Università degli studi di Pisa
Dipartimento di Fisica
Tesi di Laurea Magistrale
Single-element Silicon and Germanium
Strained Superlattices
CANDIDATO RELATORI
Berardo Mario Manzi Prof. Giuseppe Grosso
Dott. Michele Virgilio
Contents
Introduction 1
1 Geometrical description of Silicon and Germanium lattices
and superlattices 5
1.1 Geometrical structure of silicon and germanium crystals . . . 5
1.2 Tetragonal description of Si and Ge crystals . . . 7
1.3 Tetragonal cell for periodic multilayer structures . . . 9
2 The tight-binding model for lattices and superlattices 13 2.1 Tight-binding for fcc lattice . . . 13
2.1.1 Spin-orbit coupling . . . 18
2.2 TB model for tetragonal cells . . . 23
2.3 Strain eects in multilayer structures . . . 25
3 Single-element silicon and germanium strained superlattices 29 3.1 Description of the single-element SSL supercell . . . 30
3.2 Silicon single-element strained superlattices . . . 33
3.3 Germanium single-element strained superlattices . . . 42
Conclusions 49 A Tight-binding Hamiltonian matrix elements 51 A.1 Hamiltonian interaction terms for the fcc structure . . . 51
Introduction
Since the invention of the rst transistors, silicon and germanium have been of fundamental importance in the developement of electronic devices. Silicon in particular, due to its great availability and cheapness, is the uncontested leader in the eld of microelectronics and integrated circuits. Complemen-tary metal oxide semiconductor (CMOS) architectures, which dominate elec-tronic circuit implementations, are totally or almost totally silicon based.
Another important area of application of semiconducting materials is op-tics. In this context silicon and germanium technology is commonly substi-tuted by materials, such as GaAs and InP, with more appropriate properties. In particular, both silicon and germanium are indirect gap semiconductors and optical transitions from the valence bands to the conduction bands are forbidden. This represents an obstacle for the development of silicon-based lasers and other modern optical devices.
Strain engineering has shown to be a solution to overcome the limitations of silicon and germanium crystals. Much eort has been spent over years to understand strain eects applied to these structures. Both theoretical works [1, 2, 3] and experiments [4, 5] investigated the consequences of stress and strain on the band structures of semiconductors.
An example of crystal structures which exploits strain eects are SiGe heterostructures. These structures are obtained by deposition of layers of dierent Si/Ge-based materials on a thick bulk substrate, using fabrication techniques such as Molecular Beam Epitaxy (MBE) [6] or Chemical Vapor Deposition (CVD) [7]. Strain is introduced by the lattice mismatch of the materials, due to the dierent lattice parameters of the corresponding bulk crystals. For instance, the lattice parameter of elemental silicon, forced to match a germanium substrate, diers by about 4% from its bulk value. The consequent strain eects are large compared to strain eects in heterostruc-tures composed of other semiconductors, e. g. GaAs and AlAs, for which the lattice mismatch is at least an order of magnitude smaller. The properties of SiGe heterostructures have been intensively studied by many authors. A pioneering ab initio treatment of interface eects present when dierent SiGe alloys are forced to match has been proposed by C. G. Van de Walle and R. M. Martin [8].
Introduction by L. Esaki and R. Tsu [10]. These cystal structures are obtained by alter-nating n atomic layers of a given semiconductor material with m layers of a second type, with n+m usually very small. SLs of particular importance for silicon-based devices are Sin/Gem SLs, whose features were thoroughly
in-vestigated in the literature. Electronic and optical properties were inspected changing the number of layers and the substrate on which the SLs are grown. Works like reference [11] showed that for Si/Ge superlattices with n+m ≥ 10, grown on germanium or Ge-rich substrates, direct gaps can be achieved by band folding processes. However, most of these structures could not realize optical gains comparable with the commonly used direct gap semiconductor compounds of group III-V.
Recently, a new type of silicon multilayer structure has been proposed by M. Hunag et al. [12]. These single element strained superlattices (SSLs) are silicon membranes on which quantum dots of dierent lattice parameter are applied periodically, locally changing the strain conditions (see gure 1). The alternating unstrained and strained silicon atomic layers simulate a
two-Figure 1: (a) A top-view scan-ning electron microscope (SEM) image of 1D strained superlat-tices fabricated by depositing Ge quantum dots (dark area) on Si (001) nanoribbons [12]. (b) The supercell structure of a Si strain SL containing two unit cells of unstrained Si and two unit cells of strained Si [13].
component SL crystal without interface eects due to chemical composition. The potential of these structures has been investigated theoretically by Z. Liu and co-workers [13], who described such a SSLs under dierent strain con gurations for dierent sizes of the primitive cell. Their work is based on rst-principles calculations in the framework of density functional theory (DFT).
The present work investigates electronic properties of these single element SSLs of silicon and germanium exploiting the tight-binding (TB) framework. The TB model, rst proposed by F. Bloch in 1928, consists in expanding the crystal states in linear combinations of atomic orbitals. Evaluated among these states, the one-electron hamiltonian operator can be approximated with a nite hamiltonian matrix whose size depends on the number of or-bitals per atom considered. The matrix elements describing the interaction
between the atoms in the crystal are rewritten in terms of two-center inte-grals, using a prescription introduced by J. C. Slater and G. F. Koster [14]. In the semi-empirical approach, the two-center integrals and the diagonal matrix elements of the hamiltonian are taken as empirical parameters.
The TB model used in this thesis exploits a sp3d5s∗parametrization, rst
proposed by J.-M. Jancu, R. Scholz, F. Beltram, F. Bassani [17]. Many stud-ies have been performed using this parametrization and their results were in good agreement with experimental data. For instance, a thorough inspection of optical properties of Si/Ge superlattices grown on germanium or Ge-rich substrates, performed in the TB framework, is presented in reference [18].
The present work is a rst application of the TB model to the above mentioned single element SSLs and provides the rst results on germanium single-element SSLs. Results on the electronic band structures and on the localizations of the electronic states are displayed, with special emphasis given to the possibility to achieve direct gaps.
The thesis is structured as follows. Chapter 1 begins with a review of the fundamental geometries involved in diamond-like crystal structures. The primitive cell of the face centered cubic (fcc) lattice and the fundamental cell in reciprocal space, or momentum space (the Brillouin zone), are de ned. Afterwards, a tetragonal lattice description is introduced, which is more suit-able for multilayer structures. The last section of this chapter is dedicated to the basic notions of strain and its eect on the tetragonal crystal structures. Chapter 2 presents the theoretical framework used in the computation of the energy bands. The TB model is derived starting from the one electron hamiltonian. Afterwards, spin-orbit interaction is added as a perturbation to the TB hamiltonian. In fact, especially for germanium based semicon-ductor materials, the spin-orbit splitting of the topmost valence bands is small but not negligible. The chapter ends with the generalization of the TB framework to strained multilayer structures. A detailed discussion of the parametrizations used and their limits is presented before the results are displayed.
Chapter 3 represents the core of this work. In the introductory section, the fundamental cell used for single-element SSLs is described. The cell is shown to be composed of two subcells with an equal number of layers. The rst one is an unstrained bulk cell of either silicon or germanium. The second one is made of the same material but under dierent strain conditions along the growth axis. Strain in the orthogonal plane is considered to be negligible. The electronic properties of silicon and germanium crystals which exploit this geometry are investigated varying the total number of layers (i. e. the length of the cell) and the applied strain. The rest of the chapter is divided in two sections, one dedicated to silicon and one to germanium single-element SSLs. The behaviour of the energy band structure of silicon SSLs and, in partic-ular, of the top valence bands and bottom conduction bands is investigated. While the maximum of the topmost valence band of silicon SSLs is found to
Introduction be at the origin of the Brilloiun zone for any strain-size con guration, the po-sition of the minima of the conduction band are strongly dependent on both cell length and strain. In fact, even in bulk silicon crystals, it can be obtained that the conduction minima degeneracy decreses under uniaxial strain, i. e. the energy of the minium along the growth axis shifts with respect to the energy of the minima along the orthogonal directions [8]. The energy dier-ence depends on the type of strain, compressive or tensile. In single-element SSLs, the interplay of splitting eect and dierent con nement conditions of the electronic states due to the varying cell size produces a great variety of important band structure pro les. In particular, for large silicon SSL cells subject to large compressive strain along the growth direction a direct gap is obtained. The direct gap is mainly a consequence of the folding of the energy bands along the growth direction in the origin of the Brillouin zone. In fact, for silicon SSL cells with many layers, the lowest conduction band is shown to be at. The eective importance for this direct gap can only be determined by further investigation on the optical matrix elements.
The last section of this chapter is dedicated to the study of the properties of germanium based SSLs. Following the same approach used for silicon, the electronic band structure is investigated as a function of the cell size and the applied strain. From analysis of the germanium bulk energy band structure it is expected that, for large tensile strain along the growth direction, the shape of the lowest conduction band changes enough to let the minimum occur at the origin of the Brillouin zone, where the maximum of the topmost valence band occurs. Such a result has been obtained for large germanium single-element SSL cells under large tensile strain, thus achieving a direct gap. This gap is shown to be genuine, i. e. not due to folding eects, and relevant for optical purposes.
The concluding chapter resumes the results obtained in this thesis, and proposes the direction for future investigation on the presented single-element SSLs.
Chapter 1
Geometrical description of
Silicon and Germanium lattices
and superlattices
The rst important feature required to adequately describe the superlattice optical and electronic properties is its crystalline structure. The geometrical description of superlattices is a generalization of the bulk crystals description. In this chapter I will illustrate the dierent possible geometries involved in silicon-germanium superlattices, starting from the bulk Bravais lattice and the corresponding Brillouin zone. Once xed this part I will change the description to one more useful for multilayer structures with a given growth direction. In the last section I will de ne the most general crystal geometry used in the following chapters, introducing dierent lattice parameters to adequately consider strain eects.
1.1 Geometrical structure of silicon and germanium
crystals
The Bravais lattices of silicon and germanium crystals have the diamond face centered cubic (fcc) structure. This means that the crystal structure can be described by the three primitive vectors:
~t1 = a 2(0, 1, 1), ~t2 = a 2(1, 0, 1), ~t3= a 2(1, 1, 0) (1.1)
and by the basis vectors: ~
d1 = (0, 0, 0), d~2=
a
4(1, 1, 1), (1.2)
where a is the edge of the cubic conventional cell, as in gure 1.1(a). In other words, the bulk crystal structures consists of two interpenetrating face
Geometrical description of Silicon and Germanium lattices and superlattices
t
1t
2t
3a
a
(a)Γ
L
X
W
K
U
k
xk
zk
y (b)Figure 1.1: (a) Conventional cell of edge a of the fcc crystal. The primitive cell generated by the three vectors ~t1, ~t2and ~t3 is displayed inside the cube. (b) Brillouin zone of the fcc
lattice. High symmetry points of the Brillouin zone are indicated by capital letters.
centered cubic (fcc) lattices. In this way, the atom at the origin has the four rst nearest neighbours at ~ u1 = ~d2= a 4(1, 1, 1), ~u2 = ~d2− ~t1= a 2(1, −1, −1), ~ u3 = ~d2− ~t2= a 2(−1, 1, −1), ~u4 = ~d2− ~t3= a 2(−1, −1, 1). (1.3) Conversely, the atom in ~d2 has nearest neighbours at:
~v1 = (0, 0, 0), ~v2 = a 2(0, 1, 1), ~v3 = a 2(1, 0, 1), ~v4 = a 2(1, 1, 0). (1.4) The parameter a assumes the values aSi = 5.43Å and aGe = 5.6563Å for
Silicon and Germanium, respectively [19, 20]. For completeness, I write down the alloy parameter a(x) for Si1−xGex, where x can assume any value
0 ≤ x ≤ 1,obtained by the interpolating formula [21, 22]:
a(x) = aSi+ 0.200326x(1 − x) + (aGe− aSi)x2. (1.5)
The ~tivectors de ne a net called the direct lattice, and every vector of the
lattice can be written as a combination of the primitive vectors (1.1). There is another important space useful for the computation of periodic quantities, the so called reciprocal or dual space. The primitive vectors {~gi} in the
reciprocal space are combinations of the three primitive vectors {~ti} de ned
as:
1.2 Tetragonal description of Si and Ge crystals
From this relation it is easy to derive the explicit expressions of the ~gi.
Observing that ~g1 is orthogonal both to ~t2 and ~t3 or, in other words, it is
parallel to ~t2× ~t3, and noticing that ~g1· ~t1 = 2π, I obtain:
~ g1 =
2π
Ω~t2× ~t3, (1.7)
where Ω = ~t1· ~t2× ~t3 is the volume of the primitive cell in the direct space.
In an analogous manner the other two vectors are derived: ~ g2 = 2π Ω~t3× ~t1, ~g3 = 2π Ω~t1× ~t2. (1.8)
In the case of the fcc lattice (Ω = a3/4) the primitive vectors of the reciprocal
lattice are: ~g1= 2π a (−1, 1, 1), ~g2 = 2π a (1, −1, 1), ~g3= 2π a (1, 1, −1), (1.9) de ning a body centered cubic (bcc) lattice of edge 4π/a.
Most of the physical quantities are evaluated in the so called rst Bril-louin zone de ned as the cell in reciprocal space with the property that any point of the cell is closer to the chosen lattice point than to any other. The construction of such a zone is very simple: taking as reference point the ori-gin, I draw the planes which bisect the lines joining the nearest neighbours to the origin. The volume enclosed by the intersection of these planes is the Brillouin zone. In this way the Brillouin zone of the fcc lattice is the truncated octahedron shown in gure 1.1(b). As shown in the gure, some points result to be very important for their symmetry properties and they usually take conventional names. For the fcc Brillouin zone they are:
Γ = (0, 0, 0), X = 2π a (1, 0, 0), L = π a(1, 1, 1), W = 2π a (1, 1 2, 0), K = 2π a ( 3 4, 3 4, 0), U = 2π a (1, 1 4, 1 4). (1.10) It is worth to notice that points equivalent by symmetry to the ones listed above will be called with the same name. For instance, the point 2π/a(1, 0, 0) and the equivalent points 2π/a(0, 1, 0), 2π/a(0, 0, 1), 2π/a(0, 0, −1), 2π/a(0, −1, 0) and 2π/a(−1, 0, 0) are called X.
1.2 Tetragonal description of Si and Ge crystals
The primitive vectors de ned in the previous section determine the prim-itive cell containing the smallest number of atoms allowed by the crystal symmetry. However, according to speci c problems, it is not always the
Geometrical description of Silicon and Germanium lattices and superlattices most useful choice. For instance, in the case of multilayer structures grown along the (001) axis, it is convenient to adopt the three vectors:
~ τ1 = a 2(1, −1, 0), ~τ2= a 2(1, 1, 0), ~τ3 = a(0, 0, 1), (1.11) which de ne a tetragonal cell with twice the volume (Ω = a3/2) and twice
the atoms of the fcc primitive cell. The atoms in the tetragonal cell are located in the positions:
~ d1= (0, 0, 0), d~2= a 4(1, 1, 1), ~ d3= a 2(1, 0, 1), ~ d4= a 4(1, −1, 3). (1.12) The new cell is represented in gure 1.2. For what concerns the rst nearest
τ2 τ1 τ3 d2 d3 d4 d1
Figure 1.2: The tetragonal cell described in the text, generated by the vectors ~τ1, ~τ2 and ~τ3.
The ~diindicate the positions of
the atoms in the cell.
neighbours of a given atom in the cell, their coordinates are still the same as in the cubic description but they are disposed in such a way that each atom of a given layer has two nearest neighbours on the following layer and two on the previous one, e.g. ~u1, ~u4 on layer z = a/4 and ~u2, ~u3 on layer
z = −a/4for the atom in the origin.
The construction of the rst Brillouin zone is straightforward. The new fundamental vectors in reciprocal space are:
~h1 = 2π a (1, −1, 0), ~h2 = 2π a (1, 1, 0), ~h3= 2π a (0, 0, 1). (1.13) In other words, the unit cell in dual space is still a tetragonal cell. Figure 1.3 shows the Brillouin zone of the tetragonal cell inside the previous octahedral Brillouin zone. It is worth to notice that some of the previous high symmetry points are the same as in fcc Brillouin zone, like Xk. However, this point is
no longer equivalent to the cell edge point X⊥ along the growth direction,
1.3 Tetragonal cell for periodic multilayer structures
Figure 1.3: Brillouin zone of the tetragonal cell, shown inside the Brillouin zone of the fcc crystal. High symme-try points are reported. Comparison with g-ure 1.1(b) illustrates how the geometrical de-scription in reciprocal space changes.
X
||L
k
xk
yk
zX
⊥1.3 Tetragonal cell for periodic multilayer
struc-tures
Multilayer structures, like superlattices, are generalizations of the above de-scribed crystals. In modern deposition techniques, like molecular beam epi-taxy (MBE) [6], these structures are grown layer by layer and very dierent kinds of materials can be grown together introducing controlled strain and atomic composition eects. The structures analyzed in this thesis are crys-tals grown on a thick unstrained substrate, made of Si1−xGex alloy.
A complete treatment of strain in elasticity theory can be found in Ref. [23]. For the purpose of the present work, I will introduce some of the notations which will be used in the following. One of the most impor-tant de nition is the strain tensor εij, which links the coordinate system
x0y0z0 of a deformed volume element to the undeformed coordinate system xyz. In other words the relations:
x0 = εxxx + εxyy + εxzz,
y0 = εyxx + εyyy + εyzz,
z0 = εzxx + εzyy + εzzz,
(1.14) de ne the strain tensor εij. In the interest of the present work, strain axes
will be taken coincident with a xyz coordinate system having the z axis parallel to ~τ3 and the xy plane parallel to the planes generated by ~τ1 and ~τ2
Geometrical description of Silicon and Germanium lattices and superlattices the crystal and the strain tensor can be written in diagonal form:
ε = εxx 0 0 0 εyy 0 0 0 εzz . (1.15)
The speci c relationships between the tensor components depend on the stresses acting on the volume. Two cases will be of interest in the following: epitaxial growth with in-plane biaxial strain and orthogonal relaxation, and uniaxial strain along the z-direction without relaxation in the orthogonal plane.
Semiconductor layers grown on top of a thick substrate with dierent lattice parameter will minimize the elastic energy of the whole structure, allowing a (almost) perfect matching in the in-plane directions (xy plane). This matching produces strain deformations in both the xy plane and the orthogonal direction, according to the ratio:
η = −εk ε⊥
, (1.16)
where εk = εxx = εyy and ε⊥ = εzz. One of the consequences of these
deformations is the change of lattice parameter: the in-plane parameter ak
of the grown layers equals the bulk lattice parameter as of the substrate,
while the orthogonal lattice parameter a⊥changes according to relation [21]:
a⊥ = a0 1 − 2c12 c11 a k a0 − 1 , (1.17)
where c12 and c11 are stiness constants.
Structures subject to uniaxial strain which are not allowed to relax in the orthogonal directions can be described by a one-component strain tensor, where the only non-zero component is εzz. The lattice parameter ak does not
change while the lattice parameter az along the z direction can be calculated
from:
az = a0(εzz+ 1). (1.18)
In both cases, for what concerns the crystal structures, the tetragonal cell is now deformed. The lattice parameter in the in-plane directions is dierent from the perpendicular one and the fundamental vectors become:
~t1=
ak
2 (1, −1, 0), ~t2= ak
2 (1, 1, 0), ~t3 = a⊥(0, 0, 1). (1.19) In an analogous manner the ~di vectors will change.
Starting from the structure of the tetragonal cell, larger cells can be obtained increasing the number of layers of the unit cell. Moreover, these layers can be chosen to be made of dierent materials, e. g. a given co-herent repetition of silicon atoms and of germanium atoms. However, when
1.3 Tetragonal cell for periodic multilayer structures
dierent materials are present in the same structures a few considerations are mandatory. First, dierent materials mean dierent lattice parameters in the corresponding bulk crystals, and, thus, the above mentioned strain re-lations have to be correctly implemented. A second consideration concerns the interface between two materials: if the i-th layers is of one type (e. g. silicon) and the (i + 1)-th of another type (e. g. germanium), the lattice parameter a⊥ has to be chosen appropriately. In this work, I will take the
interface lattice parameter (aint)⊥ equal to the average of the orthogonal
lattice parameters of the neighbour materials: (aint)⊥=
(ai)⊥+ (ai+1)⊥
2 . (1.20)
For instance, taking a cell analogous to gure 1.2 but with six layers, three made of Silicon and three of Germanium, and supposing they are grown on a bulk silicon substrate, the new basis set becomes:
~ d1 = (0, 0, 0), d~4 = ( a|| 4 , a|| 4 , a(Si)⊥ 2 + (aint)⊥ 4 ), ~ d2 = ( a|| 4 , a|| 4 , a(Si)⊥ 4 ), d~5 = (0, 0, a(Si)⊥ 2 + (aint)⊥ 4 + a(Ge)⊥ 4 ), ~ d3 = ( a|| 2 , 0, a(Si)⊥ 2 ), ~ d6 = ( a|| 4 , a|| 4 , a(Si)⊥ 2 + (aint)⊥ 4 + a(Ge)⊥ 2 ), (1.21)
where, for the bulk silicon substrate, a|| = a
(Si) ⊥ = aSi, (aint)⊥ = (a (Si) ⊥ + a(Ge)⊥ )/2and a (Ge)
⊥ is de ned by equation (1.18) with a0 = aGe. Notice that
adding layers in a tetragonal cell is equivalent to add new basis vector to the set 1.21. For the sake of simplicity, the subscript of each vector will be used to indicate also the layer (see gure 1.2, i.e. layer 1 is the layer identi ed by vector ~d1.
The inspection of the previous example leads to another important con-sideration. When it is necessary to preserve periodic boundary conditions, I expect the total number of layers of a generic supercell to be multiple of 4, otherwise a sort of discontinuity is arti cially introduced in the crystal. For example, in the crystal structure described by the vectors (1.21) the atomic positions on layer 6 are the same as the positions on layer 2. Layer 7 made of the same material as layer 1 and with atoms in the same positions as the ones on layer 3, has to be described by a further basis vector ~d7. Repeating this
reasoning, a 12 vector basis set is required to properly account for periodic boundary conditions. Once the periodic boundary conditions are satis ed, the new total length L of the cell becomes:
L = N X i=1 (a⊥)i 4 , (1.22)
Geometrical description of Silicon and Germanium lattices and superlattices where (a⊥)i is the perpendicular lattice parameter between the i-th and
(i + 1)-th layers. The vector ~t3 also needs to be rede ned and becomes:
~t3 = L(0, 0, 1). (1.23)
The Brillouin zone is obtained in similar manner as described in section 1.2, with the substitutions a → ak for the in-plane directions and a → L for the
orthogonal direction. Obviously, the total cell length L increases with the number of layers in the material and, consequently its volume in direct space also increases. Conversely, the volume of the Brillouin zone reduces. This will lead to important eects which are discussed in the following chapters.
Chapter 2
The tight-binding model for
lattices and superlattices
The main tool used in this thesis for the evaluation of band structures and op-tical properties is the semi-empirical tight-binding (TB) model. This chapter illustrates this method and the Hamiltonian used in the following to describe the electronics and optics of silicon, germanium and their alloys. The follow-ing sections are structured as follows: rst, the tight-bindfollow-ing Hamiltonian for bulk crystal structures with the fcc lattice is derived. Then, the method is speci ed to the case of crystal with tetragonal lattice. Spin-orbit and strain eects on the electronic band structure are also discussed.
2.1 Tight-binding for fcc lattice
The band structure of a crystal, as any other physical system, should be derived, at least in principle, solving the Schrödinger equation:
Hψ = Eψ (2.1)
with H being the many body non-relativistic Hamiltonian:
H =X i |~pi|2 2m + X I | ~PI|2 2MI −X i,I zIe2 |~ri− ~RI| +1 2 X i6=j e2 |~ri− ~rj| +1 2 X I6=J zIzJe2 | ~RI− ~RJ| , (2.2) where the lower case subscripts refer to electrons, the upper case to nuclei, m, ~p, ~r, e are the mass, momentum, position and charge of the electrons, respectively, zI is the charge in unit of e of the I-th nucleus. This task is
really demanding in practice, and, generally, impossible to achieve even with the modern computing machines, due to the huge number of electrons and nuclei to be considered. Some approximation has to be used. The rst step, as described in reference [24], is to reduce the many electron problem to a
The tight-binding model for lattices and superlattices single particle equation, with the nuclei supposed to be xed. This can be obtained conceptually by approximations like the Hartree-Fock theory or the Density Functional Theory. The resulting Hamiltonian has the form:
H = |~p|
2
2m+ V (~r), (2.3)
where V (~r) must be an appropriately made potential, as indicated by the Hartree-Fock or Density Functional theories. In the case of crystals, the potential is invariant under translations of direct lattice vectors, i.e.:
V (~r + ~tn) = V (~r), (2.4)
and (2.3) is the basic hamiltonian of the band theory. The model consists of expanding the crystal eigenvectors as a linear combination of Bloch sums:
|ψ(~k, ~r)i =X
i,µ
ciµ(~k) |Φiµ(~k, ~r)i (2.5)
where the Bloch sums Φiµ are de ned as:
|Φiµ(~k, ~r)i = √1 N X ~tm ei~k·(~tm+ ~dµ)|φ(a) iµ(~r − ~dµ− ~tm)i . (2.6)
In this last relation N is the number of unit cells in the crystal. The atomic orbitals {φ(a)
iµ }are an orthonormal set centered on every atom:
hφ(a)jν(~r − ~dν− ~tn)|φ(a)iµ(~r − ~dµ− ~tm)i = δijδµνδmn, (2.7)
where δij is the usual Kronecker delta. Inserting equations (2.3) and (2.5)
in equation (2.1), I obtain: X
i,µ
ciµ(~k)H |Φiµ(~k, ~r)i = E(~k)
X
i,µ
ciµ|Φiµ(~k, ~r)i (2.8)
and, multiplying both members of this equation by hΦjν(~k, ~r)|, the Schrödinger
equation becomes: X
i,µ
hΦjν(~k, ~r)| H |Φiµ(~k, ~r)i ciµ= E(~k)
X
i,µ
ciµ(~k) hΦjν(~k, ~r)|Φiµ(~k, ~r)i .
(2.9) At this point it is important to notice that the orthogonality of the {φ(a)
iµ}
implies the orthogonality of the Bloch sums, and the previous equation sim-pli es:
X
i,µ
2.1 Tight-binding for fcc lattice
This equation can now be solved, if the matrix element in the left-hand side is known. To compute it I write it explicitly in terms of the atomic orbitals:
Hijµν(~k) = hΦjν(~k, ~r)| H |Φiµ(~k, ~r)i =
1 N
X
~tm,~tn
hφ(a)jν(~r − ~dν− ~tn)| e−i~k·(~tn+ ~dν)Hei~k·(~tm+ ~dµ)|φ(a)iµ(~r − ~dµ− ~tm)i
Because of the translational invariance of the Hamiltonian, I can choose ~tn = 0, drop the sum over ~tn and the factor 1/N. Furthermore, I assume
that the crystal potential in equation (2.3) can be expressed as a sum over atomic-like potentials, centered at the atomic positions (~tl+ ~dλ):
V = X
~tl, ~dλ
Va(~r − ~dλ− ~tl). (2.11)
With the above assumptions, the previous equation becomes: X ~ tm ei~k·(~tm+ ~dµ− ~dν)hφ(a) jν(~r − ~dν)| |~p|2 2m + Va(~r − ~dν) |φ(a)iµ (~r − ~dµ− ~tm)i + X ~tm ei~k·(~tm+ ~dµ− ~dν)hφ(a) jν (~r − ~dν)| V 0 (~r) |φ(a)iµ(~r − ~dµ− ~tm)i
where Va(~r − ~dν) is the contribution to the potential (2.11) of the atom
in tm = 0 and position ~dν, and V0(~r) is the remaining part of the crystal
potential (2.11). The rst two terms of the above expression represent the Hamiltonian of the atom at ~dν and ~tm= 0, for which it holds:
|~p|2 2m + Va(~r − ~dν) |φ(a)jν (~r − ~dν)i = Ejν0 |φ (a) jν(~r − ~dν)i . (2.12)
I have thus for the matrix element:
Hijµν(~k) = hΦjν(~k, ~r)| H |Φiµ(~k, ~r)i = Ejν0 δijδµν+ X ~tm ei~k·(~tm+ ~dµ− ~dν)hφ(a) jν (~r − ~dν)| V 0 (~r) |φ(a)iµ(~r − ~dµ− ~tm)i . (2.13)
The rst term in this expression is the atomic eigenvalue of the atom in position dν, while the second term is the potential which describes the
in-teraction between φ(a)
jν (~r − ~dν) and all the other atoms in the crystal. Some
further approximations are necessary at this point, but before introducing them, I notice that the ~tm = 0, ν = µ contribution to the second term of
equation (2.13), i.e.:
The tight-binding model for lattices and superlattices called the crystal eld integral, can be added to the atomic eigenvalue E0
jν.
This means that the new atomic on site energy Ejν contains contributions
due to the symmetry of the crystal structure in which the atom is embedded. The terms in Hµν
ij (~k)with ~tn6= 0have to be appropriately approximated.
As a rst approximation, three center integrals, which involve two orbitals centered on two dierent sites, with the potential centered on a third site, can be considered negligible, due to the localized nature of the atomic orbitals. The second important consideration, also connected with the localized na-ture of the atomic orbitals, allows to limit the sum in equation (2.13) to a small number of nearest neighbours. Second or even third nearest neighbours could in principle be used but increasing the order of neighbours increases the number of parameters necessary for the tight-binding description. In the present case, following reference [17], a rst nearest neighbours approxima-tion will be used. The sum in equaapproxima-tion (2.13) becomes:
X ~tI,µ6=ν ei~k·(~tI+ ~dµ− ~dν)hφ(a) jν(~r − ~dν)| Va(~r − ~dµ− ~tI) |φ (a) iµ(~r − ~dµ− ~tI)i (2.15)
where the tI are the nearest neighbours of the atom in dν. Even with this
approximations the matrix element Hij,µν(~k) is still de ned as an in nite
matrix in the indices i,j, since the sum in (2.15) is extended to all possible atomic orbitals {φ(a)
i }. The diagonalization of such a matrix is quite hard to
achieve and not really useful in practical cases. The set of atomic orbitals has to be reduced to a small number which appropriately describes most of the physical content. Once a basis is chosen the energy integrals or Slater-Koster parameters:
Ei,j =
Z
φ(a)jν (~r − ~dν)Va(~r − ~dµ− ~tI)φ(a)iµ(~r − ~dµ− ~tI)d~r (2.16)
can be expressed in terms of a small number of independent two-center in-tegrals [14]:
Vssσ, Vspσ, Vppσ, Vppπ, Vsdσ, Vpdσ, Vpdπ,
Vddσ, Vddπ, Vddδ, Vs∗sσ, Vs∗pσ, Vs∗dσ,
where Vabξ are integrals evaluated among two atomic orbitals of type a and
b, and ξ refers to the component of angular momentum around the direction joining the two atoms. To clarify the meaning of the convention used to iden-tify these integrals, I take, for instance, two neighbouring s orbitals. Being the two orbitals spherically symmetric the total angular momentum can only be zero. However, considering two p orbitals, two overlap con gurations can be obtained: the orbitals can overlap along the direction joining the atoms or in the orthogonal direction. In the rst case, the bound is said to be of type σ, while in the second, it is of type π. For d-orbitals, another bound is
2.1 Tight-binding for fcc lattice
possible and called a δ bound. These integrals are used in the tight-binding approach as parameters and are evaluated either analytically, or numerically, or semiempirically.
In the special case of group IV semiconductors with the zinc blende struc-ture, dierent models have been proposed. The simplest model consists in the rst neighbours tight-binding approximation with a sp3 basis. This choice
is justi ed by the outer electron con guration of the atoms considered and, even in its simplicity, it describes some of the electronic and optical proper-ties of the considered materials. However, more accurate models are available in the literature, with a higher number of orbitals. A thorough discussion is presented by Jancu, Scholz, Beltram and Bassani in the already mentioned work of reference [17]. They introduced the sp3d5s∗ parametrization which
will be used in the following of this thesis since it adequately describes most of the interesting features of silicon, germanium and their alloys. With this choice the basis set consists of the 10 atomic orbitals for each atom in the primitive cell:
s, px, py, pz, dxy, dxz, dyz, dx2−y2, d3z2−r2, s∗,
each with double degeneracy, due to spin multiplicity. The introduction of the excited s∗ states, as discussed in reference [16], is fundamental to obtain
the indirect character of the band gap. In fact, the s∗ orbitals couple with
the p-like conduction states of the diamond structure crystals, lowering the energies near the L and X points of the Brillouin zone.
On this basis set equation (2.13), in the rst neighbours approximation, can be written as:
H(~k) =H
11 H12
H21 H22
(2.17) where the index µ = {1, 2} indicates the atom in position dµ, and the Hµν(~k)
are 10x10 matrices. H11(~k) and H22(~k) are diagonal matrices containing
the on site energies of the atoms in the cell position ~dµ. In fact, the nearest
neighbours of the atom in position d1 are on the lattice individuated by the
atom in position ~d2, and vice versa. The on site energies are parameters
labelled with the orbital they refer to:
Es(a), E(a)p , Ed(a), Es(a)∗ .
The matrix elements hi,j = (H11)i,j are thus:
h1,1 = Es(a), h2,2= h3,3= h4,4 = Ep(a),
h5,5 = h6,6 = h7,7= h8,8= h9,9 = Ed(a),
The tight-binding model for lattices and superlattices H12(~k) = (H21(~k))∗ is the matrix containing the interaction terms, i.e. equa-tion (2.15). Each term of this matrix consists of two parts: the phase factors gi and the Slater-Koster parameters Ei,j. The gi arise from dierent
combi-nations of signs and exponential factors entering the sum (2.15). The four possible combinations for the diamond structure are:
g1 = 4(cos x cos y cos z − i sin x sin y sin z)
g2 = 4(− cos x sin y sin z + i sin x cos y cos z)
g3 = 4(− sin x cos y sin z + i cos x sin y cos z)
g4 = 4(− sin x sin y cos z + i cos x cos y sin z)
(2.18) where x = kxa/4, y = kya/4 and z = kza/4. These factors have to be
multiplied by the energy integrals listed in table 2.1. The parameters l,m and n entering the Slater-Koster integrals are the cosine directors of the vectors joining two nearest neighbours. The explicit derivation of the matrix elements is given in appendix A.1. Equation (2.10) can now be solved, giving the band structures of silicon and germanium which are shown in gures 2.1a and 2.1b, respectively. Notice the zero of the energy in these gures as in the following of this work has been taken at the maximum of the topmost valence band.
Most of the bulk features of silicon and germanium can be deduced from the inspection of these gures. For instance, the minimum of the conduction band is positioned along the Γ − X direction (∆ direction) in silicon and at Lin germanium, as expected from experiments [25]. However, the topmost valence bands are degenerate at Γ, since, until this point, no eect breaking the crystal symmetry has been considered. The rst important term is spin-orbit coupling which introduces band splitting, as shown in the next section.
2.1.1 Spin-orbit coupling
The spin-orbit coupling in crystals can be written as (see e. g. reference [26]):
HSO= α~L · ~S, (2.19)
where ~L is the orbital angular momentum operator, ~S = 1
2~~σ is the spin
operator, ~σ are the Pauli matrices and α is the strength of the coupling. This term can be treated as a perturbation to the tight-binding Hamiltonian HT B de ned in equation (2.3) because in general small with respect to the
interband energies. The term ~L· ~S can also be rewritten in terms of the total angular momentum ~J = ~L + ~S:
~
L · ~S = 1 2
~J2− ~L2− ~S2, (2.20)
whose expectation value becomes: h~L · ~Si = ~
2
2.1 Tight-binding for fcc lattice L Γ X W K L W X U=K -10 0 10 Energy (eV)
C.B.
V.B.
L’3 L1 L3 Γ’25 Γ15 Γ’2 X4 X1 (a) L Γ X W K L W X U=K -10 0 10 Energy (eV)C.B.
V.B.
Γ’25 Γ15 Γ’2 X4 X1 L’3 L1 L3 (b)Figure 2.1: Band structures of (a) silicon and (b) germanium, without spin-orbit interac-tion. The regions corresponding to the conduction bands and to the valence bands are identi ed by C.B. and V.B., respectively. The energy zero is xed at the maximum of the topmost valence band. The notation used for the representations of the simple group is shown in the near gap region. Notice the abrupt change from U to K in the right part of the gures.
The tight-binding model for lattices and superlattices
Table 2.1: Slater-Koster parameters in terms of two-center integrals, with l,m and n being the director cosines of of the position vector between two nearest neighbour atoms. Terms not reported in this table can be obtained by cyclic permutations.
Es,s Vssσ Es,x lVspσ Ex,x l2Vppσ+ (1 − l2)Vppπ Ex,y lmVppσ− lmVppπ Ex,z lnVppσ− lnVppπ Es,xy √ 3lmVsdσ Es,x2−y2 1 2 √ 3(l2− m2)V sdσ Es,3z2−r2 [n 2− 1 2(l 2+ m2)]V sdσ Ex,xy √ 3l2mVpdσ+ m(1 − 2l2)Vpdπ Ex,yz √ 3lmnVpdσ− 2lmnVpdπ Ex,zx √ 3l2nVpdσ+ n(1 − 2l2)Vpdπ Ex,x2−y2 1 2 √ 3l(l2− m2)V pdσ+ l(1 − l2+ m2)Vpdπ Ey,x2−y2 1 2 √ 3m(l2− m2)V pdσ− m(1 + l2− m2)Vpdπ Ez,x2−y2 1 2 √ 3n(l2− m2)V pdσ− n(l2− m2)Vpdπ Ex,3z2−r2 l[n 2−1 2(l 2+ m2)]V pdσ− √ 3ln2Vpdπ Ey,3z2−r2 m[n 2−1 2(l2+ m2)]Vpdσ− √ 3mn2Vpdπ Ez,3z2−r2 n[n 2−1 2(l2+ m2)]Vpdσ+ √ 3n(l2+ m2)Vpdπ Exy,xy 3l2m2Vddσ+ (l2+ m2− 4l2m2)Vddπ+ (n2+ l2m2)Vddδ Exy,yz 3lm2nVddσ + ln(1 − 4m2)Vddπ+ ln(m2− 1)Vddδ Exy,zx 3l2mnVddσ + mn(1 − 4l2)Vddπ+ mn(l2− 1)Vddδ Exy,x2−y2 3 2lm(l 2− m2)V ddσ+ 2lm(m2− l2)Vddπ+ 1 2lm(l 2− m2)V ddδ Eyz,x2−y2 3 2mn(l2− m2)Vddσ − mn[1 + 2(l2− m2)]Vddπ+ mn[1 + 12(l2− m2)]V ddδ Ezx,x2−y2 3 2nl(l 2− m2)V ddσ+ nl[1 − 2(l2− m2)]Vddπ− nl[1 − 12(l2− m2)]V ddδ Exy,3z2−r2 √ 3lm[n2−12(l2+ m2)]Vddσ− 2 √ 3lmn2Vddπ+ √ 3 2 lm(1 + n 2)V ddδ Eyz,3z2−r2 √ 3mn[n2− 1 2(l 2+ m2)]V ddσ+ √ 3mn(l2+ m2− n2)V ddπ− √ 3 2 mn(l 2+ m2)V ddδ Ezx,3z2−r2 √ 3ln[n2−12(l2+ m2)]Vddσ+ √ 3ln(l2+ m2− n2)V ddπ− √ 3 2 mn(l2+ m2)Vddδ Ex2−y2,x2−y2 3 4(l2− m2)2Vddσ+ [l2+ m2− (l2− m2)2]Vddπ+ [n2+ 14(l2− m2)2]V ddδ Ex2−y2,3z2−r2 √ 3 2 (l2− m2)[n2− 1 2(l2+ m2)]Vddσ+ √ 3n2(m2− l2)V ddπ+ √ 3 4 (1 + n2)(l2− m2)Vddδ E3z2−r2,3z2−r2 [n2− 12(l2+ m2)]2Vddσ+ 3n2(l2+ m2)Vddπ+ 3 4(l 2+ m2)2V ddδ
2.1 Tight-binding for fcc lattice
where j, l and s are the quantum numbers of the three operators ~J2, ~L2 and ~S2, respectively. The atomic orbital basis set adopted in the previous
section has to be doubled to account for the spin states, becoming: s↑, p↑x, p↑y, p↑z, dxy↑ , d↑xz, d↑yz, d↑x2−y2, d ↑ 3z2−r2, s ∗↑, s↓, p↓x, p↓y, p↓z, dxy↓ , d↓xz, d↓yz, d↓x2−y2, d ↓ 3z2−r2, s ∗↓,
for each atom in the primitive cell. In general, the only spin-orbit coupling considered involves p orbitals, since the contribution of the d orbitals is small compared [27]. To obtain the contribution of spin-orbit to the Hamiltonian matrix H(~k), the p orbitals have to be written in terms of the eigenstates Ψj,jz of the total angular momentum and its z component Jz, using the
Clebsch-Gordan decomposition. The derivation is straightforward and it will be omitted in this context. The non-zero matrix elements calculated in this manner are:
hp↑x| HSO|p ↑ yi = −iλ, hp↑x| HSO|p ↓ zi = λ, hp↑y| HSO|p ↓ zi = −iλ, hp↓x| HSO|p ↓ yi = iλ, hp↓x| HSO|p ↑ zi = −λ, hp↓y| HSO|p ↑ zi = −iλ, (2.22)
where λ = α~2/2. With the new basis and the o-diagonal matrix elements
listed above, the Hamiltonian of equation (2.17) becomes the following 40x40 matrix: H(~k) = H↑↑11 H 11 ↑↓ H 12 0 H↓↑11 H 11 ↓↓ 0 H 12 H21 0 H↑↑22 H↑↓22 0 H21 H↓↑22 H 22 ↓↓ . (2.23)
As an example, the eects of the spin-orbit splitting on the near gap band structure of Germanium are shown in gure 2.2.
From the model described until here some important properties can be deduced. These include eective masses and band gaps of the bulk materials. The electronic eective mass tensor m∗
ij, in units of the electron mass me, is
de ned as the reciprocal of the curvature of the energy bands: 1 m∗me ij = 1 ~2 ∂2E ∂ki∂kj (2.24)
Tables 2.2-2.4 compare the values derived from the present TB model, with parameters taken from reference [17], with experimental values and with the values derived from a TB model using the more complex parametrization of reference [28]. The band gaps are in good agreement with experimental data,
The tight-binding model for lattices and superlattices L Γ X -4 -2 0 2 4 Energy (eV) (a) L Γ X -4 -2 0 2 4 Energy (eV) HH LH SO (b)
Figure 2.2: Germanium band structure in the near gap region. The sixfold degenerate top valence bands of gure (a) split into a fourfold degenerate band and a double degenerate band of gure (b). In the second gure the convention used to indicate the top valence bands is also shown.
Table 2.2: Eective masses at the conduction minimum in Silicon. Values from the TB model of reference [28] and experimental data [29] are also reported.
Si TB model of ref. [28]Present TB model Exp.[29]
m∆l 0.71 0.900 0.9163
m∆t 0.22 0.197 0.1905
Table 2.3: Eective masses at the conduction minimum in Germanium. For convenience, the mass at Γ is also reported. Comparison with reference [28] and with experimental values from references [30], [31] are shown.
Ge TB model of ref. [28]Present TB model [30]; [31]Exp.
mLl 1.363 1.594 1.588; 1.74
mLt 0.084 0.082 0.08152; 0.079
mΓ 0.038 0.038
Table 2.4: Energy gaps of Silicon and Germanium as obtained from the present TB model. The experimental values are from reference [25]. Units are in eV.
TB Exp.
Si 1.170 1.155
Ge 0.845 0.740
but the parametrization of [17] used here is not able of correctly predict the eective masses near a band minimum. In the next chapter I will show how these limitations can be overcome. The tight-binding method derived in
2.2 TB model for tetragonal cells
this section has to be implemented for the tetragonal symmetry described in section 1.2, since this will be the geometry studied in this work.
2.2 TB model for tetragonal cells
To adapt the tight-binding model to a crystal structure with tetragonal sym-metry, it is convenient to built two dimensional Bloch sums de ned in terms of the atomic orbitals on each layer:
|Ωµ,i(~q, ~r)i = 1 pN|| X ~ τ||m ei~q·(~τ||m+ ~d||µ)|φ(a) iµ (~r − ~dµ− ~τm)i , (2.25)
where ~q is the two-dimensional component of the ~k vector (~k = (~q, kz)), N||
is the total number of cells in the plane orthogonal to the kz direction and
the sum runs over all vectors in the two-dimensional lattice. The derivation of the Hamiltonian matrix for the two-dimensional lattice is similar to that of the previous section and leads to the following expression:
Hµνij = hΩν,j(~q, ~r)| H |Ωµ,i(~q, ~r)i = δµνδijEjν+ X ~ τ||m ei~q·(~τ||m+ ~d||µ− ~d||ν)hφ(a) jν(~r − ~dν)| V 0(~r) |φ(a) iµ (~r − ~dµ− ~τm)i . (2.26)
As previously done in the case of fcc crystals, the sum can be limited to the nearest neighbours. Every atom in the tetragonal cell has two nearest neigh-bours on the following layer and two on the previous one. Consequently, each interaction matrix will involve only two nearest neighbours of the con-sidered atom. Furthermore, to recover periodicity along the z-direction, an interaction matrix between the last and rst atom in the cell times the Bloch factor ekza⊥ is introduced. The resulting Hamiltionian becomes:
H = H11 H12 0 H14e−ikza⊥ H21 H22 H23 0 0 H32 H33 H34 H41eikza⊥ 0 H43 H44 , (2.27)
where kz is the momentum and a⊥the lattice parameter in the growth
direc-tion. Since until here no strain eect has been introduced in the TB model, a⊥= a||= a. The explicit derivation and expression of the interaction
ma-trices are reported in appendix A.2. The new phase factors γi entering the
interaction terms are:
γ1= 2 cos(x||+ y||),
γ2= 2i sin(x||+ y||),
γ3= 2 cos(x||− y||),
γ4= 2i sin(x||− y||),
The tight-binding model for lattices and superlattices L Γ X⊥ X -2 -1 0 1 2 3 4 Energy (eV) (a) L Γ X⊥ X -2 -1 0 1 2 3 4 Energy (eV) (b)
Figure 2.3: The band gap region of Silicon computed for the octahedral Brillouin zone (a) and for the tetragonal zone (b). Notice that the bands along the Γ-X direction are doubled.
where x|| = qxa||/4, y|| = qya||/4. The block diagonal matrices Hµµ are the
same as discussed in section 2.1, while spin-orbit coupling can be included doubling the matrix as in equation (2.23). From now on I will consider the Hµν of equation (2.27) to be 20x20 matrices, i.e:
(H)µµ=H µµ ↑↑ H µµ ↑↓ Hµµ↓↑ H µµ ↓↓ , for ν = µ, (2.29) (H)µν =H µν ↑↑ 0 0 Hµν↓↓ , for ν 6= µ, (2.30)
where the 10x10 matrices Hµµ
ss0 are de ned analogously as the block matrix
elements in equation (2.23). In this way, spin-orbit coupling is implicitly contained in equation (2.27).
Figure 2.3 shows how the reduced Brillouin zone aects the energy bands. The X⊥= a2π
⊥(0, 0, 1)point has been folded into Γ and all the energy bands
of the octahedral cell are displayed in half the Brillouin zone in the Γ − X direction. There is no physical meaning for this eect, being the folding merely geometrical. However, it is important to distinguish such geometrical folding eects from physical strain eects as will be discussed in the following. The tetragonal cell description allows a simple generalization to multi-layer structures. Starting from Hamiltonian (2.27), a block diagonal matrix
2.3 Strain eects in multilayer structures
can be added for each layer. For each diagonal element in the matrix the cor-rect interaction terms have to be considered. The most general Hamiltonian matrix with rst neighbour interactions will be of the tridiagonal form:
H = H11 H12 0 . . . 0 H1ne−ikza⊥ H21 H22 H23 . . . 0 0 0 ... ... ... ... ... ... ... ... ... ... ... 0 0 . . . H(n−1)(n−2) H(n−1)(n−1) H(n−1)n Hn1eikza⊥ 0 . . . 0 Hn(n−1) Hnn , (2.31) where n is the number of layers in the cell. A cell constructed in this way is called a supercell. Supercells up to hundreds of layers can be obtained in this way. However, the complete description of strained structures needs a further step which will be discussed in the following section.
2.3 Strain eects in multilayer structures
Strain eects, beside aecting the geometrical structure of a semiconduc-tor, aect also its energy bands, giving rise to a large number of interest-ing changes. These changes can be used to engineer the properties of the material. Several authors dealed with this subject, ranging from theoreti-cal analyses to experiments. One of the fundamental works concerning the derivation of deformation eects in silicon and germanium is the article of G. E. Pikus and G. L. Bir [1]-[2]. For convenience, I will report only the important conclusions of their work. Also, D. J. Paul [32] resumes in a con-cise manner the main eects of strain on the band energies. Following his approach, I will describe separately the two components of strain: hydro-static strain and uniaxial strain. The rst is the consequence of the volume modi cation when stress is applied to a material. The energy Ei of a band
iat a given k-point, shifts according to:
Ei = Ei0+ ai(εxx+ εyy+ εzz), (2.32)
where E0
i is the energy at the same k-point in the unstrained bulk crystal,
ai are the hydrostatic deformation potentials, and εiiare components of the
strain tensor (1.15). The values of ai depend on the band considered and on
the type of material.
The uniaxial component of strain splits the energy bands. Under the strain condition discussed in section 1.3, the minima along the Γ−L direction do not split and remain degenerate. The six equivalent minima in the lowest conduction band along the Γ − X direction split into four equivalent minima along the (100), (100), (010) and (010) directions (conventionally 1 stays
The tight-binding model for lattices and superlattices for −1), according to:
Ec(100)= E(100)c = Ecu−1 3Ξ ∆ u(εzz− εxx), (2.33) Ec(010) = Ec(010)= Ecu−1 3Ξ ∆ u(εzz− εyy), (2.34)
and into two equivalent minima along the (001) and (001) directions, ac-cording to: Ec(001)= E(001)c = Ecu+2 3Ξ ∆ u(εzz− εxx). (2.35) Ξ∆
u is the named deformation potential for the considered band and Ecu is
the shifted energy of the unsplitted minima under hydrostatic strain (2.32). The behaviour of the top valence bands is slightly more complicated. The uniaxial strain component and the spin-orbit eect are both active on the electronic band structure and it is necessary to consider both eects simultaneously, according to:
EHH = EAV + 1 3∆0− 1 2δE001, (2.36) ELH = EAV − 1 6∆0+ 1 4δE001+ 1 2 r ∆20+ ∆0δE001+ 9 4δE 2 001, (2.37) ESO= EAV − 1 6∆0+ 1 4δE001− 1 2 r ∆20+ ∆0δE001+ 9 4δE 2 001, (2.38)
where EAV is the average energy of the three topmost valence band at Γ,
shifted from the unstrained value E0
AV according to equation 2.32. The above
relations show how the Heavy Hole (HH), Light Hole (LH) and Split-O (SO) energy bands (see gure 2.2) change with respect to their weighted average at the Γ point. The spin-orbit splitting ∆0 is the energy dierence between
the top valence bands and the SO energy bands at Γ in the absence of strain, while the strain splitting δE001 is de ned as:
δE001= 2b(εzz − εxx), (2.39)
where b is the appropriate deformation potential. The values of the de-formation potentials and spin-orbit splitting for silicon and germanium as reported in the work of Paul, togheter with the references quoted therein are shown in table 2.5. A few comments on the behaviour of the valence bands are mandatory. The above relations (2.36-2.39) show that in the case of compressive strain along the z-axis (εzz < 0, εxx = εyy ≥ 0), δE001 > 0
and the degenerate LH and HH states split, with the light hole state raising in energy. The opposite is true for tensile strain. The comparison between the two situations and the case of unstrained Germanium is shown by the numerical results reported in gure 2.4. These strain eects have to be
in-2.3 Strain eects in multilayer structures
Table 2.5: Deformation potentials and spin-orbit splitting parameter for Silicon and Ger-manium. The values are taken from reference [32] and references quoted therein. Units are in eV.
Silicon Germanium
Theory Experiment Theory Experiment
a∆c 4.88 [33] 3.3 [34] 2.55 [33] -aLc -0.66 [33] - 1.54 [33] -av 2.46 1.8 2.55 -∆0 - 0.044 [35] 0.296 [36] -b -2.35 [8] -2.1± 0.1 [37] -2.55 [8] -2.86± 0.15 [38] Ξ∆u 9.16 [8] 8.6± 0.4 9.42 [8]
-cluded correctly in the TB model, introducing scaling laws for the model parameters. The two-center integrals can be scaled according to the law:
Vijk(d) =
d0
d ηijk
Vijk(d0), (2.40)
where d0 and d are the interatomic distances of the unstrained and strained
Γ -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 Energy (eV) Γ -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 Γ -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 ∆4 ∆2 ∆4 ∆2 ∆4 ∆2 εzz<0 ε zz=0 εzz>0 LH HH SO LH LH HH SO SO HH
Figure 2.4: Bulk germanium valence bands under dierent strain conditions. The right panel shows how the LH states are at higher energy as the HH states at Γ in compressive strained germanium. The opposite is true for tensile strained germanium, as shown in the left panel. Unstrained germanium is shown in the centre panel, where the LH and HH coincide at Γ. The computed results re ect equations (2.36-2.38).
The tight-binding model for lattices and superlattices material, respectively, and ηijk are scaling empirical parameters.
Harri-son [15] chose all exponents to be 2, but successive parametrizations [17] used more accurate parameters. In order to account for the degeneracy lift-ing of the d-states due to uniaxial strain, Jancu et al. introduced shear deformations in the Slater-Koster integrals:
Exy = Ed[1 + 2bd(εzz− εxx)], (2.41)
Exz= Eyz= Ed[1 − bd(εzz− εxx)]. (2.42)
Here bd are shear parameters. These relations are generally su cient to
reproduce the correct behaviour of the energy bands in strained materials. However, particularly for germanium, also the on-site energies should be rescaled, as discussed in reference [39]. The scaling law is similar to the expression given in (2.40): Ei(d) = d0 d ηi Ei(d0). (2.43)
The complete TB model illustrated in this chapter allows the description of a great variety of properties of dierent semiconductor devices. In the next chapter the method will be applied to strained silicon and germanium multilayer structures.
Chapter 3
Single-element silicon and
germanium strained
superlattices
The eects of strain on the energy bands of semiconductors have important consequences which can be exploited to engineer a wide range of new struc-tures. An important example of such structures are Sin/Gem superlattices.
A superlattice is composed of a periodically repeated supercell, made of n layers of silicon and m layers of germanium. The lattice parameters of both material in the plane orthogonal to the growth axis match, causing the de-formations already described in section 1.3. A lot of eort has been spent since the rst superlattices have been introduced. For instance, a recent analysis of the optical properties of selected superlattices performed in the tight-binding framework is shown in reference [18].
The aim of the present work is to examine a special type of superlat-tices, the single-element strained superlattices (SSL), where the supercell is composed of n layers of unstrained silicon or germanium and m layers of the same material, but under dierent strain conditions. In the following I shall consider the case of equal number of strained and unstrained layers (n = m). A recent experiment [12], has shown an example of how these structures can be obtained. A rst theoretical analysis has been performed by Z. Liu et al. [13], using rst-principles calculations. In this thesis I adopt the tight-binding model for both silicon and germanium single-element SSLs. The atomistic approach here exploited allows a clear and more intuitive in-terpretation of the results and sound prevision of technological interesting structures. The rst section of this chapter illustrates the speci c geometry analyzed and the de nitions used. Then, the results for the electronic struc-ture of silicon and germanium, respectively, are displayed, emphasizing the important properties which arise from the choice of geometrical and strain conditions.
Single-element silicon and germanium strained superlattices
3.1 Description of the single-element SSL supercell
The supercell used in this work is composed of 2n layers of a given semicon-ductor material; half layers are subject to strain along the growth direction, taken as the z-axis. The strain components in the xy plane are considered negligible and thus the lattice parameters in this plane equal the unstrained lattice parameters of each material. The only non zero component of the strain tensor is εzz and, consequently the only lattice parameter which varies
is a⊥. As an example, gure 3.1 shows the smallest supercell obtainable with
this construction containing four layers of unstrained material and four of strained material.
a
⊥= a
a
⊥= a(
ε
zz+1)
a
||
= a
Figure 3.1: Supercell with four layers of unstrained material and four layers of strained material. Anion positions are represented as lled dots and cation positions as blank dots.
Since single-element SSLs are composed of only one semiconductor mate-rial, no chemical composition aects the energy bands in the neighbourhood of the interfaces. Only deformation eects shift the bands of the strained lay-ers with respect to the unstrained part, according to equations (2.32-2.39). For instance, from equation (2.32) the average energy of the top valence bands in silicon is expected to be lower than the same quantity in the un-strained material for compressive strain (εzz < 0) along the growth direction;
the opposite is true if (εzz > 0). The band alignment is thus a function of
the strain component εzz, as schematically shown in gure 3.2. They are
obtained by computing separately the bulk energy levels in each material, taking as reference a common energy zero. In the present calculations the zero of the energy has been chosen as the energy at the Γ point of the top-most valence band in ustrained bulk silicon. Putting together the dierent materials illustrates schematically how the TB hamiltonian eigenvalues are expected to shift. For instance, for a bandpro le as in gure 3.2, I expect
3.1 Description of the single-element SSL supercell
Figure 3.2: Band alignment of the top valence bands in the silicon SSL described by the supercell of gure 3.1. Tensile 4% strain is con-sidered. The pro les are derived calculating sepa-rately the bulk properties of the material in the two dierent strain conditions. The unstrained valence av-erage (UVA) represents the average of the SO, HH and LH valence bands at Γin unstrained bulk sili-con. The strained valence average (SVA) is the anal-ogous quantity evaluated for a strained bulk silicon structure. 0 2 4 6 8 Length (Å) 0 0,1 0,2 0,3 Energy (eV) Unstrained Region Strained Region HH LH SVA SO HH=LH UVA SO
the maximum of the topmost valence bands to occur in the strained layers. However, other eects, such as eective masses, will in uence the energies and the band alignment as discussed below.
The combination of strain and periodicity gives rise to bandpro les as it happens in the case of multiple quantum wells, with depth depending also on the strain component εzz, and width depending on the length L of the
supercell. Indeed, if the bandpro le of gure 3.2 is periodically repeated, a series of quantum wells of width L/2 is generated. Varying strain conditions and length of the supercell many dierent eects on the energy bands, and thus on the electronic and optical properties of a single-element SSL, can be inspected. For these reasons every quantity evaluated in the following, will be shown as a function of the two variables εzz and L.
One of the most important goals to investigate is the possibility to achieve a direct gap material. A direct gap in the reciprocal space can be obtained if the minimum of the lowest conduction band and the maximum of the topmost valence band occur at the same point ~k0 in the Brillouin zone.
Such a gap has important consequences in optical applications, since photon emission or absorption are vertical processes in the Brillouin zone [24]. In fact, if I consider momentum conservation in an optical transition from an initial valence state of electron wavevector ~kv to a nal conduction state of
wavevector ~kc, induced by a radiation eld of wavevector ~q, I have:
~
Single-element silicon and germanium strained superlattices In ordinary experimental situations the wavelength of the incident radiation ranges from 1000 Å to 10000 Å, thus much larger than the lattice parameters of the crystals. In these situations, the wavevector ~q is small compared to the typical values of the wavevectors ~k in the Brillouin zone and can be neglected. Consequently, the momentum conservation reduces to ~kc = ~kv,
i.e. only vertical transitions are allowed. However, a direct gap in reciprocal space is necessary but not enough for the realization of an optical transition. In fact, from the theory of optical transitions, the probability of such a transition to occur is proportional to the matrix element:
hψc(~kc, ~r)| ei~q·~r~e · ~p |ψv(~kv, ~r)i , (3.2)
where ψc(~kc, ~r), ψv(~kv, ~r) are the conduction and valence states involved in
the transition, respectively, ~e is the polarization vector of the radiation eld and ~p is the electron momentum operator. An optical transition is possible if the selection rules for equation (3.2) allow it, e. g. the parity of the electronic states ψc(~kc, ~r), ψv(~kv, ~r) . Moreover, if the spatial overlap of the two
eigen-vectors ψc(~kc, ~r) and ψv(~kv, ~r) is small, the probability becomes irrelevant.
Thus the ideal situation for optical applications requires direct gap in the reciprocal lattice and transitions occuring in the same spatial region. This is very demanding in silicon and germanium since in the bulk materials, the minimum of the lowest conduction band occurs at ~k 6= Γ, while the maxi-mum of the topmost valence band occurs at Γ (see gures 2.1a and 2.1b). In the following I will show how strain can change this con guration.
The de nition of some quantities is mandatory to introduce the results of the following sections. The rst of these quantities is the phase indicator Pn: Pn= RL2 0 |ψn(~k0, z)|2dz RL L 2 |ψn(~k0, z)|2dz , (3.3)
where ψn(~k, ~r) is the n-th eigenvector of the TB hamiltonian, as de ned by
equation (2.5), and the integrals over the other two coordinates have already been performed over the whole width of the cell. The eigenvectors are cal-culated at a given point ~k0 of the Brillouin zone, usually a high symmetry
point or an edge of the energy bands. In the present calculations the eigen-vectors are normalized to the unity over the length of the supercell, thus the integrals in the de nition of the phase indicator are both less than one. The phase indicator measures how much a given eigenvector is con ned in the unstrained half of the supercell relatively to the strained half. The impor-tance of this quantity can be explained considering the phase indicator of the topmost valence band Pv and of the lowest conduction band Pc of a given
material. Under the conditions Pv < 1 and Pc < 1, or Pv > 1 and Pc > 1,
I expect the electrons in the conduction band and the holes in the valence bands to be con ned in the same part of the supercell. Such a con guration
3.2 Silicon single-element strained superlattices
(conventionally named type I gap) would indicate a possible direct gap in the direct lattice. In these cases, if the phase indicators are evaluated at the same ~k point, the particular strain and length condition of the super-cell could be promising for optical applications. Conversely, if Pv < 1 and
Pc> 1, or Pv < 1 and Pc> 1, a type II gap is expected.
Another important quantity is the band gap energy, already discussed in chapter 2. This quantity is simply de ned as:
Eg = Ec(min)− Ev(max), (3.4)
where E(min)
c is the energy at the minimum of the lowest conduction band
and E(max)
v the energy at the maximum of the topmost valence band.
Gen-erally, the corresponding two ~k-points will not coincide, as mentioned above. The results presented in the next two sections are obtained following a common procedure. First, the energies of the topmost valence band and of the lowest conduction band are evaluated at their maximum and minimum points, respectively, as a function of the two variables εzz and L. Once
the edge energies have been found, the eigenvectors corresponding to this eigenvalues are computed and then integrated to obtain the phase indicators (3.3). Varying εzzand L, contour plots are illustrated, showing the behaviour
of Pv and Pc. Also the band gap Eg is shown as a function of the two
variables.
3.2 Silicon single-element strained superlattices
The rst structures investigated in this work are single-element SSLs made of silicon. The aim of this section is to compare our results obtained by the atomistic tight-binding model, with the results obtained by a rst principle approach by Z. Liu and co-workers [13]. The TB approach will allow to easily obtain other properties not reported in the reference.
The computation in the TB framework needs an introductory discussion on the empirical parameters used. The work of J.-M. Jancu et al. has already been mentioned and used to obtain some bulk properties of silicon. Among these properties, the eective masses along the Γ − X direction, evaluated at the minimum of the conduction band result to be underestimated, as shown in table 2.2. The same values evaluated with the parametrization from reference [40] become:
m∆l = 0.919,
m∆t = 0.201, (3.5)
in better agreement with the experimental values. However, the parameters ηijk, bd and ηi entering equations (2.40-2.43) have been taken from
