Doi:10.1016/j.na.2005.01.058

Nonlinear Analysis 63 (2005) e1607 – e1617 Non-linear strain theory for low-dimensional B. Lassena,∗, R. Melnika, M. Willatzena, L.C. Lew Yan Voonb a Mads Clausen Institute for Product Innovation, University of Southern Denmark, Grundtvigs Allé 150, DK-6400 SZnderborg, Denmark bDepartment of Physics, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA Abstract
We present a continuous non-linear strain theory and demonstrate its plausibility on modelling low- dimensional semiconductor nanostructures. The main advantages of this theory are its non-linearity,continuous nature (computationally inexpensive), and anisotropy (includes the crystal structure). Thetheory allows us to account explicitly for internal strain and for realistic interface boundary conditions.
The discussion is restricted to cases where the strain is nearly constant on the length scale of theprimitive lattice cell except at interfaces where discontinuities are allowed. The main idea behind thetheory presented here is to include as much information on the atomic structure in the continuous limitas it is necessary in order to accurately describe nanoscale systems. To illustrate the importance ofnon-linear effects and the inclusion of internal strain, we present some simple calculation for quantumwells with diamond crystal structure oriented in the [1 1 1] direction.
᭧ 2005 Elsevier Ltd. All rights reserved.
Keywords: Strain; Inhomogeneities; Crystalline structures 1. Introduction
Many nanoscale semiconductor heterostructures of experimental and theoretical interest consist of materials with a lattice mismatch of up to 13%, for example, InAs/GaAs andSn/Ge This lattice mismatch is the driving mechanism for one of the fabrication meth-ods used to construct nanoscale devices (self-assembly). In addition, it is known that strain 0362-546X/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.na.2005.01.058 B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 strongly affects the electronic and optical properties of nanoscale structures. This fact hasimportant implications and, among other things, has allowed to construct better quantumwell lasers Strain contributions are essential in a realistic description of most self-assembled nanostructures. Until now, strain calculations for such structures have mainlybeen based on either an atomistic model (Valence Force Field) or one of the continuouslinear strain models (e.g., While atomistic models are computationally expensivemaking them difficult to apply to many important low-dimensional semiconductor struc-tures (LDSS), the conventional continuous models applied to modelling of LDSS do notinclude non-linear effects and internal strain explicitly nor do they employ the correct inter-face boundary conditions. Internal strain refers to internal displacements of sublattices notdetermined by symmetry alone In this contribution, we present a continuous non-lineartheory for modelling LDSS and discuss its applications in the context of modelling diamondquantum well nanostructures. We compare results obtained with known linear models andthose obtained on the basis of the non-linear theory.
2. Theory
In this section, we present the theory behind the non-linear strain model we propose to use for LDSS strain calculation. We start with a description of the strain theory in theatomistic framework for bulk materials. This is followed by an explanation of how such adescription can be taken to the continuous limit, i.e., how to define continuous quantitiesthat capture the physics of the problem. And finally, we show how to extend the presentedtheory to heterostructures.
2.1. Atomistic picture—bulk material The atomistic picture of crystalline materials is based on a semi-classical description of these structures, i.e., atoms are treated as point particles arranged in a periodic lattice. Thisperiodic structure is best described by the primitive lattice cell and a set of vectors wj ,where j = 1, . . . , jmax, giving the position of the atoms inside the primitive lattice cell. Theprimitive lattice cell is the smallest repeated parallelepiped making up the crystal structure.
Letting v1, v2 and v3 denote the three edge vectors of the primitive lattice cell, we find thata given atomic position aij (the jth atom in the ith primitive lattice cell) can be expressedas follows: For strain energy calculations, it is not the actual position of the atoms that matters but rather the vectors connecting the different atoms. These vectors are called position vectorsand they are given as follows: Xkj (aij ) = alp − aij , B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 i.e., the vector with aij as base point and alp as end point. The reason why we do not includeindices l and p on the left-hand side is that we need the following property later on: theindex k is used to label the position vectors such that for all i, i ∈ {1, . . . , imax} and j ∈ {1, . . . , jmax}, where imax is the number of primitivelattice cells. This is possible because we have a periodic structure.
In principle, we should account for all the position vectors when we calculate the strain energy, but because the forces between atoms are short ranged we only need to take intoaccount a small subset. We label these such that k = 1, . . . , kmax. Usually, the positionvectors of the nearest or second nearest neighbors are enough to capture the physics of theproblem.
The total energy before any deformation is now given by where Vj is the energy contribution from the jth atom in the primitive lattice cell. It isa function of the standard inner product in R3 between all possible combinations of theposition vectors with the specified atom as origin, i.e., all possible combinations of k and p.
A deformation of the crystal is given by an injective map mic : A → R3, where A is the set of initial atomic positions and the position vectors are transformed according to this is just the position vectors of the deformed crystal. As long as we do not include allposition vectors in our model we cannot treat all possible deformations. This is because adeformation can bring atoms, that before the deformation had a negligible influence on theenergy (the position vector between them was not included in the energy), close enoughtogether such that the contribution cannot be neglected anymore. The rule is that we needto include as many position vectors as is necessary to describe the kind of deformations weare interested in.
The strain energy is then given by the total energy of the deformed crystal minus the total mic(aij )) ) − V0. Making a second-order Taylor expansion of this produces the well-known Valence ForceField model In order to pass to the continuum limit, we have to define some continuous quantities that capture the important physics of the problem. First, we specify extensional characteristicsof the material: B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 Definition 2.1 (Crystal volume). The crystal volume B is defined to be the union of all the
primitive lattice cells making up the material.
This definition may not be appropriate if one wants to describe surface effects, because the surface of the crystal volume defined in this way is too irregular. But we are not interestedin surface effects, hence Definition 2.1 is a reasonable definition of the crystal volume.
The next quantity we need to define in the context of the continuum limit is the position vectors. We saw in the last section (Eq. (3)) that the position vectors with one atom as basepoint is identical to the position vectors with another atom of the same kind (the same jindex) as base point (just parallelly translated to the other atom). This suggests the followingdefinition.
Definition 2.2 (Position vector fields). The position vector fields Xkj are defined to be the
constant vector fields, Xkj (r) = Xkj (aij ) for all r ∈ B.
With the help of these two definitions it is easy to show that the total energy before a where Vp is the volume of the primitive lattice cell. It should be noted here that we havemade no approximations going from Eqs. (4) to (7).
We need a rather complex definition of deformations. This is because we want to average out the behavior on the scale of the primitive lattice cell, but still ensure that we do not loseany physical important behavior. This is done by explicitly including the behavior of thewj vectors.
Definition 2.3 (Deformation). Let E be the set of constant vector fields on B given by the
wj vectors. A map : E → T R3, where T R3 is the tangent bundle over R3, is a deformationcompatible with the atomistic deformation for all j, j = 1, . . . , jmax and 0,B and R3 are the projection maps onto B and R3, respectively.
B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 captures the macroscopic behavior and behavior (inside the primitive lattice cell).
In order to find a good approximation of the transformation of the position vector fields Xkj in the continuous framework, we need the observation that Xkj (aij ) = wl − wj + for some dq ∈ Z, where wl is the vector giving the position of the atom inside the primitivelattice cell that Xkj (aij ) points to. The term involving the sum is just the vector with basepoint in the primitive lattice cell that includes aij and end point in the primitive lattice cellthat includes the atom that Xkj (aij ) points to. We now have Theorem 2.1. The transformation of the position vectors Xkj (aij ) is to the first order in : T B → T R3 is the push-forward operator on vector fields and uj = Proof. This is easily seen by making a first-order Taylor expansion of is in elasticity also known as the deformation gradient. This suggests that we should define the transformation of the position vector fields as follows.
Definition 2.4 (Transformation of the position vector fields). The transformation
It is now straightforward to show that, under the assumption that the transformation is nearly constant on the length scale of the primitive lattice cell, the strain energy is In relation to other strain models it should be noticed that a second-order Taylor expansionof V and some approximation of the internal strain and the strain = 12 ( I : T B → T B is the identity map, produces the continuum linear mechanical model thathas been used extensively in the context of LDSS modelling (e.g., B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 As an example, we derive the total strain energy for materials with diamond crystal structure. The primitive lattice cell is in this case given by v1 = 2a[1, 1, 0], v2 = 2a[1, 0, 1], v3 = 2a[0, 1, 1], where a is the lattice constant. There are two atoms inside the primitive lattice cell: w0 = [0, 0, 0], w1 = a[1, 1, 1] and the nearest-neighbor position vectors are given by X1 = X10 = −X11 = w1, X2 = X20 = −X21 = w1 − v1, X3 = X30 = −X31 = w1 − v2, X4 = X40 = −X41 = w1 − v3. We see that the position vectors around the first atom in the primitive lattice cell are given by the negative of the position vectors around the second atom.
In order to get an expression that we can calculate we make a second-order Taylor expansion of V and assume that only the diagonal products of the inner product contributeto the strain energy, arriving at (Xi) + u1 − Xi, Xi )2 (Xj ) + u1 − Xi , Xj )2 We are now in a position to consider the kind of heterostructures we are interested in.
Namely, structures where one material is embedded in another material (see Weassume that the two materials have the same crystal structure but they are allowed to havedifferent lattice constants. The first thing we need to do in order to find the strain energy isto define a reference configuration. We do this by shrinking or expanding the inner materialby a homogeneous deformation such that the two materials have the same crystal structureand lattice constant (and the same orientation). Next, we split the structure into two regions.
This is done according to the following rules: • i contains all the atoms of material i, • all the primitive lattice cells are included in 1 ∪ 2 and nothing else, • the interface between 1 and 2 consists of finitely many plane surfaces.
We note that the above procedure may not define the interface in a unique fashion, but further discussion of this issue falls outside of the scope of the present paper.
B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 We are now able to find the strain energy of the whole system in three steps. First, we then we find the strain energy of material two The last step is to find the strain energy resulting from the presence of the interface. This isdone by projecting the additional contribution around the interface into the interface surfacein the following manner.
Due to periodicity of the structure, the intersections of the position vectors with one specific plane surface will also be arranged in a periodic fashion. We can, as a consequence,describe the periodic structure on the interface by a 2-dimensional principal lattice cell anda set of 2-dimensional vectors ˆ wl giving the position of the intersections inside the primitive lattice cell. We denote with dnl the lth intersection inside the nth primitive lattice cell andwith Zsl(dnl) the vectors that intersect the surface at dnl, where the index s is chosen suchthat for all s = 1, . . . , sl,max, l = 1, . . . , lmax and n, n = 1, . . . , nmax, where nmax is the num-ber of 2-dimensional primitive lattice cells, lmax is the number of intersections inside theprimitive lattice cell and sl,max is the number of position vectors intersecting the surface atthe lth intersection inside the primitive lattice cell. This is similar to what we did for the B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 3-dimensional crystal structure. In addition to this we denote by ˆ vectors of the atom that has Zsl(dnl) as origin. Again we have that for all t = 1, . . . , tl,max, l = 1, . . . , lmax and n, n = 1, . . . , nmax, where tl,max is the numberof position vectors around the atom that has Zsl(dnl) as origin. In order to get a structurethat lives on the interface surface we parallelly translate Zsl(dnl) and ˆ that they have dnl as origin. If we now write down the contribution of the interface to thestrain energy and take this to the continuum limit in the same manner as before, we arriveat where Ap(r) is the area of the 2-dimensional interface primitive lattice cell (it is piecewiseconstant), lr,max is the number of intersections inside the 2-dimensional primitive latticecell (it is also piecewise constant) and the subscript m on the functions taken from material m. The interface contribution Vint,l,m is a function of theinner product between all possible combinations of s and t.
The strain energy of the system is now given by V = V1 + V2 + Vint. It is minima of the strain energy of the system with respect to and uj that give equilibriumstrain tensors according to (there might be more than one minimum) 3. Quantum well results
In this section, we present results of an implementation of the above optimization problem for a quantum well, i.e., a thin plate with a barrier material on both sides (see Thesystem is allowed to be deformed only in the direction of the well (perpendicular to theplate). We have chosen to investigate two material systems, Silicon in Germanium and Tinin Germanium, both with a diamond crystal structure.
The model has the following non-linearities: it is • non-linear in the definition of the strain − I) = 12(dT + d + dT d ), − I d and I d : B → R3 is given by I d(r) = r, and • non-linear with respect to the internal strain u1. Notice that u1 is the only internal strain B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 Fig. 2. The structure of a quantum well.
Because of that we have chosen to compare three different models.
This is the model we described in the previous section.
The u1, u1 terms are neglected in Eq. (14). This ensures that jV /ju1 = 0 gives a linearrelationship between u1 and elements of (in situations where jV /ju1 = 0 exists).
• Model 3 (Linear in u1 and d): The u1, u1 terms are neglected in Eq. (14) and only the linear terms with respect tod in Eq. (22) are included. This model is just the continuum mechanical model that ispredominantly used at present in the LDSS modelling context Even though we do not allow deformations perpendicular to the well, the well material will still be strained because of the way we defined the reference configuration (the wellmaterial is, in the reference configuration, deformed by a homogeneous deformation). Thestrain perpendicular to the well calculated with model 2 has the form where a is the lattice constant of the barrier material and b is the lattice constant of the wellmaterial. It has the form is know as the lattice mismatch. It can be shown that any strain tensor element calculated with model 2 or 3 will be proportional to the strainperpendicular to the well hence, the difference between model 2 and model 3 willalways be given by In order to compare the non-linear model (model 1) with models 2 and 3 we have madea series of calculations. The results of those calculations are given in and The B. Lassen et al. / Nonlinear Analysis 63 (2005) e1607 – e1617 Table 1Silicon in Germanium oriented in the [1 1 1] direction The numbers give the strain and internal strain inside the well in the direction of the well (the barrier material Table 2Tin in Germanium oriented in the [1 1 1] direction The numbers give the strain and internal strain inside the well.
lattice mismatch of the Silicon–Germanium system was 4.2% whereas it was −12.8% forthe Tin–Germanium system.
From these results we can draw the following conclusions. First, the error due to the is equal to /2. Second, the error in the internal strain u1 due to the linear approximation of u1 can be higher than the lattice mismatch . Andthird, the error in the strain due to the linear approximation of u1 is not higher than /2 atleast for the systems under investigation here. There is, however, no reason why this shouldbe the case for an arbitrary material system.
4. Conclusion
In this paper, we have presented a non-linear strain theory for low-dimensional semicon- ductor heterostructures that is continuous in nature but derived from the atomistic frame-work and, thus, contains the best of both worlds. Hence, our strain theory includes physicallyplausible interface boundary conditions. It also explicitly includes the quantity known asinternal strain. We have, in addition to this, presented results of strain calculations using thetheory developed on two different quantum well systems: Silicon in Germanium and Tinin Germanium. The obtained results have shown that non-linearities can give results thatdiffer from the linear case close to /2% and that the non-linearities in the internal strainare just as important as the non-linearities in the strain itself.
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