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Structures of glide-set 90° partial dislocation cores in diamond cubic semiconductors
ARTICLE IN PRESS
Physica B 340–342 (2003) 990–995
Structures of glide-set 90 partial dislocation cores in diamond cubic semiconductors S.P. Beckmana,b, D.C. Chrzana,b,* a
Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA b Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract Two core reconstructions of the 90 partial dislocations in diamond cubic semiconductors, the so-called single- and double-period structures, are often found to be nearly degenerate in energy. This near degeneracy suggests the possibility that both core reconstructions may be present simultaneously along the same dislocation core, with the domain sizes of the competing reconstructions dependent on temperature and the local stress state. To explore this dependence, a simple statistical mechanics-based model of the dislocation core reconstructions is developed and analyzed. Predictions for the temperature-dependent structure of the dislocation core are presented. r 2003 Elsevier B.V. All rights reserved. PACS: 71.55.Cn; 61.72.Lk Keywords: Silicon; Dislocation; Thermodynamics
1. Introduction Advances in computational hardware and algorithms now allow the direct computation of collections of near 1000 atoms using ab initio electronic structure total energy techniques. Consequently, it is now possible to study the properties of extended defects using high-quality theories rooted firmly in quantum mechanics. One set of defects that has drawn considerable attention is the structure of dislocation cores in diamond cubic and zincblende materials.
*Corresponding author. E-mail address: [email protected] (D.C. Chrzan).
It is important to understand the structure of the dislocation cores for a number of reasons. First, it is impossible to separate the electronic properties and the structural properties of a dislocation core. If one wants to understand fully the role that dislocations play in electro-optical devices, one must know the structure of the dislocation cores. Second, while a perfect, defectfree core may not be electro-optically active, the defects along that same core may be electrooptically active. The structure and properties of these defects are necessarily defined by the ground state structure of the dislocation core. In order to predict the properties of the defects, one needs to know the structure of the perfect dislocations. Third, the structure of the dislocation core is often related closely to the kinetics of dislocation
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.09.192
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
motion. A clear understanding of dislocation core structures may lead to a better understanding of dislocation mobilities, in turn leading to improved control of the dynamics of dislocations. Studies to date have yielded substantial information concerning dislocations in diamond cubic and zincblende materials [1]. It is now accepted widely that the cores of glide set dislocations in these materials are dissociated. Specifically, the ða=2Þ/1 1 0S dislocations dissociate into two Shockley partials with Burgers vectors ða=6Þ/1 1 2% S: Since the dislocations lie predominantly along /1 1 0S directions, this implies that the partial dislocations have their Burgers vectors either 90 or 30 to their line direction. To date, it appears that a consensus has been reached concerning the ground state structure of the 30 partial. However, the structure of the 90 partial remains a topic of ongoing research. Present work focuses on the relative stability of two proposed core reconstructions: the so-called single-period (SP) and double-period (DP) cores (see Fig. 1). The SP core was propose independently by Hirsch [2] and Jones [3] in the late 1970s. More recently, Bennetto et al. [4] proposed the existence of the DP core structure. Computing the energy differences between competing core structures is a nontrivial task, as the two reconstructions typically differ in energy by only a small amount. In Si, the energy difference between the reconstruction modes has been calculated, by multiple methods, to be ( [5]. It is known that approximately 55 meV=A these values are sensitive to the stress environment to the extent that the relative stability of the two core reconstructions can be reversed by the application of hydrostatic stresses [6–8]. In GaAs, it is believed that the relative energy difference between the two most probable reconstructions is ( [9]. These energy on the order of 10 meV=A differences are certainly approaching the computational limits of the best available ab initio electronic structure total energy techniques, especially when one considers the constraints on accuracy arising from imposed boundary conditions. Further, if the energy scales separating the two dislocation core types are truly of the order of ( there is the very real possibility that at meV/A;
991
(a)
(b)
(c) Fig. 1. Slip plane view of three possible core structures in a diamond cubic semiconductor. Panel (a) depicts the SP core. Panel (b) depicts the DP core. Panel (c) depicts the SP core with a direction switching defect. The highlighted bonds are used to define the thermodynamic model.
temperature both types of reconstructions will be present along a single dislocation with their relative concentrations determined by the temperature and the defects and excitations available to the differing core structures. Finally, one has the possibility that dislocation core structure may
ARTICLE IN PRESS 992
S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
depend on extrinsic defects. The immense complexity of dislocations within Si has lead one author to describe the number of variations along the dislocation cores as ‘‘bottomless’’ [10]. It is often true that in order to garner insight into a physical system, a theorist will construct a model system meant to mimic the desired physics as closely as possible, while still yielding to solution. The Ising model for magnetism is a prototypical example. In this model, the electronic degrees of freedom that give rise to spin–spin interactions, i.e. the exchange interaction arising from the solution of the many-body electron problem, is reduced to a single parameter describing the interaction amongst spins on isolated sites. Further, the degrees of freedom of the spins are reduced to where the spins may point in only two distinct directions: ‘‘up’’ and ‘‘down’’. The Ising model, while not a completely faithful model of magnetism, allows remarkable insight into the properties of magnetic systems. A similar model for the dislocation core reconstruction might also help to sort through the encountered bottomless complexity of the dislocation cores. This paper presents one attempt at developing a thermodynamic model of the temperature dependence of the core structure of the 90 partial dislocations in diamond cubic semiconductors. This initial model has two goals: (1) to predict the temperature dependence of the domain sizes expected for the SP and DP cores; (2) to help in developing an understanding of the influence of temperature on dislocation core structures.
2. Thermodynamic model Fig. 1 displays the proposed structures for the SP reconstruction, the DP reconstruction, and the structures of a direction switching defect described by Nunes et al. [11]. The structures are shown looking down upon the slip plane. The shaded area corresponds to the stacking fault. Only some of the bonds have been sketched. The sketched bonds will help to define the thermodynamic model. Consider Fig. 1a. This figure depicts a region of perfectly reconstructed SP structure. Note that the bonds associated with the reconstruction tilt to
the right as one moves up the page. In contrast, Fig. 1b shows that the sketched bonds alternate between those that tilt to the right, and those that tilt to the left. In addition, in panel (b) the bonds alternate between two different rows. Based on this observation, the two core reconstructions can be described in terms two interpenetrating onedimensional spin lattices. The first set of spins, referred to as tilt spins, are those corresponding to the diagrammed bonds. These spins, denoted st ; take on the value st ¼ 1 for a spin that tilts up and to the right, and st ¼ 1 for a spin that tilts up and to the left. The model is assumed to contain L of these spins, one for each site. In addition, a second spin is associated with each site. This spin is used to indicate the row in which the spin is found. The spin is denoted by sr ; with the r corresponding to whether or not the spin in the next site rises one row, stays on the same row, or falls a row in comparison to the spin at the present site. These spins, consequently, take on the values sr ¼ 1; sr ¼ 0; and sr ¼ 1 for spins that fall, stay on the same plane, or rise, respectively. The model contains L of these spins as well, one for each site. PThe spins at a site are then defined by the pair j ¼ ðst ðjÞ; sr ðjÞÞ and the structure of the dislocation is then defined by a sequence of pairs. For example, the perfect SP structure is given by the pairs fyð1; 0Þð1; 0Þð1; 0Þyg where the dots indicate that one has L identical pairs. Similarly, the DP structure is defined by the sequence fyð1; 1Þð1; 1Þð1; 1Þð1; 1Þyg where the dots indicate that the pattern is repeated (in this case L=2 times). The notation thus allows the presumed two ground states to be described easily. The notation also allows one to describe a number of defects (though not all) in an at least approximate way. For example, the direction switching defect [11] shown in Fig. 1c can be described by the sequence fyð1; 0Þð1; 0Þð1; 0Þð1; 0Þyg; where dots to the left indicate a repetition of the ð1; 0Þ pattern and dots to the right the ð1; 0Þ pattern. Similarly, one can describe the families of kinks identified for the SP core by Nunes et al. [11]. In fact, the model allows a simple description of many of the defects allowed by symmetry for each of the cores, though certainly not all are
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
LR kink DSD
P P ing to the configuration j ¼ ð1; 1Þ and jþ1 ¼ ð1; 0Þ; defined to be O# 14 is given by
DP Domain SP Domain
O# 14 ¼ 12 ð1 st ðjÞÞ 12 sr ðjÞ2 ð1 sr ðjÞÞ
σ t +1 −1 −1 +1 +1 −1 +1 −1 +1 +1 +1 σ r 0 0 +1 0 0 −1 +1 −1 0 0 0 site pair
12 ð1 þ st ðj þ 1ÞÞ ð1 sr ðj þ 1Þ2 Þ:
Fig. 2. A typical dislocation configuration and its spin variable description. This dislocation contains a direction switching defect (DSD), a left–right kink (LR), a DP domain and an SP domain. The tilt and rise spins are denoted by st and sr ; respectively.
properly represented. Fig. 2 depicts a typical configuration, and the spins representing it. Within this description, a DP core is simply the most highly kinked SP core [4]. The next task at hand is specification of the required ‘‘exchange’’ parameters for the model. The assignment process is explained here only briefly. The assignment begins by noting that to distinguish between SP and DP structures, one must consider the interactions between all spins at site j and all spins at site j þ 1: Since each site can exist in one of six states, this implies that there are 36 configurations of spins for which one needs to define energies. One begins by defining operators that identify the relevant combinations of spins, call these operators O# n where n refers to the configuration in question (n runs from 1 to 36). O# n ðjÞ returns the value 0 if the spin configuration at sites j and j þ 1 are not the same as the nth configuration, and returns 1 if the spins are configured correctly. Letting en define the energy of configuration n; one then writes the total energy, Etot ; of the dislocation core as Etot ¼
36 X L X n¼1
O# n ðjÞen :
993
ð1Þ
j¼1
It should be noted that in this present form, the model does not account for long-ranged elastic interactions, the effects of the stacking fault, nor for overall dislocation line length or orientation. The operators O# n ðjÞ are easily constructed by inspection. For example, the operator correspond-
ð2Þ
Similar expressions can be constructed for the remaining 35 other combinations. The energy assignments for the configurations with nonzero energy are depicted in Table 1. The SP core, configurations f7; 8; 9; 28; 29; 30g; is chosen as the reference state with Etot ¼ 0: The energy of the DP core relative to the SP core is given by g per site. The energy of the various defects employ the notation used by Nunes et al. [11]. For simplicity, it is assumed that ELR ¼ ERL : The parameter EDSD is the energy of a direction
Table 1 en assignments n
st ðjÞ
sr ðjÞ
st ðj þ 1Þ
sr ðj þ 1Þ
en
1 2 3 4 5 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 31 32 33 34 35 36
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
ERR ERR Z ELR ELR g EDSD EDSD EDSD Z ERR ERR g ELR ELR ELR ELR g ERR ERR Z EDSD EDSD EDSD ELR ELR ELR Z ERR ERR
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
switching defect for the SP reconstruction. The parameter ERR represents a configuration that is a sum of a left–right kink, i.e. a configuration similar to that for n ¼ 4; and a direction switching defect and an additional interaction energy between the direction switching defect and the left–right kink, defined to be D: Consequently, ERR ¼ ELR þ EDSD þ D: The parameter Z represents the energy of the DP core plus a direction switching defect. Its energy is taken to be Z ¼ g þ EDSD þ D: Substituting all these values and the proper definition of the operators O# n gives the following expression for the total energy of the dislocation core (assuming periodic boundary conditions): Etot ðjÞ ¼ 12 EDSD þ 12 ½sr ðjÞ2 ð2EDSD þ DÞ EDSD st ðjÞst ðj þ 1Þ þ 12 ðELR gÞsr ðjÞsr ðj þ 1Þ þ 12 ½ðg ELR Þsr ðj þ 1Þ2 þ D þ 2ELR sr ðjÞ2 ; Etot ¼
L X
Etot ðjÞ:
ð3Þ ð4Þ
j¼1
Note that this expression reduces to the correct P limits. Under the P circumstances in which all j ¼ ð1; 0Þ or all j ¼ ð1; 0Þ; the total energy is identically zero as it should be for the SP core. In the limit that the configuration is that of a perfect DP core, the expression reduces gL; again the expected result. This expression, however, neglects the effects of the companion partial dislocation, the force exerted by the stacking fault on the dislocation, and any external stresses applied to the dislocation. 3. Results The constructed model has not yet yielded to exact solution. Therefore, standard Monte Carlo methods are used to explore the predicted structure as a function of temperature, T; and the energy difference per site between the DP and SP cores, g: Note that go0 corresponds to the double period core as the ground state, g > 0 yields a SP core ground state. The parameters ELR and D are taken from the calculations of Nunes et al. for Si and chosen to be ELR ¼ 0:12 eV and D ¼ 0:385 eV
[11,12]. The parameter EDSD was chosen to be equal to 0:1 eV: This value is much less than one expects from existing calculations. The primary reason for choosing this value is to allow rapid equilibration of the model. With this choice of parameters, the concentration of direction switching defects will be substantially overestimated. However, the presence of direction switching defects is not expect to alter substantially the relative populations of double and single period reconstructed domains. The parameter g was varied between 70:1 eV; and the temperature dependence was investigated over the range of temperatures from 600 to 1800 K: The relaxation times for the spin system are quite long. To help accelerate computational speed, systems of size L ¼ 128 were investigated. The expectation value of the domain size for both SP and DP domains was computed as a function of T: The calculations employed a kinetic Monte Carlo algorithm so that a spin was flipped at every step. Results were then averaged over roughly a data set obtained by running the algorithm for 103 flips per site. 80 (a) 60
single period double period
40 20 0 domain size [sites]
994
15 (b) 10 5 0 80 (c) 60 40 20 0
200
400 600 800 temperature [K]
1000
Fig. 3. Results from the model. Panels (a), (b) and (c) are for g ¼ 0:1 eV; g ¼ 0:0 eV; and g ¼ 0:1 eV; respectively.
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
Fig. 3 contains domain size results for three values of g: 0:1; 0; and 0:1 eV: These values correspond to energy differences between competing core structures of roughly 27; 0 and ( respectively. A domain is defined here 27 meV=A; as a continuous region of the core that is either purely SP reconstructed or purely DP reconstructed (allowing for the presence of direction switching defects). Examination of this figure reveals third interesting results. First, under the circumstances in which one core structure is only slightly favored over another, the room temperature domain sizes are substantially larger than the thickness of a typical high resolution TEM foil. Consequently, the structure of the core should be accessible to TEM experiments. Second, in the circumstance in which the SP core is only slightly favored energetically, the domain size is strongly dependent on temperature, with the average domain size shrinking to approximately 10 sites at 500 K: In contrast, the domain size for the DP core falls much more slowly, suggesting that the SP core has more available low-energy excitations. The third observation is that even under circumstances in which the core structures are degenerate, the average domain size is approximately seven sites. ( roughly oneThis corresponds to roughly 30 A; half the thickness of a typical high-resolution TEM foil.
4. Conclusion In this paper, an analysis of a simple spin model describing the energetics of 90 partial dislocations in diamond cubic materials has been introduced, and a study of its basic properties has been presented. The model reveals that very small
995
( energy differences, of the order of 25 meV=A; yield a dominant ground state. Under these same conditions, the expected domain sizes are typically much larger than the thickness of a foil used in high-resolution electron microscopy experiments, suggesting that it is in principle possible to observe a single domain of the dominant core structure.
Acknowledgements This work is sponsored by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DEAC03-76SF00098. The authors also acknowledge supercomputer time provided by the National Energy Research Scientific Computing Center.
References [1] H. Alexander, H. Teichler, in: K.A. Jackson, W. Schroeter (Eds.), Handbook of Semiconductor Technology, Vol. 1, Wiley-VCH, New York, 2000, pp. 291–376. [2] P.B. Hirsch, J. Phys. (Orsay) 40 (1979) 27. [3] R. Jones, J. Phys. (Orsay) 40 (1979) 33. [4] J. Bennetto, R.W. Nunes, D. Vanderbilt, Phys. Rev. Lett. 79 (1997) 245. [5] R.W. Nunes, D. Vanderbilt, Phys. Rev. Lett. 85 (2000) 3540. . [6] N. Lehto, S. Oberg, Phys. Rev. Lett. 80 (1998) 5568. [7] K. Lin, D.C. Chrzan, Comput. Model. Eng. Sci. 3 (2002) 201. [8] K. Lin, D.C. Chrzan, Mater. Sci. Eng. A A319–321 (2001) 115. [9] S.P. Beckman, et al., J. Phys.: Condens. Matter 48 (2002) 12 673. [10] V.V. Bulatov, Scripta Mater. 45 (2001) 1247. [11] R.W. Nunes, J. Bennetto, D. Vanderbilt, Phys. Rev. B 57 (1998) 10 388. [12] R.W. Nunes, J. Bennetto, D. Vanderbilt, Phys. Rev. Lett. 77 (1996) 1516.
Physica B 340–342 (2003) 990–995
Structures of glide-set 90 partial dislocation cores in diamond cubic semiconductors S.P. Beckmana,b, D.C. Chrzana,b,* a
Department of Materials Science and Engineering, University of California, Berkeley, CA 94720, USA b Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Abstract Two core reconstructions of the 90 partial dislocations in diamond cubic semiconductors, the so-called single- and double-period structures, are often found to be nearly degenerate in energy. This near degeneracy suggests the possibility that both core reconstructions may be present simultaneously along the same dislocation core, with the domain sizes of the competing reconstructions dependent on temperature and the local stress state. To explore this dependence, a simple statistical mechanics-based model of the dislocation core reconstructions is developed and analyzed. Predictions for the temperature-dependent structure of the dislocation core are presented. r 2003 Elsevier B.V. All rights reserved. PACS: 71.55.Cn; 61.72.Lk Keywords: Silicon; Dislocation; Thermodynamics
1. Introduction Advances in computational hardware and algorithms now allow the direct computation of collections of near 1000 atoms using ab initio electronic structure total energy techniques. Consequently, it is now possible to study the properties of extended defects using high-quality theories rooted firmly in quantum mechanics. One set of defects that has drawn considerable attention is the structure of dislocation cores in diamond cubic and zincblende materials.
*Corresponding author. E-mail address: [email protected] (D.C. Chrzan).
It is important to understand the structure of the dislocation cores for a number of reasons. First, it is impossible to separate the electronic properties and the structural properties of a dislocation core. If one wants to understand fully the role that dislocations play in electro-optical devices, one must know the structure of the dislocation cores. Second, while a perfect, defectfree core may not be electro-optically active, the defects along that same core may be electrooptically active. The structure and properties of these defects are necessarily defined by the ground state structure of the dislocation core. In order to predict the properties of the defects, one needs to know the structure of the perfect dislocations. Third, the structure of the dislocation core is often related closely to the kinetics of dislocation
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.09.192
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
motion. A clear understanding of dislocation core structures may lead to a better understanding of dislocation mobilities, in turn leading to improved control of the dynamics of dislocations. Studies to date have yielded substantial information concerning dislocations in diamond cubic and zincblende materials [1]. It is now accepted widely that the cores of glide set dislocations in these materials are dissociated. Specifically, the ða=2Þ/1 1 0S dislocations dissociate into two Shockley partials with Burgers vectors ða=6Þ/1 1 2% S: Since the dislocations lie predominantly along /1 1 0S directions, this implies that the partial dislocations have their Burgers vectors either 90 or 30 to their line direction. To date, it appears that a consensus has been reached concerning the ground state structure of the 30 partial. However, the structure of the 90 partial remains a topic of ongoing research. Present work focuses on the relative stability of two proposed core reconstructions: the so-called single-period (SP) and double-period (DP) cores (see Fig. 1). The SP core was propose independently by Hirsch [2] and Jones [3] in the late 1970s. More recently, Bennetto et al. [4] proposed the existence of the DP core structure. Computing the energy differences between competing core structures is a nontrivial task, as the two reconstructions typically differ in energy by only a small amount. In Si, the energy difference between the reconstruction modes has been calculated, by multiple methods, to be ( [5]. It is known that approximately 55 meV=A these values are sensitive to the stress environment to the extent that the relative stability of the two core reconstructions can be reversed by the application of hydrostatic stresses [6–8]. In GaAs, it is believed that the relative energy difference between the two most probable reconstructions is ( [9]. These energy on the order of 10 meV=A differences are certainly approaching the computational limits of the best available ab initio electronic structure total energy techniques, especially when one considers the constraints on accuracy arising from imposed boundary conditions. Further, if the energy scales separating the two dislocation core types are truly of the order of ( there is the very real possibility that at meV/A;
991
(a)
(b)
(c) Fig. 1. Slip plane view of three possible core structures in a diamond cubic semiconductor. Panel (a) depicts the SP core. Panel (b) depicts the DP core. Panel (c) depicts the SP core with a direction switching defect. The highlighted bonds are used to define the thermodynamic model.
temperature both types of reconstructions will be present along a single dislocation with their relative concentrations determined by the temperature and the defects and excitations available to the differing core structures. Finally, one has the possibility that dislocation core structure may
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S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
depend on extrinsic defects. The immense complexity of dislocations within Si has lead one author to describe the number of variations along the dislocation cores as ‘‘bottomless’’ [10]. It is often true that in order to garner insight into a physical system, a theorist will construct a model system meant to mimic the desired physics as closely as possible, while still yielding to solution. The Ising model for magnetism is a prototypical example. In this model, the electronic degrees of freedom that give rise to spin–spin interactions, i.e. the exchange interaction arising from the solution of the many-body electron problem, is reduced to a single parameter describing the interaction amongst spins on isolated sites. Further, the degrees of freedom of the spins are reduced to where the spins may point in only two distinct directions: ‘‘up’’ and ‘‘down’’. The Ising model, while not a completely faithful model of magnetism, allows remarkable insight into the properties of magnetic systems. A similar model for the dislocation core reconstruction might also help to sort through the encountered bottomless complexity of the dislocation cores. This paper presents one attempt at developing a thermodynamic model of the temperature dependence of the core structure of the 90 partial dislocations in diamond cubic semiconductors. This initial model has two goals: (1) to predict the temperature dependence of the domain sizes expected for the SP and DP cores; (2) to help in developing an understanding of the influence of temperature on dislocation core structures.
2. Thermodynamic model Fig. 1 displays the proposed structures for the SP reconstruction, the DP reconstruction, and the structures of a direction switching defect described by Nunes et al. [11]. The structures are shown looking down upon the slip plane. The shaded area corresponds to the stacking fault. Only some of the bonds have been sketched. The sketched bonds will help to define the thermodynamic model. Consider Fig. 1a. This figure depicts a region of perfectly reconstructed SP structure. Note that the bonds associated with the reconstruction tilt to
the right as one moves up the page. In contrast, Fig. 1b shows that the sketched bonds alternate between those that tilt to the right, and those that tilt to the left. In addition, in panel (b) the bonds alternate between two different rows. Based on this observation, the two core reconstructions can be described in terms two interpenetrating onedimensional spin lattices. The first set of spins, referred to as tilt spins, are those corresponding to the diagrammed bonds. These spins, denoted st ; take on the value st ¼ 1 for a spin that tilts up and to the right, and st ¼ 1 for a spin that tilts up and to the left. The model is assumed to contain L of these spins, one for each site. In addition, a second spin is associated with each site. This spin is used to indicate the row in which the spin is found. The spin is denoted by sr ; with the r corresponding to whether or not the spin in the next site rises one row, stays on the same row, or falls a row in comparison to the spin at the present site. These spins, consequently, take on the values sr ¼ 1; sr ¼ 0; and sr ¼ 1 for spins that fall, stay on the same plane, or rise, respectively. The model contains L of these spins as well, one for each site. PThe spins at a site are then defined by the pair j ¼ ðst ðjÞ; sr ðjÞÞ and the structure of the dislocation is then defined by a sequence of pairs. For example, the perfect SP structure is given by the pairs fyð1; 0Þð1; 0Þð1; 0Þyg where the dots indicate that one has L identical pairs. Similarly, the DP structure is defined by the sequence fyð1; 1Þð1; 1Þð1; 1Þð1; 1Þyg where the dots indicate that the pattern is repeated (in this case L=2 times). The notation thus allows the presumed two ground states to be described easily. The notation also allows one to describe a number of defects (though not all) in an at least approximate way. For example, the direction switching defect [11] shown in Fig. 1c can be described by the sequence fyð1; 0Þð1; 0Þð1; 0Þð1; 0Þyg; where dots to the left indicate a repetition of the ð1; 0Þ pattern and dots to the right the ð1; 0Þ pattern. Similarly, one can describe the families of kinks identified for the SP core by Nunes et al. [11]. In fact, the model allows a simple description of many of the defects allowed by symmetry for each of the cores, though certainly not all are
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
LR kink DSD
P P ing to the configuration j ¼ ð1; 1Þ and jþ1 ¼ ð1; 0Þ; defined to be O# 14 is given by
DP Domain SP Domain
O# 14 ¼ 12 ð1 st ðjÞÞ 12 sr ðjÞ2 ð1 sr ðjÞÞ
σ t +1 −1 −1 +1 +1 −1 +1 −1 +1 +1 +1 σ r 0 0 +1 0 0 −1 +1 −1 0 0 0 site pair
12 ð1 þ st ðj þ 1ÞÞ ð1 sr ðj þ 1Þ2 Þ:
Fig. 2. A typical dislocation configuration and its spin variable description. This dislocation contains a direction switching defect (DSD), a left–right kink (LR), a DP domain and an SP domain. The tilt and rise spins are denoted by st and sr ; respectively.
properly represented. Fig. 2 depicts a typical configuration, and the spins representing it. Within this description, a DP core is simply the most highly kinked SP core [4]. The next task at hand is specification of the required ‘‘exchange’’ parameters for the model. The assignment process is explained here only briefly. The assignment begins by noting that to distinguish between SP and DP structures, one must consider the interactions between all spins at site j and all spins at site j þ 1: Since each site can exist in one of six states, this implies that there are 36 configurations of spins for which one needs to define energies. One begins by defining operators that identify the relevant combinations of spins, call these operators O# n where n refers to the configuration in question (n runs from 1 to 36). O# n ðjÞ returns the value 0 if the spin configuration at sites j and j þ 1 are not the same as the nth configuration, and returns 1 if the spins are configured correctly. Letting en define the energy of configuration n; one then writes the total energy, Etot ; of the dislocation core as Etot ¼
36 X L X n¼1
O# n ðjÞen :
993
ð1Þ
j¼1
It should be noted that in this present form, the model does not account for long-ranged elastic interactions, the effects of the stacking fault, nor for overall dislocation line length or orientation. The operators O# n ðjÞ are easily constructed by inspection. For example, the operator correspond-
ð2Þ
Similar expressions can be constructed for the remaining 35 other combinations. The energy assignments for the configurations with nonzero energy are depicted in Table 1. The SP core, configurations f7; 8; 9; 28; 29; 30g; is chosen as the reference state with Etot ¼ 0: The energy of the DP core relative to the SP core is given by g per site. The energy of the various defects employ the notation used by Nunes et al. [11]. For simplicity, it is assumed that ELR ¼ ERL : The parameter EDSD is the energy of a direction
Table 1 en assignments n
st ðjÞ
sr ðjÞ
st ðj þ 1Þ
sr ðj þ 1Þ
en
1 2 3 4 5 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 31 32 33 34 35 36
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1
ERR ERR Z ELR ELR g EDSD EDSD EDSD Z ERR ERR g ELR ELR ELR ELR g ERR ERR Z EDSD EDSD EDSD ELR ELR ELR Z ERR ERR
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
switching defect for the SP reconstruction. The parameter ERR represents a configuration that is a sum of a left–right kink, i.e. a configuration similar to that for n ¼ 4; and a direction switching defect and an additional interaction energy between the direction switching defect and the left–right kink, defined to be D: Consequently, ERR ¼ ELR þ EDSD þ D: The parameter Z represents the energy of the DP core plus a direction switching defect. Its energy is taken to be Z ¼ g þ EDSD þ D: Substituting all these values and the proper definition of the operators O# n gives the following expression for the total energy of the dislocation core (assuming periodic boundary conditions): Etot ðjÞ ¼ 12 EDSD þ 12 ½sr ðjÞ2 ð2EDSD þ DÞ EDSD st ðjÞst ðj þ 1Þ þ 12 ðELR gÞsr ðjÞsr ðj þ 1Þ þ 12 ½ðg ELR Þsr ðj þ 1Þ2 þ D þ 2ELR sr ðjÞ2 ; Etot ¼
L X
Etot ðjÞ:
ð3Þ ð4Þ
j¼1
Note that this expression reduces to the correct P limits. Under the P circumstances in which all j ¼ ð1; 0Þ or all j ¼ ð1; 0Þ; the total energy is identically zero as it should be for the SP core. In the limit that the configuration is that of a perfect DP core, the expression reduces gL; again the expected result. This expression, however, neglects the effects of the companion partial dislocation, the force exerted by the stacking fault on the dislocation, and any external stresses applied to the dislocation. 3. Results The constructed model has not yet yielded to exact solution. Therefore, standard Monte Carlo methods are used to explore the predicted structure as a function of temperature, T; and the energy difference per site between the DP and SP cores, g: Note that go0 corresponds to the double period core as the ground state, g > 0 yields a SP core ground state. The parameters ELR and D are taken from the calculations of Nunes et al. for Si and chosen to be ELR ¼ 0:12 eV and D ¼ 0:385 eV
[11,12]. The parameter EDSD was chosen to be equal to 0:1 eV: This value is much less than one expects from existing calculations. The primary reason for choosing this value is to allow rapid equilibration of the model. With this choice of parameters, the concentration of direction switching defects will be substantially overestimated. However, the presence of direction switching defects is not expect to alter substantially the relative populations of double and single period reconstructed domains. The parameter g was varied between 70:1 eV; and the temperature dependence was investigated over the range of temperatures from 600 to 1800 K: The relaxation times for the spin system are quite long. To help accelerate computational speed, systems of size L ¼ 128 were investigated. The expectation value of the domain size for both SP and DP domains was computed as a function of T: The calculations employed a kinetic Monte Carlo algorithm so that a spin was flipped at every step. Results were then averaged over roughly a data set obtained by running the algorithm for 103 flips per site. 80 (a) 60
single period double period
40 20 0 domain size [sites]
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15 (b) 10 5 0 80 (c) 60 40 20 0
200
400 600 800 temperature [K]
1000
Fig. 3. Results from the model. Panels (a), (b) and (c) are for g ¼ 0:1 eV; g ¼ 0:0 eV; and g ¼ 0:1 eV; respectively.
ARTICLE IN PRESS S.P. Beckman, D.C. Chrzan / Physica B 340–342 (2003) 990–995
Fig. 3 contains domain size results for three values of g: 0:1; 0; and 0:1 eV: These values correspond to energy differences between competing core structures of roughly 27; 0 and ( respectively. A domain is defined here 27 meV=A; as a continuous region of the core that is either purely SP reconstructed or purely DP reconstructed (allowing for the presence of direction switching defects). Examination of this figure reveals third interesting results. First, under the circumstances in which one core structure is only slightly favored over another, the room temperature domain sizes are substantially larger than the thickness of a typical high resolution TEM foil. Consequently, the structure of the core should be accessible to TEM experiments. Second, in the circumstance in which the SP core is only slightly favored energetically, the domain size is strongly dependent on temperature, with the average domain size shrinking to approximately 10 sites at 500 K: In contrast, the domain size for the DP core falls much more slowly, suggesting that the SP core has more available low-energy excitations. The third observation is that even under circumstances in which the core structures are degenerate, the average domain size is approximately seven sites. ( roughly oneThis corresponds to roughly 30 A; half the thickness of a typical high-resolution TEM foil.
4. Conclusion In this paper, an analysis of a simple spin model describing the energetics of 90 partial dislocations in diamond cubic materials has been introduced, and a study of its basic properties has been presented. The model reveals that very small
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( energy differences, of the order of 25 meV=A; yield a dominant ground state. Under these same conditions, the expected domain sizes are typically much larger than the thickness of a foil used in high-resolution electron microscopy experiments, suggesting that it is in principle possible to observe a single domain of the dominant core structure.
Acknowledgements This work is sponsored by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under Contract No. DEAC03-76SF00098. The authors also acknowledge supercomputer time provided by the National Energy Research Scientific Computing Center.
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