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Local density of states in dislocation and impurity cages
Journal of Non-CrystallineSolids 59 & 60 (1983) 89-92 North-HollandPublishingCompany
89
LOCAL DENSITY OF STATES IN DISLOCATION AND IMPURITY CAGES* Kiyoshi KAWAMURAand Ryoji OHIRA Department of Materials Science, Hiroshima University, Higashisenda-machi, Naka-ku, Hiroshima 730, Japan The local density of states (LDOS) is computed at a site which is enclosed on a square l a t t i c e by four edge dislocations or impurities. Sharp peaks appear near the bounds of energy band, which are attributed to standing waves with nodes on the impurity sites or around the dislocation cores. Electronic structure in amorphous semiconductors is discussed on this simple model. I. INTRODUCTION In amorphous substances, the long range order is lost while the short range order remains.
To describe this nature, i t was proposed that an amorphous
structure can be regarded as a crystal containing in i t highly dense dislocations.
This idea, the dislocation model of amorphous structure, was proposed
by GilmanI and Li 2 and examined by Koizumi and Ninomiya3'4 on calculating the radial d i s t r i b u t i o n function and the scattering intensity function.
However,
we have no idea about the influence of dense dislocations on wave character of electrons.
In Section 2, we consider l a t t i c e s which contain four edge disloca-
tions as i l l u s t r a t e d in Figure l as well as l a t t i c e s with four substitutional impurities at (±L,±L).
The former w i l l be called dislocation cages and the
l a t t e r , impurity cages. Size of a cage is represented by L.
From the study
of the local density of states (LDOS) computed in the tight-binding approximat i o n , we deduce electronic states within these cages. The theory obtained for such simple models w i l l be applied in Section 3 to discuss the electronic structure in amorphous semiconductors. 2. ANALYSIS OF LDOS FOR MODELLATTICES Suppose that an electron of s-character hops among the sites in Figure I. The transfer matrix is assumed constant and has a nonvanishing value only between pairs of sites, each of which are connected by a line.
The tight-binding
equation for the l a t t i c e Green function G(n,n') has been solved numerically. Then, LDOS at the s i t e n is evaluated from the imaginary part of G(~,n). *The present work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. 0022-3093/83/0000-0000/$03.00 © 1983 North-Holland/Physical Society of Japan
K. Kawamura, R. Ohira / Local density o f states
90
mllllmlmlll
m_mmm_mllm
0.5
i
-4.0
I
i
0.0
4.G
FIGURE 2
FIGURE 1 A dislocation cage of the size L=3.
LDOS at the center of the cage in Figure I .
In Figure 2, LDOS at the center of a cage is shown f o r L=3 as a function of the energy scaled by ]Tp(T represents the t r a n s f e r m a t r i x ) .
The bounds of the
spectrum l i e at ±41TI in the present model and LDOS e x h i b i t s o s c i l l a t i o n . p a r t i c u l a r , there is a sharp peak in the v i c i n i t y
In
of the upper bound. Above the
peak in energy, LDOS is suppressed appreciably from i t s value in the perfect lattice
and goes to zero continuously at the upper bound.
LDOS w i l l be c a l l e d the band t a i l
from now.
b i t s n e i t h e r a sharp peak nor a band t a i l .
This part of small
Around the lower bound, LDOS e x h i A s i m i l a r computation has been done
f o r LDOS at the center of i m p u r i t y cages, in which p o t e n t i a l due to i m p u r i t i e s is negative and l o c a l i z e d only at i m p u r i t y s i t e s .
For a large value of IV/TI
LDOS is symmetric with respect to E=O and sharp peaks and band t a i l s served near the both bounds of the energy band.
are ob-
With decreasing value of IVl
LDOS becomes asymmetric; the sharp peak near the upper bound disappears when IV/Tl~l,whereas the peak near the lower bound remains at the same p o s i t i o n . We have found t h a t the p o s i t i o n Ep of the sharp peak varies with the size of the cage approximately according to the r u l e
4L = 2~(4-1Ep/TI) - I / 2 ,
(I)
which holds not only for the dislocation cage but also for the impurity cage. To discuss the sharp peak near the spectral bounds, we can borrow the concept of the standing wave of Lifshitz 5. An electron belonging to a level near the lower bound of the energy band has in a perfect crystal large wavelength represented by the quantity on the right-hand side of (1).
On the other hand, the
wave function of an electron with E~4ITI has nearly equal values but opposite signs on the neighboring two sites.
The spatially varying amplitude has the
wavelength given by the quantity on the right-hand side of (1).
Therefore, the
relation (1) suggests that there stand stationary waves which have nodes at the
K. Kawarnura, R. Ohira / Local density o f states
91
impurity sites or around the dislocation axes from certain reasons. An impurity with the negative value of potential is a center of a repulsive force for electrons with E>O and excludes them from the impurity site when IVI is much larger than the band width.
This explanation is consistent with
the observation that the sharp peak near the upper bound of the spectrum disappears when IVl is small.
Electrons with long wavelength cannot have ampli-
tude on the impurity sites either, when electrons are attracted by the impurities.
This is because the wave function of a state within the band should be
orthogonal to the localized wave function of a bound state.
Since the bound
state exists for any value of IVi in our two dimensional l a t t i c e , this explanation is consistent with the fact that the sharp peak of LDOS near the lower bound does not move apparently with the value of V. When we go along an arbitrary path which encloses a single dislocation, we always passes over odd number of sites.
Such a structure of a dislocated l a t -
tice is incompatible with the rapidly varying wave function with E=41Ti.
A
similar situation has been discussed by Yonezawaand Cohen6, when they study the influence of odd membered rings on the energy spectrum of antibonding states in tetrahedrally coordinated semiconductors.
Thus, electrons in the
energy states near the upper bound of the energy band are excluded from the dislocation cores, whereas electrons with E=-4ITI, which have large wavelength, can exist around the dislocation cores. 3. DISCUSSIONS The preceding
theory on the cages of impurities and edge dislocations is
applied in this section to the discussion on electronic states in amorphous semiconductors.
In the following discussions, the dislocation model is employ-
ed as for the structural model of amorphous substances. 3.1. Influence of topologically singular structure of dislocations. In the preceding section, LDOSwas described as a function of energy.
How-
ever, energy is not an appropriate variable to describe the influence of the topologically singular structure of dislocations in general l a t t i c e s .
In fact,
the exclusion of electrons from the core region discussed above is the same phenomenon as the shadow7 of a dislocation observed in the scattering theoretical solution; the amplitude in the shadow region is proportional to cos(k'b/2), where k is the wave vector of the incident wave and ~ is the Burgers vector. Therefore, the dislocation excludes an electron with the wave vector k for which k'b takes the value around ~. In Si and Ge, the bottom of the conduction band is located in such a region. Then, the electrons in the v i c i n i t y of the lower bound of the conduction band may be excluded from the dislocation cores and form standing wave states.
K. Kawamura, R. Ohira / Local density of states
92
Although the dislocation in the preceding section can be a model of the odd membered ring of Yonezawaand Cohen6, the existence of odd memberedrings is not the necessary condition for the appearenceof the standing wave discussed above. This is because the wave number is concerned with the translational motion among unit cells instead of the change of a wave function within a unit cell.
Therefore, electrons in the bonding states are also excluded from dislo-
cation cores, when their wave number satisfies k'b=~. 3.2. Influence of dangling bonds on extended electronic states. According to the study of Jones8, some conduction band states are excluded from the dislocation core, since they should be orthogonal to the localized dangling bond states.
The same orthogonality of the band electron to the bound
state yields the reduction of LDOS near the lower bound of the energy band in the impurity cage discussed above. Therefore, the cage of impurities which attract electrons may be a model of the cage of dangling bonds. The influence of the dangling bond states may be dominant for electrons with large wavelength. In an amorphous semiconductor with dangling bonds, electrons near the r-point are expected to form the standing wave states. 3.3. Binary alloy model and dislocation mode] for amorphous substances. Since the structures of LDOS near the upper bound of spectrum in Figures l and 2 have commoncharacter, to investigate the influence of the electronic states of short wavelength, the dislocation may be replaced with a line of impurities with repulsive force.
I f this is allowed, we can discuss electrons
of short wavelength in amorphous semiconductors of topologically singular structure with the help of knowledge about the random binary alloy.
REFERENCES I) J.J. Gilman, J. Appl. Phys. 44 (1973) 675. 2) J.C.M. Li, Distinguished Lectures in Materials Science (Macel-Decker, New York,1974) 3) H. Koizumi and T. Ninomiya, J. Phys. Soc. Japan 44 (1978) 898. 4) H. Koizumi and T. Ninomiya, J. Phys. Soc. Japan 49 (1980) I022. 5) I.M. Lifshitz, Advan. Phys. 13 (1964) 483. 6) F. Yonezawaand M.H. Cohen, Fundamental Physics of Amorphous Semiconductors, ed. F. Yonezawa, in: Springer Series in Solid-State Sciences, Voi.25, eds. M. Cardona and P. Fulde (Springer-Verlag, Berlin 1981) ppl19-144. 7) K. Kawamura, Z. Physik 21 (1982) 201. 8) R. Jones, Phil. Mag. 35 (1977) 57.
89
LOCAL DENSITY OF STATES IN DISLOCATION AND IMPURITY CAGES* Kiyoshi KAWAMURAand Ryoji OHIRA Department of Materials Science, Hiroshima University, Higashisenda-machi, Naka-ku, Hiroshima 730, Japan The local density of states (LDOS) is computed at a site which is enclosed on a square l a t t i c e by four edge dislocations or impurities. Sharp peaks appear near the bounds of energy band, which are attributed to standing waves with nodes on the impurity sites or around the dislocation cores. Electronic structure in amorphous semiconductors is discussed on this simple model. I. INTRODUCTION In amorphous substances, the long range order is lost while the short range order remains.
To describe this nature, i t was proposed that an amorphous
structure can be regarded as a crystal containing in i t highly dense dislocations.
This idea, the dislocation model of amorphous structure, was proposed
by GilmanI and Li 2 and examined by Koizumi and Ninomiya3'4 on calculating the radial d i s t r i b u t i o n function and the scattering intensity function.
However,
we have no idea about the influence of dense dislocations on wave character of electrons.
In Section 2, we consider l a t t i c e s which contain four edge disloca-
tions as i l l u s t r a t e d in Figure l as well as l a t t i c e s with four substitutional impurities at (±L,±L).
The former w i l l be called dislocation cages and the
l a t t e r , impurity cages. Size of a cage is represented by L.
From the study
of the local density of states (LDOS) computed in the tight-binding approximat i o n , we deduce electronic states within these cages. The theory obtained for such simple models w i l l be applied in Section 3 to discuss the electronic structure in amorphous semiconductors. 2. ANALYSIS OF LDOS FOR MODELLATTICES Suppose that an electron of s-character hops among the sites in Figure I. The transfer matrix is assumed constant and has a nonvanishing value only between pairs of sites, each of which are connected by a line.
The tight-binding
equation for the l a t t i c e Green function G(n,n') has been solved numerically. Then, LDOS at the s i t e n is evaluated from the imaginary part of G(~,n). *The present work was supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture. 0022-3093/83/0000-0000/$03.00 © 1983 North-Holland/Physical Society of Japan
K. Kawamura, R. Ohira / Local density o f states
90
mllllmlmlll
m_mmm_mllm
0.5
i
-4.0
I
i
0.0
4.G
FIGURE 2
FIGURE 1 A dislocation cage of the size L=3.
LDOS at the center of the cage in Figure I .
In Figure 2, LDOS at the center of a cage is shown f o r L=3 as a function of the energy scaled by ]Tp(T represents the t r a n s f e r m a t r i x ) .
The bounds of the
spectrum l i e at ±41TI in the present model and LDOS e x h i b i t s o s c i l l a t i o n . p a r t i c u l a r , there is a sharp peak in the v i c i n i t y
In
of the upper bound. Above the
peak in energy, LDOS is suppressed appreciably from i t s value in the perfect lattice
and goes to zero continuously at the upper bound.
LDOS w i l l be c a l l e d the band t a i l
from now.
b i t s n e i t h e r a sharp peak nor a band t a i l .
This part of small
Around the lower bound, LDOS e x h i A s i m i l a r computation has been done
f o r LDOS at the center of i m p u r i t y cages, in which p o t e n t i a l due to i m p u r i t i e s is negative and l o c a l i z e d only at i m p u r i t y s i t e s .
For a large value of IV/TI
LDOS is symmetric with respect to E=O and sharp peaks and band t a i l s served near the both bounds of the energy band.
are ob-
With decreasing value of IVl
LDOS becomes asymmetric; the sharp peak near the upper bound disappears when IV/Tl~l,whereas the peak near the lower bound remains at the same p o s i t i o n . We have found t h a t the p o s i t i o n Ep of the sharp peak varies with the size of the cage approximately according to the r u l e
4L = 2~(4-1Ep/TI) - I / 2 ,
(I)
which holds not only for the dislocation cage but also for the impurity cage. To discuss the sharp peak near the spectral bounds, we can borrow the concept of the standing wave of Lifshitz 5. An electron belonging to a level near the lower bound of the energy band has in a perfect crystal large wavelength represented by the quantity on the right-hand side of (1).
On the other hand, the
wave function of an electron with E~4ITI has nearly equal values but opposite signs on the neighboring two sites.
The spatially varying amplitude has the
wavelength given by the quantity on the right-hand side of (1).
Therefore, the
relation (1) suggests that there stand stationary waves which have nodes at the
K. Kawarnura, R. Ohira / Local density o f states
91
impurity sites or around the dislocation axes from certain reasons. An impurity with the negative value of potential is a center of a repulsive force for electrons with E>O and excludes them from the impurity site when IVI is much larger than the band width.
This explanation is consistent with
the observation that the sharp peak near the upper bound of the spectrum disappears when IVl is small.
Electrons with long wavelength cannot have ampli-
tude on the impurity sites either, when electrons are attracted by the impurities.
This is because the wave function of a state within the band should be
orthogonal to the localized wave function of a bound state.
Since the bound
state exists for any value of IVi in our two dimensional l a t t i c e , this explanation is consistent with the fact that the sharp peak of LDOS near the lower bound does not move apparently with the value of V. When we go along an arbitrary path which encloses a single dislocation, we always passes over odd number of sites.
Such a structure of a dislocated l a t -
tice is incompatible with the rapidly varying wave function with E=41Ti.
A
similar situation has been discussed by Yonezawaand Cohen6, when they study the influence of odd membered rings on the energy spectrum of antibonding states in tetrahedrally coordinated semiconductors.
Thus, electrons in the
energy states near the upper bound of the energy band are excluded from the dislocation cores, whereas electrons with E=-4ITI, which have large wavelength, can exist around the dislocation cores. 3. DISCUSSIONS The preceding
theory on the cages of impurities and edge dislocations is
applied in this section to the discussion on electronic states in amorphous semiconductors.
In the following discussions, the dislocation model is employ-
ed as for the structural model of amorphous substances. 3.1. Influence of topologically singular structure of dislocations. In the preceding section, LDOSwas described as a function of energy.
How-
ever, energy is not an appropriate variable to describe the influence of the topologically singular structure of dislocations in general l a t t i c e s .
In fact,
the exclusion of electrons from the core region discussed above is the same phenomenon as the shadow7 of a dislocation observed in the scattering theoretical solution; the amplitude in the shadow region is proportional to cos(k'b/2), where k is the wave vector of the incident wave and ~ is the Burgers vector. Therefore, the dislocation excludes an electron with the wave vector k for which k'b takes the value around ~. In Si and Ge, the bottom of the conduction band is located in such a region. Then, the electrons in the v i c i n i t y of the lower bound of the conduction band may be excluded from the dislocation cores and form standing wave states.
K. Kawamura, R. Ohira / Local density of states
92
Although the dislocation in the preceding section can be a model of the odd membered ring of Yonezawaand Cohen6, the existence of odd memberedrings is not the necessary condition for the appearenceof the standing wave discussed above. This is because the wave number is concerned with the translational motion among unit cells instead of the change of a wave function within a unit cell.
Therefore, electrons in the bonding states are also excluded from dislo-
cation cores, when their wave number satisfies k'b=~. 3.2. Influence of dangling bonds on extended electronic states. According to the study of Jones8, some conduction band states are excluded from the dislocation core, since they should be orthogonal to the localized dangling bond states.
The same orthogonality of the band electron to the bound
state yields the reduction of LDOS near the lower bound of the energy band in the impurity cage discussed above. Therefore, the cage of impurities which attract electrons may be a model of the cage of dangling bonds. The influence of the dangling bond states may be dominant for electrons with large wavelength. In an amorphous semiconductor with dangling bonds, electrons near the r-point are expected to form the standing wave states. 3.3. Binary alloy model and dislocation mode] for amorphous substances. Since the structures of LDOS near the upper bound of spectrum in Figures l and 2 have commoncharacter, to investigate the influence of the electronic states of short wavelength, the dislocation may be replaced with a line of impurities with repulsive force.
I f this is allowed, we can discuss electrons
of short wavelength in amorphous semiconductors of topologically singular structure with the help of knowledge about the random binary alloy.
REFERENCES I) J.J. Gilman, J. Appl. Phys. 44 (1973) 675. 2) J.C.M. Li, Distinguished Lectures in Materials Science (Macel-Decker, New York,1974) 3) H. Koizumi and T. Ninomiya, J. Phys. Soc. Japan 44 (1978) 898. 4) H. Koizumi and T. Ninomiya, J. Phys. Soc. Japan 49 (1980) I022. 5) I.M. Lifshitz, Advan. Phys. 13 (1964) 483. 6) F. Yonezawaand M.H. Cohen, Fundamental Physics of Amorphous Semiconductors, ed. F. Yonezawa, in: Springer Series in Solid-State Sciences, Voi.25, eds. M. Cardona and P. Fulde (Springer-Verlag, Berlin 1981) ppl19-144. 7) K. Kawamura, Z. Physik 21 (1982) 201. 8) R. Jones, Phil. Mag. 35 (1977) 57.