nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Strain and displacement tensors along different growth directions

Author: Stefan Birner

If you want to obtain the input files that are used within this tutorial, please check if you can find them in the installation directory.
If you cannot find them, please submit a Support Ticket.
-> 1DstrainN11.in / *_nnp.in - input file for the nextnano³ and nextnano++ software
-> 2DstrainN11.in / *_nnp.in - input file for the nextnano³ and nextnano++ software
-> 3DstrainN11.in / *_nnp.in - input file for the nextnano³ and nextnano++ software
 

AlAs/InAs/AlAs structure on InP

• This input file simulates a InP/InAs/AlAs structure grown pseudomorphically on InP: 1DstrainN11.in
  The structure is grown pseudomorphically on InP, i.e. the InAs is compressively strained, the AlAs is tensilely strained. The growth direction [N11] is along x, the interfaces are in the (y,z) plane.
• This tutorial examines the strain tensors in the crystal and simulation coordinate system as well as the corresponding displacement tensors.
  The following parameters are used:
   

  	InP (substrate)	InAs	AlAs
  Lattice constant a (nm)	0.58697	0.60583	0.56611
  Elastic constant c11 (GPa)	101.1	83.29	125.0
  Elastic constant c12 (GPa)	56.1	45.26	53.4
  Elastic constant c44 (GPa)	45.6	39.59	54.20

   

• [100] growth direction
  (y direction along [010])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 1 0 0  ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 0  ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...

  The strain tensors in the crystal and simulation system are identical. Off-diagonal strain components e_ij are zero. The hydrostatic strain is defined as the trace of the strain tensor (e_hydro=e_xx+e_yy+e_zz). It is useful to talk about parallel strain (e_||=e_yy=e_zz) and perpendicular (e_{_|_}=e_xx) strain with respect to the interface coordinate system.
     Biaxial strain (in plane of interface):
        InAs: e_|| = e_yy = e_zz = ( a_substrate - a_layer ) / a_layer = -0.0311308   (3.1 % lattice mismatch)
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} = e_xx = - D¹⁰⁰ e_|| = - 2 (c₁₂/c₁₁) e_|| = 0.0338332
  For [100] growth direction the strain tensors and the displacement tensors coincide.
   

  [100]	exx=e_|_	eyy=ezz=e||	ehydro
  InAs	0.338332E-001	-0.311308E-001	-0.284285E-001
  AlAs	-0.314829E-001	0.368480E-001	0.422130E-001

   

• [011] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 0 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...

The strain tensors in the crystal and simulation system are not identical any more. Off-diagonal strain components e_ij are zero apart from e_yz^cr. The hydrostatic strain is independent of coordinate system.
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D⁰¹¹e_|| = - (c₁₁+3c₁₂-2c₄₄) / (c₁₁+c₁₂+2c₄₄) e_|| = 0.0209642
  Crystal system: InAs: e_yy^cr = e_zz^cr = - ??? e_|| = -0.00508332
                          InAs: e_yz^cr = - (c₁₁+2c₁₂) / (c₁₁+c₁₂+2c₄₄) e_|| = 0.0260475
  For [011] growth direction the strain tensors and the displacement tensors coincide.
  Here it is interesting to note that e_xx^cr = e_||.
   

  [011]	exxsim=e_|_	eyysim=ezzsim=e||	ehydro
  InAs	0.209642E-001	-0.311308E-001	-0.412975E-001
  AlAs	-0.227152E-001	0.368480E-001	0.509807E-001

  [011]	exxcr	eyycr=ezzcr	eyzcr
  InAs	-0.311308E-001	-0.508332E-002	0.260475E-001
  AlAs	0.368480E-001	0.706638E-002	-0.297816E-001

   

• [111] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 1 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...

The strain tensors in the crystal and simulation system are not identical. Off-diagonal strain components e_ij are still zero in the simulation system but not in the crystal system.
  Simulation system:
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D¹¹¹e_|| = - 2(c₁₁+2c₁₂-2c₄₄) / (c₁₁+2c₁₂+4c₄₄) e_|| = 0.0177374
  Crystal system: InAs: e_xx^cr = e_yy^cr = e_zz^cr = - (-4c₄₄) / (c₁₁+2c₁₂+4c₄₄) e_|| = -0.0148414
                          InAs: e_xy^cr = e_xz^cr = e_yz^cr = - (c₁₁+2c₁₂) / (c₁₁+2c₁₂+4c₄₄) e_|| = 0.0162894
  
  For [111] growth direction the strain tensors and the displacement tensors coincide.
   

  [111]	exxsim=e_|_	eyysim=ezzsim=e||	ehydro
  InAs	0.177374E-001	-0.311308E-001	-0.445243E-001
  AlAs	-0.202721E-001	0.368480E-001	0.534238E-001

  [111]	exxcr=eyycr=ezzcr	exycr=exzcr=eyzcr
  InAs	-0.148414E-001	0.162894E-001
  AlAs	0.178079E-001	-0.190400E-001

   

• [211] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 2 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...

Off-diagonal strain components e_ij are not zero any more in the simulation system. For [211] growth direction the strain tensors and the displacement tensors do not coincide any more because the displacement tensors are not symmetric any more. The diagonal entries still coincide e_ii=u_ii.
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D²¹¹e_|| = - (D²¹¹=???) e_|| = 0.0228540
   

  [211]	exxsim=e_|_	eyysim=ezzsim=e||	exzsim	ehydro		uzxsim=duz/dx	uxzsim=dux/dz
  InAs	0.228540E-001	-0.311308E-001	-0.101187E-001	-0.394077E-001		-0.202373E-001 = 2exzsim	0
  AlAs	-0.234699E-001	0.368480E-001	0.623832E-002	0.502260E-001		0.124766E-001 = 2exzsim	0

  [211]	exxcrr	eyycr=ezzcr	exycr=exzcr	eyzcr	uxycr=uxzcr	uyxcr=uzxcr	uyzcr=uzycr
  InAs	0.143990E-001	-0.269034E-001	0.156100E-001	0.422749E-002	0.227649E-001	0.845498E-002	0.422749E-002 = eyzcr
  AlAs	-0.924548E-002	0.297358E-001	-0.186356E-001	-0.711220E-002	-0.230467E-001	-0.142244E-001	-0.711220E-002 = eyzcr

  One can see that the displacement tensor for the simulation system is rather nice as the following off-diagonal components are zero:
  - u_xy^sim, u_xz^sim, u_yz^sim  are always zero for 1D structures grown along the x axis of the simulation system.
  - u_yx^sim, u_zy^sim  are zero in this particular example.
  Here one should recall the definiton of the strain tensor: e_ij = 1/2 (u_ij + u_ji) = 1/2 (du_i/d_j + du_j/d_i)
  The strain tensor is thus always symmetric but the displacement tensor only for growth directions along [001], [011], [111]. In fact, for these three directions it is also diagonal in the simulation coordinate system. In this particular example, u_xy^cr is not equal to u_yx^cr.
   

• [311] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 3 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...
See comments for [211] growth direction
  
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D³¹¹e_|| = - (D³¹¹=???) e_|| = 0.0277907
   

  [311]	exxsim=e_|_	eyysim=ezzsim=e||	exzsim	ehydro		uzxsim=duz/dx	uxzsim=dux/dz
  InAs	0.277907E-001	-0.311308E-001	-0.111833E-001	-0.344710E-001		-0.223666E-001 = 2exzsim	0
  AlAs	-0.267795E-001	0.368480E-001	0.714093E-002	0.469164E-001		0.142819E-001 = 2exzsim	0

  [311]	exxcrr	eyycr=ezzcr	exycr=exzcr	eyzcr	uxycr=uxzcr	uyxcr=uzxcr	uyzcr=uzycr
  InAs	0.257044E-001	-0.300877E-001	0.110373E-001	0.104316E-002	0.189451E-001	0.312949E-002	0.104316E-002 = eyzcr
  AlAs	-0.207193E-001	0.338179E-001	-0.141397E-001	-0.303010E-002	-0.191891E-001	-0.909030E-002	-0.303010E-002 = eyzcr

   

• [411] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 4 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...
See comments for [211] growth direction
  
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D⁴¹¹e_|| = - (D⁴¹¹=???) e_|| = 0.0302133
   

  [411]	exxsim=e_|_	eyysim=ezzsim=e||	exzsim	ehydro		uzxsim=duz/dx	uxzsim=dux/dz
  InAs	0.302133E-001	-0.311308E-001	-0.986895E-002	-0.320484E-001		-0.197379E-001 = 2exzsim	0
  AlAs	-0.285575E-001	0.368480E-001	0.646539E-002	0.451384E-001		0.129308E-001 = 2exzsim	0

  [411]	exxcrr	eyycr=ezzcr	exycr=exzcr	eyzcr	uxycr=uxzcr	uyxcr=uzxcr	uyzcr=uzycr
  InAs	0.296003E-001	-0.308243E-001	0.820438E-002	0.306496E-003	0.151828E-001	0.122598E-002	0.306496E-003 = eyzcr
  AlAs	-0.253540E-001	0.352462E-001	-0.109788E-001	-0.160176E-002	-0.155505E-001	-0.640704E-002	-0.160176E-002 = eyzcr

   

• [511] growth direction
  (y direction along [01-1])
  $domain-coordinates
 ...
 hkl-x-direction-zb     = 5 1  1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = 0 1 -1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...
See comments for [211] growth direction
  
     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D⁵¹¹e_|| = - (D⁵¹¹=???) e_|| =  0.0314570
   

  [511]	exxsim=e_|_	eyysim=ezzsim=e||	exzsim	ehydro		uzxsim=duz/dx	uxzsim=dux/dz
  InAs	0.314570E-001	-0.311308E-001	-0.847870E-002	-0.308047E-001		-0.169574E-001 = 2exzsim	0
  AlAs	-0.295219E-001	0.368480E-001	0.563906E-002	0.441740E-001		0.112781E-001 = 2exzsim	0

  [511]	exxcrr	eyycr=ezzcr	exycr=exzcr	eyzcr	uxycr=uxzcr	uyxcr=uzxcr	uyzcr=uzycr
  InAs	0.312618E-001	-0.310333E-001	0.648318E-002	0.975671E-004	0.124785E-001	0.487835E-003	0.975671E-004 = eyzcr
  AlAs	-0.275593E-001	0.358666E-001	-0.889403E-002	-0.981324E-003	-0.128815E-001	-0.490662E-002	-0.981324E-003 = eyzcr

   

• [321] growth direction
  (y direction along [-111])
  $domain-coordinates
 ...
 hkl-x-direction-zb     =  3 2 1 ! Miller indices of x coordinate axis [1 0 0]
 hkl-y-direction-zb     = -1 1 1 ! Miller indices of y coordinate axis [0 1 0]
 growth-coordinate-axis = 1 0 0
 ...

     Biaxial strain (in plane of interface):
        InAs: e_||=e_yy^sim = e_zz^sim = ( a_substrate - a_layer ) / a_layer = -0.0311308   (same as for [100])
     Uniaxial strain (perpendicular to interface):
        InAs: e_{_|_} =e_xx^sim = - D³²¹e_|| = - (D³²¹=???) e_|| = 0.0223345
   

  [321]	exxsim=e_|_	eyysim=ezzsim=e||	exysim	exzsim	ehydro	uyxsim=duy/dx	uzxsim=duz/dx	uxysim=uxzsim
  InAs	0.223345E-001	-0.311308E-001	-0.871721E-002	0.139475E-002	-0.399271E-001	-0.174344E-001 = 2exysim	0.278950E-002 = 2exzsim	0
  AlAs	-0.232538E-001	0.368480E-001	0.528954E-002	-0.515392E-003	0.504421E-001	0.105791E-001 = 2exysim	-0.103078E-002 = 2exzsim	0

  [321]	exxcrr	eyycr	ezzcr	exycr	exzcr	eyzcr
  InAs	0.116554E-001	-0.221557E-001	-0.294269E-001	0.209935E-001	0.968697E-002	0.394774E-002
  AlAs	-0.681357E-002	0.232809E-001	0.339748E-001	-0.247292E-001	-0.115866E-001	-0.626491E-002

  [321]	uxycr	uyxcr	uxzcr	uzxcr	uyzcr	uzycr
  InAs	0.285242E-001	0.134627E-001	0.142621E-001	0.511185E-002	0.448757E-002	0.340790E-002
  AlAs	-0.291077E-001	-0.203506E-001	-0.145538E-001	-0.861942E-002	-0.678354E-002	-0.574628E-002

   

• Simulation system
  Now we plot the above results for [N11] growth directions. Note that [100] is identical to ["infinity"11] growth direction. All plotted quantities are given in the simulation system.
  
  For [011], [111] and [100], the off-diagonal strain tensor component e_xz^sim is zero. For the "high symmetry growth directions" the strain and displacement tensors are diagonal. (But this does not necessarily mean that shear strain is absent. Shear strain is only absent if the strain tensor is diagonal in the crystal system, i.e. only for [100] growth direction.)
  The hydrostatic strain e_hy=Tr(e_ij) is always negative because we have a volume reduction (compressive strain). It has the largest value in magnitude for [111] and the smallest in magnitude for [100]. It shows exactly the same behaviour as the curve for e_{_|_}^sim, i.e. e_{_|_}^sim is a measure for the volume deformation in the growth direction.
  The strain in the plane of interface e_||^sim is constant for all growth directions.
  Note that the highest value for offdiagonal strain e_xz occurs for [311] growth direction (highest shear strain of any orientation).
• Crystal system
  Now we plot the above results for [N11] growth directions. Note that [100] is identical to ["infinity"11] growth direction. All plotted quantities are given in the crystal system.
  
  The hydrostatic strain e_hy=Tr(e_ij) is identical to the plot of the simulation system. (The trace of a matrix is independent of coordinate system.) It is always negative because we have a volume reduction (compressive strain). It has the largest value in magnitude for [111] and the smallest in magnitude for [100].
  For [111] growth direction, e_xx^cr=e_yy^cr=e_zz^cr and e_xy^cr=e_xz^cr=e_yz^cr.
  All off-diagonal components are zero only for [100] growth direction (no shear strain). For all other orientations we have shear strain. Shear strain (with respect to the crystal system) is responsible for piezoelectric effects.
  The off-diagonal components e_xy^cr=e_xz^cr are zero for [011] growth direction but the off-diagonal component e_yz^cr is not.
• Here the same picture is plotted again (crystal system) but this time including the important components of the displacement tensor that are responsible for the fact that the displacement tensor is not symmetric any more for [211], [311], [411] and [511].
  The strain tensor is defined as: e_ij = 1/2 (u_ij + u_ji) = 1/2 (du_i/d_j + du_j/d_i)
  Thus:

  exycr = 1/2 (uxycr+uyxcr).

  For the high symmetry growth direction [100], [011], [111] the displacement tensor is symmetric because

  uxycr=uyxcr=exycr.

  

• Strain has important effects:
  - piezoelectric fields: See Tutorial on piezoelectricity.
  - shifts and splittings of conduction and valence bands: See Tutorial on deformation potentials.
  - strain changes the k.p Hamiltonian
• Strain can be used to tailor the electronic and optical properties of heterostructures.