It is shown that the elastic field due to nonuniform temperature or a coherently misfitting inclusion in a semi-infinite region can be derived simply from the corresponding field in an infinite region. This follows from the work of Mindlin and Cheng [J. Appl. Phys. 21, 931 (1950)] but it is not necessary to calculate the thermoelastic potential itself. In particular, the displacement of the free surface is the same as that of the equivalent plane in an infinite solid, increased by a factor of 4(1−ν). The change in volume associated with the distortion of the surface is reduced by a factor of 2(1+ν)/3 from the free expansion of the inclusion. A rectangular inclusion is used to illustrate the theory.

Issue Section:
Technical Papers
Keywords:
inclusions, thermal expansion, thermoelasticity
Topics:
Displacement, Thermal expansion, Thermoelasticity, Slabs, Stress, Wire
1.
Boley, B. A., and Weiner, J. H., 1997, Theory of Thermal Stresses, Dover, Mineola, NY.
2.
Nowacki, W., 1986, Thermoelasticity, Pergamon, Oxford, UK, 2nd edition.
3.
Hu
,
S. M.
1989
, “
Stress From a Parallelepipedic Thermal Inclusion in a Halfspace
,”
J. Appl. Phys.
,
66
, pp.
2741
–
2743
.
4.
Hu
,
S. M.
1990
, “
Stress From Isolation Trenches in Silicon Substrates
,”
J. Appl. Phys.
,
67
, pp.
1092
–
1101
.
5.
Freund
,
L. B.
2000
, “
The Mechanics of Electronic Materials
,”
Int. J. Solids Struct.
,
37
, pp.
185
–
196
.
6.
Jain
,
S. C.
,
Maes
,
H. E.
,
Pinardi
,
K.
, and
De Wolf
,
I.
1996
, “
Stresses and Strains in Lattice-Mismatched Stripes, Quantum Wires, Quantum Dots, and Substrates in Si Technology
,”
J. Appl. Phys.
,
79
, pp.
8145
–
8165
.
7.
Glas
,
F.
1987
, “
Elastic State of the Thermodynamic Properties of Inhomogeneous Epitaxial Layers: Application to Immiscible III-V Alloys
,”
J. Appl. Phys.
,
62
, pp.
3201
–
3208
.
8.
Glas
,
F.
1991
, “
Coherent Stress Relaxation in a Half Space: Modulated Layers, Inclusions, Steps, and a General Solution
,”
J. Appl. Phys.
,
70
, pp.
3556
–
3571
.
9.
Pinnington
,
T.
,
Sanderson
,
A.
,
Tiedje
,
T.
,
Pearsall
,
T. P.
,
Kasper
,
E.
, and
Presting
,
H.
1992
, “
Ambient Pressure Scanning Tunneling Microscope Imaging of Hydrogen-Passivated Si/Ge Multilayers
,”
Thin Solid Films
,
222
, pp.
259
–
264
.
10.
Chen
,
H.
,
Feenstra
,
R. M.
,
Piva
,
P. G.
,
Goldberg
,
R. D.
,
Mitchell
,
I. V.
,
Aers
,
G. C.
,
Poole
,
P. J.
, and
Charbonneau
,
S.
1999
, “
Enhanced Group-V Intermixing in InGaAs/InP Quantum Wells Studied by Cross-Sectional Scanning Tunneling Microscopy
,”
Appl. Phys. Lett.
,
75
, pp.
79
–
81
.
11.
Gosling
,
T. J.
, and
Willis
,
J. R.
1995
, “
Mechanical Stability and Electronic Properties of Buried Strained Quantum Well Arrays
,”
J. Appl. Phys.
,
77
, pp.
5601
–
5610
.
12.
Faux
,
D. A.
,
Downes
,
J. R.
, and
O’Reilly
,
E. P.
1996
, “
A Simple Method for Calculating Strain Distributions in Quantum-Wire Structures
,”
J. Appl. Phys.
,
80
, pp.
2515
–
2517
.
13.
Faux
,
D. A.
,
Downes
,
J. R.
, and
O’Reilly
,
E. P.
1997
, “
Analytic Solutions for Strain Distributions in Quantum-Wire Structures
,”
J. Appl. Phys.
,
82
, pp.
3754
–
3762
.
14.
Grundmann
,
M.
,
Stier
,
O.
, and
Bimberg
,
D.
1995
, “
InAs/GaAs Pyramidal Quantum Dots: Strain Distribution, Optical Phonons, and Electronic Structure
,”
Phys. Rev. B
,
52
, pp.
11969
–
11981
.
15.
Downes
,
J. R.
,
Faux
,
D. A.
, and
O’Reilly
,
E. P.
1997
, “
A Simple Method for Calculating Strain Distributions in Quantum Dot Structures
,”
J. Appl. Phys.
,
81
, pp.
6700
–
6702
.
16.
Pryor
,
C.
,
Kim
,
J.
,
Wang
,
L. W.
,
Williamson
,
A.
, and
Zunger
,
A.
1998
, “
Comparison of Two Methods for Describing the Strain Profiles in Quantum Dots
,”
J. Appl. Phys.
,
83
, pp.
2548
–
2554
.
17.
Davies
,
J. H.
1998
, “
Elastic and Piezoelectric Fields Around a Buried Quantum Dot: A Simple Picture
,”
J. Appl. Phys.
,
84
, pp.
1358
–
1365
.
18.
Davies
,
J. H.
1999
, “
Quantum Dots Induced by Strain
,”
Appl. Phys. Lett.
,
75
, pp.
4142
–
4144
.
19.
Legrand
,
B.
,
Grandidier
,
B.
,
Nuys
,
J. P.
,
Stie´venard
,
D.
,
Ge´rard
,
J. M.
, and
Thierry-Mieg
,
V.
1998
, “
Scanning Tunneling Microscopy and Scanning Tunneling Spectroscopy of Self-Assembled InAs Quantum Dots
,”
Appl. Phys. Lett.
,
73
, pp.
96
–
98
.
20.
Chiu
,
Y. P.
1977
, “
On the Stress Field Due to Initial Strains in a Cuboid Surrounded by an Infinite Elastic Space
,”
ASME J. Appl. Mech.
,
44
, pp.
587
–
590
.
21.
Chiu
,
Y. P.
1978
, “
On the Stress Field in Surface Deformation in a Half Space With a Cuboidal Zone in Which Initial Strains are Uniform
,”
ASME J. Appl. Mech.
,
45
, pp.
302
–
306
.
22.
Mindlin
,
R. D.
, and
Cheng
,
D. H.
1950
, “
Nuclei of Strain in the Semi-Infinite Solid
,”
J. Appl. Phys.
,
21
, pp.
926
–
930
.
23.
Mindlin
,
R. D.
, and
Cheng
,
D. H.
1950
, “
Thermoelastic Stress in the Semi-Infinite Solid
,”
J. Appl. Phys.
,
21
, pp.
931
–
933
.
24.
Sen
,
B.
1951
, “
Note on the Stresses Produced by a Nuclei of Thermoelastic Strain in a Semi-Infinite Elastic Solid
,”
Q. Appl. Math.
,
8
, pp.
365
–
369
.
25.
Goodier
,
J. N.
1937
, “
On the Integration of the Thermo-Elastic Equations
,”
Philos. Mag.
,
23
, pp.
1017
–
1032
.
26.
Timoshenko, S. P., and Goodier, J. N., 1970, Theory of Elasticity, McGraw-Hill, New York, 3rd edition.
27.
Westergaard, H. M., 1952, Theory of Elasticity and Plasticity, Harvard University Press, Cambridge, MA.
28.
Barber
,
J. R.
1987
, “
Thermoelastic Distortion of the Half-Space
,”
J. Therm. Stresses
,
10
, pp.
221
–
228
.
29.
Barber, J. R., 2002, Elasticity, 2nd Ed., Kluwer, Dordrecht, The Netherlands.
30.
Nowacki
,
W.
,
1954
, “
Thermal Stresses in Anisotropic Bodies (I)
,”
Arch. Mech. Stos. (Arch. Mech.)
6
, pp.
481
–
492
.
31.
Hieke
,
M.
1955
,
Z. Angew. Math. Mech.
,
35
, pp.
285
–
294
.
32.
Garcia Blanco
,
S.
,
Glidle
,
A.
,
Davies
,
J. H.
,
Aitchison
,
J. S.
, and
Cooper
,
J. M.
2001
, “
Electron Beam Induced Densification of Ge-Doped Flame Hydrolysis Silica for Waveguide Fabrication
,”
Appl. Phys. Lett.
,
79
, pp.
2889
–
2891
.
33.
Johnson, K. L., 1985, Contact Mechanics, Cambridge University Press, Cambridge, UK.
34.
Davies
,
J. H.
,
Bruls
,
D. M.
,
Vugs
,
J. W. A. M.
, and
Koenraad
,
P. M.
2002
, “
Relaxation of a Strained Quantum Well at a Cleaved Surface
,”
J. Appl. Phys.
,
91
, pp.
4171
–
4176
.
35.
Myklestad
,
N. O.
1942
, “
Two Problems of Thermal Stress in the Infinite Solid
,”
ASME J. Appl. Mech.
,
9
, pp.
A136–A143
A136–A143
.
36.
Faivre
,
G.
1964
, “
De´formations de Cohe´rence d’un Pre´cipite´ Quadratique
,”
Phys. Status Solidi
,
35
, pp.
249
–
259
.
37.
Seo
,
K.
, and
Mura
,
T.
1979
, “
The Elastic Field in a Half Space due to Ellipsoidal Inclusions With Uniform Dilational Eigenstrains
,”
ASME J. Appl. Mech.
,
46
, pp.
568
–
572
.
38.
Glas
,
F.
2001
, “
Elastic Relaxation of Truncated Pyramidal Quantum Dots and Quantum Wires in a Half Space: An Analytical Calculation
,”
J. Appl. Phys.
,
90
, pp.
3232
–
3241
.
39.
Pearson
,
G. S.
, and
Faux
,
D. A.
2000
, “
Analytical Solutions for Strain in Pyramidal Quantum Dots
,”
J. Appl. Phys.
,
88
, pp.
730
–
736
.
40.
Lita
,
B.
,
Goldman
,
R. S.
,
Phillips
,
J. D.
, and
Battacharya
,
P. K.
1999
, “
Interdiffusion and Surface Segregation and Stacked, Self-Assembled InAs/GaAs Quantum Dots
,”
Appl. Phys. Lett.
,
75
, pp.
2797
–
2799
.
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