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Single electron charging in parallel coupled quantum dots
Superlattices and Microstructures, Vol. 20, No. 3, 1996
Single electron charging in parallel coupled quantum dots A. S. Adourian, C. Livermore, R. M. Westervelt Department of Physics and Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A.
K. L. Campman, A. C. Gossard Materials Department, U. C. Santa Barbara, Santa Barbara, CA 93110, U.S.A.
(Received 20 May 1996) We report low-temperature conductance measurements in the Coulomb blockade regime on two nominally identical tunnel-coupled quantum dots in parallel defined electrostatically in the two-dimensional electron gas of a GaAs/AlGaAs heterostructure. At low interdot tunnel coupling we find that the conductance measured through one dot is sensitive to the charge state of the neighboring dot. At larger interdot coupling the conductance data reflect the role of quantum charge fluctuations between the dots. As the interdot conductance approaches 2e2 / h, the coupled dots behave as a single large dot. c 1996 Academic Press Limited
Key words:
1. Introduction During the past few years there has been considerable interest in the electronic transport properties of quantum dots in the Coulomb blockade (CB) regime [1–4]. At very low temperatures the energy associated with the addition of a single electron to the quantum dot exceeds the thermal energy, and the conductance through the dot is regulated by Coulomb charging effects. For a quantum dot formed in a semiconductor electron gas system, the occupancy and conductance can be tuned by adjusting the electrostatic potential of a surface gate which is capacitively coupled to the quantum dot. In a single quantum dot this leads to CB oscillations in the conductance as a function of the voltage on the surface gate. Recently, the transport properties of more complicated systems consisting of two or more coupled quantum dots in the CB regime have been experimentally [5–10] and theoretically [11–17] explored. Novel effects due to the interaction between the coupled quantum dots are anticipated, and are expected to be observable in the conductance spectra. Figure 1 is a schematic layout of the gates which define the device used in this study. The device consists of two nominally identical adjacent quantum dots of lithographic size 0.8 µm × 0.5 µm. The dots are defined by seven independently tunable Cr:Au Schottky gates on a GaAs/Al0.3 Ga0.7 As heterostructure containing a ˚ beneath the surface. The 2DEG low temperature sheet two-dimensional electron gas (2DEG) located 520 A carrier density and mobility are n s = 3.5 × 1011 cm−2 and µ = 5.2 × 105 cm2 Vs−1 , respectively. The gates define five separately tunable quantum point contacts (QPCs). The outer QPCs are used to measure the conductance of each dot, while the inner QPC controls the interdot tunnel-coupling between the dots. Voltages applied to the side gates can be used to change the potential of each dot. 0749–6036/96/070411 + 07 $18.00/0
c 1996 Academic Press Limited
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Superlattices and Microstructures, Vol. 20, No. 3, 1996 B Source
C
A
1
Dot
Dot
1
2
2
F
D
Drain E Fig. 1. Schematic diagram of parallel dot device. Each dot has a lithographic area 0.5 × 0.8 µm2 .
The sample is mounted on the mixing chamber of a He dilution refrigerator with a base temperature of 30 mK. The device is carefully shielded from external electromagnetic radiation to minimize photon-assisted tunneling events [18, 19]. The differential conductance of the sample is measured by applying a small a.c. bias voltage of 10 µV at low frequency ( f = 11 Hz) across the source and drain reservoirs and recording the current with a current preamplifier and a lock-in amplifier. To measure the total capacitance and single-particle level spacing of a single dot, a finite d.c. drain-source bias Vbias was applied to dot 1. For these measurements, gates C, F, and gate 2 which define dot 2 are grounded, and dot 1 is well isolated from the 2D reservoir by setting the inner QPC so that G BE = 0, where G ij is defined as the conductance of the QPC defined by gates i and j. The QPCs used to measure dot 1 are adjusted to the tunneling regime so that both G AB and G DE ∼ = 0.02e2 / h. Figure 2A plots the locations of the differential conductance peaks measured as a function of Vbias and V1 , showing the rhombus-shaped Coulomb blockade region of zero conductance [1–4, 20]. The single-dot total capacitance is determined by the vertical height of the CB region and is measured to be C6 ∼ = 275 aF, corresponding to a charging energy U = e2 /C6 ∼ = 582 µeV. Shown in Fig. 2B is the single-dot differential conductance dI /dVbias versus Vbias for V1 = −2.016 V. The additional peaks on either side of the CB region are due to tunneling through single particle levels on the dot, and correspond to single particle level separations of 1E ∼ 50 µeV. Figure 3 shows the single-dot conductance of dot 1 as a function of the gate voltage V1 in equilibrium at zero d.c. drain-source bias, with gates C, F, and 2 grounded. The periodicity of the CB peaks is determined by the capacitance between dot 1 and gate 1, which from Fig. 3 we measure to be C11 = 22 aF. Similar measurements using the other gates and dot 2 yield gate capacitances Cij between dot i and gate j of C12 = 5 aF, C21 = 5 aF, and C22 = 23 aF. If the conductance through dot 1 is monitored while both side gates 1 and 2 are swept together, we find C1,1+2 = 27 aF, and similarly for dot 2 C2,1+2 = 28 aF, as would be expected from simple addition of the individual capacitances of each dot to both gates. Fitting the peaks shown in Fig. 3 to a
Superlattices and Microstructures, Vol. 20, No. 3, 1996
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0.10 A 0.08 dI/dVbias (e2/h)
400 Vbias (µV)
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–400
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–800 –2.021
–2.019 –2.017 Gate voltage V1 (V)
–2.015
0.00 –800
–400
0 Vbias (µV)
400
800
Fig. 2. A, Gray-scale representation of conductance peaks in the plane of bias voltage Vbias and gate voltage V1 for dot 1. Dark areas correspond to high conductance values. B, Single-dot differential conductance d I /d Vbias versus Vbias for V1 = −2.016 V.
Conductance (e2/h)
0.10
0.05
0.00 –1.73
–1.72
–1.71 –1.70 Gate voltage V1 (V)
–1.69
Fig. 3. Single-dot conductance versus gate voltage V1 for dot 1 at zero drain-source bias.
thermally broadened lineshape [1–4, 20, 21], we extract the peak width 1V1 ∼ = 0.18 mV. Using the relation k B T = eC11 1V1 /2C6 , we find an electron temperature Te ∼ = 8 µeV. = 90 mK, so that k B T ∼ Measurements of the coupled quantum dot system were performed by energizing all gates, and setting dot 1 to be conducting and dot 2 nonconducting. This was accomplished by adjusting G AB , G DE ∼ = 0.02e2 / h for dot 1, and G BC , G EF = 0 for dot 2. We first investigate the regime where the two dots are well isolated by setting VBE = −503 mV, which corresponds to an interdot conductance of G BE ∼ = 0.2e2 / h when the influence of the other gates is taken into account. Shown in Fig. 4A are a set of curves of the conductance through dot 1 as a function of the voltage V1 on side gate 1 at incremented values of the voltage V2 on side gate 2. Figure 4B is a gray scale representation of the same data. The pattern of measured conductance peaks exhibits two periodicities, one in each direction of gate voltage. The smaller scale disruptions in peak positions between adjacent data traces are due to charging events in the sample which occur on a time scale shorter than the time required to complete a single data trace.
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Superlattices and Microstructures, Vol. 20, No. 3, 1996 A
Gate voltage V2 (V)
Gate voltage V2 (V)
–2.01
B
–2.01 –2.02 –2.03 –2.04
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 4. A, Conductance traces for parallel dot device at G int ≈ 0.2e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 4A.
For the case of very weak interdot conductance G BE ∼ = 0.2e2 / h appropriate for Fig. 4, capacitive interaction between the dots dominates, and we invoke a classical capacitance model to gain an understanding of the structure in the data. The equivalent circuit of the model is illustrated in Fig. 5A. As reported previously [7], minimizing the electrostatic energy of this model system, subject to the constraint that the number of electrons on each dot is independently quantized, yields a charging diagram such as the one schematically shown in Fig. 5B. Each unit cell corresponds to a different charge state of the coupled dot system, and the boundaries between adjacent cells represent energy degeneracy points where U (N1 , N2 ) = U (N1 ±1, N2 ±1). The shape of the unit cell depends on the capacitance values used in the model. The periodicity of the charging diagram in the V1 direction reflects single electron charging of dot 1. Changes in the occupancy of dot 2, reflected by the periodicity in the V2 direction, lead to discontinuous jumps in the electrostatic potential which influences dot 1 and result in the periodic abrupt shifts in the phase of the CB peak spectra seen in Figs. 4A and 4B. It should be emphasized that this capacitive model is inappropriate for the regime of finite interdot tunneling, where quantum charge fluctuations govern the behavior of the system and the number of electrons on each dot is not well defined. Recent calculations [11, 12] have predicted that as the interdot tunnel conductance approaches 2e2 / h the coupled dot system behavior should resemble that of a single large composite dot. Comparing the model charging diagram with Fig. 4B we see that the large conductance maxima in the data correspond to a change in the occupancy of the conducting dot (dot 1) of one electron. Also visible in the data are the smaller conductance peaks corresponding to boundaries where the occupancy of the nonconducting dot (dot 2) changes by one electron. These smaller conductance peaks are not expected from simple Coulomb blockade theory. One possible mechanism leading to the appearance of the peaks which reflect a change in the occupancy of dot 2 is inelastic cotunneling [12, 22, 23]. In this process an electron tunnels from the source reservoir to dot 1 while an electron simultaneously tunnels from dot 1 to dot 2, satisfying the charge degeneracy of dot 2; the electron then escapes from dot 2 to the drain reservoir by a similar process, resulting in a non-zero measured current through dot 1. The boundaries corresponding to an internal redistribution of charge are not present in the data because a conductance maximum requires a change in the total number of electrons (N1 + N2 ). Shown in Fig. 6 are conductance plots similar to Fig. 4, but with the interdot tunnel conductance increased to G BE ∼ = 0.9e2 / h, corresponding to approximately one-half mode of conductance connecting the two dots. As before, the data exhibit periodicities in each gate voltage direction. However, there are two marked
Superlattices and Microstructures, Vol. 20, No. 3, 1996 B
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C12 C22 N1 V1
C11
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(1,3)
V2
(0,1) (0,0)
(1,2) (1,1) (1,0)
(2,2) (2,1) (2,0)
V1
Fig. 5. A, Equivalent circuit used in capacitive charging model. Cij is the capacitance between gate i and dot j. Nı represents the number of electrons on dot i. B, Charging diagram for capacitive coupling indicating (N1 , N2 ) charge states. Dark boundaries correspond to changes in the charge state of dot 1. Dashed boundaries correspond to internal redistribution of charge.
differences when compared with the data of Fig. 4A, B: (1) the boundaries corresponding to a change in the charge state of the nonconducting dot (dot 2) of one electron have become more prominent, and (2) the splitting between adjacent columns of unit cells corresponding to even and odd total electron number N1 + N2 , denoted schematically by the dashed lines in Fig. 5B, has increased. In this regime of finite interdot tunnel conductance the purely capacitive charging model introduced above is not applicable, because quantum charge fluctuations between the two dots allow for non-integer values of N1 and N2 . Recent calculations [11, 12] have shown that these quantum charge fluctuations lower the ground state energy of those states with an internal polarization; for instance, with V1 = V2 , interdot tunneling reduces the electrostatic energy of the states with odd total electron number N1 + N2 . Whereas in the weak interdot tunneling regime the ‘odd’ electron is well confined to one dot or the other, in the finite interdot tunneling case this ‘odd’ electron can spread across both dots, reducing the polarization and thus the energy of the system. Qualitatively, this leads to the increased splitting between adjacent columns of unit cells seen in Fig. 6A, B [11, 12]. The increase in the magnitude of the conductance maxima corresponding to changes in the charge state of the nonconducting dot (dot 2) seen in Fig. 6 is also a result of increased quantum charge fluctuations between the dots, by which an electron on dot 1 can tunnel to dot 2 and satisfy the charge degeneracy condition on dot 2. Figure 7A, B shows conductance data for the case where there is approximately one mode coupling the two dots, G BE ∼ = 1.9e2 / h. In this strong coupling regime, the two dots behave essentially as a single large dot, with only the total electron number N1 + N2 and the capacitance of the large dot to each of the gates determining the CB characteristics. This behavior is predicted in the theory of Refs [11] and [12], which calculate the coupled dot system should behave as a single large composite dot for interdot conductance of exactly 2e2 / h. From this data we measure the capacitance of this large combined dot to each of the gates of Cdot,1 ∼ = 25 aF and Cdot,2 ∼ = 27 aF. In conclusion, we have observed the evolution of the CB spectra of a parallel coupled quantum dot system as the interdot tunnel conductance is varied. In the regime of finite interdot tunnel conductance quantum fluctuations of charge between the two dots become important, as reflected in the conductance data. For the case of two well isolated dots, a classical capacitance model reproduces the features seen in the CB data. In particular, the addition of an electron to the non-conducting dot is seen to strongly influence the conductance measured through the conducting dot. Acknowledgements—We thank J. M. Golden, B. I. Halperin, and K. A. Matveev for useful discussions, and C. H. Crouch and D. Ralph for experimental assistance. This work was supported at Harvard by NSF grant NSF-DMR-95-01438, by ONR grant N00014-95-1-0866, and in part by the MRSEC program of the NSF under award #DMR-94-00396 and at UCSB under grant AFOSR F 49620-94-1-0158.
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Superlattices and Microstructures, Vol. 20, No. 3, 1996 A
Gate voltage V2 (V)
Gate voltage V2 (V)
–2.01
B
–2.01 –2.02 –2.03 –2.04
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 6. A, Conductance traces for parallel dot device at G int ≈ 0.9e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 6A.
–2.01
–2.01 Gate voltage V2 (V)
Gate voltage V2 (V)
A
–2.02 –2.03 –2.04
B
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 7. A, Conductance traces for parallel dot device at G int ≈ 1.9e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 7A.
References [1] C. W. J. Beenakker and H. van Houten, in Solid State Physics, Edited by H. Ehrenreich and D. Turnbull, Academic Press, San Diego, (1991). [2] M. A. Kastner, Rev. Mod. Phys. 64, 849 (1992). [3] H. Grabert and M. H. Devoret, eds. Single Charge Tunneling. NATO ASI Ser. B, Physics. Vol. 294, Plenum Press, New York, (1992). [4] D. V. Averin and K. K. Likharev, in Mesoscopic Phenomena in Solids. Edited by B. L. Altshuler, P. A. Lee and R. A. Webb, North-Holland Publishers, Elsevier: Amsterdam, (1991). [5] F. R. Waugh, M. J. Berry, D. J. Mar, R. M. Westervelt, K. C. Campman and A. C. Gossard, Phys. Rev. Lett. 75, 705 (1995).
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[6] F. R. Waugh, M. J. Berry, C. H. Crouch, C. Livermore, D. J. Mar, R. M. Westervelt, K. C. Campman and A. C. Gossard, Phys. Rev. B 53, 1413 (1996). [7] F. Hofmann, T. Heinzel, D. A. Wharam, J. P. Kotthaus, G. Bohm, W. Klein, G. Trankle and G. Weimann, Phys. Rev. B 51, 13872 (1995). [8] L. W. Molenkamp, K. Flensberg and M. Kemerink, Phys. Rev. Lett. 75, 4282 (1995). [9] M. Kemerink and L. W. Molenkamp, Appl. Phys. Lett. 65, 1012 (1994). [10] R. H. Blick, R. J. Haug, J. Weis, D. Pfannkuche, K. v. Klitzing and K. Eberl, Phys. Rev. B 53, 7899 (1996). [11] J. M. Golden and B. I. Halperin, Phys. Rev. B 53, 3893 (1996). [12] K. A. Matveev, L. I. Glazman and H. U. Baranger, Phys. Rev. B 53, 1034 (1996). [13] C. A. Stafford and S. Das Sarma, Phys. Rev. Lett. 72, 3590 (1994). [14] G. Klimeck, G. Chen and S. Datta, Phys. Rev. B 50, 2316 (1994). [15] I. M. Ruzin et al. Phys. Rev. B 45, 13469 (1992). [16] C. Y. Fong et al. Phys. Rev. B 46, 9538 (1992). [17] G. W. Bryant, Phys. Rev. B 48, 8024 (1993). [18] J. M. Martinis and M. Nahum, Phys. Rev. B 48, 18316 (1993). [19] J. M. Hergenrother, J. G. Liu, M. T. Tuominen, D. C. Ralph and M. Tinkham, Phys. Rev. B 51, 9407 (1995). [20] E. B. Foxman et al. Phys. Rev. B 47, 10020 (1993). [21] C. W. J. Beenakker, Phys. Rev. B 44, 1646 (1991). [22] D. V. Averin and Yu. V. Nazarov, Phys. Rev. Lett. 65, 2446 (1990). [23] D. V. Averin and A. A. Odnitsov, Phys. Lett. A 140, 251 (1989).
Single electron charging in parallel coupled quantum dots A. S. Adourian, C. Livermore, R. M. Westervelt Department of Physics and Division of Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A.
K. L. Campman, A. C. Gossard Materials Department, U. C. Santa Barbara, Santa Barbara, CA 93110, U.S.A.
(Received 20 May 1996) We report low-temperature conductance measurements in the Coulomb blockade regime on two nominally identical tunnel-coupled quantum dots in parallel defined electrostatically in the two-dimensional electron gas of a GaAs/AlGaAs heterostructure. At low interdot tunnel coupling we find that the conductance measured through one dot is sensitive to the charge state of the neighboring dot. At larger interdot coupling the conductance data reflect the role of quantum charge fluctuations between the dots. As the interdot conductance approaches 2e2 / h, the coupled dots behave as a single large dot. c 1996 Academic Press Limited
Key words:
1. Introduction During the past few years there has been considerable interest in the electronic transport properties of quantum dots in the Coulomb blockade (CB) regime [1–4]. At very low temperatures the energy associated with the addition of a single electron to the quantum dot exceeds the thermal energy, and the conductance through the dot is regulated by Coulomb charging effects. For a quantum dot formed in a semiconductor electron gas system, the occupancy and conductance can be tuned by adjusting the electrostatic potential of a surface gate which is capacitively coupled to the quantum dot. In a single quantum dot this leads to CB oscillations in the conductance as a function of the voltage on the surface gate. Recently, the transport properties of more complicated systems consisting of two or more coupled quantum dots in the CB regime have been experimentally [5–10] and theoretically [11–17] explored. Novel effects due to the interaction between the coupled quantum dots are anticipated, and are expected to be observable in the conductance spectra. Figure 1 is a schematic layout of the gates which define the device used in this study. The device consists of two nominally identical adjacent quantum dots of lithographic size 0.8 µm × 0.5 µm. The dots are defined by seven independently tunable Cr:Au Schottky gates on a GaAs/Al0.3 Ga0.7 As heterostructure containing a ˚ beneath the surface. The 2DEG low temperature sheet two-dimensional electron gas (2DEG) located 520 A carrier density and mobility are n s = 3.5 × 1011 cm−2 and µ = 5.2 × 105 cm2 Vs−1 , respectively. The gates define five separately tunable quantum point contacts (QPCs). The outer QPCs are used to measure the conductance of each dot, while the inner QPC controls the interdot tunnel-coupling between the dots. Voltages applied to the side gates can be used to change the potential of each dot. 0749–6036/96/070411 + 07 $18.00/0
c 1996 Academic Press Limited
412
Superlattices and Microstructures, Vol. 20, No. 3, 1996 B Source
C
A
1
Dot
Dot
1
2
2
F
D
Drain E Fig. 1. Schematic diagram of parallel dot device. Each dot has a lithographic area 0.5 × 0.8 µm2 .
The sample is mounted on the mixing chamber of a He dilution refrigerator with a base temperature of 30 mK. The device is carefully shielded from external electromagnetic radiation to minimize photon-assisted tunneling events [18, 19]. The differential conductance of the sample is measured by applying a small a.c. bias voltage of 10 µV at low frequency ( f = 11 Hz) across the source and drain reservoirs and recording the current with a current preamplifier and a lock-in amplifier. To measure the total capacitance and single-particle level spacing of a single dot, a finite d.c. drain-source bias Vbias was applied to dot 1. For these measurements, gates C, F, and gate 2 which define dot 2 are grounded, and dot 1 is well isolated from the 2D reservoir by setting the inner QPC so that G BE = 0, where G ij is defined as the conductance of the QPC defined by gates i and j. The QPCs used to measure dot 1 are adjusted to the tunneling regime so that both G AB and G DE ∼ = 0.02e2 / h. Figure 2A plots the locations of the differential conductance peaks measured as a function of Vbias and V1 , showing the rhombus-shaped Coulomb blockade region of zero conductance [1–4, 20]. The single-dot total capacitance is determined by the vertical height of the CB region and is measured to be C6 ∼ = 275 aF, corresponding to a charging energy U = e2 /C6 ∼ = 582 µeV. Shown in Fig. 2B is the single-dot differential conductance dI /dVbias versus Vbias for V1 = −2.016 V. The additional peaks on either side of the CB region are due to tunneling through single particle levels on the dot, and correspond to single particle level separations of 1E ∼ 50 µeV. Figure 3 shows the single-dot conductance of dot 1 as a function of the gate voltage V1 in equilibrium at zero d.c. drain-source bias, with gates C, F, and 2 grounded. The periodicity of the CB peaks is determined by the capacitance between dot 1 and gate 1, which from Fig. 3 we measure to be C11 = 22 aF. Similar measurements using the other gates and dot 2 yield gate capacitances Cij between dot i and gate j of C12 = 5 aF, C21 = 5 aF, and C22 = 23 aF. If the conductance through dot 1 is monitored while both side gates 1 and 2 are swept together, we find C1,1+2 = 27 aF, and similarly for dot 2 C2,1+2 = 28 aF, as would be expected from simple addition of the individual capacitances of each dot to both gates. Fitting the peaks shown in Fig. 3 to a
Superlattices and Microstructures, Vol. 20, No. 3, 1996
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800
0.10 A 0.08 dI/dVbias (e2/h)
400 Vbias (µV)
B
0
–400
0.06 0.04 0.02
–800 –2.021
–2.019 –2.017 Gate voltage V1 (V)
–2.015
0.00 –800
–400
0 Vbias (µV)
400
800
Fig. 2. A, Gray-scale representation of conductance peaks in the plane of bias voltage Vbias and gate voltage V1 for dot 1. Dark areas correspond to high conductance values. B, Single-dot differential conductance d I /d Vbias versus Vbias for V1 = −2.016 V.
Conductance (e2/h)
0.10
0.05
0.00 –1.73
–1.72
–1.71 –1.70 Gate voltage V1 (V)
–1.69
Fig. 3. Single-dot conductance versus gate voltage V1 for dot 1 at zero drain-source bias.
thermally broadened lineshape [1–4, 20, 21], we extract the peak width 1V1 ∼ = 0.18 mV. Using the relation k B T = eC11 1V1 /2C6 , we find an electron temperature Te ∼ = 8 µeV. = 90 mK, so that k B T ∼ Measurements of the coupled quantum dot system were performed by energizing all gates, and setting dot 1 to be conducting and dot 2 nonconducting. This was accomplished by adjusting G AB , G DE ∼ = 0.02e2 / h for dot 1, and G BC , G EF = 0 for dot 2. We first investigate the regime where the two dots are well isolated by setting VBE = −503 mV, which corresponds to an interdot conductance of G BE ∼ = 0.2e2 / h when the influence of the other gates is taken into account. Shown in Fig. 4A are a set of curves of the conductance through dot 1 as a function of the voltage V1 on side gate 1 at incremented values of the voltage V2 on side gate 2. Figure 4B is a gray scale representation of the same data. The pattern of measured conductance peaks exhibits two periodicities, one in each direction of gate voltage. The smaller scale disruptions in peak positions between adjacent data traces are due to charging events in the sample which occur on a time scale shorter than the time required to complete a single data trace.
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Superlattices and Microstructures, Vol. 20, No. 3, 1996 A
Gate voltage V2 (V)
Gate voltage V2 (V)
–2.01
B
–2.01 –2.02 –2.03 –2.04
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 4. A, Conductance traces for parallel dot device at G int ≈ 0.2e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 4A.
For the case of very weak interdot conductance G BE ∼ = 0.2e2 / h appropriate for Fig. 4, capacitive interaction between the dots dominates, and we invoke a classical capacitance model to gain an understanding of the structure in the data. The equivalent circuit of the model is illustrated in Fig. 5A. As reported previously [7], minimizing the electrostatic energy of this model system, subject to the constraint that the number of electrons on each dot is independently quantized, yields a charging diagram such as the one schematically shown in Fig. 5B. Each unit cell corresponds to a different charge state of the coupled dot system, and the boundaries between adjacent cells represent energy degeneracy points where U (N1 , N2 ) = U (N1 ±1, N2 ±1). The shape of the unit cell depends on the capacitance values used in the model. The periodicity of the charging diagram in the V1 direction reflects single electron charging of dot 1. Changes in the occupancy of dot 2, reflected by the periodicity in the V2 direction, lead to discontinuous jumps in the electrostatic potential which influences dot 1 and result in the periodic abrupt shifts in the phase of the CB peak spectra seen in Figs. 4A and 4B. It should be emphasized that this capacitive model is inappropriate for the regime of finite interdot tunneling, where quantum charge fluctuations govern the behavior of the system and the number of electrons on each dot is not well defined. Recent calculations [11, 12] have predicted that as the interdot tunnel conductance approaches 2e2 / h the coupled dot system behavior should resemble that of a single large composite dot. Comparing the model charging diagram with Fig. 4B we see that the large conductance maxima in the data correspond to a change in the occupancy of the conducting dot (dot 1) of one electron. Also visible in the data are the smaller conductance peaks corresponding to boundaries where the occupancy of the nonconducting dot (dot 2) changes by one electron. These smaller conductance peaks are not expected from simple Coulomb blockade theory. One possible mechanism leading to the appearance of the peaks which reflect a change in the occupancy of dot 2 is inelastic cotunneling [12, 22, 23]. In this process an electron tunnels from the source reservoir to dot 1 while an electron simultaneously tunnels from dot 1 to dot 2, satisfying the charge degeneracy of dot 2; the electron then escapes from dot 2 to the drain reservoir by a similar process, resulting in a non-zero measured current through dot 1. The boundaries corresponding to an internal redistribution of charge are not present in the data because a conductance maximum requires a change in the total number of electrons (N1 + N2 ). Shown in Fig. 6 are conductance plots similar to Fig. 4, but with the interdot tunnel conductance increased to G BE ∼ = 0.9e2 / h, corresponding to approximately one-half mode of conductance connecting the two dots. As before, the data exhibit periodicities in each gate voltage direction. However, there are two marked
Superlattices and Microstructures, Vol. 20, No. 3, 1996 B
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415 V2
C12 C22 N1 V1
C11
(0,2)
N2 Cint C21
(2,3)
(1,3)
V2
(0,1) (0,0)
(1,2) (1,1) (1,0)
(2,2) (2,1) (2,0)
V1
Fig. 5. A, Equivalent circuit used in capacitive charging model. Cij is the capacitance between gate i and dot j. Nı represents the number of electrons on dot i. B, Charging diagram for capacitive coupling indicating (N1 , N2 ) charge states. Dark boundaries correspond to changes in the charge state of dot 1. Dashed boundaries correspond to internal redistribution of charge.
differences when compared with the data of Fig. 4A, B: (1) the boundaries corresponding to a change in the charge state of the nonconducting dot (dot 2) of one electron have become more prominent, and (2) the splitting between adjacent columns of unit cells corresponding to even and odd total electron number N1 + N2 , denoted schematically by the dashed lines in Fig. 5B, has increased. In this regime of finite interdot tunnel conductance the purely capacitive charging model introduced above is not applicable, because quantum charge fluctuations between the two dots allow for non-integer values of N1 and N2 . Recent calculations [11, 12] have shown that these quantum charge fluctuations lower the ground state energy of those states with an internal polarization; for instance, with V1 = V2 , interdot tunneling reduces the electrostatic energy of the states with odd total electron number N1 + N2 . Whereas in the weak interdot tunneling regime the ‘odd’ electron is well confined to one dot or the other, in the finite interdot tunneling case this ‘odd’ electron can spread across both dots, reducing the polarization and thus the energy of the system. Qualitatively, this leads to the increased splitting between adjacent columns of unit cells seen in Fig. 6A, B [11, 12]. The increase in the magnitude of the conductance maxima corresponding to changes in the charge state of the nonconducting dot (dot 2) seen in Fig. 6 is also a result of increased quantum charge fluctuations between the dots, by which an electron on dot 1 can tunnel to dot 2 and satisfy the charge degeneracy condition on dot 2. Figure 7A, B shows conductance data for the case where there is approximately one mode coupling the two dots, G BE ∼ = 1.9e2 / h. In this strong coupling regime, the two dots behave essentially as a single large dot, with only the total electron number N1 + N2 and the capacitance of the large dot to each of the gates determining the CB characteristics. This behavior is predicted in the theory of Refs [11] and [12], which calculate the coupled dot system should behave as a single large composite dot for interdot conductance of exactly 2e2 / h. From this data we measure the capacitance of this large combined dot to each of the gates of Cdot,1 ∼ = 25 aF and Cdot,2 ∼ = 27 aF. In conclusion, we have observed the evolution of the CB spectra of a parallel coupled quantum dot system as the interdot tunnel conductance is varied. In the regime of finite interdot tunnel conductance quantum fluctuations of charge between the two dots become important, as reflected in the conductance data. For the case of two well isolated dots, a classical capacitance model reproduces the features seen in the CB data. In particular, the addition of an electron to the non-conducting dot is seen to strongly influence the conductance measured through the conducting dot. Acknowledgements—We thank J. M. Golden, B. I. Halperin, and K. A. Matveev for useful discussions, and C. H. Crouch and D. Ralph for experimental assistance. This work was supported at Harvard by NSF grant NSF-DMR-95-01438, by ONR grant N00014-95-1-0866, and in part by the MRSEC program of the NSF under award #DMR-94-00396 and at UCSB under grant AFOSR F 49620-94-1-0158.
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Gate voltage V2 (V)
Gate voltage V2 (V)
–2.01
B
–2.01 –2.02 –2.03 –2.04
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 6. A, Conductance traces for parallel dot device at G int ≈ 0.9e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 6A.
–2.01
–2.01 Gate voltage V2 (V)
Gate voltage V2 (V)
A
–2.02 –2.03 –2.04
B
–2.02 –2.03 –2.04 –2.05
–2.05 –2.03 –2.02 Gate voltage V1 (V)
–2.03 –2.02 Gate voltage V1 (V)
Fig. 7. A, Conductance traces for parallel dot device at G int ≈ 1.9e2 / h. The conductance scale is arbitrary. B, Gray-scale representation of conductance data in Fig. 7A.
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