- Email: [email protected]
Comparison of electronic transport through triple electric-barrier structures and triple magnetic-barrier structures
2 February
1998
PHYSICS
Physics Letters A 238 (1998)
ELSEYIER
LETTERS
A
185-191
Comparison of electronic transport through triple electric-barrier structures and triple magnetic-barrier structures Yong Guo a*b,‘,Bing-Lin Gu ‘, Zhi-Qiang
Li b, Yoshiyuki Kawazoe b
a Department of Physics, Tsinghua University, Beijing 100084, China h Institute for Materials Research, Tohoku University, Sendai 980-77. Japan Received
3 I July 1997; revised manuscript
received 4 November Communicated
1997; accepted by A. Lagendijk
for publication
20 November
I997
Abstract We present a detailed comparison of electron transport through triple electric-barrier structures and triple magnetic-barrier structures. Tunneling through triple electric-barrier structures is mainly determined by the momentum along the tunneling direction, while in triple magnetic-barrier structures transport strongly depends not only on structural parameters but also the transverse momentum. It is confirmed that both the transmission coefficient and the conductance are dramatically reduced with increasing height or width of the middle barrier in triple magnetic-barrier structures. @ 1998 Elsevier Science B.V. PACS: 73.40.Gk; 03.65.Ge Ke~nwr&‘dstTunneling; Electric barrier; Magnetic
barrier: Wave-vector
filtering
1. Introduction Tunneling of electrons in double- and multiplebarrier electric structures has been extensively studied both theoretically and experimentally due to its potential applications. Many novel features such as sharp current peaks, negative differential resistance, intrinsic bistability, etc., have been discovered. The electricbarrier (EB) structure can be generated by, e.g., the energy band discontinuity or offset in heterostructures. Recently, the electron motion in the magnetic-barrier (MB) structure has caused a vast research interest. Experimentally, an inhomogeneous magnetic field on nanometer scale can be realized with the creation of ’ Correspondence address: search, Tohoku University, [email protected].
Institute for Sendai 980-77,
Materials ReJapan. E-mail:
03759601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. frIs0375-9601(97)00911-0
magnetic dots [ 11, the patterning of ferromagnetic materials [ 21, and the deposition of superconducting materials on conventional heterostructures [ 31. Theoretically, Peeters and Matulis [4] investigated the two-dimensional electron gas (2DEG) motion under the influence of a magnetic step, magnetic well, and magnetic barrier. Transport of electrons in more complicated magnetic barriers [ 5-71 and periodically arranged magnetic barriers [8,9] was also studied. The features of wave-vector filtering and magnetic minibands were determined. Ramaglia et al. [ lo] considered the effect of a local magnetic field on the tunneling current through a thick potential barrier and found that the magnetic field was localized strictly within the potential barrier, which led to resonances that were centered within the barrier. Miiller [ I 1 ] analyzed an infinite strip of a 2DEG in a magnetic field that varies linearly across the strip and showed that the
186
Y GUI et ~1. /Physics
Letters A 238 (1998)
system has a time-reversal asymmetry and that charge flow takes place only in the direction perpendicular to the field gradient. The transport of electrons in a curved 2DEG was investigated by Foden et al. [ 121 and the problem of a 2DEG in a smooth magnetic barrier geometry of different shape was considered by Calvo [ 131. Moreover, transport of electrons in a unidirectional weak magnetic field modulation was also investigated [ 14-161. Almost at the same time, Carmona et al. [ 171, Ye et al. [ 181 and Izawa et al. [ 191 realized periodic lateral magnetic fields experimentally and observed oscillatory magnetoresistance due to a commensurability effect between the classical cyclotron diameter and the period of magnetic modulation. Those experimental techniques opened up the way to experiments in alternating magnetic fields with periods in the nanometer regime. Very recently there has been increased interest [ 20,211 in tunneling through triple-barrier structures since these structures offer the possibility of studying the tunnel-coupled states of a double quantum well and its potential applications. It is well known that the features of electrons tunneling through an asymmetric multiple-barrier electric structure are greatly different from those of electrons tunneling through a symmetric multiple-barrier electric structure [ 22,231. It is also well known that the electron’s behavior in MB structures is drastically different from that in the EB case [ 5-71. Electron transport through the latter structure is an inherently two-dimensional process and exhibits complicated resonant tunneling features. It can be expected that there are some novel effects on the transport of electrons in triple EB structures and triple MB structures. Therefore, in this work we systematically investigate the transport properties of electrons in the triple EB structure and in the triple MB structure. Similarities and differences between electronic transport in these two cases are presented. The asymmetric effect that is introduced by changing the widths of the two quantum wells in the triple EB structure or by changing the height or the width of the middle barrier in the triple MB structure is also examined. 2. Theory With the method developed in Ref. [24], one can calculate the transmission coefficient for electrons tunneling through a triple EB structure as depicted in
IRS-191
2
3-t-
4
‘f
5
-
78
6
Fig. I. (a) A triple electric-barrier (EB) structure; (b) a triple magnetic-barrier (MB) structure.
Fig. la. cl0 and d are the height and the width of the electric barriers. WI and w2 are the widths of the left and right wells, respectively. Here we focus our attention on electron transport through a triple MB structure as shown in Fig. lb. BI, B2, and B3 are the heights of the left barrier, the middle barrier, and the right barrier, respectively. dl, d2, and d3 are their respective widths. The Schrijdinger equation for a 2DEG in the (x, y) plane with a magnetic field B along the z direction in each region reads +#‘+eAi)‘lD(x..v) (i=
1,2 ,...,
=Elf/(x,y) 8),
(1)
where m* is the effective mass of the electron, and A; is the Landau vector potential which is taken in the Landau gauge A; = (0, Ai( x) (0). For convenience, in the MB case, we express all quantities in dimensionless units by using the cyclotron frequency wc = eBo/m* and the magnetic length IB = ,/?$& ( BO= 0.1 T) . For GaAs m* can be taken as O.O67m, (m, is the free electron mass). The coordinate r is in units of ls, A; in units of Bols, and the energy E in units of b,. The wave function can be assumed to be $(x, y) = e”vy$(n), where kj is the wave vector of the electron in the y direction. So we obtain the following ID Schrijdinger equation, 2
- [Ai
-t k,]* + 2E
The function V(x) = [Ai terpreted as a k!-dependent
$(x)
= 0.
(2)
+ kF]*/2 can be inelectric potential. In
K Guo et al. /Physics
Letters A 238 (1998) IRS-191
region 1 and region 8, the wave functions are free electron wave functions. which can be written as @I(X) = ei&ix + reeiklx, ‘and Ys(x) = reiksx, where k;=J2E[Ai +k,12 (i= 1,8). In region 2 to region 7, the wave function Ijl(X) can be written as a linear combination of Hermitian functions [ 71
p;(X) =exp(-it?)[CLJ:(&i)
+DiUF([i)]
i = 2,3,4,5,6,7,
(3)
where ai=-,
eBi m’ i = 2, i = 3, i = 4,
= 2dl + 2d2 - fik!/eBz,
i = 5,
= 2dl + 2d2 + hky/eB3,
i = 6,
= 2dl + 2d2 + 2d3 - fik,/eB3,
i= 7,
(4)
and C,, Di are arbitrary constants. U/ and U,’ in Eq. (3) are Hermitian functions. Matching the wave functions at the edges of the barriers, the transmission amplitude r and the reflection amplitude Y can be obtained. Then, the transmission coefficient of electrons tunneling through the triple MB nanostructure is given by
187
3. Results and discussions In this section we discuss our numerical results for electron transport through triple EB and MB structures at zero temperature. Fig. 2 shows numerical results for the transmission coefficient for electrons tunneling through one symmetric and two asymmetric triple EB structures without a magnetic field applied. The asymmetry of the structures are introduced by the unequal widths of the two quantum wells within barriers. The configuration is depicted in Fig. la. One can clearly see that there are two typical and very sharp peaks for electrons tunneling through the symmetric triple EB structure, and each peak at resonance reaches a value of unity. However, the resonant peaks fall off rapidly for electrons tunneling through the asymmetric triple EB structure. An interesting result to notice is that three resonant peaks appear for electrons tunneling through the asymmetric structure where the width of the right well is twice as large as that of the left one, and the diminishing degree of resonant peaks becomes smaller and smaller from the low energy region to the high energy region. We can easily explain these phenomena by using the concepts of complete resonant tunneling and incomplete resonant tunneling [23]. It is well known that coherent resonant tunneling will occur if the incident energy E of the particle is equal to a quasibound level Eo in the quantum well. The transmission coefficient reaches its maximum at resonance. When E is not equal to Eo, it will decrease rapidly. So the transmission coefficient has a sharp peak near E = Eo. For the symmetric triple EB structure, there are two quasibound levels in each well in
T( E, k,) = $i2. 1.0
In the ballistic regime, the conductance can be derived as the electron flow averaged over half the Fermi surface [ 5,251,
L
I
I
I
w,=w,=40A ----- w,=40A, w,=8oA ......... w,=40A, w,=zoA -
0.8 :
5 ‘?J 0.6 .u, E :
d=20A,
t-&=0.3eV
r/2 G = Go
.I
T(EF, &sin4)
cos4d4,
(6)
-T/2
0.1
where 4 is the angle of incidence relative to the x direction; EF the Fermi energy; GO = e2m*vF//fi2 with I the length of the structure in the y direction and UF the Fermi velocity.
Energy
Fig. 2. Transmission coefficient
0.2
(eV)
versus incident energy through
one symmetric and two asymmetric triple EB structures n magnetic field applied.
K Guo et al./Physics
188
the quantum region. Since the quasibound levels of the two quantum wells are aligned in this case, electrons can perform a complete resonant tunneling through the structure, whereas they can not through the asymmetric triple EB structure due to the nonmatching of the quasibound levels between the two quantum wells. Therefore, in the transmission spectrum, the feature of complete resonant tunneling in the symmetric structure and that of incomplete resonant tunneling in the asymmetric structure should be visible. Figs. 3a-3c present the transmission coefficient under the influence of a uniform transverse magnetic 1
1.0,
(a) 5
0.8 ;
j.‘:
-k=O
c; / :: ----- kY=5.0x107/m : :: k;=_5,Ox, O'/m ., ;: :_: : ~ :j : ; :: : .: * i
1
..-: 0.6 E 2 0.4 s F 0.2 1 0.0
i
L k=O ----- kY=5.0x10'/m k;=_5.Ox, 0'/,7,
-
B=iOT
0.0
1.0
:
! (c)
0.8 r
.$ 0.6 .E 0.4 -
8=1 OT w =40A,w,=20A dhA,U,=0.3eV
Energy (ev) Fig. 3. (a)-(c)
Transmission coefficient versus incident energy
through one symmetric and two asymmetric triple EB structures for different values of the wave vector k, under the influence of an applied transverse magnetic field.
Letters A 238 (1998) 185-191
field applied to the barrier region in the symmetric and asymmetric triple EB structures. It is evident that the transmission coefficient at resonance is reduced for electron transport through the symmetric structure. A comparison of the dashed curve in Fig. 2 (at zero magnetic field) with the curves in Fig. 3b (at non-zero magnetic field) shows that the first and the third peaks are enhanced whereas the second peak is reduced. From Figs. 3a-3c we can also see that in comparison to the transmission peaks for the ky = 0 case, the transmission peaks for the ky > 0 case shift to the right while the transmission peaks for the ky < 0 case shift to the left. These phenomena can be explained as follows. The transverse magnetic field not only changes the shape of the effective potential of the structure but also the symmetry of the effective potential. In the presence of a magnetic field, the quasibound levels in the two quantum wells are changed correspondingly. The change of the quasibound levels results in a shift in the peak position and a change in the peak value. We notice that for the case where no magnetic field is applied, a separation of variables is possible which reduces the problem to a one-dimensional resonant tunneling process, while the tunneling process is inherently a two-dimensional process under the influence of an applied transverse magnetic field. However, in the EB case, introducing the transverse momentum in the transmission coefficient does not appreciably change the basic feature of resonant tunneling. In Figs. 4a-4c the numerical results are presented for one triple MB structure which is composed of three identical magnetic barriers, and two triple MB structures consisting of three barriers with unequal heights or unequal widths. The structural parameters are set to be BI = Bj = 0.1 T, d1 = d3 = 1, (a) d2 = 1, B2 = 0.1 T, (b) d2 = I, B2 = 0.2 T, and (c) d2 = 2, B2 = 0.1 T. The configuration is depicted in Fig. lb. In all of our plots, solid, dashed, and dotted curves represent the transmission coefficient for kJ = 0, ky = 0.7, and ky = -0.7, respectively. It is evident that for electron transport through the MB structure of identical barriers, two peaks appear and each peak at resonance reaches unity value. This implies that electrons can be transported through the structure completely at a certain incident energy. This feature is very similar to that for electrons tunneling through the symmetric triple EB structure without electric and magnetic fields applied. In Fig. 4a we can also see that in comparison
L Guo et al. /Physics
0.0
0.5
1.0
Energy
1.5
Letters A 238 (1998) 185-191
2.0
(Aq
Fig. 4. (a)-(e) Transmission coefficient versus incident energy through one triple MB structure of three identical barriers, one triple MB structure consisting of three barriers with unequal heights, and one triple MB structure consisting of three barriers with unequal widths. For (a), (b) and (c), the structural parameters are set to be BI = B3 = 0.1 T, dl = dj = 1, and (a) & = 0.1 T, d2 = I, (b) B2 = 0.2 T, d2 = I, (c) B2 = 0.1 T, d? =2.
to the peaks for the kj = 0.0 case, the transmission peaks become sharper and shift to a higher energy for electrons with k, = 0.7. A noticeable result is that the transmission coefficient for electrons with the same k,.-value transported through the triple MB structure, in which building barriers are of unequal heights (see Fig. 4b) or unequal widths (see Fig. 4c), is drastically reduced. The degree of decrease depends strongly on the value and the orientation of the wave-vector k,.
189
The transmission coefficient is greatly suppressed for the ky 3 0 cases. The degrees of decrease are also much different for different MB structures. The higher or the wider the middle magnetic barrier, the more the transmission coefficient is reduced. These phenomena can be explained from the point of view of the effective potential of the corresponding structure [ 51. The shape of the effective potential is widely different for electrons with different kF in different triple MB structures. In the case of the triple MB structure of identical barriers, the effective potential is a symmetric triple barrier and a double-well electric structure for ky > 0, so similar tunneling features should be present in the transmission coefficient as electrons tunnel through the symmetric triple EB structure; for k,. < 0 the effective potential is a triple-well structure. In this case the tunneling is due to virtual states above the three wells (a similar analysis was carried out by Matulis et al. [ 51 for the double MB case). For the triple MB cases with the middle barrier higher or wider than the side barriers, the effective potential of the structure is still triple-barrier for kx > 0 and triple-well for k?, < 0. However, its shape is greatly changed (i.e. a triple barrier consisting of three barriers with unequal heights or unequal widths, or a triple well consisting of three wells with unequal depths or unequal widths). Therefore, the feature of variations of the transmission peaks is somewhat similar to electrons tunneling through the asymmetric triple EB case. However, the variations of the transmission coefficient exhibit strongly wave-vector-dependent features. Tunneling is an inherently two-dimensional process and exhibits complicated resonant tunneling features. The results indicate that triple MB structures of unidentical building barriers possess stronger wave-vector filtering. In order to further discuss the effect introduced by the structural composition, we display in Figs. 5a and 5b the conductance through one triple MB structure which consists of three identical magnetic barriers, and four triple MB structures in which the height or the width of the middle magnetic barrier is set to different values. It is seen that although there are a few very sharp peaks in the transmission spectrum, we can not see very sharp resonant peaks in the conductance. For the structure of identical building blocks, we see a shoulder structure and a weak and wide peak in our considered Fermi energy range. With an increment of
190
E Guo et al/Physics ~-1.61
I
,’ ,’ ”
,I’
;--. ...___ ._.__________....-~-~‘~ B,=B,=O.lT d,=d?=d,=l B,=O.lT ----- B,=0.2T ......... B,_0,3T
~-1.61
I
Letters A 238 (1998) 185-191
be explained as changing of the orbits of moment electrons. Finally, we would like to point out that although our consideration of electron tunneling through a rectangular triple MB structure gives only a qualitative picture, these features should be present in the more realistic cases with barriers of smooth shape [ 181 which can be divided into N(> 1) segments by using the approximated method derived in Ref. [6]. The basic properties of electron transport do not depend on the actual shape of the magnetic barrier, but on the presence of barriers in the potential V(x).
4. Conclusions
ii
,’
o.o”+-0.5
’
1.0
0.0
Fermi
Energy
1.5
20
(lb,)
Fig. S. (a) Conductance through one triple MB structure of identical barriers and two triple MB
structures consisting of three
barriers with unequal heights and equal widths. (b) Conductance through one triple MB structure and two triple MB structures consisting of three magnetic barriers with unequal widths and equal heights.
the height or the width of the middle barrier, obvious resonant peaks appear, and the conductance is drastically reduced. The conductance in the lower Fermi energy decreases faster than that in the higher Fermi energy. Moreover, the “threshold” Fermi-energy values shift to the right when the height or the width of the middle barrier is increased. For MB structures, the effective potential V(x) = [A;(x) + ky12/2 depends upon the vector potential A;(x) which is related with the magnetic field by B(s) = dA;(x)/dx. A slight change of the function B(x) may result in a drastic variation of the effective potential, and therefore renders both the transmission coefficient and the conductance sensitive to the structural parameters. Since the conductance is calculated by the averaging of the transmission coefficient T( E, k?) over half the Fermi surface. when the height or the width of the middle magnetic barrier is increased, the amplitude of the vector potential Ai and thus the height or the width of the middle barrier in the effective potential V(x) increases. This yields a reduction in the conductance. Semiclassically, the reduction of the conductance can
In conclusion, there exist great differences between electrons tunneling through a triple EB structure and a triple MB structure. For the triple EB case, the problem of the electron’s motion can be reduced to a one-dimensional problem, and electrons with different wave vector k, possess similar properties of resonant tunneling. Under the influence of an applied transverse magnetic field to the barrier region, introducing the transverse momentum in the transmission coefficient results in a shift of the peak position and in a change in the peak value. However, the basic feature of resonant tunneling is not greatly changed. For the triple MB case, the characteristics of electron transport strongly depends on not only the structural parameters but also the value and the orientations of the wave vector k,. The latter makes the tunneling an inherently two-dimensional process. There still exist great differences for electron transport through symmetric and asymmetric triple EB or MB structures. The triple MB structure consisting of unidentical barriers possess stronger wave-vector filtering than the structure of identical magnetic barriers. We believe that the study presented here will be helpful to device designing and device applications.
Acknowledgement Two of us (B.-L.G and Y.G.) would like to acknowledge partial support from the High Technology Research and Development Program of RR. China.
Y Guo et ol./Physics
Letters A 238 (1998) 185-191
References M.A. McCord, D.D. Awschalom, Appt. Phys. Lett. 57 (1990) 2153. M.L. Leadbeater et al., J. Appl. Phys. 69 (1991) 4689; K.M. Krishnan, Appl. Phys. Lett. 61 (1992) 2365; W. Van Roy et al., J. Magn. Magn. Mater. 121 (1993) 197; R. Yagi. Y. lye, J. Phys. Sot. Jpn. 62 ( 1993) 1279. 131 A.K. Geim. Pis’ma, Zh. Eksp. Teor. Fiz. 50 ( 1989) 359 I JETP Lett. 50 ( 1990) 3891; S.J. Bending, K. von klitzing, K. Ploog, Phys. Rev. Lett. 65 (1990) 1060. 141 EM. Peeters, A. Matulis, Phys. Rev. B 48 ( 1993) 15166. 151 A. Matulis. FM. Peeters, P Vasilopoulos, Phys. Rev. Ien. 72 (1994) 1518. 161 J.Q. You, Lide Zhang, PK. Ghosh, Phys. Rev. B 52 (1995) 17243. 171 Yong Guo, Binglin Gu, Wenhui Duan, Yu Zhang, Phys. Rev. B 55 (1997) 9314. 181 J.Q. You, Lide Zhang, Phys. Rev. B 54 (1996) 1526. 191 IS. Ibrahim, FM. Peeters, Phys. Rev. B 52 ( 1995) 17321. IlO1 V.M. Ramaglia et al., Phys. Rev. B 43 (1991) 2201; V.M. Ramaglia, E Ventriglia, J. Phys. Condens. Matter 3 (1991) 4881.
191
1111 J.E. Mtiller, Phys. Rev. Lea. 68 (1992) 385. f121 C.L. Foden, M.L. Leadbeater, J.H. Burroughes, and M. Pepper, J. Phys. Condens. Matter 6 (1994) L127. 1131 M. Calve. Phys. Rev. B48 ( 1993) 2365; J. Phys. Condens. Matter 6 (1994) 3329; Phys. Rev. B 51 ( 1995) 2268. [I41 FM. Peeters, P Vasilopoulos, Phys. Rev. B 47 ( 1993) 1466; F? Vasilopoulos, EM. Peeters. Superlatt. Microstruct. 7 ( 1990) 393. [I51 DP Xue, G. Xiao, Phys. Rev. B 45 (1992) 5986. [I61 X.G. Xu, S.E. Ulloa, Solid State Commun. 82 ( 1992) 945. 1171 H.A. Carmona et al., Phys. Rev. Lett. 74 ( 1995) 3009. [I81 PD. Ye et al., Phys. Rev. Lea. 74 ( 1995) 3013. I191 S. Izawa, S. Katsumoto, A. Endo, Y. lye, J. Phys. Sot. Jpn. 64 (1995) 706. [201 PM. Martin et al., Semicond. Sci. Technol. 9 (1994) 493. [211 L.D. Macks, .%A. Brown, R.P. Starrett, et al., Physica B 201 ( 1994) 374; L.D. Macks et al., Phys. Rev. B 54 (1996) 4857. 1221 S.S. Allen, S.L. Richardson, Phys. Rev. B SO ( 1994) I 1693. 1231 Yong Guo, Binglin Gu, Wenhui Duan, Z. Phys. B 102 ( 1997) 217. u41 Yong Guo, You-cheng Li. Xiao-jun Kong. Cheng-wen Wei, Phys. Rev. B 50 (1994) 17249. [=I M. Biittiker, Phys. Rev. Len. 59 ( 1986) 1761.
1998
PHYSICS
Physics Letters A 238 (1998)
ELSEYIER
LETTERS
A
185-191
Comparison of electronic transport through triple electric-barrier structures and triple magnetic-barrier structures Yong Guo a*b,‘,Bing-Lin Gu ‘, Zhi-Qiang
Li b, Yoshiyuki Kawazoe b
a Department of Physics, Tsinghua University, Beijing 100084, China h Institute for Materials Research, Tohoku University, Sendai 980-77. Japan Received
3 I July 1997; revised manuscript
received 4 November Communicated
1997; accepted by A. Lagendijk
for publication
20 November
I997
Abstract We present a detailed comparison of electron transport through triple electric-barrier structures and triple magnetic-barrier structures. Tunneling through triple electric-barrier structures is mainly determined by the momentum along the tunneling direction, while in triple magnetic-barrier structures transport strongly depends not only on structural parameters but also the transverse momentum. It is confirmed that both the transmission coefficient and the conductance are dramatically reduced with increasing height or width of the middle barrier in triple magnetic-barrier structures. @ 1998 Elsevier Science B.V. PACS: 73.40.Gk; 03.65.Ge Ke~nwr&‘dstTunneling; Electric barrier; Magnetic
barrier: Wave-vector
filtering
1. Introduction Tunneling of electrons in double- and multiplebarrier electric structures has been extensively studied both theoretically and experimentally due to its potential applications. Many novel features such as sharp current peaks, negative differential resistance, intrinsic bistability, etc., have been discovered. The electricbarrier (EB) structure can be generated by, e.g., the energy band discontinuity or offset in heterostructures. Recently, the electron motion in the magnetic-barrier (MB) structure has caused a vast research interest. Experimentally, an inhomogeneous magnetic field on nanometer scale can be realized with the creation of ’ Correspondence address: search, Tohoku University, [email protected].
Institute for Sendai 980-77,
Materials ReJapan. E-mail:
03759601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. frIs0375-9601(97)00911-0
magnetic dots [ 11, the patterning of ferromagnetic materials [ 21, and the deposition of superconducting materials on conventional heterostructures [ 31. Theoretically, Peeters and Matulis [4] investigated the two-dimensional electron gas (2DEG) motion under the influence of a magnetic step, magnetic well, and magnetic barrier. Transport of electrons in more complicated magnetic barriers [ 5-71 and periodically arranged magnetic barriers [8,9] was also studied. The features of wave-vector filtering and magnetic minibands were determined. Ramaglia et al. [ lo] considered the effect of a local magnetic field on the tunneling current through a thick potential barrier and found that the magnetic field was localized strictly within the potential barrier, which led to resonances that were centered within the barrier. Miiller [ I 1 ] analyzed an infinite strip of a 2DEG in a magnetic field that varies linearly across the strip and showed that the
186
Y GUI et ~1. /Physics
Letters A 238 (1998)
system has a time-reversal asymmetry and that charge flow takes place only in the direction perpendicular to the field gradient. The transport of electrons in a curved 2DEG was investigated by Foden et al. [ 121 and the problem of a 2DEG in a smooth magnetic barrier geometry of different shape was considered by Calvo [ 131. Moreover, transport of electrons in a unidirectional weak magnetic field modulation was also investigated [ 14-161. Almost at the same time, Carmona et al. [ 171, Ye et al. [ 181 and Izawa et al. [ 191 realized periodic lateral magnetic fields experimentally and observed oscillatory magnetoresistance due to a commensurability effect between the classical cyclotron diameter and the period of magnetic modulation. Those experimental techniques opened up the way to experiments in alternating magnetic fields with periods in the nanometer regime. Very recently there has been increased interest [ 20,211 in tunneling through triple-barrier structures since these structures offer the possibility of studying the tunnel-coupled states of a double quantum well and its potential applications. It is well known that the features of electrons tunneling through an asymmetric multiple-barrier electric structure are greatly different from those of electrons tunneling through a symmetric multiple-barrier electric structure [ 22,231. It is also well known that the electron’s behavior in MB structures is drastically different from that in the EB case [ 5-71. Electron transport through the latter structure is an inherently two-dimensional process and exhibits complicated resonant tunneling features. It can be expected that there are some novel effects on the transport of electrons in triple EB structures and triple MB structures. Therefore, in this work we systematically investigate the transport properties of electrons in the triple EB structure and in the triple MB structure. Similarities and differences between electronic transport in these two cases are presented. The asymmetric effect that is introduced by changing the widths of the two quantum wells in the triple EB structure or by changing the height or the width of the middle barrier in the triple MB structure is also examined. 2. Theory With the method developed in Ref. [24], one can calculate the transmission coefficient for electrons tunneling through a triple EB structure as depicted in
IRS-191
2
3-t-
4
‘f
5
-
78
6
Fig. I. (a) A triple electric-barrier (EB) structure; (b) a triple magnetic-barrier (MB) structure.
Fig. la. cl0 and d are the height and the width of the electric barriers. WI and w2 are the widths of the left and right wells, respectively. Here we focus our attention on electron transport through a triple MB structure as shown in Fig. lb. BI, B2, and B3 are the heights of the left barrier, the middle barrier, and the right barrier, respectively. dl, d2, and d3 are their respective widths. The Schrijdinger equation for a 2DEG in the (x, y) plane with a magnetic field B along the z direction in each region reads +#‘+eAi)‘lD(x..v) (i=
1,2 ,...,
=Elf/(x,y) 8),
(1)
where m* is the effective mass of the electron, and A; is the Landau vector potential which is taken in the Landau gauge A; = (0, Ai( x) (0). For convenience, in the MB case, we express all quantities in dimensionless units by using the cyclotron frequency wc = eBo/m* and the magnetic length IB = ,/?$& ( BO= 0.1 T) . For GaAs m* can be taken as O.O67m, (m, is the free electron mass). The coordinate r is in units of ls, A; in units of Bols, and the energy E in units of b,. The wave function can be assumed to be $(x, y) = e”vy$(n), where kj is the wave vector of the electron in the y direction. So we obtain the following ID Schrijdinger equation, 2
- [Ai
-t k,]* + 2E
The function V(x) = [Ai terpreted as a k!-dependent
$(x)
= 0.
(2)
+ kF]*/2 can be inelectric potential. In
K Guo et al. /Physics
Letters A 238 (1998) IRS-191
region 1 and region 8, the wave functions are free electron wave functions. which can be written as @I(X) = ei&ix + reeiklx, ‘and Ys(x) = reiksx, where k;=J2E[Ai +k,12 (i= 1,8). In region 2 to region 7, the wave function Ijl(X) can be written as a linear combination of Hermitian functions [ 71
p;(X) =exp(-it?)[CLJ:(&i)
+DiUF([i)]
i = 2,3,4,5,6,7,
(3)
where ai=-,
eBi m’ i = 2, i = 3, i = 4,
= 2dl + 2d2 - fik!/eBz,
i = 5,
= 2dl + 2d2 + hky/eB3,
i = 6,
= 2dl + 2d2 + 2d3 - fik,/eB3,
i= 7,
(4)
and C,, Di are arbitrary constants. U/ and U,’ in Eq. (3) are Hermitian functions. Matching the wave functions at the edges of the barriers, the transmission amplitude r and the reflection amplitude Y can be obtained. Then, the transmission coefficient of electrons tunneling through the triple MB nanostructure is given by
187
3. Results and discussions In this section we discuss our numerical results for electron transport through triple EB and MB structures at zero temperature. Fig. 2 shows numerical results for the transmission coefficient for electrons tunneling through one symmetric and two asymmetric triple EB structures without a magnetic field applied. The asymmetry of the structures are introduced by the unequal widths of the two quantum wells within barriers. The configuration is depicted in Fig. la. One can clearly see that there are two typical and very sharp peaks for electrons tunneling through the symmetric triple EB structure, and each peak at resonance reaches a value of unity. However, the resonant peaks fall off rapidly for electrons tunneling through the asymmetric triple EB structure. An interesting result to notice is that three resonant peaks appear for electrons tunneling through the asymmetric structure where the width of the right well is twice as large as that of the left one, and the diminishing degree of resonant peaks becomes smaller and smaller from the low energy region to the high energy region. We can easily explain these phenomena by using the concepts of complete resonant tunneling and incomplete resonant tunneling [23]. It is well known that coherent resonant tunneling will occur if the incident energy E of the particle is equal to a quasibound level Eo in the quantum well. The transmission coefficient reaches its maximum at resonance. When E is not equal to Eo, it will decrease rapidly. So the transmission coefficient has a sharp peak near E = Eo. For the symmetric triple EB structure, there are two quasibound levels in each well in
T( E, k,) = $i2. 1.0
In the ballistic regime, the conductance can be derived as the electron flow averaged over half the Fermi surface [ 5,251,
L
I
I
I
w,=w,=40A ----- w,=40A, w,=8oA ......... w,=40A, w,=zoA -
0.8 :
5 ‘?J 0.6 .u, E :
d=20A,
t-&=0.3eV
r/2 G = Go
.I
T(EF, &sin4)
cos4d4,
(6)
-T/2
0.1
where 4 is the angle of incidence relative to the x direction; EF the Fermi energy; GO = e2m*vF//fi2 with I the length of the structure in the y direction and UF the Fermi velocity.
Energy
Fig. 2. Transmission coefficient
0.2
(eV)
versus incident energy through
one symmetric and two asymmetric triple EB structures n magnetic field applied.
K Guo et al./Physics
188
the quantum region. Since the quasibound levels of the two quantum wells are aligned in this case, electrons can perform a complete resonant tunneling through the structure, whereas they can not through the asymmetric triple EB structure due to the nonmatching of the quasibound levels between the two quantum wells. Therefore, in the transmission spectrum, the feature of complete resonant tunneling in the symmetric structure and that of incomplete resonant tunneling in the asymmetric structure should be visible. Figs. 3a-3c present the transmission coefficient under the influence of a uniform transverse magnetic 1
1.0,
(a) 5
0.8 ;
j.‘:
-k=O
c; / :: ----- kY=5.0x107/m : :: k;=_5,Ox, O'/m ., ;: :_: : ~ :j : ; :: : .: * i
1
..-: 0.6 E 2 0.4 s F 0.2 1 0.0
i
L k=O ----- kY=5.0x10'/m k;=_5.Ox, 0'/,7,
-
B=iOT
0.0
1.0
:
! (c)
0.8 r
.$ 0.6 .E 0.4 -
8=1 OT w =40A,w,=20A dhA,U,=0.3eV
Energy (ev) Fig. 3. (a)-(c)
Transmission coefficient versus incident energy
through one symmetric and two asymmetric triple EB structures for different values of the wave vector k, under the influence of an applied transverse magnetic field.
Letters A 238 (1998) 185-191
field applied to the barrier region in the symmetric and asymmetric triple EB structures. It is evident that the transmission coefficient at resonance is reduced for electron transport through the symmetric structure. A comparison of the dashed curve in Fig. 2 (at zero magnetic field) with the curves in Fig. 3b (at non-zero magnetic field) shows that the first and the third peaks are enhanced whereas the second peak is reduced. From Figs. 3a-3c we can also see that in comparison to the transmission peaks for the ky = 0 case, the transmission peaks for the ky > 0 case shift to the right while the transmission peaks for the ky < 0 case shift to the left. These phenomena can be explained as follows. The transverse magnetic field not only changes the shape of the effective potential of the structure but also the symmetry of the effective potential. In the presence of a magnetic field, the quasibound levels in the two quantum wells are changed correspondingly. The change of the quasibound levels results in a shift in the peak position and a change in the peak value. We notice that for the case where no magnetic field is applied, a separation of variables is possible which reduces the problem to a one-dimensional resonant tunneling process, while the tunneling process is inherently a two-dimensional process under the influence of an applied transverse magnetic field. However, in the EB case, introducing the transverse momentum in the transmission coefficient does not appreciably change the basic feature of resonant tunneling. In Figs. 4a-4c the numerical results are presented for one triple MB structure which is composed of three identical magnetic barriers, and two triple MB structures consisting of three barriers with unequal heights or unequal widths. The structural parameters are set to be BI = Bj = 0.1 T, d1 = d3 = 1, (a) d2 = 1, B2 = 0.1 T, (b) d2 = I, B2 = 0.2 T, and (c) d2 = 2, B2 = 0.1 T. The configuration is depicted in Fig. lb. In all of our plots, solid, dashed, and dotted curves represent the transmission coefficient for kJ = 0, ky = 0.7, and ky = -0.7, respectively. It is evident that for electron transport through the MB structure of identical barriers, two peaks appear and each peak at resonance reaches unity value. This implies that electrons can be transported through the structure completely at a certain incident energy. This feature is very similar to that for electrons tunneling through the symmetric triple EB structure without electric and magnetic fields applied. In Fig. 4a we can also see that in comparison
L Guo et al. /Physics
0.0
0.5
1.0
Energy
1.5
Letters A 238 (1998) 185-191
2.0
(Aq
Fig. 4. (a)-(e) Transmission coefficient versus incident energy through one triple MB structure of three identical barriers, one triple MB structure consisting of three barriers with unequal heights, and one triple MB structure consisting of three barriers with unequal widths. For (a), (b) and (c), the structural parameters are set to be BI = B3 = 0.1 T, dl = dj = 1, and (a) & = 0.1 T, d2 = I, (b) B2 = 0.2 T, d2 = I, (c) B2 = 0.1 T, d? =2.
to the peaks for the kj = 0.0 case, the transmission peaks become sharper and shift to a higher energy for electrons with k, = 0.7. A noticeable result is that the transmission coefficient for electrons with the same k,.-value transported through the triple MB structure, in which building barriers are of unequal heights (see Fig. 4b) or unequal widths (see Fig. 4c), is drastically reduced. The degree of decrease depends strongly on the value and the orientation of the wave-vector k,.
189
The transmission coefficient is greatly suppressed for the ky 3 0 cases. The degrees of decrease are also much different for different MB structures. The higher or the wider the middle magnetic barrier, the more the transmission coefficient is reduced. These phenomena can be explained from the point of view of the effective potential of the corresponding structure [ 51. The shape of the effective potential is widely different for electrons with different kF in different triple MB structures. In the case of the triple MB structure of identical barriers, the effective potential is a symmetric triple barrier and a double-well electric structure for ky > 0, so similar tunneling features should be present in the transmission coefficient as electrons tunnel through the symmetric triple EB structure; for k,. < 0 the effective potential is a triple-well structure. In this case the tunneling is due to virtual states above the three wells (a similar analysis was carried out by Matulis et al. [ 51 for the double MB case). For the triple MB cases with the middle barrier higher or wider than the side barriers, the effective potential of the structure is still triple-barrier for kx > 0 and triple-well for k?, < 0. However, its shape is greatly changed (i.e. a triple barrier consisting of three barriers with unequal heights or unequal widths, or a triple well consisting of three wells with unequal depths or unequal widths). Therefore, the feature of variations of the transmission peaks is somewhat similar to electrons tunneling through the asymmetric triple EB case. However, the variations of the transmission coefficient exhibit strongly wave-vector-dependent features. Tunneling is an inherently two-dimensional process and exhibits complicated resonant tunneling features. The results indicate that triple MB structures of unidentical building barriers possess stronger wave-vector filtering. In order to further discuss the effect introduced by the structural composition, we display in Figs. 5a and 5b the conductance through one triple MB structure which consists of three identical magnetic barriers, and four triple MB structures in which the height or the width of the middle magnetic barrier is set to different values. It is seen that although there are a few very sharp peaks in the transmission spectrum, we can not see very sharp resonant peaks in the conductance. For the structure of identical building blocks, we see a shoulder structure and a weak and wide peak in our considered Fermi energy range. With an increment of
190
E Guo et al/Physics ~-1.61
I
,’ ,’ ”
,I’
;--. ...___ ._.__________....-~-~‘~ B,=B,=O.lT d,=d?=d,=l B,=O.lT ----- B,=0.2T ......... B,_0,3T
~-1.61
I
Letters A 238 (1998) 185-191
be explained as changing of the orbits of moment electrons. Finally, we would like to point out that although our consideration of electron tunneling through a rectangular triple MB structure gives only a qualitative picture, these features should be present in the more realistic cases with barriers of smooth shape [ 181 which can be divided into N(> 1) segments by using the approximated method derived in Ref. [6]. The basic properties of electron transport do not depend on the actual shape of the magnetic barrier, but on the presence of barriers in the potential V(x).
4. Conclusions
ii
,’
o.o”+-0.5
’
1.0
0.0
Fermi
Energy
1.5
20
(lb,)
Fig. S. (a) Conductance through one triple MB structure of identical barriers and two triple MB
structures consisting of three
barriers with unequal heights and equal widths. (b) Conductance through one triple MB structure and two triple MB structures consisting of three magnetic barriers with unequal widths and equal heights.
the height or the width of the middle barrier, obvious resonant peaks appear, and the conductance is drastically reduced. The conductance in the lower Fermi energy decreases faster than that in the higher Fermi energy. Moreover, the “threshold” Fermi-energy values shift to the right when the height or the width of the middle barrier is increased. For MB structures, the effective potential V(x) = [A;(x) + ky12/2 depends upon the vector potential A;(x) which is related with the magnetic field by B(s) = dA;(x)/dx. A slight change of the function B(x) may result in a drastic variation of the effective potential, and therefore renders both the transmission coefficient and the conductance sensitive to the structural parameters. Since the conductance is calculated by the averaging of the transmission coefficient T( E, k?) over half the Fermi surface. when the height or the width of the middle magnetic barrier is increased, the amplitude of the vector potential Ai and thus the height or the width of the middle barrier in the effective potential V(x) increases. This yields a reduction in the conductance. Semiclassically, the reduction of the conductance can
In conclusion, there exist great differences between electrons tunneling through a triple EB structure and a triple MB structure. For the triple EB case, the problem of the electron’s motion can be reduced to a one-dimensional problem, and electrons with different wave vector k, possess similar properties of resonant tunneling. Under the influence of an applied transverse magnetic field to the barrier region, introducing the transverse momentum in the transmission coefficient results in a shift of the peak position and in a change in the peak value. However, the basic feature of resonant tunneling is not greatly changed. For the triple MB case, the characteristics of electron transport strongly depends on not only the structural parameters but also the value and the orientations of the wave vector k,. The latter makes the tunneling an inherently two-dimensional process. There still exist great differences for electron transport through symmetric and asymmetric triple EB or MB structures. The triple MB structure consisting of unidentical barriers possess stronger wave-vector filtering than the structure of identical magnetic barriers. We believe that the study presented here will be helpful to device designing and device applications.
Acknowledgement Two of us (B.-L.G and Y.G.) would like to acknowledge partial support from the High Technology Research and Development Program of RR. China.
Y Guo et ol./Physics
Letters A 238 (1998) 185-191
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