- Email: [email protected]
zincblende crystals
J. Phys. Chem Solids Voi 59. No. I. pp. 21-26, 1998 1997 Elsevier Science Ltd Printed in Great Britain. All rights resewed 0022-3697i98 $19.00 + 0.00
0
PII: s0022-3697(9300120-0
Pergamon
POISSON’S
RATIOS
IN DIAMOND/ZINCBLENDE
GARNET B. ERDAKOS
and SHANG-FEN
CRYSTALS
REN*
Department of Physics, Illinois State University, Normal, IL 61790-4560, USA (Received 3 February 1997; accepted 10 March 1997)
Abstract-The Poisson’s ratios of crystals with diamond/zincblende structures are studied by using a simple method which considers the elastic energy to be due to bond lengths and bond angles. The relationships for two Poisson’s ratios, “its and Ye,, are calculated with three slightly different models and compared with the experimental data. This investigation of the materials’ macroscopic properties of Poisson’s ratios reveals their behavior at the microscopic level. 0 1997 Elsevier Science Ltd. All rights reserved. Keywords: D. crystal structure, D. elastic properties, D. mechanical properties, D. microstructure
1. INTRODUCTION
mainly shearing strains at the microscopic level in these materials. Let us first define the problem precisely. Assume that the tensile stress is applied along the [ 1lo] direction of a diamond/zincblende crystal, and the stabilized strains in the [llO], [ilO] and [OOl] directions are 6,ro, 61io and 8o01.Two Poisson’s ratios can be defined to describe the deformation as
When a solid is subjected to a tensile stress, it usually extends lengthwise and contracts transversely. This behavior is described by the Poisson’s ratio, a number that is used to characterize a material’s elastic properties. The Poisson’s ratio of a solid subjected to a tensile stress is defined as the negative ratio of stabilized lateral to lengthwise strain, and the negative sign is introduced to ensure that Poisson’s ratios are usually positive numbers. The phenomenon of a negative Poisson’s ratio, i.e. a solid expanding transversely to an applied tensile stress, is counter-intuitive but does occur [l]. Materials with negative Poisson’s ratios have been investigated by many different models [2-71, but studies of the Poisson’s ratios in diamond/zincblende structures, some of which have negative Poisson’s ratios, are rare [8]. Since the mechanical properties of a material are influenced significantly by its Poisson’s ratios [9, lo], and because most semiconductor devices are made with materials in the diamond/zincblende structure, such a study can be of practical use. Here we report our study of Poisson’s ratios in diamond/zincblende crystals by using a simple method. In this method the total energy of the crystal is considered as the sum of the elastic energy owing to bond lengths and bond angles, and the stabilized strains under the stress keep the total energy minimum [3]. By using this method, our investigation of the Poisson’s ratios of diamond/ zincblende structures proceeded with three slightly different models, and the results have an interesting comparison with the available experimental data. This investigation of the macroscopic property of Poisson’s ratios in diamond/zincblende crystals reveals the fact that a stress applied along the [ 1lo] direction would cause
The experimental Poisson’s ratios, vii0 and Ye,. of cubic crystals can be derived from the experimental data of elastic constants c, ,, c L2and C~ as [2] R - 2q4 yi’o= R+2cd4
and
‘hc44 ‘O”‘= c,,(R+24’ where R=c,,+c12
,
1+2% (
>
The experimental data of elastic constants c , ,, c 12and C~ [ 11, 121 and the Poisson’s ratios derived from them for diamond/zincblende crystals are listed in Table 1. It can be seen that the majority of materials have ~00~values close to 0.5, and all vllo values are almost zero with a few negative numbers. Next, we investigated theoretically the relationship for the Poisson’s ratios by a simple method.
*Authorto whom correspondenceshould be addressed. 21
G. B. ERDAKOSand S.-F. BEN
22
Table 1. The experimental data of elastic constants, cl,, c 12and CM[ 11,121,for diamond/zincblende c~stak, an$the Poisson’s ratios, or,,, and Ye,, derived [2] from them, where the units for elastic constants are 10 dyn cm Material
cl1
Diamond Si Ge Ge AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb ZnSe ZnTe CdTe HgTe
10.764 16.577 12.40 12.853 13.2 12.5 8.769 14.050 11.84 8.834 10.22 8.329 6.592 8.72 7.13 5.38 5.361
Cl2
c44
1.252 6.393 4.13 4.826 6.30 5.34 4.341 6.203 5.37 4.023 5.76 4.526 3.563 8.24 4.07 3.74 3.660
5.774 7.962 6.83 6.680 6.15 5.42 4.076 7.033 5.91 4.322 4.60 3.959 2.996 3.92 3.12 2.018 2.123
2. THEORETICAL MODELS AND RESULTS We begin by choosing
an arbitrary atom 0 in a crystal
with diamoncl/zincblende
structure
origin of the coordinate [ilO] and [OOl] directions
system (Fig. 1). The [l lo], in the crystal are taken as the
_?, j, and i directions neighbor model,
distance interactions
and take it as the
respectively,
and the second-
is taken as the unit length. between
neighbor are considered.
In this
atoms up to the second
The nearest neighbors
of atom
0 are labeled A, B, C and D in Fig. 1 and the secondneighbors
are labeled E to Q. Three strain terms, 6,, 8z
VII0
VCQl
Experimental method
0.0076 0.0623 0.0043 0.0253 0.0460 0.1010 0.0389 0.0246 0.0215 0.0308 0.0154 0.0011 0.0253 0.0245 0.0245 -0.0146 -0.0269
0.115 0.362 0.332 0.366 0.455 0.384 0.476 0.43 1 0.444 0.441 0.555 0.543 0.527 0.557 0.557 0.705 0.701
Brillouin scattering ultrasound ultrasound ultrasound estimation estimation ultrasound ultrasound ultrasound ultrasound ultrasound ultrasound ultrasound Brillouin scattering resonance not available not available
its strained position is (cl + 6 ,)/2,0, for the second-neighbor
Year and reference 1975, [ 1l] 1964, ill1 1971, [ll] 1963, [ll] 1975, [ll] 1975, [l l] 1960, [ll] 1981, [I 11 1973, [ll] 1975, [I 11 1966, [I21 1963, [12] 1975, [12] 1975, [12] 1963, [12] 1973, [12] 1975, [12]
J2( 1 + 63)/4), and
atom, J, its original position is
(1 ,O,O)and its strained position is (1 + 6 ,,O,O).The changes in bond length, Ad, and bond angle, A& with strain can
be calculated from these atomic positions. When the crystal is subjected to a tensile stress along the [ 1lo] direction, the change of its total energy is considered as the sum of the change of two types of elastic energy: the change of bond-length elastic energy and the change of bond-angle elastic energy. Each bondlength change will cause an energy change as
and &, are introduced to describe the strain in the .?, 9 and i directions positions
respectively,
(6)
and the original and strained
of each neighboring
atom can be determined
with respect to atom 0. For example,
for the nearest-
neighbor atom, A, its original position is (l/2,0,
,‘2/4) and
where d is the original bond-length
and Ad is the change
of bond length, and each bond-angle change will cause an energy change as
where Ed” and E,” are the force constants with respect to the bond
length
and bond
angle,
respectively.
For
the Ad/d for the bond OJ is calculated as 6t, and the change A0 for the angle OAJ is calculated as 242(6, - &J/3. Our calculations are performed progressively in three slightly different models.
example,
2.1. Model 1 On the basis of the assumption interactions
that the bond-length
between the nearest neighbors will be much
in our first model we consider only the nearest-neighbor bondlength and both nearest-neighbor and second-neighbor bond-angles. All four nearest-neighboring bonds, six angles between four nearest neighbors, and 12 secondneighbor angles are considered. The total change in the stronger than those between the second-neighbors,
Fig. 1. The center atom 0, its four-neatest-neighbor atoms A-D and 12 second-neighbor atoms E-Q in diamond/zincblende cystal.
Poisson’s ratios in diamondhincblende
crystals
23
Relations of Poisson’s Ratios in Diamond/Zincblende Crystals 1
Ii
I 1 I I”$’
II
-0.05
0
’ 1 I,‘I’
1 ,I
1 ’ 1’
0.8
0.6
-0.15
-0.1
0.05
0.1
0.15
Fig. 2. Relationships for Poisson’s ratios ~~10and VW,in diamond/zincblende crystals. The labeled black dots are experimental data derived from measured elastic constants listed in Table 1 by using eqns (3) and (4). The dotted curve is calculated by model 1, the dashed curve by model 2, and the solid curve by model 3.
bond-length
elastic energy is calculated as
26,+i$+
$262 +&)'I
Introducing E = BIA and solving for Yeand Ye, we find that (8)
-8+117e-243t2 8 + 3696 + 243~~
(13)
16 - 72~ - 243~~ v3 = 8 + 3696 + 243~~ ’
(141
v2 = and the total change in the bond-angle elastic energy as
and
AE, = @’
+(6,-6,)2+(62-61)2]T
(9)
where we have only kept up to the squared terms of 6. Introducing the symbols the Ye = -62/81 and ~3 = -63/6r, which are actually the vii0 and ~00~defined in eqns (1) and (2) at the beginning of this paper, the total charge in elastic energy is calculated as
where A = Ed”@12and B = 4E,‘$/3. With the requirement that static equilibrium corresponds to a state of minimum energy, the derivatives of the total change in energy with respect to v2 and v3 can be set equal to zero, which gives A(16v2+8Ys)+B(-9+45v2-54v+0
(111
and A(-8+8~2+8~s)+B(54-54~zl08~s)=O
(12)
As we mentioned above, here v2 and Y, are the Poisson’s ratios vii0 and vool introduced at the beginning of this paper. The relationship for the Poisson’s ratios calculated in this model is shown as the dotted curve in Fig. 2, where values of Eare taken from about 0.04 to 0.15. It is obvious that this result does not agree well with most of the experimental data, which are the labelled black dots in Fig. 2. When E= 0, i.e. in the limited case that the angular elastic energy is eliminated, we have v2 = - 1 and ~3= 2, which is far out of the range in this figure. This calculation reveals the fact that the nearest-neighbor bond-length interactions do not play a key role in the property of Poisson’s ratios for most diamond/zincblende materials as we had expected, which leads us to the second model. 2.2.Model 2 In the second model we consider the changes in bondlength elastic energy for both nearest and second neighbors with bond-angle elastic energy totally neglected. In this case, the total change in elastic energy is
G. B. EKDAKOSand S.-F. KBN
24
calculated as AE= AEd, + AEd =
2
1 x4 TEd,”
i=l
where Edi” and Ed2” are the force constants for the firstand second-neighbor bond-lengths. Again keeping only up to squared terms of 6, the total energy change is 1 AE = -Ed,” 324 2
+ Q2 + ;(262 + 64
1 + -Ed2” s: + s; + $(6, + 62 + 264. 2
(16)
Introducing vr and v3 as defined earlier, we get AE =A [ $2 - v~)~ + f(2v2 + v,)~]
l+(-vz)2+~(l-vz-2vs)2
1,
(17)
where A = 0.5Edlnbf and B = 0.5Ed2”$. Taking the derivatives with respect to v2 and v3, as in the first calculation, and letting e = BIA, we now find that - 8 - 27~ v2= 8+81e+162e2
(18)
16+72e+81e2 v3= 8+81s+162e2’
(1%
We then attempted to understand why the nearestneighbor bond-length elastic energy, which we expect to be the most important element in the total change of the elastic energy, has almost no influence on the Poisson’s ratios compared with the second-neighbors. By looking at eqn (6), we found that there could be two possibilities, either Ed”- 0 or Ad - 0 for nearest neighbors. Since Ed” + 0 does not make sense from our understanding of the physics involved, we looked at the Ad terms, which were mentioned at the beginning of Section 2. From the terms related to nearest neighbors it is obvious that if 6 1= A2= -l/2h3, all the nearest-neighbor bond-lengths will remain constant under the stress, so the elastic energy change due to nearest-neighbor bond-lengths will be equal to zero. This can also be seen from eqns (8) and (16). This is actually the case for this structure if we notice that we already derived v2 = -ij2/6, = - 1 and v3 = -S&3, = -2 as the limiting case of E = 0 in Model 1. At this point we realized that in strained crystals of this structure, the nearest-neighbor bond-lengths would almost remain constant and the tensile stress applied along the [l lo] direction would only cause shearing strain [l] at the microscopic level. In other words, all the nearest-neighbor bond lengths will remain constant, and only the bond angles would change. Of course, the second-neighbor bond-lengths will also change. After understanding this from the first two models, we came up with the third model, which we believed would be the best approximation.
and 2.3. Model 3
The calculated result in this model is represented by the dashed curve in Fig. 2. By a simple visual analysis, the second calculation agrees better than the first one with most of the experimental data, which tells us that the bond-length elastic energy is much more important than the bond-angle elastic energy. However, to obtain values of vii0 and vmi on this curve that fall within the range of experimental data, E must take numbers with very large absolute value. For the curve in the range of vlia from -0.01 to 0, e was taken from about 16 to about 1000. To calculate the curve in the range of vii0 from 0 to 0.01, c was taken from about - 1000 to about - 17. In the limiting case that ICI- co,we have v2 = 0 and v3 = 0.5, which is a point that sits almost at the center of all the experimental data. This implies that the change of the secondneighbor bond-length elastic energy is much greater than that of the nearest neighbor from the definition of e. But this is somewhat counter-intuitive, because we all would expect that the bond-length interactions between the nearest neighbors would be much stronger than that between the second-neighbors.
In this model, we calculate the total change in elastic energy by considering only the second-neighbor bondlengths and both nearest- and second-neighbor bondangles. With the definitions in eqns (6) and (7), the total change in elastic energy is now calculated as AE = AE, + AE, = iEd’ [6: + 6; + i(6, + 62 + 263)2]
+;Ea”[;(6,+62-26j)l+(6,-S3)2+(62-S3)2]. (20)
Substituting v2 and v3 into the equation, we get AE=A
I+(-v~)~+~(I-v~-~v~)~ [
I
+B[;(l-~2+2v~)~+(l+yl)~+(v2-v~)~],
where,
as
B=(4/3)E,‘%f.
in
(21)
the first model, A = Ed”SfI2 and Taking the derivatives with respect to
Poisson’s ratios in diamond/zincbiende crystals v2 and v3 and setting equal to zero, we find A(-l+5~~+2v~)+B(-1+5v~-6v~)=O
(22)
and A(-2+2v2+4v3)+B(6-6v2+12v,)=0.
(23)
With E = B/A, our final calculation gives 5E - 3E2 Y2=Viio=
2+13E+3E2
and 1 - 2E- 3E2 v3=v@Jl= 2+ 13t+3E2’
(25)
25
for nearest neighbors is considered, which indicates that the extension strain [1] in nearest-neighboring bonds at the microscopic level in these two materials is more likely. So by deriving expressions for the Poisson’s ratios, which are macroscopic quantities, the behavior of such crystals at a microscopic level is revealed. This phenomenon actually results from the fact that the force constants between the nearest neighbors are much stronger than between the second neighbors, which makes the nearest-neighboring bond-lengths almost rigid. We notice that in lattice dynamics calculations, it gives quite good results when the second-neighbor force constant is totally neglected [14], which is consistent with our investigation on the property of Poisson’s ratios.
This final result is shown in Fig. 2 by the solid curve,
which indeed agrees the best with the experimental data.
3. DISCUSSION
We have calculated the Poisson’s ratios vi10and vml with three slightly different models. The relationships for the Poisson’s ratios are shown graphically, and the results are compared with the experimental data. Because the experimental data are not measured directly but derived from the elastic constants c I,, c I=and cM, there are some errors in these data. It is noted that cl2 is particularly sensitive to experimental error and that the accuracy is higher when ultrasonic measurements are taken [ 131. In general, it is reasonable if we estimate the error bar as the distance between two different data points for the same material, such as the two points of Ge in this figure, which are derived from two sets of experimental data by different people with different methods. We noticed that the error bar for some materials were twice as big as the one seen for Ge in Fig. 2 when estimated in this way. Considering the experimental data with such an error bar, our third model really gives a good explanation of the relationships for vi10 and ~001. From the above calculations in three slightly different models, we started from considering only the nearestneighboring bond-lengths but ended at considering only the second-neighboring bond-lengths. This gives the results vile = 0 and vml = 0.5, a point at the center of all the available experimental data in Fig. 2, if only the second-neighbor bond-length elastic energy is considered. This reveals the fact that for these materials, because of the strong resistance to change in the bond length between nearest neighbors, the stress applied along the [l lo] direction would cause shearing strain [I] at the microscopic level. On the other hand, C and Ge are two materials which are closest to our first model (the dotted curve) where the bond-length elastic energy only
4.CONCLUSION
In conclusion, we have investigated the relationships for the Poisson’s ratios vi10 and voot in diamond/zincblende crystals with a simple method that considers two types of elastic energy: the bond-length elastic energy and bondangle elastic energy. It is found that the bond-length elastic energy between second-neighbors plays the most important role in determining the Poisson’s ratios for most of these materials. This investigation of the macroscopic properties of the Poisson’s ratio reveals the fact that, at the microscopic level, the stress applied along the [ 1lo] direction would mainly cause shearing strains [l] in these materials. This result is consistent with lattice dynamics calculations of these materials, and furthers our understanding of physical properties of materials with these structures.
Achowledgemenrs-This
work was supported in part by an award from the Research Corporation.
REFERENCES 1. Love, A. E., Marhematical Theory of Elecrricity. Dover, New York, 1944. 2. Milstein, F. and Haung, K., Phys. Rev., 1978, B18 (6). 2529. 3. Ren, S. Y., Acra Physica Sinica, 1983, B32 (5). 664. 4. Evans, K. E., .I. Phys. D: Appl. Phys., 1989,22, 1870. 5. Keskar, N. R. and Chelikowsky, J. R., Nature, 1992. 358, 222. 6. Rothenburg, L., Berlin, A. A. and Bathurst, R. J., Nature, 1991,354,470. 7. Warren, T., J. Appl. Hays., 1990.67 (12). 7591. 8. Zhang, Q. and Ren, S. Y., personal communication. 9. Caddock, B. D. and Evans, K. E.. J. Phys. D: Appl. Phys., 1877,22,
1989.
10. Lakes, R., Science, 1987,235, 1039. 11. Hellwage, K.-H. (Editor in Chief), Numerical Data and Functional Relationships in Science and Technology, Vol. 17a, Physics of III-V Compounds.
Madelung/Springer-Verlag, Berlin/Heidelberg!New York. 1982.
G. B. ERDAKOS and S.-F. BEN
26
12. Hellwage, K.-H. (Editor in Chief), Numerical Data and Functional Relationships in Science and Technology, Vol. I7b,
Physics
of II-VI
and I-VII
Compounds.
MadelunglSpringer-Verlag, Berlin/Heidelberg/New York, 1982.
13. Benbattouche, N., Saunders, G. A., Lambson, E. F. and Hoenle. W., J. Phys. D: Appl. Phys., 1989.22,670. 14. Harrison, W. A., Electronic Structure and the Properties of Solid. W. H. Freeman and Company, San Francisco, CA, 1980.
0
PII: s0022-3697(9300120-0
Pergamon
POISSON’S
RATIOS
IN DIAMOND/ZINCBLENDE
GARNET B. ERDAKOS
and SHANG-FEN
CRYSTALS
REN*
Department of Physics, Illinois State University, Normal, IL 61790-4560, USA (Received 3 February 1997; accepted 10 March 1997)
Abstract-The Poisson’s ratios of crystals with diamond/zincblende structures are studied by using a simple method which considers the elastic energy to be due to bond lengths and bond angles. The relationships for two Poisson’s ratios, “its and Ye,, are calculated with three slightly different models and compared with the experimental data. This investigation of the materials’ macroscopic properties of Poisson’s ratios reveals their behavior at the microscopic level. 0 1997 Elsevier Science Ltd. All rights reserved. Keywords: D. crystal structure, D. elastic properties, D. mechanical properties, D. microstructure
1. INTRODUCTION
mainly shearing strains at the microscopic level in these materials. Let us first define the problem precisely. Assume that the tensile stress is applied along the [ 1lo] direction of a diamond/zincblende crystal, and the stabilized strains in the [llO], [ilO] and [OOl] directions are 6,ro, 61io and 8o01.Two Poisson’s ratios can be defined to describe the deformation as
When a solid is subjected to a tensile stress, it usually extends lengthwise and contracts transversely. This behavior is described by the Poisson’s ratio, a number that is used to characterize a material’s elastic properties. The Poisson’s ratio of a solid subjected to a tensile stress is defined as the negative ratio of stabilized lateral to lengthwise strain, and the negative sign is introduced to ensure that Poisson’s ratios are usually positive numbers. The phenomenon of a negative Poisson’s ratio, i.e. a solid expanding transversely to an applied tensile stress, is counter-intuitive but does occur [l]. Materials with negative Poisson’s ratios have been investigated by many different models [2-71, but studies of the Poisson’s ratios in diamond/zincblende structures, some of which have negative Poisson’s ratios, are rare [8]. Since the mechanical properties of a material are influenced significantly by its Poisson’s ratios [9, lo], and because most semiconductor devices are made with materials in the diamond/zincblende structure, such a study can be of practical use. Here we report our study of Poisson’s ratios in diamond/zincblende crystals by using a simple method. In this method the total energy of the crystal is considered as the sum of the elastic energy owing to bond lengths and bond angles, and the stabilized strains under the stress keep the total energy minimum [3]. By using this method, our investigation of the Poisson’s ratios of diamond/ zincblende structures proceeded with three slightly different models, and the results have an interesting comparison with the available experimental data. This investigation of the macroscopic property of Poisson’s ratios in diamond/zincblende crystals reveals the fact that a stress applied along the [ 1lo] direction would cause
The experimental Poisson’s ratios, vii0 and Ye,. of cubic crystals can be derived from the experimental data of elastic constants c, ,, c L2and C~ as [2] R - 2q4 yi’o= R+2cd4
and
‘hc44 ‘O”‘= c,,(R+24’ where R=c,,+c12
,
1+2% (
>
The experimental data of elastic constants c , ,, c 12and C~ [ 11, 121 and the Poisson’s ratios derived from them for diamond/zincblende crystals are listed in Table 1. It can be seen that the majority of materials have ~00~values close to 0.5, and all vllo values are almost zero with a few negative numbers. Next, we investigated theoretically the relationship for the Poisson’s ratios by a simple method.
*Authorto whom correspondenceshould be addressed. 21
G. B. ERDAKOSand S.-F. BEN
22
Table 1. The experimental data of elastic constants, cl,, c 12and CM[ 11,121,for diamond/zincblende c~stak, an$the Poisson’s ratios, or,,, and Ye,, derived [2] from them, where the units for elastic constants are 10 dyn cm Material
cl1
Diamond Si Ge Ge AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb ZnSe ZnTe CdTe HgTe
10.764 16.577 12.40 12.853 13.2 12.5 8.769 14.050 11.84 8.834 10.22 8.329 6.592 8.72 7.13 5.38 5.361
Cl2
c44
1.252 6.393 4.13 4.826 6.30 5.34 4.341 6.203 5.37 4.023 5.76 4.526 3.563 8.24 4.07 3.74 3.660
5.774 7.962 6.83 6.680 6.15 5.42 4.076 7.033 5.91 4.322 4.60 3.959 2.996 3.92 3.12 2.018 2.123
2. THEORETICAL MODELS AND RESULTS We begin by choosing
an arbitrary atom 0 in a crystal
with diamoncl/zincblende
structure
origin of the coordinate [ilO] and [OOl] directions
system (Fig. 1). The [l lo], in the crystal are taken as the
_?, j, and i directions neighbor model,
distance interactions
and take it as the
respectively,
and the second-
is taken as the unit length. between
neighbor are considered.
In this
atoms up to the second
The nearest neighbors
of atom
0 are labeled A, B, C and D in Fig. 1 and the secondneighbors
are labeled E to Q. Three strain terms, 6,, 8z
VII0
VCQl
Experimental method
0.0076 0.0623 0.0043 0.0253 0.0460 0.1010 0.0389 0.0246 0.0215 0.0308 0.0154 0.0011 0.0253 0.0245 0.0245 -0.0146 -0.0269
0.115 0.362 0.332 0.366 0.455 0.384 0.476 0.43 1 0.444 0.441 0.555 0.543 0.527 0.557 0.557 0.705 0.701
Brillouin scattering ultrasound ultrasound ultrasound estimation estimation ultrasound ultrasound ultrasound ultrasound ultrasound ultrasound ultrasound Brillouin scattering resonance not available not available
its strained position is (cl + 6 ,)/2,0, for the second-neighbor
Year and reference 1975, [ 1l] 1964, ill1 1971, [ll] 1963, [ll] 1975, [ll] 1975, [l l] 1960, [ll] 1981, [I 11 1973, [ll] 1975, [I 11 1966, [I21 1963, [12] 1975, [12] 1975, [12] 1963, [12] 1973, [12] 1975, [12]
J2( 1 + 63)/4), and
atom, J, its original position is
(1 ,O,O)and its strained position is (1 + 6 ,,O,O).The changes in bond length, Ad, and bond angle, A& with strain can
be calculated from these atomic positions. When the crystal is subjected to a tensile stress along the [ 1lo] direction, the change of its total energy is considered as the sum of the change of two types of elastic energy: the change of bond-length elastic energy and the change of bond-angle elastic energy. Each bondlength change will cause an energy change as
and &, are introduced to describe the strain in the .?, 9 and i directions positions
respectively,
(6)
and the original and strained
of each neighboring
atom can be determined
with respect to atom 0. For example,
for the nearest-
neighbor atom, A, its original position is (l/2,0,
,‘2/4) and
where d is the original bond-length
and Ad is the change
of bond length, and each bond-angle change will cause an energy change as
where Ed” and E,” are the force constants with respect to the bond
length
and bond
angle,
respectively.
For
the Ad/d for the bond OJ is calculated as 6t, and the change A0 for the angle OAJ is calculated as 242(6, - &J/3. Our calculations are performed progressively in three slightly different models.
example,
2.1. Model 1 On the basis of the assumption interactions
that the bond-length
between the nearest neighbors will be much
in our first model we consider only the nearest-neighbor bondlength and both nearest-neighbor and second-neighbor bond-angles. All four nearest-neighboring bonds, six angles between four nearest neighbors, and 12 secondneighbor angles are considered. The total change in the stronger than those between the second-neighbors,
Fig. 1. The center atom 0, its four-neatest-neighbor atoms A-D and 12 second-neighbor atoms E-Q in diamond/zincblende cystal.
Poisson’s ratios in diamondhincblende
crystals
23
Relations of Poisson’s Ratios in Diamond/Zincblende Crystals 1
Ii
I 1 I I”$’
II
-0.05
0
’ 1 I,‘I’
1 ,I
1 ’ 1’
0.8
0.6
-0.15
-0.1
0.05
0.1
0.15
Fig. 2. Relationships for Poisson’s ratios ~~10and VW,in diamond/zincblende crystals. The labeled black dots are experimental data derived from measured elastic constants listed in Table 1 by using eqns (3) and (4). The dotted curve is calculated by model 1, the dashed curve by model 2, and the solid curve by model 3.
bond-length
elastic energy is calculated as
26,+i$+
$262 +&)'I
Introducing E = BIA and solving for Yeand Ye, we find that (8)
-8+117e-243t2 8 + 3696 + 243~~
(13)
16 - 72~ - 243~~ v3 = 8 + 3696 + 243~~ ’
(141
v2 = and the total change in the bond-angle elastic energy as
and
AE, = @’
+(6,-6,)2+(62-61)2]T
(9)
where we have only kept up to the squared terms of 6. Introducing the symbols the Ye = -62/81 and ~3 = -63/6r, which are actually the vii0 and ~00~defined in eqns (1) and (2) at the beginning of this paper, the total charge in elastic energy is calculated as
where A = Ed”@12and B = 4E,‘$/3. With the requirement that static equilibrium corresponds to a state of minimum energy, the derivatives of the total change in energy with respect to v2 and v3 can be set equal to zero, which gives A(16v2+8Ys)+B(-9+45v2-54v+0
(111
and A(-8+8~2+8~s)+B(54-54~zl08~s)=O
(12)
As we mentioned above, here v2 and Y, are the Poisson’s ratios vii0 and vool introduced at the beginning of this paper. The relationship for the Poisson’s ratios calculated in this model is shown as the dotted curve in Fig. 2, where values of Eare taken from about 0.04 to 0.15. It is obvious that this result does not agree well with most of the experimental data, which are the labelled black dots in Fig. 2. When E= 0, i.e. in the limited case that the angular elastic energy is eliminated, we have v2 = - 1 and ~3= 2, which is far out of the range in this figure. This calculation reveals the fact that the nearest-neighbor bond-length interactions do not play a key role in the property of Poisson’s ratios for most diamond/zincblende materials as we had expected, which leads us to the second model. 2.2.Model 2 In the second model we consider the changes in bondlength elastic energy for both nearest and second neighbors with bond-angle elastic energy totally neglected. In this case, the total change in elastic energy is
G. B. EKDAKOSand S.-F. KBN
24
calculated as AE= AEd, + AEd =
2
1 x4 TEd,”
i=l
where Edi” and Ed2” are the force constants for the firstand second-neighbor bond-lengths. Again keeping only up to squared terms of 6, the total energy change is 1 AE = -Ed,” 324 2
+ Q2 + ;(262 + 64
1 + -Ed2” s: + s; + $(6, + 62 + 264. 2
(16)
Introducing vr and v3 as defined earlier, we get AE =A [ $2 - v~)~ + f(2v2 + v,)~]
l+(-vz)2+~(l-vz-2vs)2
1,
(17)
where A = 0.5Edlnbf and B = 0.5Ed2”$. Taking the derivatives with respect to v2 and v3, as in the first calculation, and letting e = BIA, we now find that - 8 - 27~ v2= 8+81e+162e2
(18)
16+72e+81e2 v3= 8+81s+162e2’
(1%
We then attempted to understand why the nearestneighbor bond-length elastic energy, which we expect to be the most important element in the total change of the elastic energy, has almost no influence on the Poisson’s ratios compared with the second-neighbors. By looking at eqn (6), we found that there could be two possibilities, either Ed”- 0 or Ad - 0 for nearest neighbors. Since Ed” + 0 does not make sense from our understanding of the physics involved, we looked at the Ad terms, which were mentioned at the beginning of Section 2. From the terms related to nearest neighbors it is obvious that if 6 1= A2= -l/2h3, all the nearest-neighbor bond-lengths will remain constant under the stress, so the elastic energy change due to nearest-neighbor bond-lengths will be equal to zero. This can also be seen from eqns (8) and (16). This is actually the case for this structure if we notice that we already derived v2 = -ij2/6, = - 1 and v3 = -S&3, = -2 as the limiting case of E = 0 in Model 1. At this point we realized that in strained crystals of this structure, the nearest-neighbor bond-lengths would almost remain constant and the tensile stress applied along the [l lo] direction would only cause shearing strain [l] at the microscopic level. In other words, all the nearest-neighbor bond lengths will remain constant, and only the bond angles would change. Of course, the second-neighbor bond-lengths will also change. After understanding this from the first two models, we came up with the third model, which we believed would be the best approximation.
and 2.3. Model 3
The calculated result in this model is represented by the dashed curve in Fig. 2. By a simple visual analysis, the second calculation agrees better than the first one with most of the experimental data, which tells us that the bond-length elastic energy is much more important than the bond-angle elastic energy. However, to obtain values of vii0 and vmi on this curve that fall within the range of experimental data, E must take numbers with very large absolute value. For the curve in the range of vlia from -0.01 to 0, e was taken from about 16 to about 1000. To calculate the curve in the range of vii0 from 0 to 0.01, c was taken from about - 1000 to about - 17. In the limiting case that ICI- co,we have v2 = 0 and v3 = 0.5, which is a point that sits almost at the center of all the experimental data. This implies that the change of the secondneighbor bond-length elastic energy is much greater than that of the nearest neighbor from the definition of e. But this is somewhat counter-intuitive, because we all would expect that the bond-length interactions between the nearest neighbors would be much stronger than that between the second-neighbors.
In this model, we calculate the total change in elastic energy by considering only the second-neighbor bondlengths and both nearest- and second-neighbor bondangles. With the definitions in eqns (6) and (7), the total change in elastic energy is now calculated as AE = AE, + AE, = iEd’ [6: + 6; + i(6, + 62 + 263)2]
+;Ea”[;(6,+62-26j)l+(6,-S3)2+(62-S3)2]. (20)
Substituting v2 and v3 into the equation, we get AE=A
I+(-v~)~+~(I-v~-~v~)~ [
I
+B[;(l-~2+2v~)~+(l+yl)~+(v2-v~)~],
where,
as
B=(4/3)E,‘%f.
in
(21)
the first model, A = Ed”SfI2 and Taking the derivatives with respect to
Poisson’s ratios in diamond/zincbiende crystals v2 and v3 and setting equal to zero, we find A(-l+5~~+2v~)+B(-1+5v~-6v~)=O
(22)
and A(-2+2v2+4v3)+B(6-6v2+12v,)=0.
(23)
With E = B/A, our final calculation gives 5E - 3E2 Y2=Viio=
2+13E+3E2
and 1 - 2E- 3E2 v3=v@Jl= 2+ 13t+3E2’
(25)
25
for nearest neighbors is considered, which indicates that the extension strain [1] in nearest-neighboring bonds at the microscopic level in these two materials is more likely. So by deriving expressions for the Poisson’s ratios, which are macroscopic quantities, the behavior of such crystals at a microscopic level is revealed. This phenomenon actually results from the fact that the force constants between the nearest neighbors are much stronger than between the second neighbors, which makes the nearest-neighboring bond-lengths almost rigid. We notice that in lattice dynamics calculations, it gives quite good results when the second-neighbor force constant is totally neglected [14], which is consistent with our investigation on the property of Poisson’s ratios.
This final result is shown in Fig. 2 by the solid curve,
which indeed agrees the best with the experimental data.
3. DISCUSSION
We have calculated the Poisson’s ratios vi10and vml with three slightly different models. The relationships for the Poisson’s ratios are shown graphically, and the results are compared with the experimental data. Because the experimental data are not measured directly but derived from the elastic constants c I,, c I=and cM, there are some errors in these data. It is noted that cl2 is particularly sensitive to experimental error and that the accuracy is higher when ultrasonic measurements are taken [ 131. In general, it is reasonable if we estimate the error bar as the distance between two different data points for the same material, such as the two points of Ge in this figure, which are derived from two sets of experimental data by different people with different methods. We noticed that the error bar for some materials were twice as big as the one seen for Ge in Fig. 2 when estimated in this way. Considering the experimental data with such an error bar, our third model really gives a good explanation of the relationships for vi10 and ~001. From the above calculations in three slightly different models, we started from considering only the nearestneighboring bond-lengths but ended at considering only the second-neighboring bond-lengths. This gives the results vile = 0 and vml = 0.5, a point at the center of all the available experimental data in Fig. 2, if only the second-neighbor bond-length elastic energy is considered. This reveals the fact that for these materials, because of the strong resistance to change in the bond length between nearest neighbors, the stress applied along the [l lo] direction would cause shearing strain [I] at the microscopic level. On the other hand, C and Ge are two materials which are closest to our first model (the dotted curve) where the bond-length elastic energy only
4.CONCLUSION
In conclusion, we have investigated the relationships for the Poisson’s ratios vi10 and voot in diamond/zincblende crystals with a simple method that considers two types of elastic energy: the bond-length elastic energy and bondangle elastic energy. It is found that the bond-length elastic energy between second-neighbors plays the most important role in determining the Poisson’s ratios for most of these materials. This investigation of the macroscopic properties of the Poisson’s ratio reveals the fact that, at the microscopic level, the stress applied along the [ 1lo] direction would mainly cause shearing strains [l] in these materials. This result is consistent with lattice dynamics calculations of these materials, and furthers our understanding of physical properties of materials with these structures.
Achowledgemenrs-This
work was supported in part by an award from the Research Corporation.
REFERENCES 1. Love, A. E., Marhematical Theory of Elecrricity. Dover, New York, 1944. 2. Milstein, F. and Haung, K., Phys. Rev., 1978, B18 (6). 2529. 3. Ren, S. Y., Acra Physica Sinica, 1983, B32 (5). 664. 4. Evans, K. E., .I. Phys. D: Appl. Phys., 1989,22, 1870. 5. Keskar, N. R. and Chelikowsky, J. R., Nature, 1992. 358, 222. 6. Rothenburg, L., Berlin, A. A. and Bathurst, R. J., Nature, 1991,354,470. 7. Warren, T., J. Appl. Hays., 1990.67 (12). 7591. 8. Zhang, Q. and Ren, S. Y., personal communication. 9. Caddock, B. D. and Evans, K. E.. J. Phys. D: Appl. Phys., 1877,22,
1989.
10. Lakes, R., Science, 1987,235, 1039. 11. Hellwage, K.-H. (Editor in Chief), Numerical Data and Functional Relationships in Science and Technology, Vol. 17a, Physics of III-V Compounds.
Madelung/Springer-Verlag, Berlin/Heidelberg!New York. 1982.
G. B. ERDAKOS and S.-F. BEN
26
12. Hellwage, K.-H. (Editor in Chief), Numerical Data and Functional Relationships in Science and Technology, Vol. I7b,
Physics
of II-VI
and I-VII
Compounds.
MadelunglSpringer-Verlag, Berlin/Heidelberg/New York, 1982.
13. Benbattouche, N., Saunders, G. A., Lambson, E. F. and Hoenle. W., J. Phys. D: Appl. Phys., 1989.22,670. 14. Harrison, W. A., Electronic Structure and the Properties of Solid. W. H. Freeman and Company, San Francisco, CA, 1980.