Solids whose crystal structures are based on tetrahedrally-coordinated ions may show the intriguing property of negative thermal-expansion.

The ternary AIBIIICVI semiconductors (A = Ag, Cu; B = Al, Ga, In; C = S, Se, Te), exhibit such a tetrahedral coordination (see Fig. 5.7). The coordination tetra­hedron around an anion (sulfur or selenium) consists of two monovalent and two trivalent cations. The chemical bonds within such a tetrahedron are of mixed covalent and ionic character, whereby the ionicity of the bonds is different for the AI-CVI and BIII-CVI bonds. These different interactions result in different bond lengths (RAC Ф RBC) as well as bond angles and lead to a displacement of the anions from the ideal tetrahedral site by a quantity u = lx — %l (where x is the anion x coordinate).

The linear thermal-expansion coefficients are closely related to the Gruneisen parameters у of lattice vibrations [25]. The occurrence of a negative thermal — expansion can be understood using the notation of a balance between acoustic shear and compression modes of the observed crystal structure. The Gruneisen parameters of the shear modes show a tendency to negative values, while those of the com­pression modes are positive [25-27]. Hence, the temperature dependence of the thermal expansion is determined by the degree of excitations of the various modes and can change its sign when the relative thermal-population of the modes varies.

In AIBIIIC2VI chalcopyrite-type semiconductors the thermal-expansion behaviour is described by the independent linear thermal-expansion coefficients aa and ac with

The uniaxial chalcopyrite-type structure comes with two independent Gruneisen parameters ya and yc, which are related to aa and ac according to [28]:

Ca = C [(C11 + C13H + 4^] and Ус = pm [2c13aa + 4,«c] . (5.4)

Cp Cp

Here Vm is the molar volume, Cp the molar specific-heat at constant pressure and cy are the adiabatic elastic-stiffnesses. With increasing ionicity the Gruneisen parameter should become more negative [29]. Thus, the covalent character of the chemical bond is expected to strongly affect the Gruneisen parameter.

The determination of linear thermal-expansion coefficients by dilatometry or X-ray diffraction [30-33] has shown that aa and ac vary independently with tem­perature. This is caused by the axial symmetry of the chalcopyrite-type crystal structure and the difference in strength of the Cu-CVI and BIII-CVI cation-anion bonds.

The investigation of the negative thermal-expansion is conveniently achieved by neutron powder diffraction. One aspect for the use of neutrons is again the high intensity in the diffraction pattern at high Q-values, important for an exact deter­mination of the chalcogen position. It is important to monitor the change of this position at low temperatures to describe the bond stretching during cooling. The negative thermal-expansion has been studied for several chalcopyrite-type com­pounds, whereby the focus now lies on Cu(InxGa1-x)Se2 once with high (x = 0.918) and once with low indium content (x = 0.096), to show the effect of different bond ionicities on the negative thermal-expansion. Neutron powder diffraction patterns were collected for temperatures between 1.5 K > T > 300 K and structures refined by the Rietveld method according to the previously-described sequence. The ion — icity can be calculated following Phillip’s definition [34]:

(XA-XB)2

= 1 — e 4

with XA and XB the electronegativity of the elements A and B (XCu = 1.9; XGa = 1.81; XIn = 1.78). According to Phillip’s definition the bond ionicities increase from Cu-Se (fi = 0.1002) to In-Se (fi = 0.115) and Ga-Se (f — = 0.128). Thus the ionicity of the BIII-Se cation-anion bond is increasing with increasing substitution of indium by gallium. From this it follows that with a high amount of gallium the difference in bond ionicity between the Cu-Se and BIII-Se cation-anion bond increases, which lead to an increased anisotropy.

The higher anisotropy affects the change of lattice parameters with decreasing temperature, which is stronger for the gallium-rich Cu(In, Ga)Se2 and pure CuGaSe2 than for indium-rich Cu(In, Ga)Se2 (see Fig. 5.8). Applying a third-order polynomial fit to the lattice parameters the thermal-expansion coefficients aa and ac, can be derived. The temperature at which the linear thermal-expansion becomes negative (T0), is seen to vary with the chemical composition (see Table 5.2).

The variation of T0 with chemical composition should be discussed in context with the bond ionicity of the BnI-Se bonds, which increase with increasing sub­stitution of indium by gallium. Thus, with increasing gallium content the ionicity increases and the temperature, for which the linear thermal-expansion coefficient changes its sign, increases. This is observed for Cu(InxGa1-x)Se2 with different x-values as summarized in Table 5.2.

The In/(In + Ga) ratio strongly influences the character of the covalent-ionic BIII-Se cation-anion bond, and therefore the behaviour of the linear thermal — expansion coefficients of the two lattice constants aa and ac.

Also, the x-parameter of the selenium anion as a function of temperature is strongly affected by the different bond ionicities. In the Ga-rich sample the tetragonal deformation u = 0.25-x(Se) strongly tends to zero with decreasing temperature, whereas it stays almost constant for the In-rich sample (Fig. 5.9). This effect is explained by the higher bond-ionicity for the Ga-Se bond compared to the In-Se bond.

The change of the tetragonal distortion and the anion position parameter x(Se) is reflected by the change in the average cation-anion bond distances and angles, which change markedly for the Ga-rich sample compared to In-rich sample (see

Fig. 5.10).