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Integer filling factor phases in vertical diatomic artificial molecules
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Physica E 22 (2004) 502 – 505 www.elsevier.com/locate/physe
Integer lling factor phases in vertical diatomic arti cial molecules D.G. Austinga; b;∗ , S. Taruchab; c , K. Murakib , F. Ancilottod , M. Barrancoe , A. Emperadorf , R. Mayole , M. Pie a Institute
for Microstructural Sciences M23A, National Research Council of Canada, Montreal Road, Ottawa, Ont., Canada K1A 0R6 b NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan c Department of Physics and ERATO Mesoscopic Correlation Project, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan d Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica, Universit9 a di Padova, Padova I-35131, Italy e Departament ECM, Facultat de Fisica, ; Universitat de Barcelona, Barcelona E-08028, Spain f Dipartimento di Fisica, Universit9 a di Trento, Povo I-38050, Italy
Abstract We investigate integer lling factor phases of many-N -electron vertically coupled semiconductor quantum dot arti cial molecules when the inter-dot coupling is strong and weak. The experimental results are analyzed within local-spin densityfunctional theory. Maximum density droplets composed of electrons in both bonding and anti-bonding, or just bonding states are revealed. Interesting isospin physics (i.e., isospin transitions) can occur particularly when the inter-dot coupling is reduced. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.15Mb; 85.30.Vw; 36.40.Ei; 73.20.Dx Keywords: Coupled quantum dots; Maximum density droplet; Double quantum well bilayer systems; Quantum Hall e?ects; Arti cial molecules; Local-spin density-functional theory
1. Introduction Arti cial quantum molecules (QM’s) composed of two vertically coupled semiconductor quantum dots (QDs) are the subject of much experimental and theoretical investigation [1–10]. The two QD’s in the QM’s we make are well de ned and highly symmetric [11], ∗
Corresponding author. Institute for Microstructural Sciences M23A, National Research Council of Canada, Montreal Road, Ottawa, Ont., Canada K1A 0R6. Tel.: +1-613-991-9989; fax: +1-613-952-8701. E-mail address: [email protected] (D.G. Austing).
so our vertical QM’s are well suited for the study of exotic many-body molecular states in the presence of a magnetic eld.
2. Discussion The QM devices, shown schematically in Fig. 1, are fabricated from a triple barrier structure with a central barrier thickness, b, typically between 2:5 nm (strong coupling: tunnel coupling energy, JSAS ∼ 3:5 meV) and 6:0 nm (weak coupling: JSAS ∼ 0:3 meV) [2,3].
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.055
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
Fig. 1. Schematic of vertical QM device, and cartoons of the arrangement of electrons in B and AB single particle states, and isospin Lip for certain integer lling factors.
The current (Id ) Lowing through two dots located inside circular mesas (diameter ¡ 1 m) is measured at ∼ 100 mK as a function of voltage between the substrate and top contact (Vd ∼ 0:2 mV), and voltage on the single gate (Vg ). The magnetic eld (B== ) is applied parallel to the current. The interpretation of the experimental data is based on the application of local-spin density-functional theory (LSDFT). It follows the methods comprehensively described elsewhere [8–10]. The QM is modeled by stacking two equal (“homonuclear”) QDs in the direction parallel to B== . Experimental B== –N phase diagrams for QM’s with (a) b = 2:5 nm and (b) b = 6:0 nm showing the evo-
503
Fig. 2. Experimental B== –N phase diagrams for QM structures with (a) b = 2:5 nm and (b) b = 6 nm. The variations in amplitude (arbitrary scale) of the current peaks are not discussed in this paper. The low- eld boundary of each integer lling factor phase of nite width is drawn as a bold line. In the MDDB+AB phase of (b), there are a series of features (some connected by dashed lines) that we attribute to isospin transitions.
lution of Coulomb oscillations (e?ectively the ground state electrochemical potentials) are given in Fig. 2. The range of N , the number of electrons in the QM, is between 12 and about 40. (a) is similar to a single dot phase diagram [12]. 1 We identify two threshold
1 Many features, particularly at large N (¿ 20) and large B== (¿ 8 T) of the single vertical dot phase diagram have yet to be satisfactorily explained by any model. For lateral single dots, also see Ciorga et al. [12]
504
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
lines marking the end of the Fock–Darwin single particle level crossings ( lling factor, B = 2), and the start of the spin-polarized compact maximum density droplet (MDDB ; B = 1) [13]. 2 Since JSAS is suPciently large, essentially only bonding (B) states are lled (see Fig. 1 schematic). 3 The B = 1 line originates from the N =2 singlet–triplet transition [3]. (b) is di?erent to (a) since now anti-bonding (AB) electrons play a role as the coupling is reduced. We can identify the onset of two cohabiting MDD’s- one is made of B states (MDDB ), and the other is made of AB states (MDDAB ), so we call this phase MDDB+AB ( T = 2 i.e., B = 1 + AB = 1). In the MDDB+AB phase, there are a series of features (some connected by dashed lines) that we attribute to isospin transitions since they do not appear in the phase diagrams for a single QD, or a strongly coupled QM. A cartoon of a rearrangement of a single electron between the AB and B single particle states for T = 2 is shown in Fig. 1. B== –N phase diagrams calculated by LSDFT for QM structures with (a) b = 2:5 nm and (b) b = 6 nm are given in Fig. 3. The low (high)- eld boundary of each integer lling factor phase of nite width is drawn as a bold (faint) line. Only values corresponding to (‘large’) N = 4M (M = 3–9) are meaningful here and we assume the two dots are identical. The e?ective lateral harmonic oscillator potential we take for all values of b (at 0 T) has strength, ˝!0 = KNB−1=4 , where K = 6:91 meV is deduced from a t to the onset of MDDB in Fig. 2 (a), and NB (NAB ) is the number of electrons in B (AB) states. Since for b = 2:5(6:0) nm, JSAS ¿ ˝!0 (JSAS ¡ ˝!0 ) and NB ∼ N (NB ∼ N=2 at 0 T), we found this crucial to achieve a good quantitative description of the molecular phases. 4 In the LSDFT calculations here, at least up to N = 36, electrons only occupy B single particle states. Consistent with 2
For MDD physics in single dots, see Ref. [13] For the b = 2:5 nm QM some weak features appear in the experimental phase diagram for N ¿ 25 (not resolvable in Fig. 2(a)), just to the right of the B = 2 line, that can be explained by the occupation of at most a single electron in the lowest AB state (unpublished). The single QD like shell structure at 0 T(N ¡ 18) is discussed elsewhere [3,9]. 4 Full-blown self-consistent modeling of large N vertical QM’s in a magnetic eld has not yet been reported in the literature. For small N , we have previously employed an N -independent ˝!0 [8– 10], but for the large N values of interest here, an N -dependent ˝!0 of the form we take is justi able [3]. 3
Fig. 3. Calculated B== –N phase diagrams for QM structures with (a) b=2:5 nm and (b) b=6 nm. The low (high)- eld boundary of each integer lling factor phase of nite width is drawn as a bold (faint) line. Only the values corresponding to N =12; 16; : : : ; 32; 36 are meaningful here. Two regions of di?erent isospin within the MDDB+AB phase in (b) are separated by a dashed line.
the experimental data, as the coupling is reduced, on the one hand we nd the stability of the MDDB rapidly decreases (e?ectively to zero) [5], and on the other hand an MDDB+AB phase appears in which isospin transitions can occur. Within the MDDB+AB ( T = 2)
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
phase, regions of di?erent isospin, IZ = (NB − NAB )=2, are identi ed by LSDFT, and IZ is generally found to increase with N and B== . Elsewhere, we discuss in detail the rational of the KNB−1=4 law, the onset and range of stability of all the phases, the dependence with b of the IZ transitions in the MDDB+AB phase, the “homonuclear” assumption (and whether mismatched dots [9] help stabilize the MDDB as b increases), and the T = 4 phase ( B = 2 + AB = 2) for the b = 6:0 nm QM for which LSDFT predicts isospin transitions that are ‘spin-Lip-driven’ (not shown in Fig. 3(b)) [14]. Acknowledgements This work has been performed under Grants BFM2002-01868 from DGI and 2001SGR-00064 from Generalitat of Catalunya, and supported by the Specially Promoted Research, Grant-In-Aid for Scienti c Research, and by DARPA-QUIST program (DAAD 19-01-1-0659). F.A. has been funded by CESCA-CEPBA, contract no. HPRI-1999-CT-00071. A.E. has been funded by the Spanish SecretarTUa de Estado de EducaciTon y Universidades. We are grateful for the assistance of T. Honda with processing the samples, and for useful discussions with K. Ono, H. Tamura, Y. Tokura and S. Sasaki. References [1] T. Schmidt, et al., Phys. Rev. Lett. 78 (1997) 1544; R.J. Luyken, et al., Nanotechnology 10 (1999) 14; M. Bayer, et al., Science 291 (2001) 451; K. Ono, et al., Science 297 (2002) 1313.
505
[2] D.G. Austing, et al., Physica B 249 –251 (1998) 206. [3] S. Amaha, et al., Solid State Commun. 119 (2001) 183. [4] J.J. Palacios, P. Hawrylak, Phys. Rev. B 51 (1995) 1769; J.H. Oh, et al., Phys. Rev. B 53 (1996) R13264; H. Imamura, et al., Phys. Rev. B 53 (1996) 12613; O. Mayrock, et al., Phys. Rev. B 56 (1997) 15760; H. Tamura, Physica B 249 –251 (1998) 210; Y. Asano, Phys. Rev. B 58 (1998) 1414; Y. Tokura, et al., J. Phys.: Condens. Matter 11 (1999) 6023; Y. Tokura, et al., Physica E 6 (2000) 676; G. Burkard, et al., Phys. Rev. B 62 (2000) 2581; L. Martin-Moreno, et al., Phys. Rev. B 62 (2000) R10633; T. Matsuse, et al., Eur. Phys. J. D 16 (2001) 391; S.-R.E. Yang, J. Schliemann, A.H. MacDonald, Phys. Rev. B 66 (2002) 153302. [5] J. Hu, et al., Phys. Rev. B 54 (1996) 8616. [6] M. Rontani, et al., Solid State Commun. 112 (1999) 151; M. Rontani, et al., Solid State Commun. 119 (2001) 309. [7] B. Partoens, F.M. Peeters, Physica B 298 (2001) 282; B. Partoens, F.M. Peeters, Europhys. Lett. 56 (2001) 86. [8] M. Pi, et al., Phys. Rev. B 63 (2001) 115316. [9] M. Pi, et al., Phys. Rev. Lett. 87 (2001) 066801. [10] F. Ancilotto, et al., Phys. Rev. B 67 (2003) 205311. [11] S. Tarucha, et al., Phys. Rev. Lett. 77 (1996) 3613. [12] D.G. Austing, et al., Jpn. J. Appl. Phys. 38 (1999) 372; T.H. Oosterkamp, et al., Phys. Rev. Lett. 82 (1999) 2931; M. Ciorga, et al., Phys. Rev. Lett. 88 (2002) 256804. [13] A.H. MacDonald, S.-R.E. Yang, M.D. Johnson, Aust. J. Phys. 46 (1993) 345; S.M. Reimann, et al., Phys. Rev. Lett. 83 (1999) 3270; S.-R.E. Yang, A.H. MacDonald, Phys. Rev. B 66 (2002) 041304(R). [14] D.G. Austing, et al., unpublished.
Physica E 22 (2004) 502 – 505 www.elsevier.com/locate/physe
Integer lling factor phases in vertical diatomic arti cial molecules D.G. Austinga; b;∗ , S. Taruchab; c , K. Murakib , F. Ancilottod , M. Barrancoe , A. Emperadorf , R. Mayole , M. Pie a Institute
for Microstructural Sciences M23A, National Research Council of Canada, Montreal Road, Ottawa, Ont., Canada K1A 0R6 b NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan c Department of Physics and ERATO Mesoscopic Correlation Project, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan d Istituto Nazionale per la Fisica della Materia and Dipartimento di Fisica, Universit9 a di Padova, Padova I-35131, Italy e Departament ECM, Facultat de Fisica, ; Universitat de Barcelona, Barcelona E-08028, Spain f Dipartimento di Fisica, Universit9 a di Trento, Povo I-38050, Italy
Abstract We investigate integer lling factor phases of many-N -electron vertically coupled semiconductor quantum dot arti cial molecules when the inter-dot coupling is strong and weak. The experimental results are analyzed within local-spin densityfunctional theory. Maximum density droplets composed of electrons in both bonding and anti-bonding, or just bonding states are revealed. Interesting isospin physics (i.e., isospin transitions) can occur particularly when the inter-dot coupling is reduced. ? 2003 Elsevier B.V. All rights reserved. PACS: 71.15Mb; 85.30.Vw; 36.40.Ei; 73.20.Dx Keywords: Coupled quantum dots; Maximum density droplet; Double quantum well bilayer systems; Quantum Hall e?ects; Arti cial molecules; Local-spin density-functional theory
1. Introduction Arti cial quantum molecules (QM’s) composed of two vertically coupled semiconductor quantum dots (QDs) are the subject of much experimental and theoretical investigation [1–10]. The two QD’s in the QM’s we make are well de ned and highly symmetric [11], ∗
Corresponding author. Institute for Microstructural Sciences M23A, National Research Council of Canada, Montreal Road, Ottawa, Ont., Canada K1A 0R6. Tel.: +1-613-991-9989; fax: +1-613-952-8701. E-mail address: [email protected] (D.G. Austing).
so our vertical QM’s are well suited for the study of exotic many-body molecular states in the presence of a magnetic eld.
2. Discussion The QM devices, shown schematically in Fig. 1, are fabricated from a triple barrier structure with a central barrier thickness, b, typically between 2:5 nm (strong coupling: tunnel coupling energy, JSAS ∼ 3:5 meV) and 6:0 nm (weak coupling: JSAS ∼ 0:3 meV) [2,3].
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.055
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
Fig. 1. Schematic of vertical QM device, and cartoons of the arrangement of electrons in B and AB single particle states, and isospin Lip for certain integer lling factors.
The current (Id ) Lowing through two dots located inside circular mesas (diameter ¡ 1 m) is measured at ∼ 100 mK as a function of voltage between the substrate and top contact (Vd ∼ 0:2 mV), and voltage on the single gate (Vg ). The magnetic eld (B== ) is applied parallel to the current. The interpretation of the experimental data is based on the application of local-spin density-functional theory (LSDFT). It follows the methods comprehensively described elsewhere [8–10]. The QM is modeled by stacking two equal (“homonuclear”) QDs in the direction parallel to B== . Experimental B== –N phase diagrams for QM’s with (a) b = 2:5 nm and (b) b = 6:0 nm showing the evo-
503
Fig. 2. Experimental B== –N phase diagrams for QM structures with (a) b = 2:5 nm and (b) b = 6 nm. The variations in amplitude (arbitrary scale) of the current peaks are not discussed in this paper. The low- eld boundary of each integer lling factor phase of nite width is drawn as a bold line. In the MDDB+AB phase of (b), there are a series of features (some connected by dashed lines) that we attribute to isospin transitions.
lution of Coulomb oscillations (e?ectively the ground state electrochemical potentials) are given in Fig. 2. The range of N , the number of electrons in the QM, is between 12 and about 40. (a) is similar to a single dot phase diagram [12]. 1 We identify two threshold
1 Many features, particularly at large N (¿ 20) and large B== (¿ 8 T) of the single vertical dot phase diagram have yet to be satisfactorily explained by any model. For lateral single dots, also see Ciorga et al. [12]
504
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
lines marking the end of the Fock–Darwin single particle level crossings ( lling factor, B = 2), and the start of the spin-polarized compact maximum density droplet (MDDB ; B = 1) [13]. 2 Since JSAS is suPciently large, essentially only bonding (B) states are lled (see Fig. 1 schematic). 3 The B = 1 line originates from the N =2 singlet–triplet transition [3]. (b) is di?erent to (a) since now anti-bonding (AB) electrons play a role as the coupling is reduced. We can identify the onset of two cohabiting MDD’s- one is made of B states (MDDB ), and the other is made of AB states (MDDAB ), so we call this phase MDDB+AB ( T = 2 i.e., B = 1 + AB = 1). In the MDDB+AB phase, there are a series of features (some connected by dashed lines) that we attribute to isospin transitions since they do not appear in the phase diagrams for a single QD, or a strongly coupled QM. A cartoon of a rearrangement of a single electron between the AB and B single particle states for T = 2 is shown in Fig. 1. B== –N phase diagrams calculated by LSDFT for QM structures with (a) b = 2:5 nm and (b) b = 6 nm are given in Fig. 3. The low (high)- eld boundary of each integer lling factor phase of nite width is drawn as a bold (faint) line. Only values corresponding to (‘large’) N = 4M (M = 3–9) are meaningful here and we assume the two dots are identical. The e?ective lateral harmonic oscillator potential we take for all values of b (at 0 T) has strength, ˝!0 = KNB−1=4 , where K = 6:91 meV is deduced from a t to the onset of MDDB in Fig. 2 (a), and NB (NAB ) is the number of electrons in B (AB) states. Since for b = 2:5(6:0) nm, JSAS ¿ ˝!0 (JSAS ¡ ˝!0 ) and NB ∼ N (NB ∼ N=2 at 0 T), we found this crucial to achieve a good quantitative description of the molecular phases. 4 In the LSDFT calculations here, at least up to N = 36, electrons only occupy B single particle states. Consistent with 2
For MDD physics in single dots, see Ref. [13] For the b = 2:5 nm QM some weak features appear in the experimental phase diagram for N ¿ 25 (not resolvable in Fig. 2(a)), just to the right of the B = 2 line, that can be explained by the occupation of at most a single electron in the lowest AB state (unpublished). The single QD like shell structure at 0 T(N ¡ 18) is discussed elsewhere [3,9]. 4 Full-blown self-consistent modeling of large N vertical QM’s in a magnetic eld has not yet been reported in the literature. For small N , we have previously employed an N -independent ˝!0 [8– 10], but for the large N values of interest here, an N -dependent ˝!0 of the form we take is justi able [3]. 3
Fig. 3. Calculated B== –N phase diagrams for QM structures with (a) b=2:5 nm and (b) b=6 nm. The low (high)- eld boundary of each integer lling factor phase of nite width is drawn as a bold (faint) line. Only the values corresponding to N =12; 16; : : : ; 32; 36 are meaningful here. Two regions of di?erent isospin within the MDDB+AB phase in (b) are separated by a dashed line.
the experimental data, as the coupling is reduced, on the one hand we nd the stability of the MDDB rapidly decreases (e?ectively to zero) [5], and on the other hand an MDDB+AB phase appears in which isospin transitions can occur. Within the MDDB+AB ( T = 2)
D.G. Austing et al. / Physica E 22 (2004) 502 – 505
phase, regions of di?erent isospin, IZ = (NB − NAB )=2, are identi ed by LSDFT, and IZ is generally found to increase with N and B== . Elsewhere, we discuss in detail the rational of the KNB−1=4 law, the onset and range of stability of all the phases, the dependence with b of the IZ transitions in the MDDB+AB phase, the “homonuclear” assumption (and whether mismatched dots [9] help stabilize the MDDB as b increases), and the T = 4 phase ( B = 2 + AB = 2) for the b = 6:0 nm QM for which LSDFT predicts isospin transitions that are ‘spin-Lip-driven’ (not shown in Fig. 3(b)) [14]. Acknowledgements This work has been performed under Grants BFM2002-01868 from DGI and 2001SGR-00064 from Generalitat of Catalunya, and supported by the Specially Promoted Research, Grant-In-Aid for Scienti c Research, and by DARPA-QUIST program (DAAD 19-01-1-0659). F.A. has been funded by CESCA-CEPBA, contract no. HPRI-1999-CT-00071. A.E. has been funded by the Spanish SecretarTUa de Estado de EducaciTon y Universidades. We are grateful for the assistance of T. Honda with processing the samples, and for useful discussions with K. Ono, H. Tamura, Y. Tokura and S. Sasaki. References [1] T. Schmidt, et al., Phys. Rev. Lett. 78 (1997) 1544; R.J. Luyken, et al., Nanotechnology 10 (1999) 14; M. Bayer, et al., Science 291 (2001) 451; K. Ono, et al., Science 297 (2002) 1313.
505
[2] D.G. Austing, et al., Physica B 249 –251 (1998) 206. [3] S. Amaha, et al., Solid State Commun. 119 (2001) 183. [4] J.J. Palacios, P. Hawrylak, Phys. Rev. B 51 (1995) 1769; J.H. Oh, et al., Phys. Rev. B 53 (1996) R13264; H. Imamura, et al., Phys. Rev. B 53 (1996) 12613; O. Mayrock, et al., Phys. Rev. B 56 (1997) 15760; H. Tamura, Physica B 249 –251 (1998) 210; Y. Asano, Phys. Rev. B 58 (1998) 1414; Y. Tokura, et al., J. Phys.: Condens. Matter 11 (1999) 6023; Y. Tokura, et al., Physica E 6 (2000) 676; G. Burkard, et al., Phys. Rev. B 62 (2000) 2581; L. Martin-Moreno, et al., Phys. Rev. B 62 (2000) R10633; T. Matsuse, et al., Eur. Phys. J. D 16 (2001) 391; S.-R.E. Yang, J. Schliemann, A.H. MacDonald, Phys. Rev. B 66 (2002) 153302. [5] J. Hu, et al., Phys. Rev. B 54 (1996) 8616. [6] M. Rontani, et al., Solid State Commun. 112 (1999) 151; M. Rontani, et al., Solid State Commun. 119 (2001) 309. [7] B. Partoens, F.M. Peeters, Physica B 298 (2001) 282; B. Partoens, F.M. Peeters, Europhys. Lett. 56 (2001) 86. [8] M. Pi, et al., Phys. Rev. B 63 (2001) 115316. [9] M. Pi, et al., Phys. Rev. Lett. 87 (2001) 066801. [10] F. Ancilotto, et al., Phys. Rev. B 67 (2003) 205311. [11] S. Tarucha, et al., Phys. Rev. Lett. 77 (1996) 3613. [12] D.G. Austing, et al., Jpn. J. Appl. Phys. 38 (1999) 372; T.H. Oosterkamp, et al., Phys. Rev. Lett. 82 (1999) 2931; M. Ciorga, et al., Phys. Rev. Lett. 88 (2002) 256804. [13] A.H. MacDonald, S.-R.E. Yang, M.D. Johnson, Aust. J. Phys. 46 (1993) 345; S.M. Reimann, et al., Phys. Rev. Lett. 83 (1999) 3270; S.-R.E. Yang, A.H. MacDonald, Phys. Rev. B 66 (2002) 041304(R). [14] D.G. Austing, et al., unpublished.