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Theory of room temperature quantized resistance effects in metal-a-Si:H-metal thin film structures
J O U R N A L OF
NON-C NESOLII ELSEVIER
Journal of Non-CrystallineSolids 198-200 (1996) 825-828
Theory of room temperature quantized resistance effects in metal-a-Si:H-metal thin film structures J. Hajto
B. McAuley b, A.J. Snell b, A.E. Owen b
a Department of Applied Chemical and Physical Sciences, Napier University, Edinburgh EHIO 5DT, Scotland, UK b Department of Electrical Engineering, University of Edinburgh, Edinburgh EH9 3JL, Scotland, UK
Abstract A theoretical model of room temperature quantized resistance steps is described in terms of movement of a free electron Fermi surface into a quantized k-space (associated with electron confinement in real space) under the influence of applied electric field. The model is in good accordance with experimental observations such as the non-periodicity in quanta and voltage space and gives very realistic values for the parameters of electron lifetime, Fermi wavevector and geometry. Room temperature quantization is observed when the value of free electron lifetime ( ~ 10- ~4 s) is matched with a small size for electron confinement ( ~ 7 nm or less).
1. Introduction
2. Theoretical considerations
In electroformed V - a - S i - C r devices, the possibility of room temperature ballistic transport has been suggested by Hajto et al. [1,2] and later by Yun et al. [3] and Jafar and Haneman [4]. In these devices the injection into the ballistic channel is carried out in a two terminal geometry by an externally applied electric field, which is varied. This is in contrast to split gate quantum devices (operating at ~ 4 K) in which a three terminal configuration is used [5]. In this paper we describe a simple model which avoids the complications of a full solution of the Schr~Sdinger equation but does predict the general features of electron transport in these devices.
Consider a channel (which is bounded by an infinite potential) connecting two wide regions each containing a 3-dimensional electron gas (3 DEG), the channel is of suitable dimensions to allow ballistic transport [6]. Within the channel the electron is confined in two of the channel's dimensions and we assume that it is free to move along the axis of the channel. This system can be described using the following conditions on the k values (wavevector) of the electrons in the channel:
(1) * Correspondingauthor. Tel.: +44-131 455 2288; fax: +44-131 455 2290; e-mail: [email protected].
where k x is the k value along the axis of the channel, ky and k z are the transverse components of
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0022- 309 3(96)00048-8
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
826
k to the axis of the channel, ny and n: are integers, L~ and Ly are the dimensions of the ballistic channel. Along the channel (i.e. the x direction) the electron is free to move and thus can have any energy (ballistic conduction). However, in the y and z direction the electron is confined as if in a one dimensional infinitely deep potential well. Consequently the energy of the electron in the y and z direction is quantized. A n y electron passing through the channel must have k values described by Eq. (1). In ballistic transport, electron scattering does not occur therefore the ky and k: values of the electron remain constant. From the Landauer-Btittiker formalism [7] each such set of k values is equivalent to a 1-D sub-band and has a resistance of R = h / 2 e 2 = 12906 ~ where R is the resistance, h is the Planck's constant and e is the electron charge. The ballistic condition is fulfilled if the electron transit time, tth, across the channel along the axis ( x direction) is less than the mean free electron lifetime [2], ~-, that is tth < "r. For a channel of constant length this leads to a minimum velocity or a minimum wavevector in the x direction, kth. So that k x > kth is the condition for ballistic transport. For this model it is useful to visualize this condition in a k space diagram. In Fig. 1 an infinite set of lines (only a small section of which are shown) marks the conditions on ky set by Eq. (1).
ky Electric Field, Ecr Ikth
etc
etc Fig. 1. A cross-section (in the k X - k,. plane of k space) showing the Fermi surface in contact with the ballistic sub-band at k x = kth and k,.=k z=0.
The same quantization occurs for the k~ values although not shown in Fig. 1. But instead of - o , < k x < w only values of ]kx[ > kth (the ballistic condition) are shown i.e., only those states capable of ballistic transport through the channel. Because of the similarity with the Landauer-Biittiker formalism these states are referred to as ballistic sub-bands. To prevent electrons from being scattered thermally then two adjacent quantum levels must have an energy separation A e, which is greater than kBT. So, for an electron with quantum numbers ny and n z its energy is given by
e(ny'nz)
= -~m
~
ny2 +
]
n2z + C ,
(2) where Ly and L z are the dimensions of the channel and C is a constant arising from the energy of the electron due to motion in the x direction. From Eq. (2) the minimum energy separation A e ( n y ) or A e ( n ~ ) between two adjacent states can be calculated. To prevent any thermal scattering, A e ( n y ) or A e ( n z) must be greater than kBT. From Eq. (2) this can be expressed as a condition on the dimensions of the cross-section of the ballistic channel, Ly, Lz, <76Aat T=300K. The sphere marked on Fig. 1 represents the Fermi surface of the 3 DEG. For this simple model it is assumed that if an electron occupies a common state in both the Fermi surface and one of the ballistic sub-bands then it will travel ballistically. The displacement of the Fermi sphere under the influence of applied electric field E in k space can be given by the following expression; ~k = - ( l e l / h ) E ~ ' , where ~is the mean free electron lifetime. Consider the situation when the Fermi sphere just comes into contact with the ballistic sub-bands. For simplicity we shall initially consider a field only applied in the x direction. Fig. 1 shows a cross-section of the Fermi surface when it comes into contact with the ballistic sub-band with the condition ky = k z = 0. In this case, one sub-band has been opened but the others remain closed. At this point, gk x, the displacement of the Fermi sphere has a critical value, gkcr, at which the ballistic condition is realized. It can be seen from Fig. 1 that kth = Akcr + Akfo, where kfo
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
ky
8
Electric Field, E < E > Ecr
I
827
I
I
cv'6
I
4~ +___~
',kth i
+
k,
++
~.+
t i i i i
1.7
1.8
1.9
2.0 2.1 2.2 2.3 2.4 E l e c t r i c F i e l d (107V/m)
2.5
Fig. 2. Fermi surface in contact with more sub-bands at a field greater than that in Fig. 1.
Fig. 3. Qr versus E, crosses experiment, continuous line = theory with k f = l . 4 1 X 1 0 t° m I, ~ . = 2 N 1 0 - 1 4 s, Ecr= 1.75X107 V m -1, L ~ = I A, L: = 7 5 A, 0 = 0 ° and 05=25 °.
is the radius of the Fermi sphere. For the value of ~kcr there is a corresponding critical field in the x direction, Ecr. Increasing the field will cause more overlap between the Fermi surface and the ballistic sub-bands. Fig. 2 shows how more sub-bands come into contact with the Fermi surface as the field increases. The number of sub-bands available for ballistic conduction depends upon the area of the circle of intersection defined by the plane k x = kth and the Fermi surface. From the geometry of Fig. 2 the radius of the circle of intersection r e is given by; rf [-(Skc~-~kx)2-2kf(Skor- 5kx)] 1/2, where ~k x is the displacement of the Fermi sphere in the x direction under an applied field E x. Therefore, given the radius of the circle of intersection and the channel dimensions, the number of opened sub-bands can be calculated at a given field. Note that applying a field only in the x direction will open ballistic sub-bands in symmetrical pairs after the first sub-band has been opened. A transverse field will break this symmetry and so will allow sub-bands to be opened individually. Using basic trigonometry an angled field can be split into three components and therefore a channel that is not aligned with the field can be modelled.
mental results have been transformed by subtracting a parallel tunnelling current from the original current-voltage characteristics [1-3]. Qr = [h/2 e2]/R = 12906 f l / R refers to the dimensionless quantized resistance which is equal to the number of sub-bands opened for conduction. The parameters used by the model have already been defined earlier with the exceptions of 0 and ~b which are the angles of the applied electric field to the x-z and the x-y planes. For the calculations, a typical value of kf = 1.41 × 10 j° m -t is chosen because it corresponds to a typical value in pure V. Values of r in the range 1 x 10-13 s to 1 × 10-15 s were selected as they are
=
8
~
~
6
O ::HI;I H ; ; ; H i~ ,.-a
II t~
+
+
4-
3. Comparison of the model with experiments 0
t.....a
1.8
In Figs. 3 and 4 experimental results (marked by crosses) are compared with predictions of the k space model (marked by the solid line). The experi-
2.0
2.2 Electric
2.4 2.6 F i e l d ( 10 7V/m)
~..8
Fig. 4. Qr versus E, k ~ = l . 4 1 X 1 0 1 ° m 1, 7 . = 1 . 8 X 1 0 - 1 4 s, Eer=107 V m l, L , = 3 0 , ~ , L z = 7 0 , ~ , 0 = 0 °, 05=25 °.
828
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
typical for metals. W e selected values of Ly and L~ less than 75 A to give a spacing of the energy levels, A E, which is greater than kBT at room temperature. In Fig. 3, the low values of t y = 1 - 1 0 A increase the separation of ky ballistic sub-bands in k-space to such an extent that makes the system one dimensional regarding quantization. This combined with an angled field (~b = 25 °) produces a near regular steplike structure. However, in Fig. 4 both the dimensions L~, and L~ are within the range of 1 0 - 7 5 A. This produces j u m p s that do not occur at regular voltages and are of more than one quanta (i.e. R = h / 2 e 2) in size. Results such as those in Fig. 3 are more common than those shown in Fig. 4 and have been observed experimentally in V - p + - C r devices [1-3]. It should be also emphasized that the number of open sub-bands depends quite critically on the magnitude of the free electron lifetime. Values of 7" either one magnitude larger or smaller than 10-14 s will destroy quantization by changing the magnitude of shift of Fermi sphere per unit electric field.
4. Conclusion To conclude, we have constructed a simple theoretical model for an ideal ballistic channel across
which an electric field is applied. This situation is similar to the model of V - p + - C r devices, as proposed by Hajto et al [1,2], and after accounting for a parallel tunnelling current, can give similar results. In addition to describing the general features of the devices, the model also gives very realistic values for the parameters (e.g. kf, 7", Ly and Lz). Room temperature quantization is observed when the ' w i n d o w ' value of electron lifetime ( ~ 10 -14 s) is matched with a small size for electron confinement ( ~ 70 or less).
References
[l] J. Hajto, M.J. Rose, A.J. Snell, I.S. Osborne, A.E. Owen and P.G. LeComber, J. Non-Cryst. Solids 137&138 (1991) 499. [2] J. Hajto, A.E. Owen, A.J, Snell, P.G. LeComber and M.J. Rose, Philos. Mag. 66 (1991) 349. [3] E.-J. Yun, M.F. Becker and R.M. Walser, Appl. Phys. Lett. 63 (1993) 2493. [4] M. Jafar and D. Haneman, Phys. Rev. B47 (1993) 10911. [5] D.A. Wharam, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J. Phys. C60 (1988) 848. [6] T.J. Thornton, M. Pepper, H. Ahmed, D. Andrews and G.J. Davies, Phys. Rev. Lett. 56 (1986) 1198. [7] R. Landauer, IBM J. Res. Dev. 1 (1957) 223.
NON-C NESOLII ELSEVIER
Journal of Non-CrystallineSolids 198-200 (1996) 825-828
Theory of room temperature quantized resistance effects in metal-a-Si:H-metal thin film structures J. Hajto
B. McAuley b, A.J. Snell b, A.E. Owen b
a Department of Applied Chemical and Physical Sciences, Napier University, Edinburgh EHIO 5DT, Scotland, UK b Department of Electrical Engineering, University of Edinburgh, Edinburgh EH9 3JL, Scotland, UK
Abstract A theoretical model of room temperature quantized resistance steps is described in terms of movement of a free electron Fermi surface into a quantized k-space (associated with electron confinement in real space) under the influence of applied electric field. The model is in good accordance with experimental observations such as the non-periodicity in quanta and voltage space and gives very realistic values for the parameters of electron lifetime, Fermi wavevector and geometry. Room temperature quantization is observed when the value of free electron lifetime ( ~ 10- ~4 s) is matched with a small size for electron confinement ( ~ 7 nm or less).
1. Introduction
2. Theoretical considerations
In electroformed V - a - S i - C r devices, the possibility of room temperature ballistic transport has been suggested by Hajto et al. [1,2] and later by Yun et al. [3] and Jafar and Haneman [4]. In these devices the injection into the ballistic channel is carried out in a two terminal geometry by an externally applied electric field, which is varied. This is in contrast to split gate quantum devices (operating at ~ 4 K) in which a three terminal configuration is used [5]. In this paper we describe a simple model which avoids the complications of a full solution of the Schr~Sdinger equation but does predict the general features of electron transport in these devices.
Consider a channel (which is bounded by an infinite potential) connecting two wide regions each containing a 3-dimensional electron gas (3 DEG), the channel is of suitable dimensions to allow ballistic transport [6]. Within the channel the electron is confined in two of the channel's dimensions and we assume that it is free to move along the axis of the channel. This system can be described using the following conditions on the k values (wavevector) of the electrons in the channel:
(1) * Correspondingauthor. Tel.: +44-131 455 2288; fax: +44-131 455 2290; e-mail: [email protected].
where k x is the k value along the axis of the channel, ky and k z are the transverse components of
0022-3093/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0022- 309 3(96)00048-8
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
826
k to the axis of the channel, ny and n: are integers, L~ and Ly are the dimensions of the ballistic channel. Along the channel (i.e. the x direction) the electron is free to move and thus can have any energy (ballistic conduction). However, in the y and z direction the electron is confined as if in a one dimensional infinitely deep potential well. Consequently the energy of the electron in the y and z direction is quantized. A n y electron passing through the channel must have k values described by Eq. (1). In ballistic transport, electron scattering does not occur therefore the ky and k: values of the electron remain constant. From the Landauer-Btittiker formalism [7] each such set of k values is equivalent to a 1-D sub-band and has a resistance of R = h / 2 e 2 = 12906 ~ where R is the resistance, h is the Planck's constant and e is the electron charge. The ballistic condition is fulfilled if the electron transit time, tth, across the channel along the axis ( x direction) is less than the mean free electron lifetime [2], ~-, that is tth < "r. For a channel of constant length this leads to a minimum velocity or a minimum wavevector in the x direction, kth. So that k x > kth is the condition for ballistic transport. For this model it is useful to visualize this condition in a k space diagram. In Fig. 1 an infinite set of lines (only a small section of which are shown) marks the conditions on ky set by Eq. (1).
ky Electric Field, Ecr Ikth
etc
etc Fig. 1. A cross-section (in the k X - k,. plane of k space) showing the Fermi surface in contact with the ballistic sub-band at k x = kth and k,.=k z=0.
The same quantization occurs for the k~ values although not shown in Fig. 1. But instead of - o , < k x < w only values of ]kx[ > kth (the ballistic condition) are shown i.e., only those states capable of ballistic transport through the channel. Because of the similarity with the Landauer-Biittiker formalism these states are referred to as ballistic sub-bands. To prevent electrons from being scattered thermally then two adjacent quantum levels must have an energy separation A e, which is greater than kBT. So, for an electron with quantum numbers ny and n z its energy is given by
e(ny'nz)
= -~m
~
ny2 +
]
n2z + C ,
(2) where Ly and L z are the dimensions of the channel and C is a constant arising from the energy of the electron due to motion in the x direction. From Eq. (2) the minimum energy separation A e ( n y ) or A e ( n ~ ) between two adjacent states can be calculated. To prevent any thermal scattering, A e ( n y ) or A e ( n z) must be greater than kBT. From Eq. (2) this can be expressed as a condition on the dimensions of the cross-section of the ballistic channel, Ly, Lz, <76Aat T=300K. The sphere marked on Fig. 1 represents the Fermi surface of the 3 DEG. For this simple model it is assumed that if an electron occupies a common state in both the Fermi surface and one of the ballistic sub-bands then it will travel ballistically. The displacement of the Fermi sphere under the influence of applied electric field E in k space can be given by the following expression; ~k = - ( l e l / h ) E ~ ' , where ~is the mean free electron lifetime. Consider the situation when the Fermi sphere just comes into contact with the ballistic sub-bands. For simplicity we shall initially consider a field only applied in the x direction. Fig. 1 shows a cross-section of the Fermi surface when it comes into contact with the ballistic sub-band with the condition ky = k z = 0. In this case, one sub-band has been opened but the others remain closed. At this point, gk x, the displacement of the Fermi sphere has a critical value, gkcr, at which the ballistic condition is realized. It can be seen from Fig. 1 that kth = Akcr + Akfo, where kfo
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
ky
8
Electric Field, E < E > Ecr
I
827
I
I
cv'6
I
4~ +___~
',kth i
+
k,
++
~.+
t i i i i
1.7
1.8
1.9
2.0 2.1 2.2 2.3 2.4 E l e c t r i c F i e l d (107V/m)
2.5
Fig. 2. Fermi surface in contact with more sub-bands at a field greater than that in Fig. 1.
Fig. 3. Qr versus E, crosses experiment, continuous line = theory with k f = l . 4 1 X 1 0 t° m I, ~ . = 2 N 1 0 - 1 4 s, Ecr= 1.75X107 V m -1, L ~ = I A, L: = 7 5 A, 0 = 0 ° and 05=25 °.
is the radius of the Fermi sphere. For the value of ~kcr there is a corresponding critical field in the x direction, Ecr. Increasing the field will cause more overlap between the Fermi surface and the ballistic sub-bands. Fig. 2 shows how more sub-bands come into contact with the Fermi surface as the field increases. The number of sub-bands available for ballistic conduction depends upon the area of the circle of intersection defined by the plane k x = kth and the Fermi surface. From the geometry of Fig. 2 the radius of the circle of intersection r e is given by; rf [-(Skc~-~kx)2-2kf(Skor- 5kx)] 1/2, where ~k x is the displacement of the Fermi sphere in the x direction under an applied field E x. Therefore, given the radius of the circle of intersection and the channel dimensions, the number of opened sub-bands can be calculated at a given field. Note that applying a field only in the x direction will open ballistic sub-bands in symmetrical pairs after the first sub-band has been opened. A transverse field will break this symmetry and so will allow sub-bands to be opened individually. Using basic trigonometry an angled field can be split into three components and therefore a channel that is not aligned with the field can be modelled.
mental results have been transformed by subtracting a parallel tunnelling current from the original current-voltage characteristics [1-3]. Qr = [h/2 e2]/R = 12906 f l / R refers to the dimensionless quantized resistance which is equal to the number of sub-bands opened for conduction. The parameters used by the model have already been defined earlier with the exceptions of 0 and ~b which are the angles of the applied electric field to the x-z and the x-y planes. For the calculations, a typical value of kf = 1.41 × 10 j° m -t is chosen because it corresponds to a typical value in pure V. Values of r in the range 1 x 10-13 s to 1 × 10-15 s were selected as they are
=
8
~
~
6
O ::HI;I H ; ; ; H i~ ,.-a
II t~
+
+
4-
3. Comparison of the model with experiments 0
t.....a
1.8
In Figs. 3 and 4 experimental results (marked by crosses) are compared with predictions of the k space model (marked by the solid line). The experi-
2.0
2.2 Electric
2.4 2.6 F i e l d ( 10 7V/m)
~..8
Fig. 4. Qr versus E, k ~ = l . 4 1 X 1 0 1 ° m 1, 7 . = 1 . 8 X 1 0 - 1 4 s, Eer=107 V m l, L , = 3 0 , ~ , L z = 7 0 , ~ , 0 = 0 °, 05=25 °.
828
J. Hajto et al. / Journal of Non-Crystalline Solids 198-200 (1996) 825-828
typical for metals. W e selected values of Ly and L~ less than 75 A to give a spacing of the energy levels, A E, which is greater than kBT at room temperature. In Fig. 3, the low values of t y = 1 - 1 0 A increase the separation of ky ballistic sub-bands in k-space to such an extent that makes the system one dimensional regarding quantization. This combined with an angled field (~b = 25 °) produces a near regular steplike structure. However, in Fig. 4 both the dimensions L~, and L~ are within the range of 1 0 - 7 5 A. This produces j u m p s that do not occur at regular voltages and are of more than one quanta (i.e. R = h / 2 e 2) in size. Results such as those in Fig. 3 are more common than those shown in Fig. 4 and have been observed experimentally in V - p + - C r devices [1-3]. It should be also emphasized that the number of open sub-bands depends quite critically on the magnitude of the free electron lifetime. Values of 7" either one magnitude larger or smaller than 10-14 s will destroy quantization by changing the magnitude of shift of Fermi sphere per unit electric field.
4. Conclusion To conclude, we have constructed a simple theoretical model for an ideal ballistic channel across
which an electric field is applied. This situation is similar to the model of V - p + - C r devices, as proposed by Hajto et al [1,2], and after accounting for a parallel tunnelling current, can give similar results. In addition to describing the general features of the devices, the model also gives very realistic values for the parameters (e.g. kf, 7", Ly and Lz). Room temperature quantization is observed when the ' w i n d o w ' value of electron lifetime ( ~ 10 -14 s) is matched with a small size for electron confinement ( ~ 70 or less).
References
[l] J. Hajto, M.J. Rose, A.J. Snell, I.S. Osborne, A.E. Owen and P.G. LeComber, J. Non-Cryst. Solids 137&138 (1991) 499. [2] J. Hajto, A.E. Owen, A.J, Snell, P.G. LeComber and M.J. Rose, Philos. Mag. 66 (1991) 349. [3] E.-J. Yun, M.F. Becker and R.M. Walser, Appl. Phys. Lett. 63 (1993) 2493. [4] M. Jafar and D. Haneman, Phys. Rev. B47 (1993) 10911. [5] D.A. Wharam, M. Pepper, H. Ahmed, J.E.F. Frost, D.G. Hasko, D.C. Peacock, D.A. Ritchie and G.A.C. Jones, J. Phys. C60 (1988) 848. [6] T.J. Thornton, M. Pepper, H. Ahmed, D. Andrews and G.J. Davies, Phys. Rev. Lett. 56 (1986) 1198. [7] R. Landauer, IBM J. Res. Dev. 1 (1957) 223.