Now, if Eq. 1 is used to estimate the drain current with scattering in the drain extension region and
if we take δVD = 200 mV, which is larger than the estimated 100 mV, then we find the drain current to be 1.83 mA/µm
(inset of Fig. 7). The calculated drain current is however much lower at 1.38 mA/µm!
The physics of the large reduction in drain current for the smaller values of YR−Scatt is essentially that discussed in
section III: When scattering in the channel does not effectively thermalize carriers, the current distribution is peaked at
energies above Eb , upon carriers exiting the channel. Scattering in the drain extension region then causes reflection of
electrons towards the source-end. As a result, Eb increases so as to keep the electron density in the channel approximately
Cox (VG − VS ). The drain current decreases dramatically as a result of the increase in Eb . Admittedly, this argument in
terms of Cox (VG − VS ) is over simplified but it seems to capture the essential point. The main point is that if carriers
are not relaxed upon exiting the channel (as would be the case for nano-transistors), then, the drain extension region
cannot be modeled by a simple series resistance. That is, Eq. (1) fails for nano-transistors where the channel length
is comparable to the scattering length. The effect of the drain extension region in causing a reduction in drain current
would be small in the following cases:
IEEE TRANSACTIONS IN ELECTRON DEVICES
7
2
Device A, LCh = 10 nm, Lscatt = 2.2 nm
I (mA/µm)
1.8
1.7
ballistic
1
D
ID (mA/µm)
1.9
2
1.85
1.7
1.5
1.38
0.5
series resistance
scattering
1.6
V (V)
0
0
1.5
D
0.2
0.4
0.6
1.4
1.3
10
20
YR−Scatt (nm)
30
Fig. 7. ID versus YR−Scatt for device A with scattering present only in the drain extension region from 5 nm to 30 nm. The large reduction in drain current is due to
scattering of hot carriers from the drain extension region back in to the channel. The physics of this effect is completely different from ’classical series resistance’ in MOSFETs,
which is a much smaller effect. The results obtained from the ’series resistance’ and ’scattering calculations’ (this paper) are indicated by the arrows. The electron-phonon
scattering time is five times larger than in Fig. 2. Inset: Drain current versus drain voltage in the ballistic limit, showing the drain current estimate from the series resistance
picture and from our calculation.
(i) The channel is much longer than the scattering length such that the carriers exiting the channel at the drain-end
are energetically relaxed / thermalized. Then, the modeling of the drain extension region as a simple series resistance
would be appropriate. This is seen in the right end of Fig. 7, where, upon sufficient relaxation of electrons, the decrease
in current with increase in YR−Scatt becomes much smaller.
(ii) The drain extension region rapidly flares out. Then, the probability for a scattered electron to return to the
source-end will be small due to the larger number of modes available in the drain extension region. A careful analysis
on how fast the drain extension region flares out should also take into account the role of the Miller effect.
V. Conclusions
In conclusion, we find that the potential profile, channel and scattering length scales play an important role in
determining the relative importance of scattering at different locations along the channel of a nanotransistor. In devices
where the channel length is comparable to the scattering length, the role of scattering in the drain-end (right half of
the channel) is comparable to the role of scattering in the source-end (left half of the channel), in reducing the drain
current (Fig. 2 and dashed line of Fig. 3). This is contrary to a belief that scattering is significantly more important
in the source-end of the device. The reason for the detrimental role of scattering in the drain-end are the hot carriers
in the drain-end. When the channel length is much larger than the scattering length, then scattering in the source-end
becomes relatively more important than scattering in the drain-end (solid line of Fig. 3). In this case, we stress that
it is the energetic redistribution of carriers due to scattering in the source-end to energies below the source injection
barrier (Eb ) that makes scattering in the drain-end relatively less detrimental to the drain current.
The classical series resistance picture for modeling the narrow extension regions fail for nanotransistors. The reason
for this failure are the hot carriers entering the drain extension region. A straight forward option to enable the usage of
the series resistance picture is to push the region treated as a drain series resistance further to the right, such that all
carriers entering this region are energetically relaxed. A more interesting option of altering the classical series resistance
picture to account for the hot carriers in the drain end of nanotransistors was not considered in this paper.
The relative importance of scattering in the drain-end of nanotransisors, where the channel length is comparable
or smaller than the scattering length, points to the importance of making the extension regions small. Long extension regions in nanotransistors will affect the performance (drive current) much more adversely than in long channel
transistors.
IEEE TRANSACTIONS IN ELECTRON DEVICES
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Appendix
The approach consists of solving the nonequilibrium Green’s function and Poisson’s equations. The effective mass
Hamiltonian considered is,
X h̄2 d 1 d d
1 d
d
1 d
+
+
+ V (x, y),
(2)
H=
−
2 dx mbx dx
dy mby dy
dz mbz dz
b
where (mbx , mby , mbz ) are the (x, y, z) components of the effective mass in valley b of silicon, and the potential does not
vary in the z direction. The gate oxides are treated as hard walls, the channel is extremely narrow (1.5 nm), the drain
and gate biases are smaller than 0.7 V, and the dual gate FET is perfectly symmetric in the X-direction of Fig. 1. The
first three subband energy levels in the source extension region are approximately equal to 173 meV, 691 meV (both due
to my = 0.98m0 ) and 891 meV (due to my = 0.19m0 ) above the bulk conduction band. The Fermi energy of bulk silicon
at the doping density considered (1E+20 cm−3 ) is approximately 60 meV above the conduction band. For the doping
density considered, electrons are primarily injected from the source into the first subband. At the drain end, more than
one energy level can in principle contribute to current. As only a few subbands are populated, we model transport in
these subbands in an approximate way using the 1D Schroedinger equation as outlined below. We find the spatially
dependent subband energies En (y) by solving Schroedinger’s equation at each y-cross section (y is only a parameter),
h̄2 d
1 d
− b
+ V (x, y) Ψn (x, y) = En (y)Ψn (x, y) .
(3)
2mx dx mbx dx
n = ν, b, where ν and b represent the quantum number due to quantization in the X-direction and the valley respectively.
The valley indices b are required in the calculations of the self-energies for scattering as will be discussed below. In
our calculation, we typically retain only the three lowest energy levels. Coupling between the subbands is neglected
except via phonon coupling. For the device dimensions and voltages considered, reference [22] found the approximation
of considering decoupled subbands to hold good for ultra thin body phase coherent MOSFETs. We solve the following
equations for the Green’s functions,
2
h̄2 kz2
h̄ d
1 d
E−
− −
+ En (y) Grn (y, y 0 , kz , E)
2mnz
2 dy mny dy
Z
− dy1 Σrn (y, y1 , kz , E)Grn (y1 , y 0 , kz , E) = δ(y − y 0 ) , and
(4)
2
h̄2 kz2
h̄ d
1 d
0
E−
− −
+ En (y) Gα
n (y, y , kz , E)
2mnz
2 dy mny dy
Z
Z
r
α
0
a
0
− dy1 Σn (y, y1 , kz , E)Gn (y1 , y , kz , E) = dy Σα
n (y, y1 , kz , E)Gn (y1 , y , kz , E) ,
(5)
where, α ∈>, <. mny and mnz are the effective masses of silicon in the y and z directions that give rise to subband index
n.
The self-energies, Σr,>,<
can be written as,
n
Σα
n
Σα
n,P honon
α
= Σα
n,C + Σn,P honon , where
α
= Σα
n,el + Σn,inel .
(6)
(7)
α
α
Σα
n,C is the self-energy due to the leads. The phonon self-energy Σ n,P honon consists of two terms, Σn,el due to elastic
α
and Σn,inel due to inelastic scattering. The self-energy due to the leads is non zero only at the first (source) and last
(drain) grid points because gate tunneling is neglected.
The following common approximations to calculate the phonon self-energies are used: (i) Phonon scattering is treated
only within the self-consistent Born approximation, (ii) The phonon bath is assumed to always be in equilibrium, and so
their occupation numbers are given by the Bose-Einstein distribution function with a spatially independent temperature.
(iii) The correlation between subbands n and n0 (6= n) are neglected. (iv) Scattering due to phonons is assumed to be
isotropic. That is, the scattering rate from (kz , E) to (kz0 , E 0 ) does not depend on kz and kz0 . This approximation is
computationally advantageous because the self-energies due to phonon scattering appear only as diagonal terms in Eqs.
4 and 5. One can derive from these assumptions that the self-energies due to electron-phonon scattering at grid point
yi are given by [23], [24],
p 0Z
X
mnz
1
el
√
Σα
dEz √ Gα
(8)
(y
,
E)
=
D
0 (yi , Ez , E) ,
el,n i
n,n0
Ez n
πh̄
2
n0
IEEE TRANSACTIONS IN ELECTRON DEVICES
Σ<
inel,n (yi , E)
=
9
X
i,η
Dn,n
0
X
i,η
Dn,n
0
n0 ,η
and
mnz 0
√
πh̄ 2
nB (h̄ωη )G<
n0 (yi , Ez , E
Σ>
inel,n (yi , E)
=
n0 ,η
p
Z
1
dEz √
Ez
− h̄ωη ) + (nB (h̄ωη ) + 1)G>
n0 (yi , Ez , E + h̄ωη ) ,
p
mnz 0
√
πh̄ 2
Z
(9)
1
dEz √
Ez
>
nB (h̄ωη )G>
n0 (yi , Ez , E + h̄ωη ) + (nB (h̄ωη ) + 1)Gn0 (yi , Ez , E − h̄ωη ) .
(10)
α ∈>, <, r in Eq. 8, η represents the phonon modes, and the square of the matrix elements for phonon scattering are
given by,
el
Dn,n
0
i,η
Dn,n
0
1
D2 kT
= (δν,ν 0 + )δb,b0 A 2
2
ρv
"
#
2
Df2 η h̄
Dgη
h̄
1
+ (1 − δb,b0 )
= (δν,ν 0 + ) δb,b0
2
2ρωgη
ρωf η
(11)
(12)
The contribution to elastic scattering is only from acoustic phonon scattering. The values of the deformation potential,
DA , Dgη and Df η , and phonon frequencies ωgη and ωf η are taken from [18]. ρ is the mass density, k is the Boltzmann
constant, T is the temperature and v is the velocity of sound. b and b0 are indices representing the valley. The
following scattering processes are included: acoustic phonon scattering in the elastic approximation and g-type intervalley
scattering with phonon energies of 12, 19 and 62 meV. It was verified that f-type (19, 47 and 59 meV phonon) intervalley
scattering did not significantly change our results and conclusions. This can be rationalized by noting that f-type
intervalley scattering involves subbands with energies higher than the lowest subband. In the regions, where scattering
was not included, the deformation potential was set equal to zero.
Σrinel,n can be obtained using the Kramers-Kronig relationship,
h
i
r
0
Z
Im
Σ
(y
,
E
)
i
inel,n
1
Re Σrinel,n (yi , E) =
P
dE 0
and
(13)
π
E0 − E
h
i
1
Im Σrinel,n (yi , E) =
Σ>
(yi , E) − Σ<
(14)
inel,n (yi , E) ,
2i inel,n
where P stands for the principal part of the integral. Note that the self-energies due to electron-phonon scattering
depend only on the total energy E (and not on kz ) due to the assumption of isotropic scattering.
The self-energy due to phonon scattering, has real and imaginary parts, both of which vary with energy. The imaginary
part of the electron-phonon self-energy which is central to our calculations is responsible for scattering induced broadening
of energy levels and energetic redistribution of carriers. The real part of the self-energy which contributes to the shift
of the quasi-particle energy levels, appears as a real potential (like the electrostatic potential) in the Green’s function
equations (Eqs. 4 and 5). To evaluate the importance of the real part of the self-energy in our calculations, we performed
simulations with acoustic phonon scattering in silicon, with and without the real part of the self-energy included. We
find that the drive current calculated with the real part of the self-energy set to zero in general agrees to within 2 percent
of the current calculated with the real part of the self-energy included. This result is not totally surprising because
MOSFET electrostatics tends to shift the potential profile appropriately to determine the correct charge under the gate.
In the calculations presented in this paper, the real part of the self-energy is set to zero.
In the numerical solution, we consider N uniformly spaced grid points in the Y -direction with the grid spacing equal
to ∆y. The discretized form of Eqs. 4 and 5 are:
Ai,i Grn (yi , yi0 , kz , E) + Ai,i+1 Grn (yi+1 , yi0 , kz , E) + Ai,i−1 Grn (yi−1 , yi0 , kz , E) =
δi,i0
, and
∆y
0
α
0
α
0
Ai,i Gα
n (yi , yi , kz , E) + Ai,i+1 Gn (yi+1 , yi , kz , E) + Ai,i−1 Gn (yi−1 , yi , kz , E) =
α
Σn (yi , E)Gan (yi , yi0 , kz , E) ,
(15)
(16)
where,
Ai,i
Ai±1,i
= E−
= +
h̄2 kz2
h̄2
−
− En (yi ) − Σrn (yi , kz , E) and
2mnz
mny ∆y 2
h̄2
2mnz ∆y 2
(17)
(18)
IEEE TRANSACTIONS IN ELECTRON DEVICES
10
The self-energy due to the source and drain leads contribute only to grid point 1 (left end of the source extension region)
2
and grid point N (right end of the drain extension region), and are given by [25]: Σrn,C (y1 , kz , E) = ( 2mh̄n ∆y2 )2 gs (kz , E),
z
2
<
r
Σrn,C (yN , kz , E) = ( 2mh̄n ∆y2 )2 gd (kz , E), Σ<
n,C (y1 , kz , E) = −2iIm(Σn,C (y1 , kz , E))fs (E), Σn,C (yN , kz , E) =
z
r
−2iIm(Σrn,C (yN , kz , E))fd (E), Σ>
n,C (y1 , kz , E) = 2iIm(Σn,C (y1 , kz , E))[1 − fs (E)], and
>
r
Σn,C (yN , kz , E) = 2iIm(Σn,C (yN , kz , E))[1 − fd (E)], where y1 an yN are the left (source-end) and right (drain-end)
most grid points respectively, gs (kz , E) and gd (kz , E) are the surface Green’s functions of the source and drain leads
respectively, and fs and fd are the Fermi functions in the source and drain contacts respectively.
The non equilibrium electron and current densities are calculated in both the channel and extension regions using the
algorithm for G< in [26], which avoids full inversion of the A matrix. For completeness, we state the expressions for the
electron and current densities used [26],
nn (yi , kz , E) = −iG<
n (yi , yi , kz , E)
X
e
h̄2
Jn (yi , kz , E) =
[G< (yi , yi+1 , kz , E) − G<
n (yi+1 , yi , kz , E)] .
h̄ n 2mny ∆y 2 n
Note that Eqs. 19 and 20 do not include spin and valley
point yi are given by,
p
X m n0 Z
√z
n(yi ) = 2
πh̄
2
n
p 0Z
X mn
√z
J(yi ) = 2
πh̄
2
n
(19)
(20)
degenaracies. The total electron and current densities at grid
dE
2π
Z
1
dEz √ nn (yi , Ez , E)
Ez
(21)
dE
2π
Z
1
dEz √ Jn (yi , Ez , E) ,
Ez
(22)
where the prefactor of 2 in the above equations account for two fold spin degenaracy. While the transport equations are
solved in one dimension, we solve Poisson’s equation in two dimensions. The two dimensional electron density used in
Poisson’s equation is computed from Eqs. (3) and (19) using,
n(xi , yi , kz , E) = nn (yi , kz , E)|Ψn (xi , yi )|2 .
(23)
The boundary conditions to Poisson’s and Green’s function equations are applied at the ends of the source and drain
extension regions (left and right ends of the source and drain extension regions shown in Fig. 1). In solving the the
Green’s function and Poisson’s equation, note that an applied bias corresponds to a difference in the Fermi levels used
in the source and drain regions. The electrostatic potential at the left and right most grid points of the source and drain
extension regions respectively are calculated self consistently using the boundary conditions.
Finally, we make a comment on the need for solving quantum mechanical equations to capture the essential effect of hot
carriers, described in this paper. The phase of the electron is not central to the physics described in our paper (though
the exact value of the drain current depends on it). In calculating the drain current, the quantum mechanical effects
of quantization in the X-direction and tunneling along the Y-direction (Fig. 1) can be accounted for semiclassically.
So, we feel that a method such as the Monte Carlo method approach to nanotransistors [27], which keeps track of the
details of the energetic redistribution of electrons at various spatial locations, will well describe many aspects of the role
of scattering.
References
[1]
J. Kedzierski, P. Xuan, V. Subramanian, J. Bokor, T-J King, C. Hu, and E. Anderson. A 20 nm gate-length ultra-thin body p-mosfet
with silicide source/drain. Superlattices and Microstructures, 28:445–52, 2000.
[2] A. Hokazono et. al. 14 nm gate length cmosfets utilizing low thermal budget process with poly-sige and ni salicide. In International
Electron Devices Meeting. Technical Digest, pages 639–42. IEEE , Piscataway, NJ, USA, 2002.
[3] F. Boeuf. 16 nm planar nmosfet manufacturable within state-of-the-art cmos process thanks to specific design and optimisation. In
International Electron Devices Meeting. Technical Digest, pages 637–640. IEEE , Piscataway, NJ, USA, 2001.
[4] B. Yu et. al. Finfet scaling to 10 nm gate length. In nternational Electron Devices Meeting. Technical Digest, pages 251–4. IEEE ,
Piscataway, NJ, USA, 2002.
[5] Y. Taur and T. H. Ning. Fundamentals of Modern VLSI Devices. Cambridge University Press, 1998.
[6] M. Fischetti and S. Laux. Long-range coluomb interactions in small si devices. part 1: Performance and reliability. J. Appl. Phys.,
89:1205–1231, 2001.
[7] M. Fischetti. Long-range coluomb interactions in small si devices. part 2: Effective electron mobility in thin-oxide structures. J. Appl.
Phys., 89:1232–1250, 2001.
[8] M. Fischetti and S. Laux. Monte carlo study of sub-bandgap impact ionization in silicon field-effect transistors. In IEDM Tech. Dig.,
page 305, 1995.
[9] R.-H. Yan, A. Ourmazd, K. F. Lee, D. Y. Jeon, C. S. Rafferty, and M. R. Pinto. Scaling the si metal-oxide-semiconductor field-effect
transistor into the 0.1 µm regime using vertical doping engineering. Appl. Phys. Lett., 59:3315–3317, 1991.
[10] D. J. Frank, S. E. Laux, and M. Fischetti. Monte carlo simulation of a 30 nm dual-gate mosfet: How short can si go? In IEDM Technical
Digest, pages 553–556, 1992.
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11
[11] Y. Taur, D. A. Buchanan, W. Chen, D. J. Frank, K. E. Ismail, S-H. Lo, G. A. Sai-Halasz, R. G. Viswanathan, H-J. C. Wann, S. J.
Wind, and H-S. Wong. CMOS scaling into the nanometer regime. Proc. of the IEEE, 85:486–504, 1997.
[12] H.-S. P. Wong, K. K. Chan, and Y. Taur. Self-aligned (top and bottom) double-gate mosfet with a 25 nm thick silicon channel. In
IEDM Technical Digest, pages 427–430, 1997.
[13] F. G. Pikus and K. K. Likharev. Nanoscale field-effect transistors: An ultimate size analysis. Appl. Phys. Lett., 71:3661–3663, 1997.
[14] Z. Ren, R. Venugopal, S. Datta, M. Lundstrom, D. Jovanovic, and D. J. Fossum. The ballistic nanotransistor: a simulation study. In
IEDM Technical Digest, pages 715–718, 2000.
[15] L. Chang, S. Tang, T-J. King, J. Bokor, and C. Hu. Gate length scaling and threshold voltage control of double-gate mosfets. In IEDM
Technical Digest, pages 719–722, 2000.
[16] J. R. Watling et. al. Preprint.
[17] Z. Ren and M. S. Lundstrom. Essential physics of carrier transport in nanoscale mosfets. IEEE TED, 49:133–141, 2002.
[18] M. S. Lundstrom. Fundamentals of carrier transport. Addison-Wesley Publishing Company, 1990.
[19] K. Natori. Ballistic metal-oxide-semiconductor field effect transistor. J. Appl. Phys., 76:4870, 1994.
[20] M. S. Lundstrom. Elementary scattering theory of the mosfet. IEEE Elec. Dev. Lett., 18:361–363, 1997.
[21] P. J. Price. Monte carlo calculation of electron transport in solids. Semiconductor and Semimetals, 14:249–308, 1979.
[22] R. Venugopal, Z. Ren, S. Datta, M. S. Lundstrom, and D. Jovanovic. Simulating quantum transport in nanoscale transistors: Real
versus mode-space approaches. J. Appl. Phys., 92:3730–3739, 2002.
[23] G. D. Mahan. Quantum transport equation for electric and magnetic fields. Physics Reports, 145:251, 1987.
[24] R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic. Single and multiband modeling of quantum electron transport through layered
semiconductor devices. J. Appl. Phys., 81:7845, 1997.
[25] S. Datta. Electronic Transport in Mesoscopic Systems. Cambridge University Press, 1997.
[26] A. Svizhenko et al. Two dimensional quantum mechanical modeling of nanotransistors. J. of Appl. Phys., 91:2343–2354, 2002.
[27] M. V. Fischetti and S. E. Laux. Monte carlo study of electron transport in silicon inversion layers. Phys. Rev. B, 48:2244, 1993.
PHYSICAL REVIEW B
VOLUME 58, NUMBER 8
15 AUGUST 1998-II
Conductance of carbon nanotubes with disorder: A numerical study
M. P. Anantram*
NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California 94045-1000
T. R. Govindan†
Applied Research Laboratory, P.O. Box 30, State College, Pennsylvania 16804-0030
~Received 9 March 1998; revised manuscript received 15 May 1998!
We study the conductance of carbon nanotube wires in the presence of disorder, in the limit of phasecoherent transport. For this purpose, we have developed a simple numerical procedure to compute transmission
through carbon nanotubes and related structures. Two models of disorder are considered, weak uniform disorder and isolated strong scatterers. In the case of weak uniform disorder, our simulations show that the
conductance is not significantly affected by disorder when the Fermi energy is close to the band center. Further,
the transmission around the band center depends on the diameter of these zero band-gap wires. We also find
that the calculated small bias conductance as a function of the Fermi energy exhibits a dip when the Fermi
energy is close to the second subband minima. In the presence of strong isolated disorder, our calculations
show a transmission gap at the band center, and the corresponding conductance is very small.
@S0163-1829~98!05532-5#
I. INTRODUCTION
The experimental and theoretical study of carbon nanotubes ~CNT! has recently been active because these lowdimensional materials display interesting properties both
from a fundamental physics and applications viewpoint. The
mechanical strength of CNT combined with their rich electronic properties have led to a demonstration of their proposed applications as STM tips,1 field emission sources,2 and
nanoscale devices.3–5 CNT can presently be cut to lengths
varying from tens of a nanometer to a many micrometers,
and experiments have shown promise as molecular wires.3
On the theoretical side, studies of the conductance of CNT
with single defects and a junction between tubes have generated interest,6–8 as has the low-energy excitation spectrum
in the presence of electron-electron interaction.9–13
A metallic CNT has two propagating subbands at the
Fermi energy. This can yield a maximum low bias conductance of 4e 2 /h (6.25kV). The prospect of realizing conductances close to 4e 2 /h will significantly depend on ~i! the role
of disorder and/or defects in reducing the conductance of this
low-dimensional material and ~ii! the ability to realize near
perfect contacts with macroscopic sized voltage pads. Using
numerical simulation, we study the effect of two types of
disorder. The first type of disorder is a relatively weak uniform disorder that is distributed throughout the sample. This
model has been considered previously in different contexts.14
The second type of disorder is isolated strong scatterers.
These scatterers physically correspond to lattice sites onto
which an electron cannot hop easily. We find that the two
types of disorder affect the conductance in very different
manners. We present the results of our conductance calculations in nanotubes of different lengths and diameters. We
also make suggestions to observe some of these results experimentally. The second contribution of our paper is a procedure that can be used for the numerical computation of the
transport properties of CNT with defects, T, Y , and other
junctions15–17 and CNT heterostructures. Our procedure includes the effect of semi-infinite leads in an efficient manner.
0163-1829/98/58~8!/4882~6!/$15.00
PRB 58
The Green’s-function-based transport formulation of Refs.
18–20 is employed and is applicable to devices with arbitrary disordered regions and junctions.
The paper is organized as follows. We discuss the model
and the Green’s-function method in Sec. II. This is followed
by a discussion of the numerical results in Sec. III. We conclude in Sec. IV.
II. MODEL
The electronic properties of CNT have been calculated in
the context of various approximations. We use the simplest
model, which assumes the nanotube to be an s p 2 bonded
network. The corresponding single-particle Hamiltonian
is21–23
H5
t i j c †i c j .
(i e 0i c †i c i 1 (
i, j
~2.1!
Here, e 0i is the on-site potential and t i j is the hopping parameter between lattice sites i and j. $ c †i ,c i % are the creation and
annihilation operators at site i. In the absence of defects, the
on-site potential e 0i is zero and the hopping parameter is
23.1 eV.22 We calculate the conductance of a structure that
consists of two semi-infinite perfect CNT leads separated by
a region with defects ~Fig. 1!. In the presence of defects, both
the on-site potential and the hopping parameter change.
Here, we only consider the variation in the on-site potential,
FIG. 1. A schematic representation of the structure across which
the transmission is calculated. Our calculation accounts for semiinfinite leads connected to the disordered region.
4882
© 1998 The American Physical Society
CONDUCTANCE OF CARBON NANOTUBES WITH . . .
PRB 58
e 0i → e 0i 1 d e i .
~2.2!
In the case of a uniformly distributed weak disorder, d e i is
randomly chosen from the interval 6 u e randomu at every lattice
point. Increasing e random corresponds to increasing the
amount of disorder. In the case of substitutional defects, d e i
is set to a large number at some random lattice sites. In a real
sample, d e i would be expected to have a finite spatial extent.
In this paper, the finite spatial extent is neglected and the
random component is treated as a d -function potential.
The transmission coefficient between the left and right
leads is calculated using the expression19,20
T ~ E ! 5tr~ G L G r G R G a ! .
~2.3!
The coupling of the device to the left and right leads, G L and
G R , is given by
G k ~ E ! 52 p V †k Im@ g rk ~ E !# V k ,
S
A1
B 12
O
~2.4!
O
O
O
where kPL,R. g rk (E) is the Green’s-function matrix of the
kth semi-infinite lead, G r and G a are the retarded and advanced Green’s-function matrices of the device ~including
the coupling to the semi-infinite leads!, and V k is the matrix
that couples the kth lead to the device ~disordered! region.
The trace is over the device nodes. To obtain the Green’s
functions, we solve the following equation:
~ EI2H2S rL 2S rR ! G r 5I,
~2.5!
S rk 5V †k g rk (kPL,R)V k
represents the self-energy due
where
to the semi-infinite leads and I is the identity matrix of dimension equal to the number of device lattice sites. In general, for a structure with N atoms, solving for all elements of
the Green’s function involves inverting an N3N matrix.
Computational resources limit the size of the system that can
be considered. However, by careful ordering of lattice sites
the matrix corresponding to Eq. ~2.5! is block tridiagonal
@Eq. ~2.6!#.
O
B 21
A2
B 23
O
O
O
O
O
B 32
d
d
O
O
O
O
O
d
d
d
O
O
O
O
O
d
d
d
O
O
O
O
O
d
d
B N21N
O
O
O
O
O
B N21N
AN
For this purpose we divide the structure into smaller units,
each unit typically representing one or a few rings of atoms
along the circumference of the tube. The diagonal submatrix
A i ~dimension of N i 3N i ) represents EI2H2S rL 2S rR of the
ith unit and the off-diagonal submatrix B i j ~dimension of
N i 3N j ) represents the coupling between units i and j, where
N i and N j are the number of sites in units i and j. O are
empty matrices. In the near-neighbor tight-binding scheme,
B i j is nonzero only when u i2 j u 51. Hence, there is a block
tridiagonal structure for Eq. ~2.5!. Calculating the phasecoherent transmission coefficient involves only the offdiagonal component of the Green’s function connecting the
left and right ends of the device (G rN1 ). This further reduces
the labor to compute the transmission coefficient. We solve
for G rN1 by using an efficient block tridiagonal elimination
procedure. Using this procedure, we are able to calculate the
transmission coefficient through long disordered regions.
The Green’s function g rk is calculated via an iterative
procedure.24 The matrix equation corresponding to the semiinfinite leads is the same as Eq. ~2.6!, only that the matrix is
semi-infinite, with all A i 5A5E2H1i h ~evaluated at a unit
in lead k) and B i j 5B tji 5B. The equations for G k and S rk
involve only the submatrix @ g rk # 11 , which corresponds to the
semi-infinite Green’s function of the unit in lead k that is
closest to the device region. From Eq. ~2.6!, @ g rk # 11 is given
by the following equation:24
4883
DS D S D
G r11
1
G r12
O
d
O
d
5
O
d
O
r
G 1N21
O
G 1N
O
@ g rk # 11 5
~2.6!
.
I
E2H1i h 2B t @ g rk # 11 B
.
~2.7!
The current across the device is calculated using the
Landauer-Buttiker formula,
I5
2e
\
E
dET ~ E !@ f 1 ~ E ! 2 f 2 ~ E !# ,
~2.8!
where the factor 2 accounts for spin degeneracy. f 1 (E) and
f 2 (E) are the Fermi functions of the waves incident from the
two contacts to the device. Note that in the present work, we
calculate only the phase-coherent transmission coefficient
~the effect of electron-phonon interaction is neglected! and
that temperature dependence is only via the Fermi factors of
electrons. Two important considerations in a calculation of
current are the equilibrium location of the Fermi level with
respect to the band bottom of the device when connected to
the contacts25 and the self-consistent potential profile of the
device in the presence of an applied bias. We assume the
case of reflectionless contacts20,26 and consider the scenario
where the Fermi energy can be varied with respect to the
band bottom of the CNT. The ability to vary the Fermi en-
4884
M. P. ANANTRAM AND T. R. GOVINDAN
FIG. 2. Transmission versus energy for a ~10,10! CNT with
disorder distributed over a length of 1000 Å. The significant features here are the robustness of the transmission around the zero of
energy, as the strength of disorder is increased, and the dip in transmission at energies close to the beginning of the second subband.
The inset shows energy versus wave vector for the first ~solid! and
the second subband ~dashed!; the velocity of electrons at the
minima of the solid line is zero.
ergy in a CNT has been demonstrated experimentally in
Refs. 3, 27, and 28. The potential in the device is not calculated self-consistently and we simply assume a linear drop in
the applied potential, while calculating the current versus
voltage characteristics.
III. RESULTS AND DISCUSSION
A. Weak uniform disorder
In a conventional one-dimensional chain, electrons
traverse only a single effective path across the leads and as a
result transmission is significantly altered by small amounts
of disorder.29 In comparison, electrons in a CNT can travel
around defects because of the larger number of atoms in a
cross section ~the number of modes is only two at the band
center!. An important issue is how disorder affects the conductance of CNT wires. We calculate transmission ~by this
we mean the sum of the transmission coefficient over the
incident modes, ( n T n ) as a function of both the length of the
disordered region and the magnitude of disorder using the
procedure described in Sec. II. Transmission versus energy
and conductance versus gate voltage for one configuration of
disorder are shown in Fig. 2. Transmission in a CNT has the
following features that are in common with a single-moded
one-dimensional chain: rapidly varying peaks that signify local resonances created by disorder and decrease in the average value with increasing disorder as the mismatch in the
energies of the resonances increases with increase in
disorder.14,29
We now discuss features that are typical of carbon nanotubes. Figure 2 shows a significant reduction in the transmission coefficient at energies close to the beginning of the second subband, even for weak disorder strengths. This leads to
a dip in conductance when the Fermi level is close to the
beginning of the second subband ~Fig. 3!. The origin of this
PRB 58
FIG. 3. The low bias conductance versus gate voltage for the
structure used in Fig. 2. The figure clearly shows the dip in the
conductance when the Fermi energy is close to the second subband
minima. At the lower temperature, features due to the quasibound
resonances in the disordered region are not averaged out when compared to the high-temperature case.
dip is due to low velocity electrons in the second subband
and can be understood as follows. In a perfect lattice, the
velocity (dE/dk) of electrons with the quantum number of
the second subband and with an energy close to the beginning of the second subband is nearly zero. These lowvelocity electrons are easily reflected by the smallest of disorders. Disorder causes mixing of the first and second
subbands. As a result, electrons incident in either subband at
these energies develop a large reflection coefficient ~in comparison to energies close to the band center!. Increasing the
disorder strength results in further reduction of the conductance and also results in the broadening of the dip. Subsequent to Eq. ~2.2!, we mentioned that the finite spatial extent
of d e i is neglected in our study. A model that includes the
finite spatial extent of d e i would require larger lengths of
disordered regions to see dips whose magnitude is comparable to those in Figs. 2 and 3. The results in Fig. 2 are for
one random configuration of disorder distributed over a
length of 1000 Å. We have carried out simulations over different length scales and disorder configurations and our results for the average transmission at the band center, averaged over more than a thousand disorder configurations, are
summarized in Fig. 4. The important point here is that for the
smaller disorder strengths, the average transmission of a
micrometer-long ~10,10! tube is not significantly affected by
disorder, thus demonstrating the relative robustness of transport at the band center. For disordered regions larger than
some localization length (L 0 ), the conductance of quasi-onedimensional samples has been predicted to decrease exponentially with length, g5g 0 exp(2L/L0), in the phasecoherent limit.14 For lengths shorter than the localization
length, the decrease in conductance is not given by this equation. We observe this to be the case in our simulations ~inset
of Fig. 4!. The values of L 0 corresponding to disorder
strengths of 1 eV and 1.75 eV are 3353 Å and 1383 Å,
respectively.
We also compute transmission for nanotubes of different
diameters. This study illustrates the effect of the number of
PRB 58
CONDUCTANCE OF CARBON NANOTUBES WITH . . .
FIG. 4. The conductance versus length of the ~10,10! CNT.
While for the large disorder strengths the conductance is significantly affected by disorder, the conductance is reasonably large for
the smaller values of disorder. This demonstrates the robustness of
these wires to weak uniform disorder. Inset: log~Conductance! versus length for disorder strength of 1.75 eV in a ~5,5! CNT. The solid
line ~filled circle! corresponds to the simulation and the dashed
line ~empty circle! corresponds to that obtained using
g5g 0 exp(2L/L 0 ).
atoms in a cross section of the wire. We compare transmission of the ~10,10! tube with that of ~5,5! and ~12,0! zigzag
tubes. The diameters of these tubes are 13.4 Å, 9.4 Å, and
6.7 Å, respectively. For the ~10,10! and ~5,5! tubes, the band
structures at energies close to the Fermi energy are similar.21
But the number of atoms in a unit cell of a ~5,5! tube is only
half of that in a ~10,10! tube ~they have 20 and 40 atoms,
respectively!. Figure 5 shows the average transmission versus wire length. The important point here is that in spite of
the identical transmission of a disorder-free ~10,10! and ~5,5!
tube at energies around the band center, transmission is
smaller for the ~5,5! tube in the presence of disorder. This is
because the ~5,5! tube has a smaller number of atoms around
FIG. 5. The average transmission at the band center versus disorder strength for wires of different diameter and chirality; the
transmission has been averaged over a thousand different realizations of the disorder. The main feature here is that the average
transmission decreases with a decrease in the number of atoms
along the circumference of the wire ~see text!.
4885
the circumference, thus reducing the number of paths by
which electrons can travel around defects and across the device. To support this viewpoint, we compare these results to
conductance of a 1000 Å long ~12,0! zigzag tube. We find
that transmission is in between that of the ~10,10! and ~5,5!
tubes ~Fig. 5!. This is because the ~12,0! tube has a diameter
that is in between that of the ~10,10! and ~5,5! tube, and as a
result the number of effective paths is larger than that available to a ~5,5! tube but smaller than that of a ~10,10! tube.
Recently, arm chair, zigzag, and tubes with chiralities in
between have been experimentally characterized by STM
imaging.30,31 Transport measurements of single-wall CNT at
low temperatures have so far been limited by Coulomb
blockade due to large barriers at the contact-CNT
interface.3,27 Disorder of some degree is bound to exist in
CNT samples and we believe that the variation in the linearresponse conductance with the gate potential32 and the dip in
the conductance at energies close to the crossing of the first
and second subbands can be observed in situations where the
contact resistance is not the dominant factor. The length dependence of the conductance can also be studied by varying
the length of the tube between the electrodes. One caveat is
that phonon scattering will cause an increase in the low bias
conductance in the presence of strong disorder with an increase in temperature. Our calculations are relevant at low
temperatures where phonon scattering is not significant.
B. Strong isolated defects
An electron cannot hop on to such a defect site either due
to a large mismatch in the on-site potential or weak bonds
with its neighbors ~Sec. II!. Scattering from a single defect
causes a maximum reduction in the transmission at the band
center E50. For example, the transmission of a ~10,10! tube
reduces from 2 to approximately 0.94 due to a single defect.5
We are interested in the effect of a few such defects scattered
randomly along the length of the tube. Reflection from more
than a single defect causes the creation of quasibound states
along the tube, the exact locations of which are sensitive to
the position of the defects. We find that a significant feature
that is independent of the exact location of these defects is
the opening of a transmission gap at the center of the band as
defects are added. The second feature that we see in the
simulations is that the width of the transmission gap increases with an increase in the defect density. The transmission has sharp decreases at energies corresponding to the
opening of the second subband, but this effect is relatively
weak compared to the previous case of disorder. The simulation results illustrating these features are shown in Figure 6
for a wire of length 1000 Å with ten defects scattered along
the length randomly. As a result of the transmission gap, the
low bias conductance is greatly reduced from the defect-free
case, at zero gate voltage. Conductance further depends significantly on temperature @inset of Fig. 7~a!#. In summary,
while the conductance is not significantly affected by relatively weak uniform disorder ~Figs. 3 and 4!, we find that the
conductance here is much smaller than 2e 2 /h at zero gate
voltage. Conductance increases with gate voltage, with features of resonances due to the quasibound states superimposed. These features get averaged out with an increase in
temperature. We also calculate current as a function of applied voltage by assuming a linear drop in the applied volt-
4886
M. P. ANANTRAM AND T. R. GOVINDAN
FIG. 6. The transmission versus energy for a ~10,10! CNT with
ten strong isolated scatterers sprinkled randomly along a length of
1000 Å. The main prediction here is the opening of a transmission
gap around the zero of energy. Inset: Comparison of the transmission for tubes of lengths 1000 Å ~solid! and 140 Å ~dashed! with
ten scatterers in each case. The transmission gap is larger for the
larger defect density and the sharp resonances close to the zero of
energy are suppressed with increasing defect density.
age. Transmission at each applied voltage is computed and
then we use Eq. ~2.8! to calculate the current. The main
feature in the I-V characteristic is the small increase in current with applied voltage close to the zero of applied voltage
~Fig. 8!. The experimental work in Ref. 4 measured the I-V
characteristics of a CNT rope. One of their main findings
was that the differential conductance is very small at zero
bias and that it increases with an increase in applied bias.
The qualitative features of Fig. 8 are similar but an important
difference is that the experiments were performed on a rope
of single-walled tubes, in which case it has recently been
predicted that a band gap could open due to tube-tube
interactions.33
PRB 58
FIG. 7. The conductance at T5300 K for the case in Fig. 6. The
low conductance at zero gate bias represents the transmission gap in
Fig. 6. The transmission resonances of Fig. 6 get averaged out here.
The inset compares the effect of temperature on the conductance.
Close to zero gate voltage, the conductance is clearly suppressed at
the lower temperature.
surement of conductance versus gate voltage will show a dip
in conductance when the Fermi energy is close to the opening of the second subband ~Fig. 3!. We compare the conductance of wires with varying diameters and find that the transmission ~conductance! increases with the diameter of the
tube for a given disorder strength ~Fig. 5; note that in the
absence of disorder the conductance is independent of the
tube diameter at zero gate voltage!. We attribute this to a
decrease in the number of effective paths by which an electron can traverse across the device with a decrease in the
diameter. The second type of defect considered is strong isolated scatterers. In contrast to the previous type of disorder,
this disorder creates a gap in the transmission at the band
IV. CONCLUSIONS
We present a method to calculate the phase-coherent
transmission through nanotubes using a Green’s-function
formalism that can include the effect of semi-infinite leads
and can handle many defects and junctions with relative
ease. We use this formalism to study the importance of scattering due to disorder. Two simple models of disorder are
considered and their effect on the conductance is discussed.
In the presence of weak uniform disorder, we find that the
conductance is not significantly affected by disorder and that
the wires behave as reasonably good quantum wires. For
example a micrometer-long ~10,10! CNT with a disorder
strength of 1 eV ~section II! has a conductance comparable to
0.16(e 2 /h). We predict that an experiment involving mea-
FIG. 8. The current ~shifted by 20.4 units along the current
axis! versus applied voltage for the same structure as in Fig. 6. The
dashed curve is the differential conductance, which is very small at
low applied voltages.
PRB 58
CONDUCTANCE OF CARBON NANOTUBES WITH . . .
center and a corresponding large reduction in the low bias
conductance. Such disorder would destroy the good conductance properties of the wire at the band center. The work
presented is based on numerical simulations. Of interest
could be further conductance experiments to look for features described in this paper. Carbon nanotubes provide an
unprecedented natural scenario for wires with a few modes
and a relatively small cross-sectional area. An analytical
study of the effect of disorder in these systems and the dependence of the conductance as a function of diameter and
chirality would be useful. Also of interest for future work
*Author to whom correspondence should be addressed, Electronic
address: [email protected]
†
Electronic address: [email protected]
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