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GaAs heterostructures with a graded layer
CRYSTAL GROWTH ELSEVIER
Journal of Crystal Growth 137 (1994) 667—670
Letter to the Editors
Stress calculation based on three-dimensional deformation of InGaAs/GaAs heterostructures with a graded layer Kazuo Nakajima Fujitsu Laboratories Limited, Atsug4 Morinosato-Wakamiya 10-1, Atsugi 243-01, Japan (Received 30 March 1993; manuscript received in finalform 30 November 1993)
Abstract A stress calculation model based on three-dimensional deformation of multilayer heterostructures composed of semiconductor crystals is proposed by expanding our reported two-dimensional deformation model. Using this model, the stress distribution in InGaAs/ graded InGaAs layer/GaAs heterostructures is calculated. The calculated result based on three-dimensional deformation is compared with that based on one-dimensional deformation.
The calculation of stress distribution in semiconductor heterostructures is very important to design the best crystal structures with small stress. Many researchers have reported calculation models of stress distribution on the basis of Timoshenko’s model [1]. However, an actual unit cell of isotropic semiconductor crystals deforms three-dimensionally. Therefore, the stress distribution in the semiconductor crystals should be calculated by taking account of effects of threedimensional deformation. Chu et al. [2] and Nakajima [3] have considered two-dimensional deformation of the strip and proposed stress calculation models. However, stress calculation models based on three-dimensional deformation have not been reported until now. In this work, a stress calculation model based on three-dimensional deformation of multilayer heterostructures composed of semiconductor crystals is proposed by expanding our reported two-dimensional model [31.In order to clearly know the difference between the calculated re-
suits based on the three- and one-dimensional deformations, stress distributions in InGaAs/ GaAs heterostructures with an InGaAs graded layer were calculated using this model. For the deformation of heterostructures cornposed of semiconductor crystals, the unit cell is three-dimensionally deformed, as shown in Fig. 1, where I~and 1~ are the tangential and perpen-
—~
,‘
Fx
/
1’ d
1
I
‘
/ -~~‘
a1
I
1
a,~ ~-——
—-~
Fig. 1. Unit cell of an isotropic crystal deformed three-dimensionally by bi-axial forces.
0022-0248/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0022-0248(93)E0740-X
668
K. ATakajima /Journal of Crystal Growth 137 (1994) 667—6 70
axis perpendicular to the heterointerface. Each )d
9
InGoAs ~
)d
M graded tB
GaAs
~5
Nal
N~8
Nt~
R
6
)
layer experiences a face force F1 and a moment Ml•The accurate bending moment, M1, for the multilayer heterostructure composed of many imaginary thin layers is derived on the basis of three-dimensional deformation:
)di
=
(1 —i1)R
=
F-al
f1Za~
dz
1
)b) layer/GaAs Fig. 2. Shematica)geometries of an InGaAs/graded heterostructure, (a) before and (h) after three-dimensional deformation.
dicular lengths of the unit cell before deformation, respectively, and a~’and d, are the tangential and perpendicular lengths of the unit cell after deformation, respectively. F~and F~are the face forces, where x and y are tangential axes parallel to the heterointerface. The schematic
3(1
—
R
has a width of Ni1, a length of NI,, a thickness of i~,a thermal expansion coefficient of a1, a Young modulus of F1, a Poisson ratio of t~, and a lattice constant of I,. Total number of imaginary thin layers is g. Fig. 2b shows the heterostructure after deformation. The ith perturbed imaginary thin layer after deformation has a width of Na~,a length of Na~,a thickness of d1, and a tangential 1,as shown in Fig. 2b. The lattice constant of a~ perturbed heterostructure has a curvature radius of R and a neutral plane at z = 0, where z is a
—
—
d~
~
k
k= I
11~)
—
=
I
(1) where 5 is the shift of the neutral axis of the bending heterostructure from z = 0. According to the balance of general forces and moments, we have g
I~-= 0,
~
(2)
=
g
+
~,
d.
/— I
g
M.
~,
F,
~
d 1
geometries of the lnGaAs/graded lnGaAs/ GaAs heterostructure are given in Fig. 2. The 1A, lengthandof Nl~, eachwhere layer ‘A’ before deformation N NIB 1B and i~are theislattice constants of the GaAs, graded InGaAs and InGaAs layers, respectively, and N is the number of the constituent cells of the layers. The thickness of each layer is tA, tB and t~., as shown in Fig. 2a. In this calculation, it is assumed that each layer consists of many imaginary thin layers [3—61. The ith imaginary thin layer before deformation
i—I
.
d~
~
+
=
0,
(3)
J1
1’
where the face force acting At on the a unit length of the F, ithisimaginary thin layer. interface between the top of the ith imaginary thin layer and the bottom of the (i + 1)th imaginary thin layer in the perturbed heterostructure, the tangential lengths should be equal because of their coherency: N
+ ‘
=
i’~(1 ~ F, E 1d1a~I
N ,
+ ‘~
+
2R
l,±~(1 v1±1)F,~1 _______ E1±1d1±1a~’~1 2R —
—
(1
K Nakajima /Journal of Crystal Growth 137 (1994) 667—6 70
of the ith imaginary thin layer. Using Eqs. (2) and (4), we have
l,(1
E,d,a~ g E1d1a~ ~ —
—
[~
[1
x
~=
/
tJ. =
k
1
E1d1a~
g
=
2)
1
l~)l. j
(i~
—
l~(1 v1)
~=
—
(6)
Substituting Eqs. (1) and (6) into Eq. (3), we have 1/R = R1/( R,
+
R3),
(7)
where R
~.
i=1
—
v.)
)
x(2 j=I Ed1+d~ ~,
1=1
Edk-~) k=I
=
3 ~ =
i
1
i—i
k=1
li), (8)
—
/ i—i
~dk-8)
-( /
\k1
1d~a~I ~ (2 ~ d~+ d, I~(1 v~) j 1
x(2 ~
—
E,a~
E
g
R3
i~(1 v) (1,
~i
7~•~~i— c~)i=1
~[(
2(1—L’ 1)R~
(11)
compressive ones, respectively. In order to calculate the stresses in InGaAs/ graded InGaAs/GaAs, the same values of a,, a 1, E1 and v, were used as those of our previous study [3]. The stress distribution in InGaAs/ graded InGaAs/GaAs was calculated using Eqs. (6)—(11) on the basis of three-dimensional deformation, and was compared with that calculated on the basis of one-dimensional deformation [5].
numerically solved and correct values of F,-, R, d,, a~and o-,- are determined. The stress distributions at 25°C in the
E~da’J
g
E1d1a~
g
R2 =4
E1d~
S.=—+1 l—v~ d1a~
Eqs. (6)—(10) cannot be analytically calculated 1.Therefore, are because these equations are made up they of two unknown variables d~and a~
E1d~a~
g
1 =6
F,
In these ues of theequations, strain, Si the andpositive ~ signify and thenegative tensile and val-
lkdk + i,d, — i.d~\
~
tk —
E1 ‘
—
E ‘k~ k =
lows:
E1d1a~ i(1 v)
i—i
X
concept, Eqs. (6)—(10) are given as new expressions of F, and R. The stress per unit area in the ith imaginary thin layer at the temperature T is given as fol-
i’~
,/=1
669
3]
)
j g ~ 1—1
In09Ga01As/graded InGaAs/GaAs structures were calculated on the basis of the three- and one-dimensional deformations, as shown in Figs. 3 and 4, respectively. The stress is shown as a function of the distance from the bottom of the GaAs layer. It was calculated for various thick-
(9)
~
0
Ed a~
nb’ I
-
1~(1 v1) —
2.0
at 25°C
Thickness of graded layer. a 6pm b. 8pm c 65pm
...—
E ikdk+ljdi_iJdj.
o
.D_-1\
~
/
GaAs
(10) In the traditional equations, the width of the layer is assumed to be unity and the thickness of the layer is assumed not to vary even after deformation. In this model, the width of the layer Nat’, varies in the same way as the length of the layer, Na~,and the thickness of the layer varies from i~ to d~ after deformation. On the basis of this
I
In0 9Ga01As/graded/GaAs
j—1
k=1
I
Three—dimensional
ThGJAS -
-40-
1 -
0
aLb
C
Thickness of InGoAs ~6pm
~ 200 300 400 Distance from bottom of GaAs (pm)
-
500
Fig. 3. Stress at 25°C in In0 9Ga0 1As/graded layer/GaAs structures on the basis of three-dimensional deformation.
670
K Nakajima /Journal of Crystal Growth 137 (1994) 667—670 5010
I One—dimensional
2.0
I
at 25°C C
-
.~
—-\
~
C—
0 9Ga01As/graded/GaAs
formation varies much more than that calculated on the basis of one-dimensional deformation.
Thickness of graded layer a 6pm b. 18 pm a 65 pm
As known from these results, the difference between the two results calculated on the basis of the three- and one-dimensional deformations is
I
I
In
=
>‘ -
~
-2.0
-4.0
‘b
fairly large and cannot be ignored. It is necessary to calculate the stress on the basis of three-dimensional deformation in order to determine the
InGaAs
Thickness of InGaAs . 6pm
-
I
0
“-c
I
I
I
00 200 300 400 DistanCe from bottom of GaAs (pm)
500
precise stress distribution in the semiconductor heterostructures. The author wishes to acknowledge the assistance of Mrs. Furuya.
Fig. 4. Stress at 25°C in In09Ga01As/graded layer/GaAs structures on the basis of one-dimensional deformation.
References nesses of the InGaAs graded layer. Comparing Fig. 3 with Fig. 4, the stress calculated on the basis of three-dimensional deformation is larger than that calculated on the basis of one-dimensional deformation. We also obtained the result that the perpendicular lattice constant a’ calculated on the basis of three-dimensional de-
[1] S. Timoshenko, J. Opt. Soc. & Rev. Sci. Instr. 11(1925) 233. [2] S.N.G. Chu, A.T. Macrander, K.E. Strege and W.D. Johnston, Jr., J. Appl. Phys. 57 (1985) 249. [3] K. Nakajima, J. Crystal Growth 126 (1993) 511. [4] K. Nakajima, J. Crystal Growth 113 (1991) 477. [5] K. Nakajima, J. Crystal Growth 121 (1992) 278. [6] K. Nakajima, J. AppI. Phys. 72 (1992) 5213.
Journal of Crystal Growth 137 (1994) 667—670
Letter to the Editors
Stress calculation based on three-dimensional deformation of InGaAs/GaAs heterostructures with a graded layer Kazuo Nakajima Fujitsu Laboratories Limited, Atsug4 Morinosato-Wakamiya 10-1, Atsugi 243-01, Japan (Received 30 March 1993; manuscript received in finalform 30 November 1993)
Abstract A stress calculation model based on three-dimensional deformation of multilayer heterostructures composed of semiconductor crystals is proposed by expanding our reported two-dimensional deformation model. Using this model, the stress distribution in InGaAs/ graded InGaAs layer/GaAs heterostructures is calculated. The calculated result based on three-dimensional deformation is compared with that based on one-dimensional deformation.
The calculation of stress distribution in semiconductor heterostructures is very important to design the best crystal structures with small stress. Many researchers have reported calculation models of stress distribution on the basis of Timoshenko’s model [1]. However, an actual unit cell of isotropic semiconductor crystals deforms three-dimensionally. Therefore, the stress distribution in the semiconductor crystals should be calculated by taking account of effects of threedimensional deformation. Chu et al. [2] and Nakajima [3] have considered two-dimensional deformation of the strip and proposed stress calculation models. However, stress calculation models based on three-dimensional deformation have not been reported until now. In this work, a stress calculation model based on three-dimensional deformation of multilayer heterostructures composed of semiconductor crystals is proposed by expanding our reported two-dimensional model [31.In order to clearly know the difference between the calculated re-
suits based on the three- and one-dimensional deformations, stress distributions in InGaAs/ GaAs heterostructures with an InGaAs graded layer were calculated using this model. For the deformation of heterostructures cornposed of semiconductor crystals, the unit cell is three-dimensionally deformed, as shown in Fig. 1, where I~and 1~ are the tangential and perpen-
—~
,‘
Fx
/
1’ d
1
I
‘
/ -~~‘
a1
I
1
a,~ ~-——
—-~
Fig. 1. Unit cell of an isotropic crystal deformed three-dimensionally by bi-axial forces.
0022-0248/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0022-0248(93)E0740-X
668
K. ATakajima /Journal of Crystal Growth 137 (1994) 667—6 70
axis perpendicular to the heterointerface. Each )d
9
InGoAs ~
)d
M graded tB
GaAs
~5
Nal
N~8
Nt~
R
6
)
layer experiences a face force F1 and a moment Ml•The accurate bending moment, M1, for the multilayer heterostructure composed of many imaginary thin layers is derived on the basis of three-dimensional deformation:
)di
=
(1 —i1)R
=
F-al
f1Za~
dz
1
)b) layer/GaAs Fig. 2. Shematica)geometries of an InGaAs/graded heterostructure, (a) before and (h) after three-dimensional deformation.
dicular lengths of the unit cell before deformation, respectively, and a~’and d, are the tangential and perpendicular lengths of the unit cell after deformation, respectively. F~and F~are the face forces, where x and y are tangential axes parallel to the heterointerface. The schematic
3(1
—
R
has a width of Ni1, a length of NI,, a thickness of i~,a thermal expansion coefficient of a1, a Young modulus of F1, a Poisson ratio of t~, and a lattice constant of I,. Total number of imaginary thin layers is g. Fig. 2b shows the heterostructure after deformation. The ith perturbed imaginary thin layer after deformation has a width of Na~,a length of Na~,a thickness of d1, and a tangential 1,as shown in Fig. 2b. The lattice constant of a~ perturbed heterostructure has a curvature radius of R and a neutral plane at z = 0, where z is a
—
—
d~
~
k
k= I
11~)
—
=
I
(1) where 5 is the shift of the neutral axis of the bending heterostructure from z = 0. According to the balance of general forces and moments, we have g
I~-= 0,
~
(2)
=
g
+
~,
d.
/— I
g
M.
~,
F,
~
d 1
geometries of the lnGaAs/graded lnGaAs/ GaAs heterostructure are given in Fig. 2. The 1A, lengthandof Nl~, eachwhere layer ‘A’ before deformation N NIB 1B and i~are theislattice constants of the GaAs, graded InGaAs and InGaAs layers, respectively, and N is the number of the constituent cells of the layers. The thickness of each layer is tA, tB and t~., as shown in Fig. 2a. In this calculation, it is assumed that each layer consists of many imaginary thin layers [3—61. The ith imaginary thin layer before deformation
i—I
.
d~
~
+
=
0,
(3)
J1
1’
where the face force acting At on the a unit length of the F, ithisimaginary thin layer. interface between the top of the ith imaginary thin layer and the bottom of the (i + 1)th imaginary thin layer in the perturbed heterostructure, the tangential lengths should be equal because of their coherency: N
+ ‘
=
i’~(1 ~ F, E 1d1a~I
N ,
+ ‘~
+
2R
l,±~(1 v1±1)F,~1 _______ E1±1d1±1a~’~1 2R —
—
(1
K Nakajima /Journal of Crystal Growth 137 (1994) 667—6 70
of the ith imaginary thin layer. Using Eqs. (2) and (4), we have
l,(1
E,d,a~ g E1d1a~ ~ —
—
[~
[1
x
~=
/
tJ. =
k
1
E1d1a~
g
=
2)
1
l~)l. j
(i~
—
l~(1 v1)
~=
—
(6)
Substituting Eqs. (1) and (6) into Eq. (3), we have 1/R = R1/( R,
+
R3),
(7)
where R
~.
i=1
—
v.)
)
x(2 j=I Ed1+d~ ~,
1=1
Edk-~) k=I
=
3 ~ =
i
1
i—i
k=1
li), (8)
—
/ i—i
~dk-8)
-( /
\k1
1d~a~I ~ (2 ~ d~+ d, I~(1 v~) j 1
x(2 ~
—
E,a~
E
g
R3
i~(1 v) (1,
~i
7~•~~i— c~)i=1
~[(
2(1—L’ 1)R~
(11)
compressive ones, respectively. In order to calculate the stresses in InGaAs/ graded InGaAs/GaAs, the same values of a,, a 1, E1 and v, were used as those of our previous study [3]. The stress distribution in InGaAs/ graded InGaAs/GaAs was calculated using Eqs. (6)—(11) on the basis of three-dimensional deformation, and was compared with that calculated on the basis of one-dimensional deformation [5].
numerically solved and correct values of F,-, R, d,, a~and o-,- are determined. The stress distributions at 25°C in the
E~da’J
g
E1d1a~
g
R2 =4
E1d~
S.=—+1 l—v~ d1a~
Eqs. (6)—(10) cannot be analytically calculated 1.Therefore, are because these equations are made up they of two unknown variables d~and a~
E1d~a~
g
1 =6
F,
In these ues of theequations, strain, Si the andpositive ~ signify and thenegative tensile and val-
lkdk + i,d, — i.d~\
~
tk —
E1 ‘
—
E ‘k~ k =
lows:
E1d1a~ i(1 v)
i—i
X
concept, Eqs. (6)—(10) are given as new expressions of F, and R. The stress per unit area in the ith imaginary thin layer at the temperature T is given as fol-
i’~
,/=1
669
3]
)
j g ~ 1—1
In09Ga01As/graded InGaAs/GaAs structures were calculated on the basis of the three- and one-dimensional deformations, as shown in Figs. 3 and 4, respectively. The stress is shown as a function of the distance from the bottom of the GaAs layer. It was calculated for various thick-
(9)
~
0
Ed a~
nb’ I
-
1~(1 v1) —
2.0
at 25°C
Thickness of graded layer. a 6pm b. 8pm c 65pm
...—
E ikdk+ljdi_iJdj.
o
.D_-1\
~
/
GaAs
(10) In the traditional equations, the width of the layer is assumed to be unity and the thickness of the layer is assumed not to vary even after deformation. In this model, the width of the layer Nat’, varies in the same way as the length of the layer, Na~,and the thickness of the layer varies from i~ to d~ after deformation. On the basis of this
I
In0 9Ga01As/graded/GaAs
j—1
k=1
I
Three—dimensional
ThGJAS -
-40-
1 -
0
aLb
C
Thickness of InGoAs ~6pm
~ 200 300 400 Distance from bottom of GaAs (pm)
-
500
Fig. 3. Stress at 25°C in In0 9Ga0 1As/graded layer/GaAs structures on the basis of three-dimensional deformation.
670
K Nakajima /Journal of Crystal Growth 137 (1994) 667—670 5010
I One—dimensional
2.0
I
at 25°C C
-
.~
—-\
~
C—
0 9Ga01As/graded/GaAs
formation varies much more than that calculated on the basis of one-dimensional deformation.
Thickness of graded layer a 6pm b. 18 pm a 65 pm
As known from these results, the difference between the two results calculated on the basis of the three- and one-dimensional deformations is
I
I
In
=
>‘ -
~
-2.0
-4.0
‘b
fairly large and cannot be ignored. It is necessary to calculate the stress on the basis of three-dimensional deformation in order to determine the
InGaAs
Thickness of InGaAs . 6pm
-
I
0
“-c
I
I
I
00 200 300 400 DistanCe from bottom of GaAs (pm)
500
precise stress distribution in the semiconductor heterostructures. The author wishes to acknowledge the assistance of Mrs. Furuya.
Fig. 4. Stress at 25°C in In09Ga01As/graded layer/GaAs structures on the basis of one-dimensional deformation.
References nesses of the InGaAs graded layer. Comparing Fig. 3 with Fig. 4, the stress calculated on the basis of three-dimensional deformation is larger than that calculated on the basis of one-dimensional deformation. We also obtained the result that the perpendicular lattice constant a’ calculated on the basis of three-dimensional de-
[1] S. Timoshenko, J. Opt. Soc. & Rev. Sci. Instr. 11(1925) 233. [2] S.N.G. Chu, A.T. Macrander, K.E. Strege and W.D. Johnston, Jr., J. Appl. Phys. 57 (1985) 249. [3] K. Nakajima, J. Crystal Growth 126 (1993) 511. [4] K. Nakajima, J. Crystal Growth 113 (1991) 477. [5] K. Nakajima, J. Crystal Growth 121 (1992) 278. [6] K. Nakajima, J. AppI. Phys. 72 (1992) 5213.