Strain in crystalline core-shell nanowires

The strain configuration induced by the lattice mismatch in a core-shell nanowire is calculated analytically, taking into account the crystal anisotropy and the difference in stiffness constants of the two materials. The method is applied to nanowires with the wurtzite structure or the zinc-blende structure with the hexagonal / trigonal axis along the nanowire, and the results are compared to available numerical calculations and experimental data. It is also applied to multishell nanowires, and to core-shell nanowires grown along the $<001>$ axis of cubic semiconductors.

(Dated: 18 March 2014) The strain configuration induced by the lattice mismatch in a core-shell nanowire is calculated analytically, taking into account the crystal anisotropy and the difference in stiffness constants of the two materials. The method is applied to nanowires with the wurtzite structure or the zinc-blende structure with the hexagonal / trigonal axis along the nanowire, and the results are compared to available numerical calculations and experimental data. It is also applied to multishell nanowires, and to core-shell nanowires grown along the < 001 > axis of cubic semiconductors.

I. INTRODUCTION.
Semiconductor nanowires (NWs) are often grown in the form of core-shell structures, in order to achieve better photonic and electronic properties: the active core is isolated from the surface defects and traps in order to obtain a better luminescence efficiency, sharper linewidths, longer coherence times and higher mobility, or even a better chemical stability. As the lattice parameter of the shell is generally different from that of the core, and since coherent structures are contemplated with no misfit defects at the interface, the elastic strain induced in the core and its effect on the electronic properties have to be taken into account. In turn, the built-in strain can be used as a further adjustable parameter: strain engineering can be used to lower the degeneracy in the valence band and select the type of holes with a larger spin for spintronics applications (for instance a larger spin-carrier coupling in diluted magnetic semiconductors), 1 or a smaller longitudinal mass to achieve a better mobility in transport properties. 2 The strain can also be designed to induce a built-in piezoelectric field, resulting in a faster separation of the electron-hole pairs in photovoltaic applications. 3 Finally, strain is an important parameter when engineering Si-Ge NWs to obtain direct bandgap configurations and efficient emission of light. 4 Analytical expressions exist for a core-shell structure made of elastically isotropic materials. 5 However, the crystal structure results in anisotropic elastic properties, the core and shell materials have different values of the stiffness constants, and the NW shape can deviate from the ideal cylinder with a circular base and for instance feature facetting. As a result, calculating the strain configuration in a real semiconductor core-shell NW is not an easy task: quantitative descriptions usually imply to compute numerically the local strain, either using a microscopic model such as the valence force field model, or performing a finite element treatment of the continuum elasticity theory. 6 A quantitative, fully analytical solution taking into account the crystal structure exists, and this is the purpose a) joel.cibert@neel.cnrs.fr of the present study. Starting with the well known expression for isotropic materials (and their extension for the transversely isotropic materials), we propose solutions for the most often encountered cases of zinc-blende and wurtzite semiconductors. We give analytical expressions for the strain in the core and in the shell, and for their effect on the extrema of bands, and we compare these predictions to the results of microscopic calculations and experimental data.
In most cases, deviations from the cylindrical strain configuration are found. In two typical cases (with zincblende or diamond semiconductor NWs along < 111 > and along < 001 >), we identify the resulting strain configuration to first-order in the parameter describing the cubic anisotropy and we show that these deviations from cylindrical symmetry are rather small. These two cases illustrate two non-isotropic strain configurations: warping along the NW axis, and anisotropy of the in-plane strain. Other orientations and crystal structures are expected to feature a combination of these two configurations.
To sum up our results: (i) the cylindrical approximation is surprisingly good, provided the appropriate truncation of the stiffness tensor is used; (ii) the result is exact for wurtzite NWs grown along the hexagonal axis, and a first-order treatment of the anisotropy quantitatively agrees with available numerical results for zincblende NWs; (iii) the present method is readily extended to other structures or orientations, and multishell NWs.
The paper is organized as follows: section II is a short summary of the general formalism and of results which are well-known for isotropic materials. In section III, we obtain analytical expressions of the strain configuration in wurtzite semiconductor NWs grown along the hexagonal axis; the transfer matrix approach allows us to consider both core-shell and multishell NWs. In sections IV and V, we use a perturbation method to describe the more complex strain configuration present in zinc-blende semiconductor NWs grown along the trigonal axis and along the cubic axis.
A. The core-shell system.
In this section we recall the well-known strain configuration in an infinitely long, cylinder-shaped, core-shell NW made of isotropic materials, in order to identify and illustrate the effects of the two elements of symmetry on the displacement field and the Lamé-Clapeyron-Navier equation. We consider a cylinder-shaped core (superscript or subscript c), infinitely long, with a circular cross section of radius r c , embedded in a shell (superscript or subscript s) of radius r s . We note z the NW axis, (r, θ) or (x, y) the in-plane coordinates measured from the NW axis. The two materials have the same crystal structure and the same orientation, with different values of the lattice constants a s and a c . The growth is assumed to be coherent, with no misfit defect at the interface, so that the lattice mismatch f = as−ac ac is fully accommodated by elastic strain.
The general solution involves calculating the displacement field u(r) which relates the position of any point r in the strained material to its value in the mismatched, unstrained system. The local deformation, in the vicinity of a point r, is fully described by the tensor of the derivatives of u(r), ∂ui ∂xj : the symmetric part is the strain tensor, ε ij = 1 2 ( ∂ui ∂xj + ∂uj ∂xi ), associated to elastic energy, while the antisymmetric part 1 2 ( ∂ui ∂xj − ∂uj ∂xi ) describes a local rotation. In the presence of body forces per unit volume F(r), the equilibrium condition, j ∂σij ∂xj + F i = 0, can be expressed as the Lamé -Clapeyron -Navier equation (there is one equation for each value of i and x i = x, y, z), In this equation, the c ijkl are the components of the stiffness tensor, which relates the stress tensor σ ij to the strain tensor ε kl through the Hooke's law, σ ij = kl c ijkl ε kl . The number of independent components c ijkl is determined by the symmetry properties of the material. 7 In a core-shell NW, we apply the Lamé -Clapeyron -Navier equation within each constituent; the body forces are zero, but we have to apply proper boundary conditions 8 at the surface and at the interface. A first series of conditions ensure the stability of the interface/surface: stress components applied to the surface (σ rr , σ rθ and σ rz ) vanish, and they are equal on both sides of the interface. Additional conditions state the continuity of the lattice: the displacement field u(r) must compensate for the lattice mismatch f . All these conditions are actually the same as for a thin epitaxial layer, but then the condition on the continuity of the lattice can be expressed on the in-plane strain components. 9 In addition, for an infinitely long NW, the overall translational invariance along the axis must be main-tained (and it is known also that in a NW of finite length, according to the Saint-Venant principle, this holds everywhere but for a segment of length equal to about the diameter at each end). Translational invariance means that the relative displacement of two neighboring points is independent of z, i.e., that all derivatives of u(r) are independent of z: ∂ ∂z ∂ui ∂xj = 0, hence ∂ui ∂z is a constant C i independent of r, ∂ui ∂z = C i , and u i (r) = C i z + D + u i (x, y). Note that the C x z, C y z and u z (x, y) contributions correspond to shear strains (ε xz , ε yz ) and are often excluded by symmetry. Finally, the equilibrium with respect to a translation along the NW axis requires that the longitudinal stress integrated over the NW section be zero.
Once determined the displacement field u(r) obeying the Lamé -Clapeyron -Navier equation and the boundary conditions, the strain tensor can be introduced into the so-called deformation potentials 10 and the possible piezoelectric field is calculated; the positions of the conduction and valence band edges follow.

B. Elastically isotropic material.
The solution for an infinitely long, circular core-shell structure made of elastically isotropic materials, is well known. 5 We briefly recall the main results, our goal being to examine what will remain valid if materials with a lower symmetry are involved.
The Lamé -Clapeyron -Navier equation writes which contains three equations, for x i = x, y and z, respectively. A more compact form better evidences the spherical symmetry: Here the stiffness tensor has only two independent components: the so-called Lamé coefficients µ = c ijij = c ijji and λ = c iijj for i = j, with c iiii = λ + 2µ. All other components vanish. If we omit the terms which vanish due to the invariance by translation or would correspond to axial shear strains (according to the discussion in the previous section), the Lamé -Clapeyron -Navier equation restricts to: As the strained system obviously retains the cylinder symmetry, we write the displacement field in cylindrical coordinates, keeping only the relevant variables: u r (r), , the inhomogeneous shear strain in the shell (in green), and the rest of the strain -uniform and isotropic in the plane (in blue). The lattice parameter is assumed to be smaller in the shell than in the core (f < 0). u θ = 0, with u x = u r (r) cos θ and u y = u r (r) sin θ. An in-plane dependence of u z would imply shear strain components ε rz which are excluded, hence u z (z) = Cz + D. Finally the Lamé -Clapeyron -Navier equation is reduced to equating to zero the Laplacian of the in-plane displacement, hence d 2 ur dr 2 + 1 r dur dr − 1 r 2 u r = 0, and u r (r) = Ar + B r 2 c r , with parameters A, B, C and D to be determined in each material. The non-vanishing components of the strain tensor are thus the longitudinal expansion ε zz = duz dz = C, the radial expansion ε rr = dur dr = A−B r 2 c r 2 , and the angular expansion ε θθ = ur r = A+B r 2 c r 2 . Note that B vanishes in the core (to avoid diverging terms at the axis, r = 0); also, D represents a global displacement of the core or the shell, hence D = 0. As a result, see Fig. 1, the strain (and the stress) are uniform in the core; in the shell, there is also a uniform component, and a non uniform shear component, rotating around the interface and close to it. Note that the stress component σ c zz is uniform also in the shell (the non-uniform B r 2 c r 2 terms in ε s θθ and ε s rr cancel each other when applying the Hooke's law).
The two parameters A c and C c in the core, and the three parameters A s , B s and C s in the shell, are determined from the boundary conditions. At the interface, the matching along z (written on u z or ε zz ) implies C c − C s = f , and the matching in the plane is realized simultaneously on u r and ε θθ if A c − A s − B s = f ⊥ . We identify the mismatch f in the direction of the NW axis, and the mismatch f ⊥ in the plane perpendicular to the axis: although this is not done usually -and not needed for isotropic materials -that will allow a better understanding of the result. The stress components are such that σ c rr (r c )−σ s rr (r c ) = 0 at the interface and σ s rr (r s ) = 0 at the sidewall. The other components (σ rθ and σ rz ) automatically vanish. The longitudinal stress integrated over the NW section vanishes: as both σ s zz and σ c zz are uniform, the condition is simply ησ c zz + (1 − η)σ s zz = 0 where η is the ratio of the core to NW cross-section areas (for a NW with circular cross-section, η = r 2 c /r 2 s ). A straightforward calculation then gives the complete set of strain components where The longitudinal strain ε zz (red arrows in Fig. 1) results from the lattice mismatch in the direction of the NW axis, which is shared between the core and the shell with a weight inversely proportional to their area (in a way similar to the strain distribution in a free-standing superlattice, where the lattice mismatch is shared with a weight inversely proportional to the thickness of each layer). A narrow core is fully strained to the thick shell (and a thin shell to a wide core). The main part of the in-plane lattice mismatch is accommodated by the shear strain in the shell rotating around the interface (green arrows in Fig. 1). The rest of the in-plane strain consists in a uniform in-plane strain in the core and a uniform component in the shell (blue arrows in Fig. 1): these components result from the competition between a direct effect of the lattice mismatch in the plane, and an indirect effect of the longitudinal strain. As a result, they can be quite small.
For a thin shell, η = 1, the core is unstrained, and the shell strain writes ε s zz = ε s θθ = −f , ε s rr = 2λ λ+2µ f , which is the result for a thin epitaxial layer on a plane substrate.
The previous result can be extended 11 to the case of two isotropic materials with different values of the shear modulus, but the same value of the Poisson ratio. In terms of Lamé coefficients, that means λs λc = µs µc . Complete expressions of the stress tensor are given in Ref. 11. We will generalize these expressions in the following section taking into account the crystal structure.
C. The effect on the electronic properties.
Two mechanisms affecting the electronic properties of a core-shell NW are determined by the strain configuration. (i) There is a direct effect of strain on the bands of a semiconductor; around the band edges, it is described phenomenologically by the so-called deformation potentials. For instance, in a zinc-blende semiconductor, the isotropic strain (change of volume), (ε xx + ε yy + ε zz ), induces a shift of the conduction band and an average shift of the valence band at the center of the Brillouin zone. A shear strain, such as (2ε zz − ε xx − ε yy ) induces a splitting of the valence band edge. (ii) When NWs are grown along a polar axis, they are expected to present a polarization due to the piezoelectric effect. This is the case of NWs with the wurtzite structure grown along the caxis, as well as NWs with the zinc-blende structure grown along the < 111 > axis. The relevant strain components entering the longitudinal polarization are 3 ε zz and (ε rr + ε θθ ) in the first case, and (2ε zz − ε xx − ε yy ) in the second case. In addition, confinement effects should be taken into account if the NW radius is small enough, and the confining potential is modified by these two mechanisms.
It is interesting to compare the results for a thin core and that for a thin epitaxial layer, both considered as the active medium of the structure. In both cases there is an isotropic strain and a shear strain. The isotropic strain is (ε xx +ε yy +ε zz ) or (ε rr +ε θθ +ε zz ) = 4µ λ+2µ f in both cases. The shear strain is (2ε zz − ε xx − ε yy ) = −2f 3λ+2µ λ+2µ in the thin epitaxial layer, and (2ε zz −ε rr −ε θθ ) = f 3λ+2µ λ+2µ . The same result holds for a thick core, with f replaced by ηf . Hence the ratio of the valence band splitting to the shift is (1) of opposite sign, and (2) twice smaller, in the core of a NW than in an epitaxial layer made of elastically isotropic materials. This property will be checked below in the presence of crystalline anisotropy: we will show that the factor is not exactly 2.

D. Actual semiconductor nanowires.
Our goal is to take into account the crystal structure of the semiconductors, by using the stiffness tensor with the appropriate symmetry. We will consider explicitly three cases: hexagonal (wurtzite) structure with the NW axis along the c-axis, and cubic (zinc-blende or diamond) structure with the NW axis along < 001 > or < 111 >. We ignore facetting and consider a NW with a circular cylinder shape. We will show that • In the case of a wurtzite NW grown along the sixfold axis, the transversely isotropic solution is exact.
• In the case of a zinc-blende NW grown along a trigonal axis, the transversely isotropic solution is an excellent approximation, which reproduces quantitatively the results of numerical approaches. Deviations due to the cubic anisotropy appear in the form of a warping along the axis, of three-fold symmetry, and can be found as the response of an elastically isotropic system to a distribution of body forces parallel to the NW axis.
• In the case of a zinc-blende NW grown along a tetragonal axis, a transversely isotropic approximation is proposed. Deviations with four-fold symmetry are found and calculated as the response to a distribution of body forces perpendicular to the NW axis.
The stiffness tensor for the zinc-blende structure reflects the cubic symmetry. 7 It contains three independent terms and the Voigt notation in the cubic axes is: The anisotropy is characterized by the parameter c = (c 11 −c 12 −2c 44 ). If c = 0, the energy of a tetragonal shear strain (characterized by c 11 − c 12 ) equals that of a trigonal shear strain (characterized by 2c 44 ) and the spherical symmetry is restored. Usual semiconductors have c < 0: they are harder against a trigonal stress, which directly involves a change of bond length, than against a tetragonal stress which is accommodated mainly by bond rotation. As a result, they are harder along a < 111 > direction and softer along a < 001 > direction, with < 110 > in between. 12 In the wurtzite structure, with z along the c-axis and x, y in the perpendicular plane, symmetry considerations imply identities such as c 22 = c 11 or c 66 = (c 11 − c 12 )/2, so that the stiffness tensor has five independent components: 7 It is invariant under any rotation around the c-axis.

III. HEXAGONAL SEMICONDUCTORS ALONG THE C-AXIS.
We consider a NW with the wurtzite structure, and its axis parallel to the c axis. We take the z axis along this axis, and x and y two arbitrary axes in the basal plane. Note that the lattice mismatch along the c-axis, f , and perpendicular to it, f ⊥ , may be different.

A. Calculation.
The complete Lamé -Clapeyron -Navier equation (Eq. 1) is written in Appendix A (Eq. A1). Omitting terms which vanish due to invariance by translation, we obtain: It reproduces exactly the Lamé -Clapeyron -Navier equation of an elastically isotropic material, Eq. 4. Hence, as the boundary conditions are invariant under a rotation around the NW axis (this is due to the invariance of the stiffness tensor noted above), the general solution for Eq. 9 is the same as that of Eq. 4, u z (z) = Cz and u r (r) = Ar + B r 2 c r . Furthermore, as this solution is such that the terms of Eq. A1 omitted in Eq. 9 all vanish, it is the exact solution of the complete equation, Eq. A1.
Applying the Hooke's law to the boundary conditions of a core-shell NW as in the previous section (at the interface, step in uz z = ε zz and in ur r = ε θθ to accommodate the lattice mismatch f and f ⊥ with no misfit dislocation, and equilibrium of σ rr ; at the sidewall, σ s rr (r s ) = 0; along the z-axis, ησ c zz + (1 − η)σ s zz = 0), we obtain the strain tensor by inverting a system of linear equations: A direct numerical calculation is possible, however it is interesting to write the boundary conditions using a transfer matrix method, which can be generalized to multishell NWs: 8 then, we have to solve a system of two linear equations, instead of 5 for a core-shell NW and (3n + 2) for a NW with (n − 1) shells.

Transfer matrix.
We thus consider a multishell NW made of a core of radius r 0 , and several layers of radius r i and lattice mismatch f i and f ⊥i with respect to the core material, with a uniform stiffness tensor over the whole NW. The radius of the last layer, i = s, is the NW radius. The relative cross section area of each layer is η i = Within each material, we define a matrix M(ρ) relating the relevant components of displacement and stress to the A, B, C parameters, with ρ = r r0 : In the general case, the values of the stiffness constants are specific to the material which makes the layer i, and accordingly there is a matrix M i appropriate to each material.
The boundary condition at the interface between layers i and (i + 1), at ρ = ρ i , is Eq. 12 establishes a relation of recurrence from the parameters on the inner side of the interface, to those on the outer side. Repeating Eq. 12 from shell to shell, we obtain the set of parameters (C i , A i , B i ) as a function of those of the core, (C 0 , A 0 , with B 0 =0). The two core parameters are finally determined by the two boundary conditions on the stress at the sidewall and along z.
The first boundary conditions, σ rr = 0 at the sidewall, is written using the projection t P r = 0 0 1 , The last condition, on σ zz integrated over the NW cross section, is with t P iz = c 33 2c 13 0 written with the values of stiffness constants appropriate to the material in layer i. Combining Eq. 12 to 14 we obtain a set of two linear equations for A 0 and C 0 . With this assumption, the continuity of σ rr at the interface at r i (last line of Eq. 11, multiplied by ρ 2 i ) is Adding these equations for all interfaces, including the surface for which the right-hand member is zero, and The second condition is that the integral of σ zz over the NW cross-section vanishes: Hence the two sums must vanish independently Another simple result is obtained for the strain along the axis. The first line of Eq. 11 or 12, The recurrence on the in-plane strain is not as simple. Indeed the transfer matrix in Eq. 12 is It is worth however to write the result for the simple core-shell NW. Simplifying the notation, with η = η 0 , where 3. Uniform stiffness tensor.
If all materials have the same values of stiffness constants (all χ i = 1), all M i matrices are identical. The recurrence relation (Eq. 12) is simply so that and finally, using Eq. 16 The strain configuration is thus: It can be applied to a multishell structure such as in Ref. 13.
In the case of the simple core-shell NW, we recover the usual expressions, Eq. 5: where A comparison between the expressions for ε zz in Eq. 18-19 and Eq. 25-26 illustrates the effect of a different hardness of the two materials: in the sharing of lattice mismatch, the weight is defined by the area ratio multiplied by the hardness ratio. In particular, for a thin layer (η ≈ 1), the strain in the core is multiplied by χ, while for a thick layer (η ≪ 1), the strain in the shell is divided by χ.
Note also that Eq. 24 can be used to describe a continuous distribution in a NW, just by replacing the discrete sums by integrals: An interesting consequence is that the strain in the core of a core-shell NW (for instance, GaAs-Ga 1−x Al x As) or in a multishell NW, is determined by the composition integrated over the shell(s), and not by the exact distribution within the shell(s).

Summary and electronic properties.
To sum up, the strain configuration in a core-shell NW grown along the hexagonal direction of wurtzite crystal is transversely isotropic. It is given by Eq. 25-26 if the two materials have the same hardness, and Eq. 18-19 for a hardness ratio χ = 1. An explicit expression, Eq. 24, also exists for a multishell NW if the stiffness constants are identical over the NW.
The potential configuration for the bottom of the conduction band and the top of the valence band near the center of the Brillouin zone is obtained from these expressions using the Bir-Pikus phenomenological coupling. 10 In the core, the non-vanishing strain components are ε zz and 1 2 (ε rr + ε θθ ) so that the Bir-Pikus Hamiltonian 10 has only diagonal elements in the usual basis quantized along the c-axis. Note however that the resulting matrix elements may be of the same order as the other terms describing the top of the valence band and the excitons (spin-orbit coupling, crystal field splitting and exchange terms). In the shell (s), the in-plane shear strain to mix the valence band states initially quantized along the c-axis. The piezoelectric effect is described by an axial polarization, determined by the two strain components ε zz and 1 2 (ε rr + ε θθ ). There is no coupling to the in-plane shear strain.
The present study also confirms that GaN-InN multiquantum-well NWs 13 should indeed feature no built-in piezoelectric field perpendicular to the QWs, but an inplane shear-strain different from well to well.
B. Application to real systems.

GaN-AlN nanowires.
The strain in the core of single GaN-AlN core-shell NWs grown by plasma-assisted molecular beam epitaxy was measured by resonant x-ray diffraction, Raman spectroscopy and high resolution transmission electron microscopy: 14 for unrelaxed NWs, it favorably compares to the results of a microscopic calculation using the valence-force-field model, and to a macroscopic calculation assuming uniform strain along the c-axis and vanishing strain in the plane.
The stiffness constants of GaN and AlN 15-17 are quite similar, hence we take χ = 1. The lattice mismatch is slightly anisotropic, f ⊥ = −2.5% and f = −4.0%. The present calculation predicts a uniform strain in the core, ε c zz = f ⊥ (1 − η) along the NW axis and Note that the small value of the in-plane strain is due to a compensation between the Poisson effect of the longitudinal mismatch f and the direct effect of the in-plane mismatch f ⊥ , see f ⊥ + B s in Eq. 26.

ZnO nanowires.
ZnO is such that c 11 − c 12 < c 13 : 18 for an isotropic lattice mismatch, the Poisson effect prevails in the in-plane strain. ZnO cores are often associated to a strongly mismatched shell and in this case the structure is no more coherent. A moderate mismatch exists in ZnO-(Zn,Mg)O. According to a synchrotron x-ray study of polycrystalline wurtzite (Zn,Mg)O, 19 it is strongly anisotropic, with f ⊥ and f opposite in sign: this is attributed to a change in the ionicity. As a result (Fig. 3), the core experiences a significant shear strain with a ten times smaller volume change. In other words, the c/a ratio, which represents the deviation from "ideal" wurtzite, is changed at almost constant volume. Note that a non linear character of the piezoelectric effect has been measured in CdTe 20 and predicted for other semiconductors as well. 21 As it is attributed to a dependence of the piezoelectric coefficient on the hydrostatic strain, this non-linear character should not show up in a ZnO-(Zn,Mg)O NW.

IV. CUBIC SEMICONDUCTORS ALONG THE < 111 > AXIS.
We now consider NWs of semiconductors with the zincblende or diamond structure, grown along a trigonal axis. The (111) plane is known to be isotropic with respect to some mechanical properties, so that the cylindrical approximation is quite natural for such NWs. We use it first, and compare its results to data known for real systems. However the shear strain present in the shell gives rise to warping, with a 3-fold symmetry, which is calculated analytically in section IV C 2.

Calculation.
If the parameter c is not zero, the stiffness tensor must be calculated in the relevant axes. It can be done on the c ijkl tensor, or directly in the Voigt notation using the rotation rules described in Ref. 12 The six components are not independent since they can be expressed using the three coefficients c 11 , c 12 and c 44 relevant for the cubic symmetry: 6 The stiffness tensor reflects the threefold symmetry of the trigonal axis: it is quite similar to that of the wurtzite structure along the c-axis. However there is a set of additional terms,c 14 . To better understand these terms, we can write the stiffness matrix in the e r , e θ , e z axes, rotated with respect to the previous one by an angle θ around the < 111 > (or e z ) axis: The trigonal symmetry of thec 14 terms is clear, as noted in Ref 23. Note that these contributions average to zero over a complete 2π-turn. Moreover, they are quite small: for instance in GaAs,c 14 c11+2c12 = −0.05. The complete Lamé -Clapeyron -Navier equation in the x, y, z basis, Eq. 1, is written in Appendix A. Omitting terms excluded by the invariance by translation, we obtain: The effect of thec 14 terms will be described in Section IV C 2. Ignoring these terms for a while, the equation is the same as in the wurtzite case. Then the solution is obtained by replacing the c ij in Eq. 18-19 by thec ij and their expression (Eq. 29). The result is identical to Eq. 18, but with f = f ⊥ = f , and If the stiffness constants are identical in the two materials, we recover the same expression as above (Eq. 5), with : B s = −f c 11 + 2c 12 c 11 + c 12 + 2c 44 (f + B s ) = f −c 12 + 2c 44 c 11 + c 12 + 2c 44 (33) In the core, the strain corresponds to a uniform hydrostatic strain ε hydro = ε zz +ε rr +ε θθ and a uniform trigonal shear strain ε shear = 2ε zz − ε rr − ε θθ . It should be kept in mind that the axis used in these expression are x = [110], y = [112], z = [111]; in the cubic axes, x ′ , y ′ , z ′ , the previous results means, for the core, ε

Excitons.
Finally, we consider the exciton energy in the core of a core-shell NW, in the absence of confinement effects. In a strained semiconductor, it is expected at E X = E 0 X − (a ′ + a) ε hydro ± 1 2 b ε shear (for a tetragonal shear strain) or E X = E 0 X − (a ′ + a) ε hydro ± 1 2 d √ 3 ε shear (for a trigonal shear strain). The coefficient a ′ describes the coupling of conduction electrons to strain, and a, b, d describe the coupling of holes (Bir-Pikus Hamiltonian 10 ). Using Eq. 33 we obtain The sign + is for the exciton formed with the light hole (moment ± 1 2 along the NW axis), the sign -for the heavy hole (± 3 2 ) exciton. A more complete analysis is given at the end of section IV C 2.
B. Application to real systems.

GaAs-based nanowires.
The calculation for a GaAs-Ga 0.65 Al 0.35 As NW with hexagonal cross section, using the valence force field model, Fig. 3c of Ref. 24, fully agrees ( Fig. 4) with the present value ε c zz = (1 − η)f . The in-plane strain is "four times smaller", 24 which also agrees with the ratio 0.22 obtained in the present calculation using the stiffness constants of (Ga,Al)As, 25 with χ = 1.
In GaAs-GaP NWs, the lattice mismatch is 3.6%, and the stiffness constants differ by a factor χ ≈ 1.10 to 1.17. 26 NWs with either a circular or a hexagonal cross sections have been modeled by Grönqvist et al. 6 using both the valence force field model and a finite element treatment of the continuum elasticity theory. Other coreshell configurations with hexagonal cross-sections are described in Ref. 27. In the case of a cylindrical crosssection, it confirms the present result that the axial strain is uniform in the core and in the shell, and that the inplane strain is also uniform in the core. Using the appropriate values of the area ratio, and the stiffness constant values of GaP 26 with an average ratio χ = 1.14 for GaAs-GaP), we calculate the solid lines shown in Fig. 4, in good agreement with numerical calculations. Note the small but visible bowing which is due to the different values of the stiffness constants in GaAs and GaP.

InAs-based nanowires.
Similar results are obtained in InAs-InP NWs. They favorably compare (Fig. 6) with the results of numerical calculations. 22 We will come back to this system in the section on warping (IV C 2).
In GaAs-InAs NWs, the approximation of a constant Poisson ratio is not reasonable and a direct inversion of the full matrix (Eq. 10 where the c ij have been replaced by thec ij , Eq. 29), or the equivalent transfer matrix method, should be used.

ZnTe nanowires.
Photoluminescence and cathodoluminescence have been measured on ZnTe-(Zn,Mg)Te core-shell NWs, 1 with a peak at 2.31eV, i.e., a 60 meV redshift with respect to the exciton in bulk ZnTe; this is a large shift, larger than usually observed in strained 2D layers. In bare ZnTe NWs, 28  In this section we discuss the two simplifying assumptions which allow us to derive the previous analytical expressions: (1) NWs have a circular cross section, and (2) in the NWs with the zinc-blende structure, the deviation from cylindrical symmetry is small.

Facets.
Most of the numerical calculations consider NWs with an hexagonal cross-sections, and actual NWs exhibit more or less well-defined facets. The present calculation does not reproduce the inhomogeneity of the in-plane strain which is calculated for a hexagonal NW, but it was already noted 33   The cylindrical symmetry is exact in the case of NWs with the wurtzite structure, with the c-axis along the NW. It is not for NWs with the zinc-blende structure. As a result, the shell is warped, as evidenced in the numerical treatment of Ref. 6. We now describe the analytical calculation of this additional contribution.
Indeed, when calculating the stress corresponding to the cylindrical strain configuration, additional components appear through thec 14 terms in the stiffness tensor: for instance, at the interface in the y-direction (x = 0, y = r c ), a stress component normal to the interface and surface, σ yz =c 14 (ε xx − ε yy ) + 2c 44 ε zy , takes a finite value if we use the strain of Eq. 18 and 32, or 25 and 33. We thus expect an additional strain to appear, ε zy = −c 14 2c44 (ε xx − ε yy ) where (ε xx − ε yy ) is taken from Eq. 18 and 32 or 25 and 33: it vanishes in the core (where B c = 0), but not in the shell where a nonuniform shear strain (ε s θθ − ε s rr ) exists. With c < 0, and f < 0 (case of GaAs-GaP, InAs-GaAs, CdTe-ZnTe core- 8. (a) Warping body forces in the shell of a core-shell NW with f < 0, grown along < 111 >, due to the trigonal symmetry of the zinc-blende structure. The shell is pushed upward and downward according to the red arrows. The tetrahedron of the atomic structure is shown in green. = −0.82%. The radial dependence of the displacement along the axis (top), the radial-axial shear strain (middle) and the tangential-axial shear strain (bottom) are shown for a NW with finite shell radius (left column)and infinite shell radius (right column). The trigonal symmetry appears through the sin 3θ or cos 3θ factor, as indicated.
shell NWs, not ZnTe-(Zn,Mg)Te), we expect a positive ε zy , i.e., the shell is pushed upward, towards [111]. Note that other non-vanishing stress components are obtained by re-introducing these warping terms into the calculation of the stress, so that they are of second order iñ c 14 .
Using the rotated stifffness tensor, Eq. 30, and forcing the stress component σ rz to be zero (and neglecting a contribution of second order inc 14 ), we obtain ε zr =c 14 2c44 (ε θθ −ε rr ) sin 3θ: the shell is alternately pushed upward and downward, with the expected trigonal symmetry (Fig. 8).
To calculate the complete strain distribution, we must re-calculate the displacement field thanks to the Lamé -Clapeyron -Navier equation.
When introducing the cylindrical solution u r (r) = Ar + B r 2 c r , u θ = 0, u z (z)) into the complete equation, Eq. 31, thec 14 terms vanish everywhere but in the third equation for the shell. There, 2 ∂ 2 ux ∂x∂y + In a treatment to first order inc 14 (i.e, in the cubic anisotropy c), we have to find an additional displacement δu which is solution of the Lamé -Clapeyron -Navier equation for the transversely isotropic NW, with no lattice mismatch (they are already compensated) but with body forces F x = 0, F y = 0, F z =c 14 8B s r 2 c sin 3θ r 3 in the shell.
Thus, δu is the response of an isotropic system to an axial shear strain 8 of trigonal symmetry (∼ sin 3θ).
The solution is δu x = 0, δu y = 0 and δu z such that in the core and in the shell, respectively. The result is of trigonal symmetry and can be written, respectively (see Appendix B for details): where we have used α c −3 = 0 (no diverging term), and the three parameters α c 3 , α s 3 , α s −3 are determined by the boundary conditions: the non-trivial boundary conditions are that δu z and σ rz =c 14 sin 3θ(ε rr − ε θθ ) + c 44 1 2 ∂ ∂r δu z are continuous at the interface, and σ rz van-the area ratio of Ref. 22, and for a thick shell. Maps are shown in Fig. 10. Note the discontinuity of ε rz at the interface, and its fast decay while ε θz progressively increases from zero and stays finite far into the shell. There is a complete agreement with the results of numerical calculations in Ref. 22.
Apart from the presence of this additional shear strain, the other strain components are modified by terms of the order of (c 14 c11 ) 2 . Taking again GaAs parameters, we find that these second order terms are of the order of 1%×f . As a result, the change of the core strain induced by thẽ c 14 terms is negligible. Note also that the contribution of the additional shear strain to σ zz vanishes due to the cos 3θ and sin 3θ factors.

D. Summary and electronic properties.
To sum up, the strain configuration in a zinc-blende NW grown along the < 111 > axis is described by a cylindrical strain, Eq. 18 and 32 (or 25 and 33 if χ = 1), complemented by an axial shear strain ("warping"), Eq. 37.
The Bir-Pikus Hamiltonian describing the coupling of holes to strain has the same symmetry as the Luttinger Hamiltonian. When expressed in the present trigonal basis (hole states | with P = −a(ε xx + ε yy + ε zz ) In the core, apart from a small axial shear strain which takes non-vanishing values close to the interface, but remains small, the strain comprises the hydrostatic strain (ε xx +ε yy +ε zz ) and the trigonal strain (ε xx +ε yy −2ε zz ). The Bir-Pikus Hamiltonian is diagonal in the trigonal basis, with a splitting equal to 2Q; the Luttinger Hamiltonian gives the effective masses of the eigenstates: the mass along the NW axis (determining the density of states and transport properties) is m * = m0 γ1−2γ3 for the |± 3 2 holes, and m * = m0 γ1+2γ3 for the |± 1 2 holes, the mass in the plane (governing confinement) being m * = m0 γ1±γ3 . This was used in section IV A 2.
In the shell, close to the interface, the dominant contribution is the shear strain which adds non-diagonal matrix elements (R and S) to the Bir-Pikus Hamiltonian which mixes the previous states. If we consider the whole NW, the axial symmetry is preserved, so that the eigenstates in the core retain their symmetry, with some mixing expected to take place in narrow NWs. In the shell, and particularly in wide NWs, the complex structure may contribute to localization.
This deformation potential landscape is complemented by the piezoelectric effect. 22 Again, the polarization in the core is along the axis, determined by − e14 √ 3 (ε xx +ε yy − 2ε zz ) where e 14 is the unique coefficient of the piezoelectric tensor (the indices refer to the cubic axes). A complex lanscape however emerges in the shell from the presence of in-plane and axial shear strains, and of additional terms in the piezoelectric tensor written in the trigonal axes. 3,22,23 V. CUBIC SEMICONDUCTORS ALONG THE < 001 > AXIS.
By contrast to the (111) plane of the zinc-blende structure, which is quite isotropic, the (001) plane is known to be strongly anisotropic. This is obvious on the stiffness tensor written in the e r , e θ , e z axes, obtained by rotating the cubic axes by an angle θ) around z (i.e., it is written in cylindrical coordinates): The present form of the stiffness tensor identifies two contributions: • one with cylindrical symmetry (that with theĉ, note thatĉ 11 −ĉ 12 = 2ĉ 66 ); if we keep only this contribution, the strain configuration is that of Eq. 18 or Eq. 25, where the c ij 's are replaced by theĉ ij 's; FIG. 11. In-plane body forces in the shell of a < 001 > NW, with f < 0, due to the 4-fold symmetry of the zinc-blende structure. The tetrahedron of the atomic structure is shown in green.
• one, proportional to c, with the expected fourfold symmetry around z. As mentioned earlier, with c < 0, a zinc-blende crystal is softer against a pure tetragonal stress (along a cubic axis, cos 4θ = 1) than against any other stress, in particular along a < 110 > axis (cos 4θ = −1).
Omitting the terms violating the translation invariance and identifying cylindrical contributions (inĉ) and contributions due to the cubic anisotropy (proportional to c), we obtain for the Lamé -Clapeyron -Navier equation in cartesian coordinates: Inserting the cylindrical solution u r (r) = Ar + B r reveals non-vanishing contributions from the terms proportional to c. As in the previous case, in a calculation to first order in c, these terms act as body forces and generate an additional displacement field δu, proportional to c. Even if these terms look quite similar to those already encountered for the < 111 > NW (they amount to in the shell, and zero in the core (Fig. 11). We thus have to find an additional in-plane displacement δu(r, θ) which is the response of the transversely isotropic system to these forces. The relevant part of the Lamé-Clapeyron-Navier equation is a two-dimensional equation: or, defining a Poisson ratio ν =ĉ 12 c11+ĉ12 = c11+3c12−2c44 4(c11+c12) , The solution is δu r r c = ĉ c 11 +ĉ 12 B s g r (r) cos 4θ where g r and g θ are two dimensionless functions of r/r cmore precisely they are sums of five terms in ( r rc ) n with n = −1, ±3, ±5, which are given in appendix B.
To sum up, the strain configuration in a core-shell NW grown along < 001 > is given by Eq. 18, with If the materials have the same hardness (χ = 1), this reduces to Eq. 25 and This is complemented by an in-plane shear strain which writes (for χ = 1): where g θθ = 4g θ + g r , and g r , g θ , g rθ and g rr are given in Appendix B, Eq. B6 with the coefficients given in B9 for the shell and the core. , and 0.1% for ε rθ All in-plane strain components exhibit a four-fold contribution due to the crystal anisotropy. Fig. 12 shows the strain map for an InAs-InP NW with the same area ratio η as in Fig. 6 and 10, and Ref. 22, but with the NW axis along < 001 >. Fig. 13 displays the radial profiles of the in-plane displacement field, the in-plane strain components (the cylindrical contribution and the modulation in sin 4θ or cos 4θ due to cubic anisotropy), and the axial strain. The cubic contribution is negligible in the central part of the core, and remains small close to the interface; in the shell, it takes significant values, yet smaller than the cylindrical contribution. Further contributions should bring terms of higher order in 4θ, with the order of magnitude of the second order terms around c/4c 11 , i.e., again, a few % in GaAs.

VI. DISCUSSION AND CONCLUSION.
It has been recognized for a long time 12 that the (111) plane of a cubic crystal (in the present case, zinc-blende or diamond semiconductor) is isotropic -and the same property also holds for the (a, b) plane of the wurtzite structure.
In a core-shell NW, this remains valid for a core-shell NW oriented along the c-axis of the wurtzite structure. This transverse isotropy has several consequences which are reminiscent from the case of a fully isotropic material. • The longitudinal strain is decoupled from the inplane strain. It is uniform in the core and in the shell, and results from a sharing of the lattice mismatch along the c axis, inversely proportional to the cross section areas.
• The in-plane strain in the core is isotropic and uniform. It is the result of a partial compensation between the direct effect of the in-plane lattice mismatch (the c 13 f ⊥ contribution in Eq. 18) and the Poisson effect from the longitudinal mismatch (the (c 11 − c 12 )f contribution). The simple result obtained for a fully isotropic material (a factor of − 1 2 in the (shear strain) / (isotropic strain) ratio when comparing the NW to the thin layer) must be adapted to the relevant stiffness constants. In the case of a GaN-AlN NW, the compensation is reinforced by the different values of the lattice mismatch in the two directions, so that the in-plane strain in the core is reduced by one order of magnitude.
• Actually the main part of the in-plane lattice mismatch is accommodated by the in-plane shear strain, which rotates around the interface so that the circular symmetry is maintained. The fact that this strain is restricted to the vicinity of the interface is a consequence of the Saint-Venant principle.
The strain distribution in a NW oriented along the < 111 > axis of a semiconductor with the zinc-blende (or diamond) structure is more complex. Shear strains and shear stresses are expected, and they appear in the numerical studies. They are due to the trigonal symmetry around the < 111 > axis, and more precisely to the presence of tetrahedral building blocks with a single orientation -while two orientations co-exist in the wurtzite structure. 36 The present analysis shows that these shear strain indeed exist in the shell, and that their influence on the strain in the core is small. The uniform strain, isotropic in the plane, which exists in the core can be calculated analytically using the stiffness tensor appropriate for the < 111 > orientation.
The same method gives analytical results also in the case of a NW with the zinc-blende (or diamond) structure grown along a cubic axis: then in-plane strain with 4-fold symmetry develops in addition to the cylindrical configuration.
While the present study assumes a circular basis of the NWs, numerical studies also reveals the role of facets: for a hexagonal basis, the strain in the core is not uniform in the corners of the hexagons. An analytical method has been proposed for isotropic materials in Ref. 37. Nevertheless, the comparison between the present calculation and the plateaus values from numerical studies suggests again a quantitative agreement, which can be seen as another consequence of the Saint-Venant's principle.
Finally, multishell NWs are currently proposed for applications such as the direct-bandgap emission from < 001 > Si-Ge NWs, 38 or a reduction of piezoelectric effects in wurtzite or < 111 > zinc-blende NWs. 13 The present study shows that a shear strain exists in such QWs, different from well to well. The transfer matrix method can also be used to incorporate the effects of surface stress, which may become significant in narrow NWs, 39 or of surface layers (oxide for instance), two effects which will be difficult to disentangle. The full Lamé -Clapeyron -Navier equations are written for the three crystal structures and orientations.
The present study involves Lamé -Clapeyron -Navier equations describing the response of a system which is invariant under a translation along the z-axis and isotropic in the basal xy plane, to body forces which are periodic in a rotation around the z-axis: F z = F sin 3θ for a zincblende NW along a < 111 > axis, and F r = F cos 4θ, F θ = −F sin 4θ for a zinc-blende NW along a < 001 > axis. Note that F z = F cos(θ) over the whole structure describes a uniform axial shear strain applied to the system, and F r = F sin 2θ, F θ = F cos 2θ a uniform transverse shear strain: a transfer matrix method was proposed in Ref. 8 for multishell NWs submitted to these two types of shear strain. The present study involves similar body forces distributions with a faster dependence on θ, localized in the shell: F z = F sin pθ with p = 3, and F r = F cos pθ, F θ = −F sin pθ with p = 4. Other ori-entations of the NWs will involve combinations of such body forces distributions.
We thus have to calculate a displacement field δu, solution of the Lamé -Clapeyron -Navier equation As the response of a linear, transversely isotropic system to an oscillating perturbation, the general solution is expected to show the same oscillatory behavior, in cos(pθ) or sin pθ.
The boundary conditions are the continuity of the total displacement field, u + δu, at the interface, and that the stress components acting on the interface and on the sidewall surface (σ rr , σ rθ , σ rz ) all vanish. The last condition must be achieved for the total stress, corresponding to u + δu. For the displacement field, it is sufficient to write that the additional displacement field does not break the contact which has been established by the cylindrical displacement field, hence δu = 0. Note that the symmetry of the system and that of the shear strain strongly reduce the number of parameters to be determined from boundary conditions. For instance, the condition that the integral of σ zz vanishes is automatically preserved by the oscillating character of δu.
In the absence of driving force in the basal plane, we keep δu r = 0 and δu θ = 0, and look for δu z = ϕ(r) sin pθ, with ϕ(r) obeying Eq. 35 The general solution is the sum of functions ∼ r n : n = −1 provides a particular solution which compensates for F z , and for n = ±p, the sum of derivatives vanishes. With F z =c 14 8B s r 2 c sin 3θ r 3 in the shell, we obtain δu z r c = α 3 ρ 3 + α −3 ρ −3 + α −1 ρ −1 c 14 c 44 B s sin 3θ where ρ = r rc , with α s −1 = −1 in the shell and α c −1 = 0 in the core. Also, α c −3 = 0 in the core to avoid a singularity at r = 0. The additional strain is thus and σ rz = 2c 44 δε rz +c 14 sin 3θ(ǫ rr − ε θθ ) = 3α 3 ρ 2 − 3α −3 ρ −4 + (2 − α −1 )ρ −2 c 14 B s sin 3θ The three remaining parameters α c 3 , α s 3 and α s −3 are determined by the non-trivial boundary conditions, on u z (at the interface) and σ rz (at the interface and surface). It is quite convenient to write these conditions using a transfer matrix: At the interface (ρ = 1), if we omit the difference in stiffness coefficients between the two materials: At the surface, using Eq. B2 at r = r s (ρ = 1/ √ η), and keeping only the second component of the vectors (the stress which must be zero), we obtain Hence α c 3 = α s 3 = η 2 (1 − η) and α s −3 = −1. If we assume a different hardness with a single factor χ between the stiffness coefficients of the shell with respect to those of the core material, Eq. B3 becomes The correction for χ non unity is small for the actual NW configurations considered here: with χ = 1.2 and η = 0.2, the corrective factor is 10% for α c 3 and negligible for the shell.
This result was used in the case of the < 111 > coreshell NWs and it can be extended to multishell NWs.
The problem is similar to the previous one: we have to find the response of a system with transverse isotropic character, to a body force distribution F. The body forces F represent an in-plane shear strain, with a four-fold symmetry due to the cos 4θ factor. A usual shear strain would have a cos 2θ and sin 2θ factors, as described in Ref. 8. The solution is a bit more complex than the response to axial shear because we are dealing with a 2D, not 1D, problem.
The equation to be solved, Eq. 42, is, in polar coordi-