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On the electron scattering in disclinated crystals
PhysicsLettersA 175 (1993) 65—68 North-Holland
PHYSICS LETTERS A
On the electron scattering in disclinated crystals V.A. Osipov Joint Institutefor Nuclear Research, Laboratory ofTheoretical Physics, 141980 Dubna, Moscow Region, Russian Federation
Received 18 January 1993; accepted forpublication 3 February 1993 Communicated by L.J. Sham.
We study disclination-induced electron scattering in materials with disclination vortices. Two kinds of scattering processes are shown to be ofimportance: the known deformation-potential scattering and an Aharonov—Bohm-like scattering generated by the topological natureof the disdinations. The relaxation time is calculated. The scattering process is found to depend essentially on the density of the conducting electrons aswell ason the disclination core radius.
In previous papers [1,2] we have shown that the topologically singular character of a crystal with disclination vortices can result in a new kind ofelectron scattering that is pure topological in its origin. In many respects, this Aharonov—Bohm-like topological scattering differs from the known deformationpotential scattering that takes place due to elastic strains caused by defects. Particularly, the topologically induced resistivity is found to have the characteristic oscillatory behaviour which depends on the disclination flux characterized by the Frank index. It is clear that both these kinds of scattering should be of importance at low temperatures where other possible mechanisms are less effective. In this Letter we study the transport properties of a disclinated crystal when both scattering mechanisms are taken into account. We consider a crystal perforated by linear disclinations oriented parallel to the z-axis. The density of the disclinations is supposed to be small. In this case, the scattering process on each defect can be examined separately. Linear disclinations apparently break the rotational symmetty on the planes perpendicular to the disclination lines. The scattering process is the most effectivefor electrons moving in these planes. Thus, the problem becomes in facttwo-dimensional. Note that the electronic properties of two-dimensional systems were intensively studied recently [3]. They are of importance in heterostructures, layered crystals and thin films, ElsevierScience Publishers B.V.
We will consider the problem in the framework of the gauge theory ofdislocations and disclinations for an elastic continuum [4] extended by including the electronic fields [5]. In two space dimensions, the gauge group for rotational defects is G = SO (2). The electronic fields are introduced in a gauge invariant form. In fact, we will study a simpler problem where a free electron with an effective mass m* scatters on an individual defect. The stationary Schrodinger equation for the electronic wave function takes the following form [1], / h2 (8~ i WA) 2+ V(r) )!PE(r) =EV’E(r) ~ 2m / 1
(
—
—
—
Here WA are the gauge fields associated with the disclination fields, and V(r) is the deformation potential. The electron energy E is measured relative to the bottom of the conduction band. Electron fields are supposed to be coupled minimally to the gauge fields. Note that even though we exclude the interaction between electrons and acoustic waves setting V= 0 in (1), we get the equation for a free electron interacting with the gauge field. We have shown that this fact can result in the disclination-induced Aharonov—Bohm (AB) effect for a certain kind of vortex defects [1]. One should emphasize that this effect exists only for defects which affect the rotational symmetry of a crystal. Particularly, it does not appear for free electrons in the dislocated crystals 65
Volume 175, number 1
PHYSICS LETTERS A
29 March 1993
since dislocations disturb only the translation group of a crystal. Nevertheless, the topologically singular character of dislocations can be manifested as well.
In the framework ofthe gauge theory [4], the strain tensor is determined as EAB= B~öUB~B— öAB where B~,= ôa X’ + ~ W0 x~is the distortion tensor. For the
For instance, it was shown in the framework of the tight-binding model for electrons that the phase-dismatching effect can exist in dislocated crystals [6]. There is also an experimental observation of Berry’s geometrical phase in electron diffraction from a screw
vortex solution thattheSpfollowEAb= 2 (r) —2, where we g( r)have = 0r found F( r). It[1] takes ging general form,
dislocation Let us take[7]. into account the deformation potential in (1). In the isotropic case, the deformation potentialis determined as V(r)= —IGSPEAB where G is an interaction constant and E~is the strain tensor. Note that the problem becomes complex even for known dislocations. This follows from the fact that the strain fields generated by the topological defects should be found from the strongly nonlinear equations. There are only several known results describing the far field of dislocations. Often, the deformarion potential near the dislocation line is modelled in the Gaussian-type form proposed first in ref. [8]: 2
2
V(r, O)—~ V0exp(—r lb ), bis the Burgers vector. In this case, the deformation potential can be considered as a short-range one simplifying the analysis of the scattering problem. In many respects, the problem becomes similar to that of forthe point defects. Fortunately, in the framework gauge theory we have found the exact solution for the disclination vortex that gives us the explicit expression for the deformation potential in a wide space region. In cylindrical coordinates (r, 0) the static vortex-like solution has the following form [1 ]~ —
11vA\
X ~
L’j
\
I
fl\
)r~) cos~l’u~,
X2(X4) =F(r) sin(vO)
,
g( t) =g
1 (t) =N0 cosh [ ~ cosh1(l/t)+~xl], 1(1/t)], It~l ~ 1 =g2(t)=N0cos[~cos (4)
where NO=2%,/i~7~T~, t=r/r 0, r0=~,/27g~A/4B~ defines the core radius, 1=0, 1, 2. Note that the deformation potential for the disclination vortex does not depend on 0. In this case, the ansatz
~PE(r)=
can be used. Thus, the stationary Schrodinger equation is rewritten as / 1 d d ~ v ‘~ 2m* \ (~— r + ‘~ 2 / + V( r) )uJ~(r) r 2u~(r) r r , r (5) =k where k2 = 2m*E/h2. For v = 0 we get the habitual equation describing electron scattering in a certain radial potential V(r). This problem arises for linear dislocations as well as for point defects on the plane. On the other hand, setting in (5) G = 0, we get the AB-like problem which has been exactly solved. The . scattering amplitude is found to be [1] — ~—
— f~(
Wr(X8)=0,
)_.
~—
~
—~—
1
—i{v}O i(v+1/2)(O+,~)
e
where { v} = v — [v] and the brackets
sin(xv) sin(~0)
6
‘
[
]
indicate the
W 0(XB)= W(r)= v/r,
(3)
where x’ is the state vector of the elastic fields. The Frank vector for such a solution2 is normal thevplane ( r). Noteto that is in w = (0, w), and rotvW= w5 It is interesting that fact the 0, Frank index = w/2x. the strain fields generated by this vortex were shown to be similar to those for an individual wedge disclination [9]. 66
e u~(r)—~=, j=0, ±1, ±2 V
(2)
and
~
integer part. One can see that (6) has a similar behaviour to be compared with the Coulomb scattering in two space dimensions (see, e.g., ref. [3]). Particularly, theClearly, scattering amplitude becomes singular when 0—* 0. small-angle scattering will be of importance in this case. Note, however, that the relaxation time does not contain any singularities. Taking into account (4) one can write the general expression for the radial potential in (5) as
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
with U0,,, will be localized lem isenergies actuallyE< two_dimensional. Let us consider the since the prob-
2 [ ~ cosh 1 (l/t)] U( t) = G K1 cosh +K 2, t~1 , 2/t = G K, cos2 [ cos ‘(lit) + ~t1] —
—
~
+K
2,
t~ 1,
—
(7)
2/I
where K 2h2/2m*r~ 1 =DGdimensional and K2=K2(j) = (j— v) D= 2B/3A are positive parameters, =4~.+j~)/3(A+2jz). One can see that the potential consists of two parts. The first part comes from the deformation potential while the second one, proportional to 1(2, comes from the kinetic term. This part contains the Frank index and can be called the
scattered states with E> U,~,,,,. We will use the phaseshift method (see, e.g., ref. [10]). The scattering process is supposed to be elastic. In the presence of a short-range potential only a contribution to the s-wave (j=0) will in be (5), remarkably affected. The asymptotic form of the wave function for the scattered electron can be written as (8)
~(r)—7(f~+fdP)exp(ikr),
where f~ takes the form (6), and exp[i(d~,+2~~)] sin ô
fdp =
0. topological one. Previously [1,2], we have considered the Dirichiet boundary problem for which only the branch with 1=2 is found to exist. There is, however, another type of the boundary problem. Namely, one can consider the homogeneous Neumann boundary conditions (see details in ref. [4]). In this case, all three branches are found to be valid [9]. One can see, however, that only the branch with 1=0 is continuous. Here, we restrict our attention to the case 1=0. The form of potential (7) depends essentially on the model parameters. It is convenient to introduce the dimensionless parameter a=K1/6K2, so that DGm*r~
a_3(Jv)2h2.
Note that in (9) there is in addition to d0 a phase shift of ~ = — vit. One can see that fdp does not depend on the scatteringangle 0. In many respects, this result is similar to that for two_dimensional electrons scattered by a charged impurity when the effects of a core are taken into account (see, e.g., ref. [10]). There is, however, an important difference. The Coulomb interaction and the hard-core scatteringcan be considered independently. In our case, the deformation potential itself depends on the gauge fields. We are interested here in the transport properties of disclinated materials. Let us calculate the relaxation time which is determined to be r’=JP(O)(l_coso)dO,
For 1=0 the potential (7) tends to the constant U,,,,,,=G(l—~D)when t-*~. Let us consider three physically interesting regions for a: (1) a ~ 1. In this limit, the contribution from the topological part in (7) is decisive. As a result, the potential becomes repulsive. It decreases continuously with increasing t while t’~t,,,,,,,,. Then, the potential slightly increases reaching the value U0,,. Note that t,....~tends to infinity when a tends to zero. The potential well is extremely shallow, lies far from the disclination line, and does not affect the scattering process. (2) a— 1. When a-+l, tm,,, moves to the region t— 1. The depth ofthe potential well increases though the well remains shallow. We will study this region in detail by using the short-range approximation for the deformation potential. Clearly, the electron states
(9)
(10)
where P( 0) = (~k/m*)If( 0)12 is the scattering probability. In our case, f(0) =f~+fdP. Hence, the relaxation time related with disclination vortices can be written as ‘=r~+~,’+r;,~, where
(11)
v),
(12)
~_-i
=
2 (it
4h r~=
‘~I~ sin
~fldIsSifl2ôo,
(13)
and r; comes from the interference term. We will not consider it here. Note that n~jj,denotes the density of disclinations. 67
Volume 175, number 1
PHYSICS LETTERS A
One can see that the contribution from the deformation potential becomes remarkable. This follows from the appearance of the potential well which defines the value of the phase shift ö0. In the general case, or, =O~ ( k, U0, R) where U0 is the depth of the well, and R is the effective radius of the interaction which has a value close to the width of the well. One can find from (7) that 312— 3cr— 1] (14) Ka U0 = -j-~—[(1 + cr) As a 1, the depth of the potential well is small. In this case, there exists a discrete level E 0 located close to the top of the well. As a result, one obtains that sin ~ 1, i.e., resonance scattering will be of importance. The remarkable role ofresonance scattering in semiconductors with neutral point defects has first been shown by Sclar in ref. [11]. He demonstrated that neutral impurities, that can grasp an electron on the shallow level, are very effective in the scattering process. However, the phase shift becomes again small in the case of deep electron levels. This fact will be of interest in the next paragraph. (3) Let us consider the limit a ~ 1. The analysis of (7) shows that the potential well becomes deep (see also (14)), and t,,...~is located in the region t < 1. In this case, the lowest discrete level E0 for an electron lies deep as well. Thus, the electron states with E< U,,,,, will be strongly localized in the region R r0. The relaxation time for scattered states takes the form (13), but the phase shift is small. In this case, the role of the AB-like topological scattering becomes a°ain~rincinal D ~ It is important to note that the parameter a depends on the value of the interaction constant G. The fact is that this constant can be directly connected with the Fermi energy EF. In two space dimensions, it can be estimated as G =EF [2]. On the other hand, E~=2ith ip/gm*, where pis the surface density of the conducting electrons and g denotes the degeneracy 2. of electron a(j=0)=2xDpr~/3gv One can seelevels. that theHence value of a depends drastically on p and r 0. More precisely, there is a characteristic dimensionless parameter pr~.For a concrete defect with fixed values ofthe Frank index v and r0, the parameter a is entirely determined by the density of .
~
68
29 March 1993
conducting electrons. Thus, in experiment, one can study all three regions by simply changing this density. It is of interest that the relaxation time is found to have a specific dependence on the density of conducting electrons. It has a minimal value when prr,~ 1. In other regions, the contribution from the deformation potential diminishes and, as a result, the relaxation time increases. It should be noted that a range) canchange be realized heterostructures (see, e.g., controlled ofthat theinin electron density (in a wide ref. [3]). Note also our case the conductivity depends on the temperature only via p( T). In metals p does not depend on the temperature whereas in semiconductorsthis dependence should be taken into account. In conclusion, we have considered the problem of electron scatteringby disclination vortices. The scattering process is found to depend essentially on the density of the conducting electrons as well as on the core radius r0. We have shown that the topological contribution to the scattering becomes principal for small and high electron densities. At ~T~r, — 1 resonance scattering is of importance. The experimental investigation of disclinated-induced scattering will be of most interest in materials for which the density ofthe conducting electrons can vary in a wide range.
References [1] V.A. Osipov, Phys. Lett. A 164 (1992) 327. [2] V.A. Osipov, Gauge theory of dislocations and disdinations planarA. elastic systems, Phys. AElectronic (1993), in press. of [31for T. Ando, Fowler and F.J. Stern, properties two-dimensional systems, Rev. Mod. Phys. 54 (1982) 437. [4] A. Kadi~andD.G.B. Edelen, A gauge theory of dislocations and disclinations, in: Lecture notes in physics, Vol. 174, eds. H. Araki, J. Ehlers, K. Hepp, R. Rippenhahn, H.A. Weidenrnuller andi. Zittarz (Springer, Berlin, 1983). [5] V.A. Osipov, PhysicaA 175 (1991)369. [6] K. Kawamura, Z. Physik B. 29 (1978)101; 30 (1978) 1. [7] D.M. 2863. Bird and A.R. Preston, Phys. Rev. Lett. 61(1988) [8) M. Ortenberg, Phys. Status Solidi (b) 60 (1973) 273. [9] R.V. Konoplich, E.V. Seivanov and V.P. Zhukov, Phys. Lett. [10] B.K. Ridley, Quantum processes in semiconductors (Clarendon, Oxford, 1982). [11] N. Sclar, Phys. Rev. 104 (1956) 1548.
PHYSICS LETTERS A
On the electron scattering in disclinated crystals V.A. Osipov Joint Institutefor Nuclear Research, Laboratory ofTheoretical Physics, 141980 Dubna, Moscow Region, Russian Federation
Received 18 January 1993; accepted forpublication 3 February 1993 Communicated by L.J. Sham.
We study disclination-induced electron scattering in materials with disclination vortices. Two kinds of scattering processes are shown to be ofimportance: the known deformation-potential scattering and an Aharonov—Bohm-like scattering generated by the topological natureof the disdinations. The relaxation time is calculated. The scattering process is found to depend essentially on the density of the conducting electrons aswell ason the disclination core radius.
In previous papers [1,2] we have shown that the topologically singular character of a crystal with disclination vortices can result in a new kind ofelectron scattering that is pure topological in its origin. In many respects, this Aharonov—Bohm-like topological scattering differs from the known deformationpotential scattering that takes place due to elastic strains caused by defects. Particularly, the topologically induced resistivity is found to have the characteristic oscillatory behaviour which depends on the disclination flux characterized by the Frank index. It is clear that both these kinds of scattering should be of importance at low temperatures where other possible mechanisms are less effective. In this Letter we study the transport properties of a disclinated crystal when both scattering mechanisms are taken into account. We consider a crystal perforated by linear disclinations oriented parallel to the z-axis. The density of the disclinations is supposed to be small. In this case, the scattering process on each defect can be examined separately. Linear disclinations apparently break the rotational symmetty on the planes perpendicular to the disclination lines. The scattering process is the most effectivefor electrons moving in these planes. Thus, the problem becomes in facttwo-dimensional. Note that the electronic properties of two-dimensional systems were intensively studied recently [3]. They are of importance in heterostructures, layered crystals and thin films, ElsevierScience Publishers B.V.
We will consider the problem in the framework of the gauge theory ofdislocations and disclinations for an elastic continuum [4] extended by including the electronic fields [5]. In two space dimensions, the gauge group for rotational defects is G = SO (2). The electronic fields are introduced in a gauge invariant form. In fact, we will study a simpler problem where a free electron with an effective mass m* scatters on an individual defect. The stationary Schrodinger equation for the electronic wave function takes the following form [1], / h2 (8~ i WA) 2+ V(r) )!PE(r) =EV’E(r) ~ 2m / 1
(
—
—
—
Here WA are the gauge fields associated with the disclination fields, and V(r) is the deformation potential. The electron energy E is measured relative to the bottom of the conduction band. Electron fields are supposed to be coupled minimally to the gauge fields. Note that even though we exclude the interaction between electrons and acoustic waves setting V= 0 in (1), we get the equation for a free electron interacting with the gauge field. We have shown that this fact can result in the disclination-induced Aharonov—Bohm (AB) effect for a certain kind of vortex defects [1]. One should emphasize that this effect exists only for defects which affect the rotational symmetry of a crystal. Particularly, it does not appear for free electrons in the dislocated crystals 65
Volume 175, number 1
PHYSICS LETTERS A
29 March 1993
since dislocations disturb only the translation group of a crystal. Nevertheless, the topologically singular character of dislocations can be manifested as well.
In the framework ofthe gauge theory [4], the strain tensor is determined as EAB= B~öUB~B— öAB where B~,= ôa X’ + ~ W0 x~is the distortion tensor. For the
For instance, it was shown in the framework of the tight-binding model for electrons that the phase-dismatching effect can exist in dislocated crystals [6]. There is also an experimental observation of Berry’s geometrical phase in electron diffraction from a screw
vortex solution thattheSpfollowEAb= 2 (r) —2, where we g( r)have = 0r found F( r). It[1] takes ging general form,
dislocation Let us take[7]. into account the deformation potential in (1). In the isotropic case, the deformation potentialis determined as V(r)= —IGSPEAB where G is an interaction constant and E~is the strain tensor. Note that the problem becomes complex even for known dislocations. This follows from the fact that the strain fields generated by the topological defects should be found from the strongly nonlinear equations. There are only several known results describing the far field of dislocations. Often, the deformarion potential near the dislocation line is modelled in the Gaussian-type form proposed first in ref. [8]: 2
2
V(r, O)—~ V0exp(—r lb ), bis the Burgers vector. In this case, the deformation potential can be considered as a short-range one simplifying the analysis of the scattering problem. In many respects, the problem becomes similar to that of forthe point defects. Fortunately, in the framework gauge theory we have found the exact solution for the disclination vortex that gives us the explicit expression for the deformation potential in a wide space region. In cylindrical coordinates (r, 0) the static vortex-like solution has the following form [1 ]~ —
11vA\
X ~
L’j
\
I
fl\
)r~) cos~l’u~,
X2(X4) =F(r) sin(vO)
,
g( t) =g
1 (t) =N0 cosh [ ~ cosh1(l/t)+~xl], 1(1/t)], It~l ~ 1 =g2(t)=N0cos[~cos (4)
where NO=2%,/i~7~T~, t=r/r 0, r0=~,/27g~A/4B~ defines the core radius, 1=0, 1, 2. Note that the deformation potential for the disclination vortex does not depend on 0. In this case, the ansatz
~PE(r)=
can be used. Thus, the stationary Schrodinger equation is rewritten as / 1 d d ~ v ‘~ 2m* \ (~— r + ‘~ 2 / + V( r) )uJ~(r) r 2u~(r) r r , r (5) =k where k2 = 2m*E/h2. For v = 0 we get the habitual equation describing electron scattering in a certain radial potential V(r). This problem arises for linear dislocations as well as for point defects on the plane. On the other hand, setting in (5) G = 0, we get the AB-like problem which has been exactly solved. The . scattering amplitude is found to be [1] — ~—
— f~(
Wr(X8)=0,
)_.
~—
~
—~—
1
—i{v}O i(v+1/2)(O+,~)
e
where { v} = v — [v] and the brackets
sin(xv) sin(~0)
6
‘
[
]
indicate the
W 0(XB)= W(r)= v/r,
(3)
where x’ is the state vector of the elastic fields. The Frank vector for such a solution2 is normal thevplane ( r). Noteto that is in w = (0, w), and rotvW= w5 It is interesting that fact the 0, Frank index = w/2x. the strain fields generated by this vortex were shown to be similar to those for an individual wedge disclination [9]. 66
e u~(r)—~=, j=0, ±1, ±2 V
(2)
and
~
integer part. One can see that (6) has a similar behaviour to be compared with the Coulomb scattering in two space dimensions (see, e.g., ref. [3]). Particularly, theClearly, scattering amplitude becomes singular when 0—* 0. small-angle scattering will be of importance in this case. Note, however, that the relaxation time does not contain any singularities. Taking into account (4) one can write the general expression for the radial potential in (5) as
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
with U0,,, will be localized lem isenergies actuallyE< two_dimensional. Let us consider the since the prob-
2 [ ~ cosh 1 (l/t)] U( t) = G K1 cosh +K 2, t~1 , 2/t = G K, cos2 [ cos ‘(lit) + ~t1] —
—
~
+K
2,
t~ 1,
—
(7)
2/I
where K 2h2/2m*r~ 1 =DGdimensional and K2=K2(j) = (j— v) D= 2B/3A are positive parameters, =4~.+j~)/3(A+2jz). One can see that the potential consists of two parts. The first part comes from the deformation potential while the second one, proportional to 1(2, comes from the kinetic term. This part contains the Frank index and can be called the
scattered states with E> U,~,,,,. We will use the phaseshift method (see, e.g., ref. [10]). The scattering process is supposed to be elastic. In the presence of a short-range potential only a contribution to the s-wave (j=0) will in be (5), remarkably affected. The asymptotic form of the wave function for the scattered electron can be written as (8)
~(r)—7(f~+fdP)exp(ikr),
where f~ takes the form (6), and exp[i(d~,+2~~)] sin ô
fdp =
0. topological one. Previously [1,2], we have considered the Dirichiet boundary problem for which only the branch with 1=2 is found to exist. There is, however, another type of the boundary problem. Namely, one can consider the homogeneous Neumann boundary conditions (see details in ref. [4]). In this case, all three branches are found to be valid [9]. One can see, however, that only the branch with 1=0 is continuous. Here, we restrict our attention to the case 1=0. The form of potential (7) depends essentially on the model parameters. It is convenient to introduce the dimensionless parameter a=K1/6K2, so that DGm*r~
a_3(Jv)2h2.
Note that in (9) there is in addition to d0 a phase shift of ~ = — vit. One can see that fdp does not depend on the scatteringangle 0. In many respects, this result is similar to that for two_dimensional electrons scattered by a charged impurity when the effects of a core are taken into account (see, e.g., ref. [10]). There is, however, an important difference. The Coulomb interaction and the hard-core scatteringcan be considered independently. In our case, the deformation potential itself depends on the gauge fields. We are interested here in the transport properties of disclinated materials. Let us calculate the relaxation time which is determined to be r’=JP(O)(l_coso)dO,
For 1=0 the potential (7) tends to the constant U,,,,,,=G(l—~D)when t-*~. Let us consider three physically interesting regions for a: (1) a ~ 1. In this limit, the contribution from the topological part in (7) is decisive. As a result, the potential becomes repulsive. It decreases continuously with increasing t while t’~t,,,,,,,,. Then, the potential slightly increases reaching the value U0,,. Note that t,....~tends to infinity when a tends to zero. The potential well is extremely shallow, lies far from the disclination line, and does not affect the scattering process. (2) a— 1. When a-+l, tm,,, moves to the region t— 1. The depth ofthe potential well increases though the well remains shallow. We will study this region in detail by using the short-range approximation for the deformation potential. Clearly, the electron states
(9)
(10)
where P( 0) = (~k/m*)If( 0)12 is the scattering probability. In our case, f(0) =f~+fdP. Hence, the relaxation time related with disclination vortices can be written as ‘=r~+~,’+r;,~, where
(11)
v),
(12)
~_-i
=
2 (it
4h r~=
‘~I~ sin
~fldIsSifl2ôo,
(13)
and r; comes from the interference term. We will not consider it here. Note that n~jj,denotes the density of disclinations. 67
Volume 175, number 1
PHYSICS LETTERS A
One can see that the contribution from the deformation potential becomes remarkable. This follows from the appearance of the potential well which defines the value of the phase shift ö0. In the general case, or, =O~ ( k, U0, R) where U0 is the depth of the well, and R is the effective radius of the interaction which has a value close to the width of the well. One can find from (7) that 312— 3cr— 1] (14) Ka U0 = -j-~—[(1 + cr) As a 1, the depth of the potential well is small. In this case, there exists a discrete level E 0 located close to the top of the well. As a result, one obtains that sin ~ 1, i.e., resonance scattering will be of importance. The remarkable role ofresonance scattering in semiconductors with neutral point defects has first been shown by Sclar in ref. [11]. He demonstrated that neutral impurities, that can grasp an electron on the shallow level, are very effective in the scattering process. However, the phase shift becomes again small in the case of deep electron levels. This fact will be of interest in the next paragraph. (3) Let us consider the limit a ~ 1. The analysis of (7) shows that the potential well becomes deep (see also (14)), and t,,...~is located in the region t < 1. In this case, the lowest discrete level E0 for an electron lies deep as well. Thus, the electron states with E< U,,,,, will be strongly localized in the region R r0. The relaxation time for scattered states takes the form (13), but the phase shift is small. In this case, the role of the AB-like topological scattering becomes a°ain~rincinal D ~ It is important to note that the parameter a depends on the value of the interaction constant G. The fact is that this constant can be directly connected with the Fermi energy EF. In two space dimensions, it can be estimated as G =EF [2]. On the other hand, E~=2ith ip/gm*, where pis the surface density of the conducting electrons and g denotes the degeneracy 2. of electron a(j=0)=2xDpr~/3gv One can seelevels. that theHence value of a depends drastically on p and r 0. More precisely, there is a characteristic dimensionless parameter pr~.For a concrete defect with fixed values ofthe Frank index v and r0, the parameter a is entirely determined by the density of .
~
68
29 March 1993
conducting electrons. Thus, in experiment, one can study all three regions by simply changing this density. It is of interest that the relaxation time is found to have a specific dependence on the density of conducting electrons. It has a minimal value when prr,~ 1. In other regions, the contribution from the deformation potential diminishes and, as a result, the relaxation time increases. It should be noted that a range) canchange be realized heterostructures (see, e.g., controlled ofthat theinin electron density (in a wide ref. [3]). Note also our case the conductivity depends on the temperature only via p( T). In metals p does not depend on the temperature whereas in semiconductorsthis dependence should be taken into account. In conclusion, we have considered the problem of electron scatteringby disclination vortices. The scattering process is found to depend essentially on the density of the conducting electrons as well as on the core radius r0. We have shown that the topological contribution to the scattering becomes principal for small and high electron densities. At ~T~r, — 1 resonance scattering is of importance. The experimental investigation of disclinated-induced scattering will be of most interest in materials for which the density ofthe conducting electrons can vary in a wide range.
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