Valley splitting in triangular Si„001… quantum wells
G. Grosso
Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Piazza Torricelli 2, I-56126 Pisa, Italy
G. Pastori Parravicini
Dipartimento di Fisica and Istituto Nazionale di Fisica della Materia, Via A. Bassi 6, I-27100 Pavia, Italy
C. Piermarocchi
Institut de Physique The ´orique, Ecole Polytechnique Fe ´de ´rale, CH-1015 Lausanne, Switzerland
Received 27 February 1996
Intervalley splitting in an n -channel Si001 inversion layer is evaluated by means of a triangular quantum
well which simulates the effect of a static electric field. All the microscopic interactions are included in the
realistic tight-binding bulk Hamiltonian of the crystal to which the electric field is applied. Eigenvalues and
projected densities of states in the triangular well are evaluated from the knowledge of the Green’s function
obtained by the renormalization procedure. We find that the valley splitting increases nonlinearly as the
external field increases. S0163-18299611548-4
We analyze the effect of a static electric field on the
equivalent valleys of the lowest conduction band of a silicon
crystal. The problem has a long-lasting story since the ex-
perimental detection
1
of Zeeman and valley splitting effects
in magnetoconductivity measurements for the inversion layer
of an n -channel Si metal-oxide-semiconductor field-effect
transistor. Since then, these effects have been confirmed by a
wide literature for an exhaustive review see Ref. 2, and, not
last, by the results on quantum Hall effect.
3
An n -channel Si001 inversion layer has two degenerate
lower valleys along the
z
direction and four degenerate
higher valleys in the orthogonal directions. The lower val-
leys’ degeneracy E ( k) =E ( k) can be lifted by an electric
field, induced, for instance, by a gate voltage orthogonal to
the 001 surface. The energy levels of the electrons are
grouped in subbands, each corresponding to quantized levels
in the z direction and continua for the motion parallel to the
surface. We focus here only on the lowest ladder of subbands
originated by the electrons with longitudinal mass oriented
along the z axis.
Several theoretical papers have addressed the problem of
calculating valley splitting in silicon inversion layers see the
review in Ref. 2; most of them within the effective-mass
theory or the k–p approximation. It is evident from them that
the main difficulties are connected with intervalley interac-
tion and with a proper account of scattering mechanism at
the SiO
2
/Si interface. In particular the use of the extended
effective mass equation has appeared rather cumbersome and
inadequate to treat this subject.
We address the problem of intervalley interaction within a
localized basis framework, by the renormalization method.
In order to mimic the physics of the inversion layer, we use
a triangular silicon quantum well. This is obtained superim-
posing to a silicon crystal a triangular potential along the
direction 001. In this way the electrons within the well
have the same potential energy as in the presence of an ap-
plied uniform electric field. In the chosen model we do not
have interface scattering; however, in problems where the
scattering is relevant, this could be introduced by proper
definition of the well boundary through the parameters used
in the superlattice Hamiltonian. The search of the electronic
energies is then performed by the overall procedure outlined
in the following points.
i We start from a localized basis representation of the
bulk silicon crystal Hamiltonian H ( k). Several requirements
for this tight-binding parametrization are needed; in particu-
lar, it must include spin-orbit coupling and give an accurate
description of the energy bands and effective masses near the
fundamental gap. For this purpose we use a recently
obtained
4
accurate parametrization which includes only the
first and few selected second-neighbor interactions between
sp
3
localized orbitals in the Slater-Koster Hamiltonian.
5
Our
parametrization provides a good valence and conduction
band structure, with the inclusion of spin-orbit effects. More-
over, if necessary, it can be easily implemented by accurate
scaling laws to take into account strain effects on the crystal
structure.
ii At this point the silicon crystal is resolved as a
multilayer structure along the 001 direction by representing
the Hamiltonian H ( k) on the basis of ‘‘two-dimensional’’
Bloch sums made by layer orbitals
6
l ,
( q)( q indicates the
k
x
and k
y
components of the superlattice Bloch vector
k=( q, k
z
), l indicates the ‘‘layer,’’ and the independent
orbitals within each layer. In our case the crystal Hamil-
tonian H ( k) contains interactions up to second neighbors,
thus it is convenient to build each layer in terms of two
adjacent atomic planes in the direction 001 and the inter-
layer interactions are restricted to nearest neighbors. This
representation allows us to map exactly the three-
dimensional crystal structure into an equivalent one-
dimensional effective lattice with site energy matrices
H
l
( q)
,
'
=
l ,
( q) | H|
l ,
'
( q) , and hopping matrices
T
l
( q)
,
'
=
l ,
( q) | H|
l 1,
'
( q) ; correspondingly the
crystal Hamiltonian takes a peculiar block-tridiagonal form.
iii By shifting the diagonal site energies of Silicon
H
l
( q)
,
'
by a potential profile with chosen periodicity we
PHYSICAL REVIEW B 15 DECEMBER 1996-I VOLUME 54, NUMBER 23
54 0163-1829/96/5423/163934/$10.00 16 393 © 1996 The American Physical Society