2.3 Relaxation and Dislocations
2.3.1 Basic Dislocation Theory
Dislocations Formed by Strain
The growth of a lattice mismatched layer (such as in figure 2.2) has an associated areal elastic energy density stored at the misfit interface, Eh, given by [People and Bean, 1985]: Eh ≈2G 1 +ν 1−ν hǫ2 (2.8)
Whereh is the thickness of the strained layer,G is its bulk shear modulus,ν is Poisson’s ratio and ǫis the strain in the layer (see equation 2.2). As the thickness of
the layer increases,Eh will become large enough to break atomic bonds at the strained interface (this thickness will be discussed in section 2.3.2), and plastic deformation of the crystal will induce relaxation towards its bulk lattice constant. These deformations displace material along lines which form discontinuities in the lattice structure and break the commensurate nature of the misfit interface. The resulting discontinuity is called a
dislocationand is shown in figure 2.8 for a compressively strained interface.
(a) (b)
Figure 2.8: Simplified diagram showing (a) a compressively strained layer (in grey) and (b) partial relaxation of the layer by a dislocation. The line direction of the dislocation is directed into the plane of the page. Adapted from Hull and Bacon [2002].
Burgers Vectors
A dislocation is defined by its Burgers vector, which points along the direction of the displacement of the crystal lattice, with a magnitude equal to the displacement. Burgers vectors can be determined by tracing a closed circuit from lattice point to lattice point
in a perfect crystal and comparing it to the same circuit traced around a dislocation. In the dislocated material, the starting point will not meet the ending point, and the vector needed to close the circuit defines the direction of the Burgers vector (figure 2.9) [Cottrell, 1964].
S F S F
b
(a) (b)
Figure 2.9: Diagram showing (a) the construction of a closed circuit in a perfect crys- talline material starting at point S and finishing at point F, and (b) the failure of the same circuit to close around a dislocation. This is used to define the direction of the Burgers vector, b. Adapted from Cottrell [1964].
The orientation of the Burgers vector relative to the line direction of a dislocation gives rise to a naming convention. If they are mutually perpendicular (as in figure 2.9), the dislocation is termed anedge dislocation. When parallel, the dislocation is ascrew dislocation. Any other angle is called a mixed dislocation, for which the Burgers vector has both edge and screw components. The Burgers vector of a dislocation must be constant along the whole length of the dislocation, otherwise there will be additional displacements introduced into the crystal lattice. Burgers vectors are therefore conserved in splitting reactions, which will be discussed in section 2.5.2.
ary between regions of crystalline material that are displaced relative to each other. Therefore, dislocations cannot terminate within the lattice and must form as a closed loop, or terminate at a free surface (such as the edge of the wafer, an impurity within the crystal, or the surface of the layer). The component of a dislocation that lies at the misfit interface is termed a misfit dislocation and contributes to the relaxation of the strained layer. Threading dislocations are the non-strain relieving components of dislocations that do not lie at the misfit interface. Dislocations formed during growth are “grown into” the material, as deposited material maintains the crystal structure, preserving the presence of dislocations.
Glide of Dislocations in Silicon-Germanium
Dislocations can move under the influence of strain by a process known asglide. Glide occurs by the sliding of one plane of atoms over another, which propagates the threading dislocation component forward, and lengthens the misfit dislocation at the strained interface. This is shown in figure 2.10.
Strained Layer
Substrate Threading Dislocation
Misfit Dislocation
(111) Glide Plane
Figure 2.10: Simplified diagram showing dislocation glide which increases the length of the misfit component at the strained interface.
The glide-planes within the material are normally the set of planes with the highest density of atoms, and the direction of displacement (in which the Burgers vector points) is the direction on that plane in which the atoms are closest packed. In silicon and germanium (and all face-centred cubic and diamond crystals) the highest density of atoms are found on the {111} planes, and the closest packed atoms are along the <101> directions (see figure 2.11). Dislocations in silicon and germanium therefore have a Burgers vector given byb = a
2 <101> (where a is the lattice constant of the material), as this is the shortest atomic translation vector along the closest atomically packed direction. b has a magnitude of 3.9˚A in bulk silicon.
60° 90° ½[110]- ½ [110]- -- - - (111) Glide Plane Misfit Dislocation [110]- (a) (b) [101] [101] [001] [100] [010]
Figure 2.11: Simplified diagram showing the orientations (a) 60◦ and (b) 90◦ Burgers
vectors for a[1¯10]misfit dislocation. Taken from Nash [2005].
The misfit dislocation line directions are formed along<110>directions in silicon and germanium, where the {111} glide-planes intersect the (001) strained interface plane. Burgers vectors (along the <101> directions) form an angle of 60◦ with the
Edge dislocations (where the Burgers vector forms an angle of 90◦ with the
dislocation line) can also form for which the Burgers vector is equal to a2 <110>. Although they each relieve more strain than 60◦ dislocations, (see section 2.5) they
cannot glide along the{111} glide-planes, and can only propagate by a higher energy process known as climb. For this reason 60◦dislocations are the predominant dislocations
at lower strain. Increasing the growth temperatures has been shown to increase the numbers of 90◦ dislocations formed [Hull and Bean, 1989].
Relaxation of a Single Dislocation
Each dislocation contributes to the relaxation of the strained layer along its length, L, by the degree of displacement it causes at the misfit interface. This is determined by resolving the Burgers vector (the magnitude of which gives the total displacement) into the misfit interface plane. Relaxation along the<110>directions is given by the com- ponent of the in-plane Burgers vector,ba, which lies perpendicular to the displacement line direction, u, as shown in figure 2.12.
This component is termed the effective Burgers vector, bef f. By considering the unit cell geometry of the diamond structure in figure 2.12, the magnitude of the Burgers vector is given byb = √ax2+az2
2 , whereax is the in-plane lattice constant (see equation 2.1) and az is the out-of-plane lattice constant of the unit cell. This leads to an equation for the magnitude of the effective Burgers vector [Bugiel and Zaumseil, 1993]:
bef f = ax
2√2 (2.9)
The line density of misfit dislocations, ρM D, in a given area, A, is determined by the total length of misfit dislocations revealed that area, LT, divided by the area.
45° [101] [110]- b u bp beff [001] [010] [100] ax az
Figure 2.12: Schematic representation of the components of a Burgers vector, b, for a 60◦ dislocation orientated along [1¯10] in a single unit cell of base dimension ax and
heightaz.
Together with bef f, the total relaxation per unit area caused by misfit dislocations, δMD, can be determined:
δM D = LT
A bef f =ρM Dbef f (2.10)
With a suitable experimental technique for revealing misfit dislocations at a strained interface, it is possible to calculate the amount of relaxation per unit area using equation 2.10. However, it is assumed that all revealed dislocations are of 60◦ type.
This will not always be the case, as some dislocations in highly strained layers may be of 90◦ type [Hull and Bean, 1989]. Further complications arise in tensile strained layers,