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The role of intersubband mixing in single-electron charging of open quantum dot
Physica E 6 (2000) 414–417
www.elsevier.nl/locate/physe
The role of intersubband mixing in single-electron charging of open quantum dot D.G. Baksheyeva;∗ , O.A. Tkachenkoa; b , V.A. Tkachenkoa a Institute
of Semiconductor Physics, 13 Lavrentieva st, Novosibirsk, 630090 Russia b Novosibirsk State University, Novosibirsk, 630090 Russia
Abstract We compare the electrostatics and electron transmission through the quantum dots that have the inlet/outlet constrictions made by (i) split nger gates and (ii) overlaying nger gates. It has been found numerically that the intermode mixing is large in the former case while almost absent in the latter. We beleive this di erence is responsible for single-electron charging of the open quantum dot of new type (ii) observed in C.-T. Liang et al. (Phys. Rev. Lett. 81 (1998) 3507). Our calculations show that the measured periods of conductance oscillations agree well with the change of the total charge of the dot by one elemental charge as the gate voltages change. Conventionally, single-electron charging does not show up in the transport through open quantum dots due to high transition probability from localized states to fully open channels. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.61.−r; 73.23.Hk Keywords: Quantum dots; Single-electron charging; Conductance quantization; ID-subband mixing
1. Introduction Recently Liang et al. [1] reported striking survival of Coulomb oscillations of the conductance G of quantum dot in zero magnetic eld in the region G ¿ 2e2 =h. The e ect was observed on the quantum dot fabricated on the basis of high-mobility 2DEG in a novel way: the metal split gate formed a channel, and three ultrathin (60 nm) continuous overlaying nger gates ( ngers) were placed across the channel (Fig. 1 ), iso-
∗ Corresponding author. Tel.: +7-3832-341733; +7-3832-331080. E-mail address: [email protected] (D.G. Baksheyev)
fax:
lated from the split gate by a PMMA layer. With some depletion voltage VF at the outermost ngers, quasiperiodic oscillations of conductance G(VSG ) in wide range 0 ¡ G ¡ 6e2 =h were observed, superimposed on quantized conductance steps. The authors of Ref. [1] clearly demonstrated continuous transition of these oscillations with increasing VF from region G ¿ 2e2 =h to G ¡ e2 =h with conservation of the period. Since the orthodox theory of Coulomb blockade [2] works well in the region G ¡ e2 =h and gives conductance oscillations, the conclusion is made in Ref. [1] that the single-electron charging was observed in the presence of up to three fully transmitted 1D-subbands. However, in contrast to orthodox theory [2] and to experimental works
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 0 4 - 0
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
Fig. 1. Calculated potential for the open quantum dot from Ref. [1], VF1; 3 = −1:4 V, VF2 = 0; VSG = −0:5 V. Split gates and overlaying nger gates are shown by hatched rectangles.
[3–5] no peak narrowing with increase of resistance were observed in Ref. [1]. The oscillations remained smooth and disappeared quickly when the background conductance was suppressed. Besides, the peak spacing uctuated by several tens of percent. Thus no pronounced narrow equidistant Coulomb peaks were observed. The present work is aimed on modeling electrostatic potential of the structure from Ref. [1] and multiple-mode electron transmission through quantum dots. Calculation of the capacitance of the dot with respect to contacts, ngers, and split gates is necessary to check the correspondence between the observed oscillation period and the change of the dot charge by one electron, and to obtain a correct estimate of the charging energy. The analysis of the transverse subband contributions into total transmission for the calculated potential, both for split and continuous ngers, will help to understand why in the latter case the charging of the quantum dot takes place in the presence of fully transmitted 1D-subbands. 2. Results and discussion The electrostatic potential pro le in the device [1,6] was determined by a solution of 3D Poisson equation with local 2DEG density given by 2D Thomas–Fermi
415
approximation and a boundary condition of frozen charge at the surface states and impurities. We have found that the number of electrons in the dot changes from 80 to 140 as VSG = −0:7–0:5 V, in concord with estimates of Ref. [1]. Calculated capacitances of the dot to the gates are also close to the experimentally estimated and lay within measured period uctuations (VSG = 2:8– 4.5 mV for experimental 3:6 ± 1 mV, and VF = 13:2–22 mV for VF1 = 23:8 mV and VF3 = 25:9 mV). The capacitances obtained demonstrate the same systematic drift with VSG and VF as observed. Thus, the conclusion that each oscillation of the conductance re ects the change of the dot charge by one electron is con rmed. However, the calculated capacitance of the dot with respect to 2DEG reservoirs is 340 –370 aF for almost closed quantum dot and is doubled when three 1D-subbands are transmitted. Thus, this capacitance is almost an order higher than that to the gates and cannot be neglected, so the charging energy is e2 =2C = 0:1– 0.2 meV. Some modeling details: The Fermi level in 2DEG is EF = 5 meV (carrier density n = 1:6 × 10−11 cm−2 ). At VF = 0 the channel pinches o when VSG = −1:8 V (the same as in the experiment). When VSG = −0:7 V the nger gates raise the potential barriers in the constrictions above the Fermi level at VF = −1:4 V (experimental pinch-o voltages are VF1 = −1:9 V and VF3 = −1:7 V). We calculated potential pro le, total charge of the dot, and the capacitances in range VSG = −0:7– 0.5 V and VF = −1:2–1.4 V, which is considered to correspond to the experiment. The quantum dot width across the channel changed from 350 to 560 nm, and the depth at the center from 2.5 to 3.1 meV (counting from EF ). Practically, the width and depth of the quantum dot were found to depend on VSG only. It is of interest to note that the presence of zero-biased central nger makes the dot 0.5 meV deeper and stabilizes the depth at ∼3 meV. Along the channel axis the top of the barrier in the constriction is 0.5 meV below EF and higher, as controlled by VF . At high nger gate voltages VF = −1:3–1.4 V, the transverse potential pro le of the constriction resembles a rectangular well with the width at Fermi level ranging from 100 to 280 nm (Fig. 1, Fermi level marked by thick line). In numerical modeling of multiple-mode electron transmission through the quantum dot we used 2D-potential obtained from the calculations of elec-
416
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
Fig. 2. Characteristics of open quantum dots of types (ii) overlying nger gates and (i) split nger gates. (a,d) total transmission coecient and contributions of 1–5 1D-subbands. (b,e) transverse crosssections of the channel in the central parts of the dots and in the constrictions. (c,f) potential pro les along the channels and energies En (x) of n=1–3 1D-subbands. The gate voltages are (ii) the same as in Fig. 1, and (i) VF = −1:6 V, VSG = −0:5 V.
trostatics for the case when (i) the outermost 160 nm wide nger gates were broken with a 260 nm gap and (ii) 60 nm wide ngers were continuous (i.e. overlaying, as in Ref. [1]). By solution of 2D Schrodinger equation it was found that when 2–3 transverse modes in the constrictions become open the transmission through type (ii) quantum dot via each of the rst three 1D-subbands is independent, i.e. no mode mixing occurs. The transmission coecient dependence T (EF ) for di erent subbands is the same as for transmission through 1D system with two smooth barriers (Fig. 2a, note that only Fabry–Perot resonances are present). In contrast, in case (i) the intermode mixing is very intensive and shows up in T (EF ) dependence as sharp Fano resonances due to electron scattering on the levels of the dot (Fig. 2d).
The di erence is due to transverse cross section of the potential in the constrictions. For type (ii) quantum dot [1] it is not parabolic, rather at and wide (Fig. 2b), and 1D-subband spacing is signi cantly less than in the case when the constrictions (i) are created by split ngers, when the parabola is always narrow and deep (Fig. 2e). As a consequence, in case (ii) the lowest transverse subbands run practically in parallel to each other in both dot and constrictions (Fig. 2c), while in usual quantum dots in the constrictions the distance between 1D-subbands in the constrictions is 2–3 times larger than that inside the dot (Fig. 2f), even when the dot is open. So the transmission through dot (ii) may be considered one-dimensional in contrast with substantially two-dimensional transport in dot (i). Note that the tunneling via closed subbands in dots (ii) is much larger than in dots (i) due to small subband spacing, though transverse subbands do not mix. We argue that in the absence of intersubband mixing the transport in the low-transparency subbands is due to sequential tunneling with single-electron charging of the dot, resulting in the Coulomb oscillations observed in Ref. [1]. The open subbands transmit the electrons coherently and provide for parallel background current and quantization steps in the conductance. With each new step the single-electron charging in the opening subband ceases while it still takes place in a higher low-transparency subband, since the intersubband transition probability in the considered dot at low temperature is very small. With increasing temperature this probability grows and Coulomb oscillations disapper rst in higher subbands, as seen in Ref. [1] (Fig. 2). For the lowest subband the oscillations survive up to 0.5 K that corresponds to the condition of observation of Coulomb oscillations kT .e2 =2C = 0:2 meV. Usual quantum dot of type (i) in the open state do not allow observing the e ects of single-electron charging even at low temperatures because of strong intersubband mixing, but they demonstrate strong interference Fano resonances [4,5]. In summary, we have numerically con rmed that the conductance oscillations observed in Ref. [1] are caused by single-electron change of the dot’s charge. The conclusion is made that the e ects of single-electron charging can be observed in the open dots provided that no intersubband mixing takes place.
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
417
Acknowledgements
References
We are grateful to Chi-Te Liang for the data, and Kvon Ze Don, M.V. Entin, and A.G. Pogosov for discussions. The work is supported by the programs of Ministry of Science of Russian Federation “Physics of Solid-State Nanostructures” (grant No. 98-1102) and “Prospective Technologies and Devices for Micro- and Nanoelectronics” (No. 02.04.5.1), and by the program “Universities of Russia–Fundamental Research” (No. 1994).
[1] C.-T. Liang, M.Y. Simmons, C.G. Smith, G.H. Kim, D.A. Ritchie, M. Pepper, Phys. Rev. Lett. 81 (1998) 3507. [2] K.K. Likharev, IBM J. Res. Develop. 32 (1988) 144. [3] E.B. Foxman, P.L. McEuen, U. Meirav, N.S. Wingreen, Y. Meir, P.A. Belk, N.R. Belk, M.A. Kastner, Phys. Rev. B 47 (1993) 10 020. [4] T. Heinzel, S. Manus, D.A. Wharam, J.P. Kotthaus, G. Bohm, W. Klein, G. Trankle, G. Weimann, Europhys. Lett. 26 (1994) 689. [5] M. Persson, J. Pettersson, B. von Sydow, P.E. Lindelof, A. Kristensen, K.F. Berggren, Phys. Rev. B 52 (1995) 8921. [6] C.-T. Liang, private communication, 1999.
www.elsevier.nl/locate/physe
The role of intersubband mixing in single-electron charging of open quantum dot D.G. Baksheyeva;∗ , O.A. Tkachenkoa; b , V.A. Tkachenkoa a Institute
of Semiconductor Physics, 13 Lavrentieva st, Novosibirsk, 630090 Russia b Novosibirsk State University, Novosibirsk, 630090 Russia
Abstract We compare the electrostatics and electron transmission through the quantum dots that have the inlet/outlet constrictions made by (i) split nger gates and (ii) overlaying nger gates. It has been found numerically that the intermode mixing is large in the former case while almost absent in the latter. We beleive this di erence is responsible for single-electron charging of the open quantum dot of new type (ii) observed in C.-T. Liang et al. (Phys. Rev. Lett. 81 (1998) 3507). Our calculations show that the measured periods of conductance oscillations agree well with the change of the total charge of the dot by one elemental charge as the gate voltages change. Conventionally, single-electron charging does not show up in the transport through open quantum dots due to high transition probability from localized states to fully open channels. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.61.−r; 73.23.Hk Keywords: Quantum dots; Single-electron charging; Conductance quantization; ID-subband mixing
1. Introduction Recently Liang et al. [1] reported striking survival of Coulomb oscillations of the conductance G of quantum dot in zero magnetic eld in the region G ¿ 2e2 =h. The e ect was observed on the quantum dot fabricated on the basis of high-mobility 2DEG in a novel way: the metal split gate formed a channel, and three ultrathin (60 nm) continuous overlaying nger gates ( ngers) were placed across the channel (Fig. 1 ), iso-
∗ Corresponding author. Tel.: +7-3832-341733; +7-3832-331080. E-mail address: [email protected] (D.G. Baksheyev)
fax:
lated from the split gate by a PMMA layer. With some depletion voltage VF at the outermost ngers, quasiperiodic oscillations of conductance G(VSG ) in wide range 0 ¡ G ¡ 6e2 =h were observed, superimposed on quantized conductance steps. The authors of Ref. [1] clearly demonstrated continuous transition of these oscillations with increasing VF from region G ¿ 2e2 =h to G ¡ e2 =h with conservation of the period. Since the orthodox theory of Coulomb blockade [2] works well in the region G ¡ e2 =h and gives conductance oscillations, the conclusion is made in Ref. [1] that the single-electron charging was observed in the presence of up to three fully transmitted 1D-subbands. However, in contrast to orthodox theory [2] and to experimental works
1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 0 4 - 0
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
Fig. 1. Calculated potential for the open quantum dot from Ref. [1], VF1; 3 = −1:4 V, VF2 = 0; VSG = −0:5 V. Split gates and overlaying nger gates are shown by hatched rectangles.
[3–5] no peak narrowing with increase of resistance were observed in Ref. [1]. The oscillations remained smooth and disappeared quickly when the background conductance was suppressed. Besides, the peak spacing uctuated by several tens of percent. Thus no pronounced narrow equidistant Coulomb peaks were observed. The present work is aimed on modeling electrostatic potential of the structure from Ref. [1] and multiple-mode electron transmission through quantum dots. Calculation of the capacitance of the dot with respect to contacts, ngers, and split gates is necessary to check the correspondence between the observed oscillation period and the change of the dot charge by one electron, and to obtain a correct estimate of the charging energy. The analysis of the transverse subband contributions into total transmission for the calculated potential, both for split and continuous ngers, will help to understand why in the latter case the charging of the quantum dot takes place in the presence of fully transmitted 1D-subbands. 2. Results and discussion The electrostatic potential pro le in the device [1,6] was determined by a solution of 3D Poisson equation with local 2DEG density given by 2D Thomas–Fermi
415
approximation and a boundary condition of frozen charge at the surface states and impurities. We have found that the number of electrons in the dot changes from 80 to 140 as VSG = −0:7–0:5 V, in concord with estimates of Ref. [1]. Calculated capacitances of the dot to the gates are also close to the experimentally estimated and lay within measured period uctuations (VSG = 2:8– 4.5 mV for experimental 3:6 ± 1 mV, and VF = 13:2–22 mV for VF1 = 23:8 mV and VF3 = 25:9 mV). The capacitances obtained demonstrate the same systematic drift with VSG and VF as observed. Thus, the conclusion that each oscillation of the conductance re ects the change of the dot charge by one electron is con rmed. However, the calculated capacitance of the dot with respect to 2DEG reservoirs is 340 –370 aF for almost closed quantum dot and is doubled when three 1D-subbands are transmitted. Thus, this capacitance is almost an order higher than that to the gates and cannot be neglected, so the charging energy is e2 =2C = 0:1– 0.2 meV. Some modeling details: The Fermi level in 2DEG is EF = 5 meV (carrier density n = 1:6 × 10−11 cm−2 ). At VF = 0 the channel pinches o when VSG = −1:8 V (the same as in the experiment). When VSG = −0:7 V the nger gates raise the potential barriers in the constrictions above the Fermi level at VF = −1:4 V (experimental pinch-o voltages are VF1 = −1:9 V and VF3 = −1:7 V). We calculated potential pro le, total charge of the dot, and the capacitances in range VSG = −0:7– 0.5 V and VF = −1:2–1.4 V, which is considered to correspond to the experiment. The quantum dot width across the channel changed from 350 to 560 nm, and the depth at the center from 2.5 to 3.1 meV (counting from EF ). Practically, the width and depth of the quantum dot were found to depend on VSG only. It is of interest to note that the presence of zero-biased central nger makes the dot 0.5 meV deeper and stabilizes the depth at ∼3 meV. Along the channel axis the top of the barrier in the constriction is 0.5 meV below EF and higher, as controlled by VF . At high nger gate voltages VF = −1:3–1.4 V, the transverse potential pro le of the constriction resembles a rectangular well with the width at Fermi level ranging from 100 to 280 nm (Fig. 1, Fermi level marked by thick line). In numerical modeling of multiple-mode electron transmission through the quantum dot we used 2D-potential obtained from the calculations of elec-
416
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
Fig. 2. Characteristics of open quantum dots of types (ii) overlying nger gates and (i) split nger gates. (a,d) total transmission coecient and contributions of 1–5 1D-subbands. (b,e) transverse crosssections of the channel in the central parts of the dots and in the constrictions. (c,f) potential pro les along the channels and energies En (x) of n=1–3 1D-subbands. The gate voltages are (ii) the same as in Fig. 1, and (i) VF = −1:6 V, VSG = −0:5 V.
trostatics for the case when (i) the outermost 160 nm wide nger gates were broken with a 260 nm gap and (ii) 60 nm wide ngers were continuous (i.e. overlaying, as in Ref. [1]). By solution of 2D Schrodinger equation it was found that when 2–3 transverse modes in the constrictions become open the transmission through type (ii) quantum dot via each of the rst three 1D-subbands is independent, i.e. no mode mixing occurs. The transmission coecient dependence T (EF ) for di erent subbands is the same as for transmission through 1D system with two smooth barriers (Fig. 2a, note that only Fabry–Perot resonances are present). In contrast, in case (i) the intermode mixing is very intensive and shows up in T (EF ) dependence as sharp Fano resonances due to electron scattering on the levels of the dot (Fig. 2d).
The di erence is due to transverse cross section of the potential in the constrictions. For type (ii) quantum dot [1] it is not parabolic, rather at and wide (Fig. 2b), and 1D-subband spacing is signi cantly less than in the case when the constrictions (i) are created by split ngers, when the parabola is always narrow and deep (Fig. 2e). As a consequence, in case (ii) the lowest transverse subbands run practically in parallel to each other in both dot and constrictions (Fig. 2c), while in usual quantum dots in the constrictions the distance between 1D-subbands in the constrictions is 2–3 times larger than that inside the dot (Fig. 2f), even when the dot is open. So the transmission through dot (ii) may be considered one-dimensional in contrast with substantially two-dimensional transport in dot (i). Note that the tunneling via closed subbands in dots (ii) is much larger than in dots (i) due to small subband spacing, though transverse subbands do not mix. We argue that in the absence of intersubband mixing the transport in the low-transparency subbands is due to sequential tunneling with single-electron charging of the dot, resulting in the Coulomb oscillations observed in Ref. [1]. The open subbands transmit the electrons coherently and provide for parallel background current and quantization steps in the conductance. With each new step the single-electron charging in the opening subband ceases while it still takes place in a higher low-transparency subband, since the intersubband transition probability in the considered dot at low temperature is very small. With increasing temperature this probability grows and Coulomb oscillations disapper rst in higher subbands, as seen in Ref. [1] (Fig. 2). For the lowest subband the oscillations survive up to 0.5 K that corresponds to the condition of observation of Coulomb oscillations kT .e2 =2C = 0:2 meV. Usual quantum dot of type (i) in the open state do not allow observing the e ects of single-electron charging even at low temperatures because of strong intersubband mixing, but they demonstrate strong interference Fano resonances [4,5]. In summary, we have numerically con rmed that the conductance oscillations observed in Ref. [1] are caused by single-electron change of the dot’s charge. The conclusion is made that the e ects of single-electron charging can be observed in the open dots provided that no intersubband mixing takes place.
D.G. Baksheyev et al. = Physica E 6 (2000) 414–417
417
Acknowledgements
References
We are grateful to Chi-Te Liang for the data, and Kvon Ze Don, M.V. Entin, and A.G. Pogosov for discussions. The work is supported by the programs of Ministry of Science of Russian Federation “Physics of Solid-State Nanostructures” (grant No. 98-1102) and “Prospective Technologies and Devices for Micro- and Nanoelectronics” (No. 02.04.5.1), and by the program “Universities of Russia–Fundamental Research” (No. 1994).
[1] C.-T. Liang, M.Y. Simmons, C.G. Smith, G.H. Kim, D.A. Ritchie, M. Pepper, Phys. Rev. Lett. 81 (1998) 3507. [2] K.K. Likharev, IBM J. Res. Develop. 32 (1988) 144. [3] E.B. Foxman, P.L. McEuen, U. Meirav, N.S. Wingreen, Y. Meir, P.A. Belk, N.R. Belk, M.A. Kastner, Phys. Rev. B 47 (1993) 10 020. [4] T. Heinzel, S. Manus, D.A. Wharam, J.P. Kotthaus, G. Bohm, W. Klein, G. Trankle, G. Weimann, Europhys. Lett. 26 (1994) 689. [5] M. Persson, J. Pettersson, B. von Sydow, P.E. Lindelof, A. Kristensen, K.F. Berggren, Phys. Rev. B 52 (1995) 8921. [6] C.-T. Liang, private communication, 1999.