Spin Bottleneck in Resonant Tunneling through Double Quantum Dots
with Different Zeeman Splittings
S. M. Huang,1,2Y. Tokura,3,4H. Akimoto,1K. Kono,1J. J. Lin,2S. Tarucha,4,5and K. Ono1,4,*
1Low Temperature Physics Laboratory, RIKEN, Wako-shi, Saitama 351-0198, Japan 2Institute of Physics, National Chiao Tung University, Hsinchu 30010, Taiwan 3
NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan
4Quantum Spin Information Project, ICORP-JST, Atsugi-shi, Kanagawa 243-0198, Japan 5Department of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
(Received 16 April 2009; published 1 April 2010)
We investigated the electron transport property of the InGaAs/GaAs double quantum dots, the electron g factors of which are different from each other. We found that in a magnetic field, the resonant tunneling is suppressed even if one of the Zeeman sublevels is aligned. This is because the other misaligned Zeeman sublevels limit the total current. A finite broadening of the misaligned sublevel partially relieves this bottleneck effect, and the maximum current is reached when interdot detuning is half the Zeeman energy difference.
DOI:10.1103/PhysRevLett.104.136801 PACS numbers: 73.63.Kv, 72.25.Mk, 73.23.Hk
Electron g factors in III-V semiconductor heterostruc-tures can be tuned by changing the alloy ratio and thickness of each quantum well (QW). Novel spin-related physics in such a g factor-tuned system has attracted considerable interest in the past decade [1]. Electrical tunings of electron g factors has been demonstrated in single and coupled double QWs. The g factor tunings are accomplished by changing the position of the electron wave functions that are spatially delocalized over the regions with different g factors [2]. The g factor tuning used above is a powerful tool for the coherent manipulation of the electron spins; however, these works were done for an ensemble of spins in QWs. To realize the spin-based quantum information devices, which is one of the most ambitious applications of semiconductor spintronics, the coherent manipulation in a single spin level is necessary [3]. The quantum dot (QD) is known as a system where a number of electron as well as their spin state can be well defined and easily controlled [4]. An application of such g factor tuning to QD systems offers a novel candidate for future electric devices for coherent spin manipulation. Tuning the Zeeman splitting of each spin in a QD array has been proposed for individual addressing of spin qubits and fast gate operation between two qubits [3,5]. Selective addressing has been demon-strated so far in a double quantum dot (DQD) made from spatially homogeneous g factors with an additional micro-ferromagnet nearby. The magnet creates a10 mT field difference of external magnetic field B in each dot [6]. QDs with different g factors can offer a much larger Zeeman energy difference in the same external field with smaller spatial separation.
In this Letter, we report a novel behavior of a single spin in a g factor-tuned DQD, where Zeeman splittings differ greatly from each other in a magnetic field. We investigate this system in a simple, well-defined regime where the total electron number N in the DQD is 0 or 1. Accompanying
the theoretical calculation, we reveal that the resonant tunneling via two Zeeman-split levels is suppressed even if one pair of Zeeman-split levels is aligned. This novel spin bottleneck effect is partially relieved by a finite broad-ening of QD states owing to interdot and/or dot-electrode tunnel couplings. As a result of competition between the bottleneck and the level broadening effect, the current is maximum in the configuration where interdot level detun-ing is set to half the Zeeman energy difference.
Vertical DQDs with different g factors are formed in a submicron-scale pillar of a triple barrier structure with a surroundingTi=Au Schottky gate, as schematically shown in the left inset of Fig.1. The structure consists of a seven-layer structure from top to bottom (or from left to right in the right inset of Fig. 1), a gradiently n-doped Al0:05Ga0:95As source electrode/ Al0:30Ga0:70As (7 nm)/
In0:04Ga0:96As (7.5 nm)/ Al0:30Ga0:70As (6.5 nm)/ GaAs
(10 nm)/ Al0:30Ga0:70As (7 nm)/ gradiently n-doped Al0:05Ga0:95As drain electrode. The fabrication procedure
is the same with the previous work [7]. Measurements were performed in a dilution refrigerator at an effective electron temperature of0:1 K and in magnetic fields of up to 15 T applied perpendicular to the wells [8].
Figure 1 shows the differential conductance, dISD=dVSD, plotted as a function of source-drain voltage VSDand gate voltage VGin zero magnetic field [9]. Current steps are recognized as dark blue lines. In the positive VSD region, near the current threshold, several current peaks (not steps) appear, as marked by arrows. These peaks are due to the resonant tunneling (RT) through the ground state of the left dot and an excited state of the right dot, as shown in the right inset. These current peak lines run nearly parallel to the VG axis because the side gate capacitively
couples to the two dots almost equally, and the alignment of the two-dot levels is nearly maintained against VG.
These behaviors have also been observed in vertical PRL 104, 136801 (2010) P H Y S I C A L R E V I E W L E T T E R S 2 APRIL 2010week ending
DQDs [10]. In the first Coulomb staircase defined as the N¼ 0 threshold line and the neighboring parallel line, the current is carried by three charge configurationsðN1;N2Þ ¼ ð0;0Þ, (1,0), and (0,1), where Ni(i¼ 1, 2) is the number of
electron in each dot. In this configuration, it is simple enough to avoid electron-electron interactions in each dot. Application of the perpendicular magnetic field changes the energies of excited orbital states of the dots and shifts VSDat the RT peak lines [10]. Hereafter, we concentrate on the current peak line found around VSD 35 mV and carry out detailed measurements under various magnetic fields. Figures 2(a)–2(d) show the dISD=dVSD plots for several magnetic fields. The current peak line, which is recognized as adjacent blue and red lines, have a clear kink structure. In vertical DQDs with the same g factor, current peak lines are always straight in any magnetic field, and the kink structure has never been observed. The kink is character-ized by two values, 1and 2, as marked in Fig.2(d). Both 1and 2linearly increase with magnetic field, as shown in Figs.2(e)and2(f ). With low magnetic fields, the peak line becomes straight asymptotically. Similar kink structures are found in all other current peak lines in various positive VSD’s at high magnetic fields, as long as the width of the peak is narrow enough to resolve kinks.
In DQDs with the same g factors, both Zeeman suble-vels are aligned at the same time at a certain VSD, regard-less of the magnetic field [10]. In DQDs with different g factors, however, the alignment of each Zeeman sublevel is achieved at its own VSD, as schematically shown in Fig.3, in accordance with the conditions of the measurements. We assign four particular conditions labeled A to D in the magnetic field. Under condition A, the aligned levels for up spin for both dots is in the transport window, and the RT current is carried out by up-spin electrons. By increasing VG, i.e., by lowering the energy level of both dots, the
down-spin Zeeman sublevel in the left dot comes within
the transport window under condition B. Although even the up-spin states are aligned and the RT channel exists, the RT is Coulomb blocked once the down-spin state in the left dot is occupied. We name this suppression process the spin bottleneck. Similarly, under condition C, the occupation of the up-spin in the left dot prohibits the subsequent tunnel-ing even though the aligned down-spin channel exists. Thus, under both conditions B and C, the bottleneck chan-nel is occupied and there is no steady-state resonant cur-rent. It should be stressed that this bottleneck effect is similar to the Pauli spin blockade [11] since the stochastic single electron occupation of the bottleneck channel ulti-mately leads to a blockade in both cases.
The bottleneck can be lifted by level broadening, which is induced by finite tunnel couplings among the dots and the electrodes. The broadening couples the misaligned Zeeman sublevels and provides an escape path for the electron in the bottleneck channel and relieves the bottle-neck effect. The electron transport is carried out within the competition between the bottleneck and the escape effects.
FIG. 2 (color). (a)–(d) Differential conductance, dISD=dVSD, as a function of VSD and VG under several different magnetic
fields, showing increasing kink structure characterized by 1and 2. Magnetic field dependences of (e) 1and (f ) 2.
-100 -50 0 50 100 -1.2 -0.6 0.0 VG (V ) V SD (mV) VSD VG 50 –50 d ISD /d VSD (nS) N = 0
FIG. 1 (color). dISD=dVSDplot at 0 T. The arrows mark current peaks due to RT where the ground state of left dots is aligned to one of the excited states of the right dot. Left inset: schematic of a gated vertical DQD structure with different g factors for each dot. We apply VSDto the bottom electrode and measure the drain current from the top electrode. Right inset: potential energy landscape for the RT condition.
(gL-gR) BB A B D C gL BB VG VSD C B A D
FIG. 3 (color). Schematic diagram of RT peak line for Zeeman mismatched system, and characteristic potential landscapes A D.
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While smaller detuning in the RT channel can carry large current, smaller detuning of the bottleneck channel is required to avoid the bottleneck. As a result of the com-promise under the intermediate detuning condition D, not under B or C, the maximum current is expected.
This scenario is supported by the theoretical analysis. We evaluated the RT current through two dots with differ-ent Zeeman splittings by the Bloch equation method [12,13]. The QD states are described by five bases, j0; 0i, j "; 0i, j #; 0i, j0; "i, and j0; #i. Each dot has its own g factor, gL, and gR, and their difference is ¼ jgL gRjBB (hereafter, we assume gL< gR<0). In the calculation, we neglect the cotunneling and tunneling processes with phonon absorption or emission and assume zero tempera-ture. We fix VSDto be large enough so that the right Fermi level is far below the right-dot level R. Figure4(a)shows the differential conductance plotted as a function of L Rand L. In the figure, we set the origin of L Rat the position where the lower Zeeman (up-spin) sublevels are aligned and Lis measured from the Fermi level of the left electrode. There are two conditions labeled case I and case II. In case I, only the up-spin Zeeman sublevel of the left dot is within the transport window (scheme A in Fig.3). In case II, both the up-spin and down-spin levels of the left dot are within the transport window (schemes B D). In both cases, we obtain a current with a single peak of Lorentzian shape. In case I, the current has a peak at zero interdot detuning. In case II, the maximum current is shifted in the negative detuning direction by the amount of=2. These peak positions in both cases I and II are found to be independent of parameters such as the coupling of the left (right) dot to its neighboring electrode, LðRÞ, and interdot coupling . Note that L R and L roughly correspond to VSDand VG in the measurement and 1 and 2correspond tojgLjBB and=2, respectively. Thus, the clear kink structure in the current peak line in Fig.4(a)is similar to those in Figs.2(a)–2(d). Peak currents are plotted in Fig.4(b)as a function of. The peak current in case II vanishes for small . In contrast, the two peak currents take similar values for large , where the dot-electrode couplings limit the current. In Figs.2(a)–2(d), the current
peak heights are almost the same (10 pA), both below and above the kink structure. This corresponds to the large . Indeed, our previous studies on vertical DQDs with similar barrier thicknesses indicate 0:1 meV and LðRÞ& 0:01 meV [7]. In this sample, Lis much smaller than R [9]. In Figs. 2(a)–2(d), transition between the regions corresponds to case I, and II is not as abrupt as in Fig.4(a) due to the finite temperature. It should be noted that, although this calculation reproduces the shift of the peak and the peak height well, the calculated peak width is always larger than the peak shift =2, whereas in the measurement,=2 seems to be twice as large as the peak width at 13 T. This discrepancy might be solved if we include the cotunneling effect.
In order to convert from 1 and 2 to jgLjBB and
ðjgL gRjÞBB, the voltage drop ratio of three barriers,
i(i¼ 1, 2, 3), is required. Figure5(a)shows dISD=dVSD
at 12 T, and we mark L1-L4 for the current threshold lines from the N ¼ 1 Coulomb diamond. At the threshold L1, the Fermi energy of the left electrode is aligned with the energy level of the left dot. The current threshold several orders of magnitude smaller, marked by L2, indicates the onset of a cotunneling process where the right-dot energy level is aligned with the Fermi energy of the left electrode. Similarly, the Fermi level of the right electrode is aligned with the right (left) dot at the threshold line marked as line L3 (L4). The slopes of these threshold lines L1-L4, that is, 32, 54, 46, and 62, respectively, give i(i¼ 1, 2, 3) of
0.27, 0.19, and 0.54, respectively, at VSD 0. Figure5(b) shows the effect of microwave irradiation on the resonant current peak. With increasing microwave power, satellite current peaks appear on the left (right) side of the main peak, owing to interdot tunneling upon the absorption (emission) of microwaves. Such microwave-assisted tun-neling in a DQD has been observed in lateral and vertical dots [14,15]. At the satellite peak position, interdot detun-ing is equal to the microwave photon energy. Thus, the
FIG. 4 (color). (a) Calculated differential conductance plotted as a function of L Rand L(normalized by and jgLjBB),
where2:5 ¼ 10 ¼ 1000L¼ 100R¼ . Transport
condi-tion is categorized as two cases, I and II. We neglect a finite transition region between I and II due to small Lhere. (b) Peak
current in cases I and II as a function of normalized interdot coupling=. Other parameters are the same as in (a).
(b) 40 42 44 46 48 20 30 40 50 60 70 ISD (pA) VSD (mV) 39.3GHz no MW -50dBm -15dBm -20dBm -35dBm -45dBm -15 -10 -5 0 5 10 15 -0.75 -0.50 -0.25 0.00 VG (V) VSD (mV) 100 0.01 d ISD /d VSD (nS) (a) L1 L2 L4 L3
FIG. 5 (color). (a) Logarithmic color scale dISD=dVSDplot at 12 T. Four current threshold lines of the N¼ 1 Coulomb diamond are marked as L1-L4. The slopes of these lines gives the voltage drop ratio of the triple barrier, i(i¼ 1, 2, 3), for the
low-VSDregion as 0.27, 0.19, and 0.54. (b) Resonant current peak under 39.3 GHz microwave. Interdot microwave-assisted tunnel-ing is clearly apparent. Distance between the main and satellite peaks was used to estimate the interdot voltage drop ratio 2in high-VSDregions as 0.193. The microwave powers are labeled for each trace. Each trace is shifted by a constant value.
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distance between the main and the satellite peaks [marked as two dotted lines in Fig.5(b)] gives 2of 0.193, which is close to the value for VSD 0. Similar values are obtained for different current peaks at different VSD’s from 0 up to 100 mV, indicating that 2 is independent of VSD. On the other hand, voltage drop ratios for outer barriers i(i¼ 1,
3) are found to depend on VSD. Indeed, in Fig.1, the current threshold line from N¼ 0 at positive VSD is not straight but bends upward. This indicates decreasing 1 with in-creasing VSD. As the slope of the current threshold lines is 12, as determined from Figs. 2(a)–2(d), 1 at VSD 35 mV is smaller than 1 at VSD 0 by 12=32. Thus, i
(i¼ 1, 2, 3) is 0.10, 0.19, and 0.71, respectively, at VSD 35 mV [9]. Using these i’s as well as the slope at VSD
35 mV, we obtain jgLj of 0.89 for our In0:04Ga0:96As dot,
andjgL gRj is 0.56. Thus, jgRj for our GaAs dot is 0.33 which is consistent with that obtained in the previous work on a vertical single dot with the same 10-nm-thick GaAs well [15]. The value of 0.89 for InGaAs seems to be large compared with the value obtained in the study of bulk InxGa1xAs [16], probably owing to the increasing density
of In at the center of the well rather than at theInGaAs= AlGaAs interfaces, as is seen in the self-assembled InGaAs dot systems [17]. A stress in the thin InGaAs layer can be another possible reason for large g factor.
The width of the current peak in both regions A and D is 0:5 mV in VSD, and corresponds to0:1 meV using 2.
This width gives an upper bound for possible contributions from phonon-absorbing/emitting tunneling, and is smaller than the Zeeman energy difference in high magnetic field (*10 T). Thus, phonon absorption/emission for a mis-aligned spin sublevel in region B (C) in Fig.3is negligible. This point is also confirmed by a more detailed calculation, as in Fig.3, where the phonon effects are included [12]. As depicted in the right inset of Fig. 1, there are some un-occupied levels located a few meV’s below the aligned level in the right-dot. Thus, in addition to the RT through the two aligned levels, there is always a tunneling process with meV-phonon emission down to these low-lying levels. Such meV-phonon-emitting tunneling gives a background current of the order of 10 pA, which defines the cur-rent threshold from the N¼ 0 region, as seen in Figs.1and 2(a)–2(d). RT current peak lines are also seen for negative VSD. However, most of the peak widths are large, and no clear kink structure is observed. For negative VSD, the direction of the =2 shift would be opposite, and 1 of the kink should be smaller as it is governed by the g factor of GaAs. This may be one of the reasons why we were not able to detect clear kink structures. In Fig. 2(d), there seems to be an additional line parallel to the current threshold line from the N¼ 0 region. We found that these lines, which appeared irregularly with some magnetic fields and VSD’s, may be due to a fluctuation of the density of states in the electrode [18].
In conclusion, we measured the single electron transport through DQDs with different g factors. We found that the
resonant tunneling is suppressed even when one of the Zeeman sublevels is aligned within the transport window. Finite level broadening of QD states partially relieves this bottleneck effect and gives a current peak when level detuning is half the Zeeman energy difference.
The authors thank H. Kosaka, T. Nakaoka for fruitful discussion and S. Schneider for experimental assistance. This work was supported by the CREST-JST and RIKEN-NCTU Joint Graduate School Program.
[1] Semiconductor Spintronics and Quantum Computation edited by D. D. Awschalom, N. Samarth, and D. Loss, (Springer-Verlag, Berlin, Germany, 2002).
[2] H. W. Jiang and E. Yablonovitch, Phys. Rev. B 64, 041307 (2001); G. Salis et al., Nature (London) 414, 619 (2001); M. Poggio et al., Phys. Rev. B 70, 121305(R) (2004). [3] D. Loss and D. P. DiVincenzo, Phys. Rev. A 57, 120
(1998); D. P. DiVincenzo and D. Loss, J. Magn. Magn. Mater. 200, 202 (1999).
[4] R. Hanson et al., Rev. Mod. Phys. 79, 1217 (2007). [5] W. A. Coish and D. Loss, Phys. Rev. B 75, 161302(R)
(2007).
[6] E. A. Laird et al., Phys. Rev. Lett. 99, 246601 (2007); M. Pioro-Ladriere et al., Nature Phys. 4, 776 (2008). [7] D. G. Austing et al., Physica B (Amsterdam) 249–251,
206 (1998).
[8] We observe significant decrease of ISD at the field larger
than 5T applied parallel to the wells, due to a field induced depletion at the outer tunneling barriers. F. Ancilotto et al., Phys. Rev. B 67, 205311 (2003).
[9] The electron density in the left electrode decreases due to the microfabrication, causing an offset of interdot detun-ing [7,10,11]. Decreased electron density leads to a thicker depletion layer in the gradiently n-doped electrode, thus increasing the effective barrier thickness. Weaker electron screening decreases the electric field inside the effective barrier. This asymmetry is further enhanced with large positive VSD, as depicted in Fig.1, right inset.
[10] K. Ono et al., Physica B (Amsterdam) 314, 450 (2002); S. Amaha et al., Phys. Status Solidi C 5, 174 (2008); C. Payette et al., Physica E (Amsterdam) 40, 1807 (2008); C. Payette et al., Phys. Rev. Lett. 102, 026808 (2009). [11] K. Ono et al., Science 297, 1313 (2002); K. Ono and S.
Tarucha, Phys. Rev. Lett. 92, 256803 (2004); J. Baugh et al., Phys. Rev. Lett. 99, 096804 (2007).
[12] Y. Tokura et al., Phys. Status Solidi B 246, 740 (2009). [13] S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932
(1996); T. H. Stoof and Yu. V. Nazarov, ibid. 53, 1050 (1996).
[14] T. H. Oosterkamp et al., Nature (London) 395, 873 (1998).
[15] T. Kodera et al., Physica E (Amsterdam) 22, 518 (2004). [16] C. Weisbuch and C. Hermann, Phys. Rev. B 15, 816
(1977).
[17] N. Liu et al., Phys. Rev. Lett. 84, 334 (2000). [18] L. P. Kouwenhoven et al., Science 278, 1788 (1997).
PRL 104, 136801 (2010) P H Y S I C A L R E V I E W L E T T E R S 2 APRIL 2010week ending
