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Outline • Epitaxy in Semiconductor Crystal Growth • Elastic Description of Strain in Cubic Semiconductor Crystals • Atomistic Description of Strain • Molecular Statics and Force Fields • Keating's Valence Force Field • Stillinger-Weber Potential • Tersoff Potential • Simulation of Nanostructures • Piezoelectricity in Zincblende and Wurtzite crystals

Semiconductors InN Group IV: Si, Ge, C Group III-V: GaAs, InAs, AlAs, GaP, InP, AlP, GaN, InN, AlN, GaSb, InSb, AlSb Group II-VI: CdSe, ZnSe, ZnS, CdS, MgSe, ZnTe

Epitaxy Epitaxy: (Greek; epi "above" and taxis "in ordered manner") describes an ordered crystalline growth on a monocrystalline substrate. Homo-epitaxy (same layer and substrate material) Hetero-epitaxy (different layer and substrate material). In the Hetero-epitaxy case growth can be: Lattice Matched: same, or very close, lattice constant of layer and substratee.g. GaSb/InAs or AlAs/GaAs Lattice Mismatched: different lattice constant of layer and substrate material e.g. InP/GaAs or InN/GaN.

Lattice Matched and Mismatched Epitaxy Lattice Matched Lattice Mismatched

Lattice Mismatched Epitaxy In lattice mismatched heteroepitaxy the layer material can be made to “adapt” (can become smaller or larger) its in plane lattice constant to match that of the substrate (pseudomorphic growth). Consequently volume conservation (though volume is not perfectly conserved) dictates that the lattice constant in the growth direction needs to become larger/smaller. In this way the lattice periodicity is maintained in the growth plane, but lost in the growth direction. Lattice Mismatched

Quantum Mechanics in action In0.52Al0.48As 1D Quantum Well In0.84Ga0.16As Scanning Tunneling Microscopy AlAs AlAs In0.52Al0.48As Growth direction Taurino et al Mat Sci and Eng B, 67 (1999) 39 Nanostructures: 2D Growth, 2D Growth + etching, 3D Growth 2D Multi Quantum Wires Green: Free Carrier, Red: Confinement

Elastic Strain Semiconductors are produced by depositing liquid or gasses that when coalesce and solidify follow the crystal structure of the “seed”, usually a substrate of high crystalline quality. During this deposition, often done in very small amounts (low growth rate), as small as depositing one atomic layer at the time, if the layer material has a bulk lattice constant larger than the substrate, then the crystal will appear slightly deformed from its equilibrium state. We refer to this material as “strained”. We chose the axes vectors x,y,z arbitrarily, but need to be linearly independent. Note that while the axes vectors are chosen to be unitary (in units of the lattice constant) in the unstrained case, the strained axes are not necessarily unitary. Unstrained Strained

Unstrained Strained Elastic Strain This picture is general and valid for all types of crystals, not just simple cubic. Unstrained and strained axis can be easily related: The numerical coefficient εij define the deformation of all the atoms in the Crystal. The diagonal terms εii control the length of the axis, while the off diagonal terms εijcontrol the angles between the axis.

Elastic Strain This set of equations are in the form of a mathematical entity called Tensor. The equations define the strained position of any atom within the crystal that upon strain moves from R to R’. Unstrained and strained positions are written in terms of the old and new axis: Important: notice how the coefficients α,β,γ are the same in the unstrained and strained system Unstrained Strained

Elastic Strain We now substitute the new axis with the expressions for the distortion: After a little manipulation and taking into account the expression for R: Provided the original position and the distortion tensor are known, this expression gives a practical way of calculating the position of any atom inside a strained unit cell.

Strain Components Often there is confusion between the terms strain and distortion. In this lectures we follow the notation used in Jasprit Singh’s book, for which the strain components eij are different from the distortion components but related to them by: The final expressions for the off-diagonal terms eij are an approximation in the limit of small strain. Dilation: expresses how much the volume of the unit cell changes, and in the limit of small strain is given by: Biaxial Strain: expresses how much the unit cell is strained in the z direction compared to the x and y: Uniaxial Strain: strain in one direction only, e.g. if eij= constant and eii=0 then the strain is uniaxial in the [111]

xy xy Xx yz xz Stress Components Stress components: the force components (per unit area) that causes the distortion of the unit cell. There are 9 components: Xx, Xy, Xz, Yx, Yy, Yz, Zx, Zy, Zz Capital letters: direction of the force Subscript: direction normal to the plane on which the stress is applied (x is normal to yz, y is normal to xz, z is normal to xy, ) The number of independent components reduces when we consider that in cubic systems (like diamond or zincblende) there is no torque on the system (stress does not produce angular acceleration). Therefore Xy= Yx, Yz= Zy, Zx= Xz And we are only left with 6: Xx, Yy, Zz ; Yz, Zx, Xy

Elastic constants The stress components are connected to the strain components via the small strain elastic constants: In practice we never have to deal with all 36 elastic constants. First of all it is always the case that cij=cji which reduced the total to 21. Second in real crystals, particularly cubic, the lattice symmetry reduces the number even more. Therefore in ZB we only have 3 independent constants: c11,c12,c44 In WZ there are 5: c11,c12,c13, c33, c44

Some more definitions Elastic strain energy density for ZB: Bulk Modulus for ZB: Shear Constant for ZB:

Properties of Semiconductors ZB a B C’ c11 c12 c44 (Ǻ) (Mbar) (Mbar) (Mbar) (Mbar)(Mbar) Si 5.431 0.980 0.502 1.660 0.640 0.796 Ge 5.658 0.713 0.410 1.260 0.440 0.677 C 3.567 0.442 0.478 10.79 1.24 5.78 Ga-As 5.653 0.757 0.364 1.242 0.514 0.634 In-As 6.058 0.617 0.229 0.922 0.465 0.444 Al-As 5.662 0.747 0.288 1.131 0.555 0.547 Ga-P 5.451 0.921 0.440 1.507 0.628 0.763 In-P 5.869 0.736 0.269 1.095 0.556 0.526 Al-P 5.463 0.886 0.329 1.325 0.667 0.627 Ga-N 4.500 2.060 0.825 3.159 1.510 1.976 In-N 4.980 1.476 0.424 2.040 1.190 1.141 Al-N 4.380 2.030 0.698 2.961 1.565 2.004 Ga-Sb 6.096 0.567 0.270 0.927 0.378 0.462 In-Sb 6.479 0.476 0.183 0.720 0.354 0.341 Al-Sb 6.135 0.855 0.414 1.407 0.579 0.399

Strain in Lattice Mismatched Epitaxy Poisson ratio: is a measure of the tendency of materials to stretch in one direction when compressed in another. This ratio depends on the substrate orientation and the type of crystal. For cubic crystals including ZB: Strain: in pseudomorphic growth one can consider, independent of the substrate orientation, strain to have only two components, one parallel to the growth plane and one perpendicular. Important: in [001] growth: e = exx= eyy and e= ezz

[111] z y (1,1,1) x Strain in [111] pseudomorphically grown layers Important: in [111] growth the combination of e and e results in a strain tensor with exx = eyy = ezz and exy = exz = eyz The distortions in this case are: Important: the distortions are expressed in the basis system where x, y and z are aligned with the [100], [010] and [001] directions. Instead e and e are defined so that they relate to strain in the (111) plane and the [111] direction, respectively.

Tetrahedral Bonding In the Zincblende crystal, just like in the diamond one, atoms bond together to form tetrahedrons. Hence the individual atomic orbitals merge to form sp3 hybrid orbitals

Perspective View View from the top Zincblende Wurtzite Wurtzite While Zinblende is the preferred crystal structure of III-As, III-P and III-Sb, III-N tend to crystallize preferentially in hexagonal form. The hexagonal crystal with a two atom basis consisting of cations and anions is called Wurtzite. Two adjacent tetrahedrons overlap in the z direction in WZ but not in ZB. Hence second nearest neighbours in WZ are actually closer than in ZB at equilibrium. The modified inter-atomic forces result in a slight reduction of the interatomic distance between the first nearest neighbours.

+ - + + + The 7th elastic parameter Is a description based on 6 strain components enough to describe all deformations in a ZB or WZ crystals? The distance that the atom is displaced by is characterized by the Kleinman parameter With a the lattice constant and γ the shear strain. This results in a crystal where the atomic bonds are not all of the same length. Only 3 identical sp3 orbitals Strain in the [111]

Strain from atomic positions Given the 5 coordinates of the atoms in a tetrahedron how do we reverse engineer the strain? This become a simple system of linear equations easily solvable. The solution gives the 6 components of the strain tensor. However the deformation on the position of the yellow atom, dependant on the Keinman parameter, is still undetermined and requires a separate calculation.

The issue of local/global composition Furthermore strain is a relative property (variation of e.g. bond length compared to an initial state). Microscopists refer to strain as difference in the bond lengths compared to the host. Theorists think of strain as deformation of a material from its bulk state. Everyone else does not usually know what they are talking about!! If dealing with an alloy and if wanting to take the theorist approach, one needs to know what the lattice constant of the alloy is. But what does composition mean? It makes sense for a large uniform block, not for non uniform. We take the approach of counting atoms up to second nearest neighbour form the centre of the tetrahedron

Modelling Strain in Real Structures Because of its impact on the electronic properties strain in semiconductor nanostructures always needs to be evaluated with the highest possible accuracy. Measurements (usually involving electron microscopy analysis) are not usually sufficiently accurate, so modelling is the only viable alternative. Simple elasticity formulas are acceptable when dealing with standard cases where strains are uniform or approximating strains as uniform is acceptable, e.g. a simple quantum well. They become useless however in complex quantum well structures, wires and dots where strains are non uniform. In time several methods have been developed ranging from continuum, finite element, analytic and atomistic. Atomistic methods are now widely used for quantum dots while continuum methods are the preferred methods for quantum wells.

Molecular Dynamics • Molecular Dynamics is a computer simulation in which a starting set of atoms or molecules is made to interact for a period of time following the laws of Physics (e.g Newton’s Laws). • In Semiconductor science one can build an atomistic model of a strained crystal but if the strain is not known a priori then atoms are not going to be in their equilibrium positions. • Then their motion paths are dictated by the “force field” generated by the potential of the solid.

Potential V (r0i) Forces F=-grad V (r0i) Velocities and acceleration Evaluate the positions after Δt Repeat till Forces are low Molecular Dynamics Initial Position of the atoms r0i Often the simulation does not require very large atomic motion. For instance for calculating strain one might only want to allow small atomic displacements from the crystal structure, without atom switching. When Energy minimisation is the fundamental criterion and forces are used to direct the geometry optimisation rather than predicting the final positions, we are using a “Molecular Statics” simulation.

Rj j jki Ri i Rk k k k Valence Force Field The “force field” that is generated by the potential of the atoms in the solid can be represented as a 3 body potential. In the Keating's Valence Force Field: The Potential is the sum of the potential energy between the pairs of atoms i and j (two body), plus a term that depends on the angle between i,j and a third atom k (three body). dij0 is the unstrained bond length of atoms i and j and 0 is the unstrained bond angle (e.g. for zinc-blend cos0=-1/3), and ijk is the angle between atoms i, j and k. The local chemistry is contained in the parameters  and  , which are fitted to the elastic constants P.N. Keating, Phys. Rev. 145, 637 (1966)

Uniform: same distortion in x,y and z V(R) Non Uniform: z stretch, x,y compress (by the same amount) and viceversa R0 R Binding Energy Valence Force Field The VFF is widely used for all types of nanostructures. VFF is basically a parabolic approximation to the potential of solids Ω is the volume occupied by one atom The main limitation is that there are only 2 parameters ( and  ) but 3 elastic constants even for Zincblende!!!

Distance between 2 ions, one of which is in the central cell Progress in Valence Force Field • Anharmonicity correction: • Ability to reproduce anharmonic effects is linked to the quality of prediction of the phonon spectrum. • Some progress has been presented (e.g. Lazarenkova et al, Superlattices and Microstructures 34, 553 (2003)). • Not clear why phonon frequencies, elastic constants and mode Grüneisen parameters are not correlated (Porter et al J. Appl. Phys. 81, 96 (1997)). • For Ionicity in Zincblende to solve this problem check recent P. Han and G. Bester, Phys. Rev. B 83 174304 (2011) • Ionicityand Wurtzite: • Empirical potentials were historically developed for Si and Ge (pure covalent bonds) • III-V are mainly covalent, partially ionic. II-VI are both covalent and ionic • Only for infinite crystals or systems were the charge is uniformly distributed this it’s not a big deal. • Important in III-N WZ (Grosse and Neugebauer, PRB 63, 085207 (2001)), and can be incorporated following Ewald summation scheme (codes available). • Also check Camacho et al (Physica E, Vol. 42, p. 1361 (2010)) “application of Keating’s valence force field to non–ideal wurtzite materials”

Rj j jki Ri i Rk k k k Stillinger-Weber The “force field” that is generated by the potential of the atoms in the solid can be represented as a 3 body potential. In the Stillinger-Weber potential: The Potential is the sum of the potential energy between the pairs of atoms i and j (two body), plus a term that depends on the angle between i,j and a third atom k (three body). This in an adaptation of the well known Lennard-Jones potential used for liquefied noble gasses. This potential works very well for Si indiamond structure where the bond angle cos0=-1/3. The local chemistry is contained in the parameters A, B , p, q , a, λ and γwhich are fitted to various material properties. F. Stillinger and T. A. Weber, Phys. Rev. B 31, 5262 (1985)

Stillinger-Weber The SW is not as widely used as VFF, but it has his niche (thermodynamics of Si mainly). In a way it should perform much better than VFF as it is not a parabolic approximation to the potential of solids. Parameterisations take into account the crystal phase diagram and check that diamond is the lowest energy structure Works reasonably well for diamond-Si but not for other crystal structures.

Outline • Epitaxy in Semiconductor Crystal Growth • Elastic Description of Strain in Cubic Semiconductor Crystals • Atomistic Description of Strain • Molecular Statics and Force Fields • Keating Valence Force Field • Stillinger-Weber Potential • Tersoff Potential • Simulation of Nanostructures • Piezoelectricity in Zincblende and Wurtzite crystals

Rj j jki Ri i Rk k k k Tersoff Potential The “force field” that is generated by the potential of the atoms in the solid can be represented as a 3 body potential. In the Tersoff potential: The Potential is the sum of the potential energy between the pairs of atoms i and j (two body), multiplied times a term (bij) that depends on the angle between i,j and a third atom k (three body). The expression for bij (known as bond order) is written as to emulate the atomic coordination number Z. Hence ζ is sometimes called the pseudo-coordination. g(θ) and ω describe the angular and radial forces dependence.

j jki i ω k k g k θ θeq Tersoff Potential angular forces: resistance to bend radial forces: resistance to stretch • When fitting to Bulk Modulus g(θ) is always g(θeq)and ωijk==1 • When fitting to Shear Constant g(θ)≠ (θeq)but ωijk==1 • When fitting c44 then both g(θ) ≠ (θeq)and ωijk ≠1 • Hence the Kleinman parameter links angular and radial forces!!!

Tersoff Potential This potential describes covalent bonding and works very well for different crystal structures for group IV and despite the partial ionicity of the bond, group III-V. The local chemistry is contained in the parameters A, B , re, α, β , γ, c, d, h, n and λ, which are fitted to various material properties. J. Tersoff, Phys Rev Lett 56, 632 (1986) & Phys Rev B 39, 5566 (1989) Sayed et al, Nuclear Instruments and Methods in Physics Research 102, 232 (1995)

Tersoff Potential The TP is not as widely used as VFF, but its use is rapidly increasing as parameterizations are improved. Again it should perform much better than VFF as it is not a parabolic approximation to the potential of solids. As there are many parameters, parameterisations can take into account many things, including the crystal phase diagram, all the cohesive and elastic properties and many more. Works rather well for zincblende and diamond group IV and III-V but it is not yet optimized for thermodynamic and vibrational properties. D. Powell, M.A. Migliorato and A.G. Cullis, Phys. Rev. B 75, 115202 (2007)

DFT Tersoff DFT Range of physical shear strains Tersoff Progress in Tersoff • The Kleinman parameter • The many parameters need putting to good use. • Kleinman deformation is critical because expresses the balance between radial and angular forces (Powell et al PRB 75, 115202 (2007)) • Ionicity and Phonons • Ionicity, like VFF, is missing. • Crystal growth only possible if ionic contribution is included (Nakamura et al J. Cryst. Growth 209, 232 (2000) • Phonons are still independent of elastic constants (Powell et al, Physica E 32, 270 (2006)

Beyond Tersoff: bond order potentials • Π versus σ –bonding • Tersoff neglects Π–bonding. Is it of consequence? • Tersoff can to some extent reproduce surface reconstruction energies (Hammerschmidt, PhD thesis) • Beyond σ -bonding • It is generally possible to rewrite the bij with expressions directly obtained from tight binding. (D.G. Pettifor, “Many atom Interactions in Solids”, Springer Proceedings in Physics 48, 1990, pag 64)) • In this way the “bond order” can be explicitly obtained analytically to any order (Murdick et al, PRB 73, 045206 (2006)). • The second moment approximation is essentially equivalent to Tersoff • (Conrad and Scheerschmidt, PRB 58, 4538 (1998))

General Tips for MD • Building Models: • If possible try and use existing software • Try and guess final positions: it saves a lot of computational time • Empirical Potentials: • Codes that use VFF, SW and Tersoff are usually freely available! • IMD (Stuttgart), CPMD (IBM-Zurich) are parallel (for running on clusters) and open source • Nemo3 (Purdue)uses VFF • Always check what version of the potentials are being used!! • Molecular Statics: • Make sure that the parameters that control the length of time the simulation is running for are set to reasonable values • Build your simulation up in size to see what you can get away with in terms of system sizes and check that results do not depend on the size chosen • Strain: • Good strain algorithms exist and are freely available • If you write your own you need a nearest neighbour list. Usually MD produces one • Gridding: • Strain is first obtained onto the atomic grid. Then to use it often it needs converting to an ordered grid. One can use various methods like Gaussian smoothing or weighted average.

Floating Fixed MD of QDs using Tersoff Potential After MD Before MD εxx εyy εzz PBC

Kleinman Parameter The distance that the atom is displaced by is characterized by the Kleinman parameter With a the lattice constant and γ the shear strain. This results in a crystal where the atomic bonds are not all of the same length.

+ + + - + - + + + + Piezoelectricity In the case of a uniaxial distortion the displacement is in the [111] direction, and can still be characterized by the Kleinman parameter Strain in the [111] 4 identical sp3 orbitals Only 3 identical sp3 orbitals The displacement of cations relative to anions in III-V semiconductors results in the creation of electric dipoles in the polar direction which in ZB is the direction that lacks inversion symmetry.

Piezoelectricity in Zincblende The effect can be quantified by writing a general expression for the polarization as a function of the so called “piezoelectric coefficients” and the distortion components. Convention is: xx=1, yy=2, zz=3, yz=4, zx=5, xy=6 In ZB, for symmetry, the only non zero coefficients are e14= e25= e36 In actual fact this picture is incomplete as only includes coefficients linked to linear terms in the strain. In the past 6 years the importance of including also coeffiecients linked to quadratic terms in the strain (e.g. 𝛆xx2 or 𝛆xy𝛆xz)has been highlighted (so called non linear or second order Piezo effect). • M.A. Miglioratoet al, Phys. Rev. B 74, 245332 (2006) • L. C. Lew Yan Voon and M. Willatzen, J. Appl. Phys. 109, 031101 (2011) REVIEW • A. Beya-Wakataet al, Phys. Rev. B 84, 195207 (2011)

Zincblende Wurtzite Piezoelectricity in Wurtzite Spontaneous polarization Strain induced polarization Quadratic terms in the strain (e.g. 𝛆xx2 or 𝛆xy𝛆xz)are also important. There is still some controversy between the early accepted values of mainly for the spontaneous polarization coefficients • L. C. Lew Yan Voon and M. Willatzen, J. Appl. Phys. 109, 031101 (2011) REVIEW • J. Pal et al, Phys. Rev. B 84, 085211 (2011)

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