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Edge quantum wire structures with novel doping profiles and their electronic states
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Solid-State Electronics Vol. 42, No. 7±8, pp. 1223±1226, 1998 # 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98 $19.00 + 0.00 S0038-1101(98)00008-2
EDGE QUANTUM WIRE STRUCTURES WITH NOVEL DOPING PROFILES AND THEIR ELECTRONIC STATES M. YAMAUCHI1,3, Y. NAKAMURA1,2 and H. SAKAKI1,2 RCAST, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan Quantum Transition Project, JST, 4-7-6 Komaba, Meguro-ku, Tokyo 153, Japan 3 Mathematical Systems Institute Inc., 2-5-3 Shinjuku, Shinjuku-ku, Tokyo 160, Japan 1
2
AbstractÐWe propose and analyze a new type of edge quantum wire structure (EQWR), where an ntype AlGaAs layer is deposited on the edge plane of a GaAs/AlGaAs n±i±p±i structure where acceptors and donors are planar-doped at the center of GaAs and AlGaAs layers, respectively. Since a sharp electric ®eld generated by the sheet doping raises the potential at the center of GaAs layer, electrons supplied by extra donors in the overgrown AlGaAs layer are driven to the two corner ridges of the GaAs layer and con®ned two-dimensionally by electrostatic ®elds. Wave functions of one-dimensional electrons in these quantum wires are analyzed and found to be eciently squeezed. Features of electronic levels are clari®ed. # 1998 Published by Elsevier Science Ltd. All rights reserved
Recently, the study of one-dimensional (1D) electron transport in a narrow quantum wire has attracted much interests because their importance both in solid physics and electronics[1±6]. For example, it has been predicted that the mobilities of 1D electrons limited by impurities and roughness are extremely enhanced in the single-mode quantum wire particularly when the electron concentration is high[2,3]. To accommodate a large number of electrons in the ground subband, the cross-sectional dimensions of the wire must be 30 nm or less. One promising way to achieve this is to adopt the edge quantum wire (EQWR)[2,3,5], in which the electron supply layer such as an n-AlGaAs layer or an FET gate structure is formed on the edge surface of undoped GaAs/AlGaAs multiple quantum well (QW) structure. In this basic EQWR structure, electrons will be con®ned by the QW potential along the stacking direction of the multi-QW structure and by electrostatic ®elds (EF) in the direction normal to the edge plane. Experiments on such EQWRs have shown rapid developments[5±9]. It is expected that the introduction of dopants into the GaAs/AlGaAs multi QW layer would modify QWR structures. In this paper, we examine and analyze QWR, formed on the edge of an n±i±p±i GaAs/AlGaAs multi-layered structure, as schematically shown in Fig. 1(a). Here n-type and p-type sheet dopants are introduced at the layer center of AlGaAs and GaAs layers, respectively. Because of this doping scheme an electric ®eld will be formed parallel to z axis and raises the potential at the center of the GaAs layer. Therefore, electrons supplied from the overgrown n-AlGaAs are expected to be driven to the left and right corners (A and B) of the GaAs layer. Hence one-dimensional electron gas
(1DEG) with the 10 nm scale cross-section is expected to be formed even when the GaAs layer is thick. In this work, we analyze theoretically the electronic states for this EQWR structure of Fig. 1(a)
Fig. 1. A new edge quantum wire structure formed on the edge of an n±i±p±i GaAs/AlGaAs structures (a), where the sheet acceptors (minus in square) and donors (plus in square) are located at the layer center of GaAs and AlGaAs layer, respectively. Electrons (e in circle) will be con®ned at the top corners of GaAs layer. Here, Ln=Lb=Lz=100 nm, Ls=10 nm, and Ld=1 mm. A potential pro®le along line A±B in (a) is deformed by the sheet dopants as schematically shown in (b). Vertical electric ®eld along line C±D is strengthened as schematically shown in (c).
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M. Yamauchi et al.
and clarify the electron distribution and the ground subband wave function by solving the Poisson and SchroÈdinger equations self-consistently, following the general scheme outlined by Laux and Stern[9,10]. Structural parameters are set as follows; the thickness Lz of GaAs and that Lb of AlGaAs layers are both set to be 100 nm where the thickness Ln of an over-grown n-AlGaAs and that Ls of spacer layer are 100 nm and 10 nm, respectively. The length Ld of GaAs/AlGaAs multilayer is quite large in reality but is set at 1 mm for the reason mentioned later. In our analysis, we assume that the Fermi level at the top and bottom boundaries of Fig. 1(a) is set at the midgap of AlGaAs. The band discontinuity DEc at the heterointerface is assumed to be 220 meV. The density of Si in n-
AlGaAs is set at 1 1018 cmÿ3, while the density of residual acceptors in GaAs and AlGaAs is set at 1 1014 cmÿ3. The temperature is set at 10 K. Figure 2(a) shows the electron density distribution n(r) for the case where the sheet acceptors and sheet donors are absent. Characteristics of this type of basic EQWRs have already been reported by Stern in detail. Note that n(r) extends along the edge surface of the GaAs layer with two peaks at the side corners. This indicates that several subbands (up to the seventh) are occupied. The appearance of two side peaks are due to the attractive force of positively charged Si donors in the AlGaAs layer. When acceptors NA and donors ND are introduced to the GaAs and the AlGaAs layer by the planar doping to the level of 8.0 1011 cmÿ2,
Fig. 2. The distributions n(r) of electron density around the edge corners of a GaAs layer. The areal density of sheet acceptors and donors are set at 0 in (a), 8.0 1011 cmÿ2 in (b). Letters ``A'' and ``B'' correspond to those in Fig. 1(a).
Edge quantum wire structures
1225
Fig. 3. The square of the ground subband wave functions are plotted for the sheet acceptors and sheet donors of at 0 and 8.0 1011 cmÿ2. Letters ``A'' and ``B'' correspond to those in Fig. 1(a).
electrons are depleted at the center of the GaAs layer and n(r) splits into two well-resolved peaks, indicating that EQWRs far narrower than the GaAs layer are formed. The wire width in the z direction is 30 nm. Here, the wire width is de®ned as the width of the regions with n(r) over 1.0 1017 cmÿ3. The linear density of 1D electrons in these pairs of wire are 1.29 106 cmÿ1 and 1.29 106 cmÿ1, respectively. Figure 3(a), (b) show the squared wave functions vj1v2 of the ground subband of the two EQWR structures with ND and NA = 0 and 8.0 1011 cmÿ2, respectively. Though two peaks at the corners have nearly the same probability density, vj1v2 at the layer center of GaAs shows dierent characteristics: namely, vj1v2 almost vanishes in Fig. 3(b) whereas it has some probability in Fig. 3(a). Note that two peaks in Fig. 3(b) is more
than 40 nm apparent indicating that two wires are uncoupled. As discussed earlier, excited subbands up to the seventh are occupied in the basic EQWR with no sheet dopants, as shown in Fig. 2(a). Once the sheet dopants are introduced to 8.0 1011 cmÿ2, the level spacings are all increased. As a result 80% of the electrons (1.0 106 cmÿ1) are accommodated in the ground subband and the remaining 20% in the second subband, keeping the higher ones unoccupied. Hence, the electron transport is mostly dominated by the ground subband. This type of EQWR structure may provide several attractive properties. For example, the presence of a pair of QWRs in each GaAs layer may allow the formation of a QWR loop with which quantum interference phenomena may be explored. In addition, the interface-roughness scattering may be smaller as compared with the basic EQWRs because
1226
M. Yamauchi et al.
the energy level in hetero-junctions is less sensitive to the roughness. Moreover the formation of ohmic contacts to 1D systems would be easier because the wider GaAs will facilitates the ohmic contact formation. In summary, we have analyzed a new edge QWR structure in which electrons are con®ned twodimensionally by electric ®elds. This scheme allows the formation of QWRs, with a small cross-sectional sizes (R30 nm) even when the thickness of GaAs layer is quite large (r100 nm).
REFERENCES
1. Sakaki, H., Surf. Sci., 1992, 623, 267. 2. Sakaki, H., Jpn. J. Appl. Phys., 1980, 19, L735.
3. Motohisa, J. and Sakaki, H., Appl. Phys. Lett., 1993, 63, 1786. 4. van Wees, B. J., van Houten, H., Beenakker, C. W. J., Williamson, J. G., Kouwenhouen, L. P., van der Marel, D. and Foxon, C. T., Phys. Rev. Lett., 1988, 60, 848. 5. Yacoby, A., Stormer, H. L., Baldwin, K. W., Pfeier, L. N. and West, K. W., unpublished. 6. Ohno, Y., Nakamura, Y., Foley, M., Someya, T., Noda, T. and Sakaki, H., Phys. Rev. B, 1995, 52, 619. 7. Nakamura, Y., Tsuchiya, M., Koshiba, S., Noge, H. and Sakaki, H., Appl. Phys. Lett., 1994, 64, 2552. 8. Yamauchi, M., Nakaumura, Y., Kadoya, Y., Sugawara, H. and Sakaki, H., Jpn. J. Appl. Phys., 1996, 35, 1886. 9. Laux, S. E. and Stern, F., Appl. Phys. Lett., 1986, 49, 91. 10. Laux, S. E., Proc. 5th Int. Conf. Numerical Analysis of Semiconductor Devices and Integrated Circuits (NASECODE V), ed. J. J. H. Miller, Boole Press, Dublin, 1987, p. 270.
Solid-State Electronics Vol. 42, No. 7±8, pp. 1223±1226, 1998 # 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 0038-1101/98 $19.00 + 0.00 S0038-1101(98)00008-2
EDGE QUANTUM WIRE STRUCTURES WITH NOVEL DOPING PROFILES AND THEIR ELECTRONIC STATES M. YAMAUCHI1,3, Y. NAKAMURA1,2 and H. SAKAKI1,2 RCAST, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan Quantum Transition Project, JST, 4-7-6 Komaba, Meguro-ku, Tokyo 153, Japan 3 Mathematical Systems Institute Inc., 2-5-3 Shinjuku, Shinjuku-ku, Tokyo 160, Japan 1
2
AbstractÐWe propose and analyze a new type of edge quantum wire structure (EQWR), where an ntype AlGaAs layer is deposited on the edge plane of a GaAs/AlGaAs n±i±p±i structure where acceptors and donors are planar-doped at the center of GaAs and AlGaAs layers, respectively. Since a sharp electric ®eld generated by the sheet doping raises the potential at the center of GaAs layer, electrons supplied by extra donors in the overgrown AlGaAs layer are driven to the two corner ridges of the GaAs layer and con®ned two-dimensionally by electrostatic ®elds. Wave functions of one-dimensional electrons in these quantum wires are analyzed and found to be eciently squeezed. Features of electronic levels are clari®ed. # 1998 Published by Elsevier Science Ltd. All rights reserved
Recently, the study of one-dimensional (1D) electron transport in a narrow quantum wire has attracted much interests because their importance both in solid physics and electronics[1±6]. For example, it has been predicted that the mobilities of 1D electrons limited by impurities and roughness are extremely enhanced in the single-mode quantum wire particularly when the electron concentration is high[2,3]. To accommodate a large number of electrons in the ground subband, the cross-sectional dimensions of the wire must be 30 nm or less. One promising way to achieve this is to adopt the edge quantum wire (EQWR)[2,3,5], in which the electron supply layer such as an n-AlGaAs layer or an FET gate structure is formed on the edge surface of undoped GaAs/AlGaAs multiple quantum well (QW) structure. In this basic EQWR structure, electrons will be con®ned by the QW potential along the stacking direction of the multi-QW structure and by electrostatic ®elds (EF) in the direction normal to the edge plane. Experiments on such EQWRs have shown rapid developments[5±9]. It is expected that the introduction of dopants into the GaAs/AlGaAs multi QW layer would modify QWR structures. In this paper, we examine and analyze QWR, formed on the edge of an n±i±p±i GaAs/AlGaAs multi-layered structure, as schematically shown in Fig. 1(a). Here n-type and p-type sheet dopants are introduced at the layer center of AlGaAs and GaAs layers, respectively. Because of this doping scheme an electric ®eld will be formed parallel to z axis and raises the potential at the center of the GaAs layer. Therefore, electrons supplied from the overgrown n-AlGaAs are expected to be driven to the left and right corners (A and B) of the GaAs layer. Hence one-dimensional electron gas
(1DEG) with the 10 nm scale cross-section is expected to be formed even when the GaAs layer is thick. In this work, we analyze theoretically the electronic states for this EQWR structure of Fig. 1(a)
Fig. 1. A new edge quantum wire structure formed on the edge of an n±i±p±i GaAs/AlGaAs structures (a), where the sheet acceptors (minus in square) and donors (plus in square) are located at the layer center of GaAs and AlGaAs layer, respectively. Electrons (e in circle) will be con®ned at the top corners of GaAs layer. Here, Ln=Lb=Lz=100 nm, Ls=10 nm, and Ld=1 mm. A potential pro®le along line A±B in (a) is deformed by the sheet dopants as schematically shown in (b). Vertical electric ®eld along line C±D is strengthened as schematically shown in (c).
1223
1224
M. Yamauchi et al.
and clarify the electron distribution and the ground subband wave function by solving the Poisson and SchroÈdinger equations self-consistently, following the general scheme outlined by Laux and Stern[9,10]. Structural parameters are set as follows; the thickness Lz of GaAs and that Lb of AlGaAs layers are both set to be 100 nm where the thickness Ln of an over-grown n-AlGaAs and that Ls of spacer layer are 100 nm and 10 nm, respectively. The length Ld of GaAs/AlGaAs multilayer is quite large in reality but is set at 1 mm for the reason mentioned later. In our analysis, we assume that the Fermi level at the top and bottom boundaries of Fig. 1(a) is set at the midgap of AlGaAs. The band discontinuity DEc at the heterointerface is assumed to be 220 meV. The density of Si in n-
AlGaAs is set at 1 1018 cmÿ3, while the density of residual acceptors in GaAs and AlGaAs is set at 1 1014 cmÿ3. The temperature is set at 10 K. Figure 2(a) shows the electron density distribution n(r) for the case where the sheet acceptors and sheet donors are absent. Characteristics of this type of basic EQWRs have already been reported by Stern in detail. Note that n(r) extends along the edge surface of the GaAs layer with two peaks at the side corners. This indicates that several subbands (up to the seventh) are occupied. The appearance of two side peaks are due to the attractive force of positively charged Si donors in the AlGaAs layer. When acceptors NA and donors ND are introduced to the GaAs and the AlGaAs layer by the planar doping to the level of 8.0 1011 cmÿ2,
Fig. 2. The distributions n(r) of electron density around the edge corners of a GaAs layer. The areal density of sheet acceptors and donors are set at 0 in (a), 8.0 1011 cmÿ2 in (b). Letters ``A'' and ``B'' correspond to those in Fig. 1(a).
Edge quantum wire structures
1225
Fig. 3. The square of the ground subband wave functions are plotted for the sheet acceptors and sheet donors of at 0 and 8.0 1011 cmÿ2. Letters ``A'' and ``B'' correspond to those in Fig. 1(a).
electrons are depleted at the center of the GaAs layer and n(r) splits into two well-resolved peaks, indicating that EQWRs far narrower than the GaAs layer are formed. The wire width in the z direction is 30 nm. Here, the wire width is de®ned as the width of the regions with n(r) over 1.0 1017 cmÿ3. The linear density of 1D electrons in these pairs of wire are 1.29 106 cmÿ1 and 1.29 106 cmÿ1, respectively. Figure 3(a), (b) show the squared wave functions vj1v2 of the ground subband of the two EQWR structures with ND and NA = 0 and 8.0 1011 cmÿ2, respectively. Though two peaks at the corners have nearly the same probability density, vj1v2 at the layer center of GaAs shows dierent characteristics: namely, vj1v2 almost vanishes in Fig. 3(b) whereas it has some probability in Fig. 3(a). Note that two peaks in Fig. 3(b) is more
than 40 nm apparent indicating that two wires are uncoupled. As discussed earlier, excited subbands up to the seventh are occupied in the basic EQWR with no sheet dopants, as shown in Fig. 2(a). Once the sheet dopants are introduced to 8.0 1011 cmÿ2, the level spacings are all increased. As a result 80% of the electrons (1.0 106 cmÿ1) are accommodated in the ground subband and the remaining 20% in the second subband, keeping the higher ones unoccupied. Hence, the electron transport is mostly dominated by the ground subband. This type of EQWR structure may provide several attractive properties. For example, the presence of a pair of QWRs in each GaAs layer may allow the formation of a QWR loop with which quantum interference phenomena may be explored. In addition, the interface-roughness scattering may be smaller as compared with the basic EQWRs because
1226
M. Yamauchi et al.
the energy level in hetero-junctions is less sensitive to the roughness. Moreover the formation of ohmic contacts to 1D systems would be easier because the wider GaAs will facilitates the ohmic contact formation. In summary, we have analyzed a new edge QWR structure in which electrons are con®ned twodimensionally by electric ®elds. This scheme allows the formation of QWRs, with a small cross-sectional sizes (R30 nm) even when the thickness of GaAs layer is quite large (r100 nm).
REFERENCES
1. Sakaki, H., Surf. Sci., 1992, 623, 267. 2. Sakaki, H., Jpn. J. Appl. Phys., 1980, 19, L735.
3. Motohisa, J. and Sakaki, H., Appl. Phys. Lett., 1993, 63, 1786. 4. van Wees, B. J., van Houten, H., Beenakker, C. W. J., Williamson, J. G., Kouwenhouen, L. P., van der Marel, D. and Foxon, C. T., Phys. Rev. Lett., 1988, 60, 848. 5. Yacoby, A., Stormer, H. L., Baldwin, K. W., Pfeier, L. N. and West, K. W., unpublished. 6. Ohno, Y., Nakamura, Y., Foley, M., Someya, T., Noda, T. and Sakaki, H., Phys. Rev. B, 1995, 52, 619. 7. Nakamura, Y., Tsuchiya, M., Koshiba, S., Noge, H. and Sakaki, H., Appl. Phys. Lett., 1994, 64, 2552. 8. Yamauchi, M., Nakaumura, Y., Kadoya, Y., Sugawara, H. and Sakaki, H., Jpn. J. Appl. Phys., 1996, 35, 1886. 9. Laux, S. E. and Stern, F., Appl. Phys. Lett., 1986, 49, 91. 10. Laux, S. E., Proc. 5th Int. Conf. Numerical Analysis of Semiconductor Devices and Integrated Circuits (NASECODE V), ed. J. J. H. Miller, Boole Press, Dublin, 1987, p. 270.