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Quantum transport in a disordered nanostructure in the presence of a magnetic field
Superlattices
and Microstructures,
Vol. 7 1, No. 2, 1992
241
QUANTUM TRANSPORT IN A DISORDERED NANOSTRUCTURE IN THE PRESENCE OF A MAGNETIC FIELD S. Cllaudllltri
ant1 F-3.Bantlyopadll_a~
Departtnent
of Electrical
UnivtGty Notre
Enginccritlg
of Notre
Dame,
Intliana
Dame 36556
51. Cahay Department
of Electrical University Cincinnati, (Received:
and
Compukr
Engineering
of Cincinnati OH 4.5221 Alay
19, 1991)
We have studied phase coherent electron transport in a disordered semiconductor quantum wire subjected to a weak magnetic field. Our model @eats disorder (elastic scattering) with scattering matrices and uses Weber functions to calculate the elements of the scattering matrices. The Weber functions account for the magnetic field exactly. We find that the universal conductance fluctuations and the two-terminal Landauer resistance of a disordered structure are surprisingly sensitive to whether the scatterers are attractive or repulsive and the difference between the effects of attractive and repulsive scatterers is accentuated by the presence of evanescent states.
I. Introduction Quantum transport in disordered nanostructures in the presence of a magnetic field has been studied extensively in the past [I]. The theoretical techniques used to analyze this type of transport mostly employed the recursive Green’s function method [2] or the Kadanoff-Baym-Keldysh formalism 131. Very recently, the scattering matrix method which we had developed earlier 141 was applied to study this problem 151. In the scattering matrix method, the transmission amplitudes for electron waves incident from various modes on a disordered structure are obtained from the overall scattering matrix for the structure. The transmission amplitudes are then used to calculate the conductance of the structure from the multichannel two-terminal Landauer-Biittiker formula [6]. In order to obtain the overall scattering manix of a disordered structure subjected to a magnetic field, one needs
0749-6036/92/020241+04$02.00/0
to solve the Schriidinger equation for electrons in the presence of a random time-independent elastic scattering potential and a magnetic vector potential. In Ref. 5, both the scattering potential and the magnetic vector potential were treated as perturbation to obtain the scattering matrix. The shortcoming of this approach is that a very large number of modes will have to be included if the magnetic field and/or the scattering is strong. This is required not just to model higher order Born scattetings which are appreciable when the scattering potential is strong and the Fermi level lies near a subband minima [5], but also to model effects induced by a strong magnetic field such as Landau condensation and transport by edge states. Since the size of the scattering matrix is 2M x %!I where Af is the total number of modes considered, inclusion of a large number of modes makes the scattering matrix large and difficult to manipulate computationally. In this paper, we have used a somewhat different ap-
0 1992 Academic
Press Limited
242
Superlattices
preach to treat transport in the presence of elastic scattering and a magnetic field. Instead of treating both the scattering and the magnetic field as perturbation, we treat only the scattering as perturbation while using Weber functions [7] to represent the wavefunctions of the modes. These functions are the wavefunctions of the hybrid magnetoelectronic tates in a quantum wire subjected to a magnetic field, so that the use of these functions automatically includes the effect of the magnetic field. This approach reduces the number of modes to be considered in the scattering matrices thus reducing the size of the matrix considerably. This allows us to model long structures with realistic defect concentrations thereby allowing us to study interesting phenomena whose study would have been computationally prohibitive otherwise. In the next few sections we describe the theoretical technique and conclude with results pertaining to the length dependence of a disordered structure’s resistance in various magnetic fields. We also study quenching of localization by the field and sample specific conductance fluctuations in the range of ballistic transport to strong localization. In all cases, we tind that the results have a strong dependence on whether the scatterers are attractive or repulsive and whether evanescent states (states with imaginary/complex wavevectors) are accounted for or neglected.
and Microstructures,
8“ =
Vol. 7 1, No. 2, 1992
ctkzo(y)
65)
where II’ is the width of the structure. The wavefunction d(?/) satisfies the eigenequation
a20 t $ * E _ iI!/’
IF","'
(Y -
YOY4 =
[
1
0
i(j)
where pi, is the cyclotron frequency and h k, I/o = ~ m*w,
(7)
The above equation reduces to the Weber equation [7]
- Ye), where 1 = E/hwc and < = @(?, The general solution of Equation (8) is
where
II. Theory (11)
The SchrOdinger equation describing transport in a confined quasi one-dimensional channel interspersed with e’lastic scatterers is given by
where I(r) is the r-function and D,(Z) is the parabolic cylinder function. The constants Ct and C2 are determined from the boundary conditions and normalization of the wavefunction. The boundary conditions give us
c2
where A is the magnetic vector potential, V(y) is the confining potential in the y-direction and V’(r, Y) is the elastic scattering potential. We choose the Landau gauge A = (-By,O,O)
(2)
where B is the z-directed magnetic flux density. The electrostatic potentials are given by V(Y)
= 0 =CO
IYI 5 d IYI > d
(3)
where 2rl is the width of the structure in the y-direction and ~‘.‘(P,Y) = y6(2 - ri)6(Y - Y,)
f0~ the ith
Scatterer
(4) The wavefunction $1can be written as
O,(Y = 4
c;=--=MY = 4
h(Y = -4 &(?I = -4
(12)
and the normalization of the wavefunction requires (13) The above two equations give us c’, and C2. For weak magnetic fields, the energy dispersion relation is given by [7]
Superlattices
and Microstructures,
Vol. 7 1. No. 2, 1992
243 III. Results
Using the energy dispersion relation of Equation (14) and the basis functions given by Equations (9) - (lli), we construct the scattering matrix for a disordered structure as described in Ref. 4. The scattering matrix relates the amplitudes of waves reflected into various modes to the amplitudes of incident waves. We consider both propagating modes with real wavevectors (which correspond to magnetoelectric subbands having their bottoms below the Fermi energy) and evanescent modes with complex/imaginary wavevectors whose bottoms are above the Fermi energy. Including the evanescent modes is important (even though they do not carry current) since they affect the transmission and reflection of the propagating modes that do carry current. From the overall scattering matrix, we directly obtain the transmission amplitudes for an electron with the Fermi energy incident in mode n7 and transmitting into mode n. These amplitudes I’,,, are then used in the Landauer formula ., 2 M. N.
to find the linear-response conductance. The summations in the above equation are carried out only over the propagating modes (JI,, is the number of propagating modes in the left contact and .V, is the number of propagating modes in the right contact).
10.00
I””
I””
I”‘,
I
”
In Fig. I, we show the conductance of a structure as a function of its length for different values of the magnetic field. The structure is a GaAs mesa 1000 A wide and having an impurity concentration of I .Zi.rlO’” c.~K’. The impurities are all attractive. The Fermi energy is 20 meV which corresponds to a two-dimensional carrier concentration of ‘,.f.r~lO” CX-~. The total number of modes were twenty, three-fourths of which were evanescent. The conductance decreases with increasing length almost linearly as expected in the weak localization regime. The fluctuations are sample specific and are universal conductance fluctuations. Their rms value is Y t 2//~ as expected from universal conductance Huctuations theory. In this figure we show how the conductance versus length protile is affected by the magnetic field. The lowest curve corresponds to no magnetic field and the topmost curve is for a magnetic field of 0.04 Tesla. The middle curve is for a magnetic field of 0.02 Tesla. The conductance curves shift upward with increasing magnetic field because of the quenching of localization by the field [8]. In Fig. 2, we show the rms values of the conductance fluctuations as a function of the magnetic field. The rms value was obtained from the data for twenty different impurity configurations. The Fermi energy was IO meV and there were four propagating modes along with twenty evanescent modes. In Figs. 3(a) and 3(b), we show the conductance versus magnetic field for attractive and repulsive scatterers. The structure is 1.6 /irn long. In all other aspects its parameters are the same as those of the structure for Fig 1. In Fig. 3(a), we did not include any evanescent states while in Fig 3(b) we had 40 evanescent states. The in-
”
Reld=O.O Tala Ficld=O.02 Tesla k&G‘ i i& ______________.
‘I
L----l 1.50
5.0
1.56
L.62
1.68
1.74
IIIj,l~,,,
!!
1
0.36
I
I
1
1.80
I
0.72
I
1.08
sc,I/I,I
c
1.44
1
1,,1~11,/111 I
I. The two terminal conductance versus length for magnetic fields of 0, 0.02 and 0.04 Tesla (inset:encircled portion magnified ).
0.16
0.22
,,I 0.28
,, 0.34
10
Magnetic Field(Tesla) 2. Rms values of the conductance fluctuations versus magnetic field.
244
Superlattices
and Microstructures,
Vol. 11, No. 2, 1992
of elastic scattering and in the presence of a magnetic held. We have also shown the influence of evanescent states on transport. This model will be useful in interpreting results of several low temperature transport experiments carried out in a magnetic field. Acknowledgement: This work was supported by the US Air Force Office of Scientific Research under grant number AFOSR-91-0211, by the Jesse H. Jones Foundation and by IBM.
0.M)
”
0.10
I””
1””
0.22
1”
0.34
“1
0.46
a I 0.58
”
0.70
REFERENCES
Magnetic Field(Tesla) IO.00
I’
”
I”
I’
._._._..._._.---‘-~.._
” _
I’
[I]. For a review see, for example, Y. Imry, Directions in Condensed Matter Physics, ed. G. Grinstein and G. Mazenko. (World Scientific, Singapore, 1986). pp. IO1 163; Nanostructure Physics and Fabrication, ed. M. A. Reed and W. P. Kirk, (Academic, Boston, 1989); C. W. J. Beenakker and H. van Houten, Solid State Physics, Vol. 44, ed. H. Ehrenreich and D. Turnbull, (Academic, New York, 1991), pp. I - 228.
”
_ .-.-._
_.
Kj 0.00 0.10
”
1’ 0.22
”
” / 0.34
”
” 046
”
: ’ 0.58
b ”
0.70
Magnetic Field(Tesla) 3. Conductance versus magnetic field: (a) no evanescent states included, (b) forty evanescent states are included. elusion of evanescent states accentuates the difference between attractive and repulsive scatterers. The conductance for attractive scatterers is expectedly lower beacuse of the formation of bound states splitting off a subband as discussed by Bagwell [9]. In conclusion, we have presented a study of quantum transport in semiconductor nanostructures in the presence
[2]. P. A. Lee and D. S. Fischer, Physical Review Letters, D. 1. Thouless and S. Kirkpatrick, Journal ofPhysics C, 14, 235 (1981). 47, 882 (1981);
131. S. Datta, Journal ofPhysics
C, 2, 8023 (1990).
[4]. M. Cahay, M. McLennan and S. Datta, Physical Review B, 37, 10125 (1988); M. Cahay, S. Bandyopadhyay,
M. A. Osman and H. L. Grubin, Surface Science, 228,331 (1990). [S]. H. Tamura and T. Ando, (preprint). [6]. M. Biittiker, IBM Journal of Research and DevelopA. D. Stone and A. Szafer, ibid,
ment, 32, 317, (1988); 32, 384 (1988).
[7]. S. Klama, Journal of Physics C. 20, 551 (1987). [8]. B. L. Al’tshuler, D. Khmelnitzkii, A. I. Larkin and P. A. Lee, Physical Review B, 22, 5142 (1980).
[9]. P. F. Bagwell, Physical Review B, 41, 10354 (1990).
and Microstructures,
Vol. 7 1, No. 2, 1992
241
QUANTUM TRANSPORT IN A DISORDERED NANOSTRUCTURE IN THE PRESENCE OF A MAGNETIC FIELD S. Cllaudllltri
ant1 F-3.Bantlyopadll_a~
Departtnent
of Electrical
UnivtGty Notre
Enginccritlg
of Notre
Dame,
Intliana
Dame 36556
51. Cahay Department
of Electrical University Cincinnati, (Received:
and
Compukr
Engineering
of Cincinnati OH 4.5221 Alay
19, 1991)
We have studied phase coherent electron transport in a disordered semiconductor quantum wire subjected to a weak magnetic field. Our model @eats disorder (elastic scattering) with scattering matrices and uses Weber functions to calculate the elements of the scattering matrices. The Weber functions account for the magnetic field exactly. We find that the universal conductance fluctuations and the two-terminal Landauer resistance of a disordered structure are surprisingly sensitive to whether the scatterers are attractive or repulsive and the difference between the effects of attractive and repulsive scatterers is accentuated by the presence of evanescent states.
I. Introduction Quantum transport in disordered nanostructures in the presence of a magnetic field has been studied extensively in the past [I]. The theoretical techniques used to analyze this type of transport mostly employed the recursive Green’s function method [2] or the Kadanoff-Baym-Keldysh formalism 131. Very recently, the scattering matrix method which we had developed earlier 141 was applied to study this problem 151. In the scattering matrix method, the transmission amplitudes for electron waves incident from various modes on a disordered structure are obtained from the overall scattering matrix for the structure. The transmission amplitudes are then used to calculate the conductance of the structure from the multichannel two-terminal Landauer-Biittiker formula [6]. In order to obtain the overall scattering manix of a disordered structure subjected to a magnetic field, one needs
0749-6036/92/020241+04$02.00/0
to solve the Schriidinger equation for electrons in the presence of a random time-independent elastic scattering potential and a magnetic vector potential. In Ref. 5, both the scattering potential and the magnetic vector potential were treated as perturbation to obtain the scattering matrix. The shortcoming of this approach is that a very large number of modes will have to be included if the magnetic field and/or the scattering is strong. This is required not just to model higher order Born scattetings which are appreciable when the scattering potential is strong and the Fermi level lies near a subband minima [5], but also to model effects induced by a strong magnetic field such as Landau condensation and transport by edge states. Since the size of the scattering matrix is 2M x %!I where Af is the total number of modes considered, inclusion of a large number of modes makes the scattering matrix large and difficult to manipulate computationally. In this paper, we have used a somewhat different ap-
0 1992 Academic
Press Limited
242
Superlattices
preach to treat transport in the presence of elastic scattering and a magnetic field. Instead of treating both the scattering and the magnetic field as perturbation, we treat only the scattering as perturbation while using Weber functions [7] to represent the wavefunctions of the modes. These functions are the wavefunctions of the hybrid magnetoelectronic tates in a quantum wire subjected to a magnetic field, so that the use of these functions automatically includes the effect of the magnetic field. This approach reduces the number of modes to be considered in the scattering matrices thus reducing the size of the matrix considerably. This allows us to model long structures with realistic defect concentrations thereby allowing us to study interesting phenomena whose study would have been computationally prohibitive otherwise. In the next few sections we describe the theoretical technique and conclude with results pertaining to the length dependence of a disordered structure’s resistance in various magnetic fields. We also study quenching of localization by the field and sample specific conductance fluctuations in the range of ballistic transport to strong localization. In all cases, we tind that the results have a strong dependence on whether the scatterers are attractive or repulsive and whether evanescent states (states with imaginary/complex wavevectors) are accounted for or neglected.
and Microstructures,
8“ =
Vol. 7 1, No. 2, 1992
ctkzo(y)
65)
where II’ is the width of the structure. The wavefunction d(?/) satisfies the eigenequation
a20 t $ * E _ iI!/’
IF","'
(Y -
YOY4 =
[
1
0
i(j)
where pi, is the cyclotron frequency and h k, I/o = ~ m*w,
(7)
The above equation reduces to the Weber equation [7]
- Ye), where 1 = E/hwc and < = @(?, The general solution of Equation (8) is
where
II. Theory (11)
The SchrOdinger equation describing transport in a confined quasi one-dimensional channel interspersed with e’lastic scatterers is given by
where I(r) is the r-function and D,(Z) is the parabolic cylinder function. The constants Ct and C2 are determined from the boundary conditions and normalization of the wavefunction. The boundary conditions give us
c2
where A is the magnetic vector potential, V(y) is the confining potential in the y-direction and V’(r, Y) is the elastic scattering potential. We choose the Landau gauge A = (-By,O,O)
(2)
where B is the z-directed magnetic flux density. The electrostatic potentials are given by V(Y)
= 0 =CO
IYI 5 d IYI > d
(3)
where 2rl is the width of the structure in the y-direction and ~‘.‘(P,Y) = y6(2 - ri)6(Y - Y,)
f0~ the ith
Scatterer
(4) The wavefunction $1can be written as
O,(Y = 4
c;=--=MY = 4
h(Y = -4 &(?I = -4
(12)
and the normalization of the wavefunction requires (13) The above two equations give us c’, and C2. For weak magnetic fields, the energy dispersion relation is given by [7]
Superlattices
and Microstructures,
Vol. 7 1. No. 2, 1992
243 III. Results
Using the energy dispersion relation of Equation (14) and the basis functions given by Equations (9) - (lli), we construct the scattering matrix for a disordered structure as described in Ref. 4. The scattering matrix relates the amplitudes of waves reflected into various modes to the amplitudes of incident waves. We consider both propagating modes with real wavevectors (which correspond to magnetoelectric subbands having their bottoms below the Fermi energy) and evanescent modes with complex/imaginary wavevectors whose bottoms are above the Fermi energy. Including the evanescent modes is important (even though they do not carry current) since they affect the transmission and reflection of the propagating modes that do carry current. From the overall scattering matrix, we directly obtain the transmission amplitudes for an electron with the Fermi energy incident in mode n7 and transmitting into mode n. These amplitudes I’,,, are then used in the Landauer formula ., 2 M. N.
to find the linear-response conductance. The summations in the above equation are carried out only over the propagating modes (JI,, is the number of propagating modes in the left contact and .V, is the number of propagating modes in the right contact).
10.00
I””
I””
I”‘,
I
”
In Fig. I, we show the conductance of a structure as a function of its length for different values of the magnetic field. The structure is a GaAs mesa 1000 A wide and having an impurity concentration of I .Zi.rlO’” c.~K’. The impurities are all attractive. The Fermi energy is 20 meV which corresponds to a two-dimensional carrier concentration of ‘,.f.r~lO” CX-~. The total number of modes were twenty, three-fourths of which were evanescent. The conductance decreases with increasing length almost linearly as expected in the weak localization regime. The fluctuations are sample specific and are universal conductance fluctuations. Their rms value is Y t 2//~ as expected from universal conductance Huctuations theory. In this figure we show how the conductance versus length protile is affected by the magnetic field. The lowest curve corresponds to no magnetic field and the topmost curve is for a magnetic field of 0.04 Tesla. The middle curve is for a magnetic field of 0.02 Tesla. The conductance curves shift upward with increasing magnetic field because of the quenching of localization by the field [8]. In Fig. 2, we show the rms values of the conductance fluctuations as a function of the magnetic field. The rms value was obtained from the data for twenty different impurity configurations. The Fermi energy was IO meV and there were four propagating modes along with twenty evanescent modes. In Figs. 3(a) and 3(b), we show the conductance versus magnetic field for attractive and repulsive scatterers. The structure is 1.6 /irn long. In all other aspects its parameters are the same as those of the structure for Fig 1. In Fig. 3(a), we did not include any evanescent states while in Fig 3(b) we had 40 evanescent states. The in-
”
Reld=O.O Tala Ficld=O.02 Tesla k&G‘ i i& ______________.
‘I
L----l 1.50
5.0
1.56
L.62
1.68
1.74
IIIj,l~,,,
!!
1
0.36
I
I
1
1.80
I
0.72
I
1.08
sc,I/I,I
c
1.44
1
1,,1~11,/111 I
I. The two terminal conductance versus length for magnetic fields of 0, 0.02 and 0.04 Tesla (inset:encircled portion magnified ).
0.16
0.22
,,I 0.28
,, 0.34
10
Magnetic Field(Tesla) 2. Rms values of the conductance fluctuations versus magnetic field.
244
Superlattices
and Microstructures,
Vol. 11, No. 2, 1992
of elastic scattering and in the presence of a magnetic held. We have also shown the influence of evanescent states on transport. This model will be useful in interpreting results of several low temperature transport experiments carried out in a magnetic field. Acknowledgement: This work was supported by the US Air Force Office of Scientific Research under grant number AFOSR-91-0211, by the Jesse H. Jones Foundation and by IBM.
0.M)
”
0.10
I””
1””
0.22
1”
0.34
“1
0.46
a I 0.58
”
0.70
REFERENCES
Magnetic Field(Tesla) IO.00
I’
”
I”
I’
._._._..._._.---‘-~.._
” _
I’
[I]. For a review see, for example, Y. Imry, Directions in Condensed Matter Physics, ed. G. Grinstein and G. Mazenko. (World Scientific, Singapore, 1986). pp. IO1 163; Nanostructure Physics and Fabrication, ed. M. A. Reed and W. P. Kirk, (Academic, Boston, 1989); C. W. J. Beenakker and H. van Houten, Solid State Physics, Vol. 44, ed. H. Ehrenreich and D. Turnbull, (Academic, New York, 1991), pp. I - 228.
”
_ .-.-._
_.
Kj 0.00 0.10
”
1’ 0.22
”
” / 0.34
”
” 046
”
: ’ 0.58
b ”
0.70
Magnetic Field(Tesla) 3. Conductance versus magnetic field: (a) no evanescent states included, (b) forty evanescent states are included. elusion of evanescent states accentuates the difference between attractive and repulsive scatterers. The conductance for attractive scatterers is expectedly lower beacuse of the formation of bound states splitting off a subband as discussed by Bagwell [9]. In conclusion, we have presented a study of quantum transport in semiconductor nanostructures in the presence
[2]. P. A. Lee and D. S. Fischer, Physical Review Letters, D. 1. Thouless and S. Kirkpatrick, Journal ofPhysics C, 14, 235 (1981). 47, 882 (1981);
131. S. Datta, Journal ofPhysics
C, 2, 8023 (1990).
[4]. M. Cahay, M. McLennan and S. Datta, Physical Review B, 37, 10125 (1988); M. Cahay, S. Bandyopadhyay,
M. A. Osman and H. L. Grubin, Surface Science, 228,331 (1990). [S]. H. Tamura and T. Ando, (preprint). [6]. M. Biittiker, IBM Journal of Research and DevelopA. D. Stone and A. Szafer, ibid,
ment, 32, 317, (1988); 32, 384 (1988).
[7]. S. Klama, Journal of Physics C. 20, 551 (1987). [8]. B. L. Al’tshuler, D. Khmelnitzkii, A. I. Larkin and P. A. Lee, Physical Review B, 22, 5142 (1980).
[9]. P. F. Bagwell, Physical Review B, 41, 10354 (1990).