Cooperative Lifting of Spin Blockade in a Three-Terminal Triple Quantum Dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Cooperative Lifting of Spin Blockade in a Three-Terminal Triple Quantum Dot
Takashi Kobayashi, Takeshi Ota, Satoshi Sasaki, and Koji Muraki
NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan (Dated: February 8, 2018)We report measurements of multi-path transport through a triple quantum dot (TQD) in thefew-electron regime using a GaAs three-terminal device with a separate lead attached to each dot.When two paths reside inside the transport window and are simultaneously spin-blockaded, the leakcurrents through both paths are significantly enhanced. We suggest that the transport processesin the two paths cooperate to lift the spin blockade. Fine structures in transport spectra indicatethat different kinds of cooperative mechanisms are involved, depending on the details of the three-electron spin states governed by the size of exchange splitting relative to nuclear spin fluctuations.Our results indicate that a variety of correlation phenomena can be explored in three-terminalTQDs.
PACS numbers: 73.63.Kv, 73.21.La, 73.23.hk
Electron spin in a semiconductor quantum dot (QD) isconsidered as a candidate for a solid-state qubit suitablefor scalable quantum information processing [1]. Pre-vious studies on single and double QDs (DQDs) havedemonstrated coherent manipulation of electron spins [2–9]. In these studies, the Pauli spin blockade (SB) [10],which forbids two electrons with parallel spins to occupythe same QD, was used to readout the spin state viaspin-charge conversion. The successful demonstration ofspin control and readout in single and double QDs hasraised interest in exploring larger systems involving moreQDs. Triple quantum dot (TQD) systems, which repre-sent the first test for scaling up, are attracting particularinterest, triggered by the proposal of a novel spin qubitcontrollable by purely electrostatic means [11–13], andare being intensively studied experimentally [14–26] andtheoretically [27–35].In a TQD, the charge degeneracy condition that al-lows for sequential tunneling through all three QDs isseverely limited, which makes it difficult to explore theTQD’s complex electronic structure via transport mea-surements [19, 20, 24–26]. A three-terminal TQD, witha separate lead attached to each QD, not only eliminatesthis difficulty but also opens the way for studying the richphysics engendered by the presence of multiple currentpaths involving three QDs. Interesting phenomena suchas separation of spin-entangled electron pairs [27] andvarious interference effects [28–31] are predicted. How-ever, previous experiments on three-terminal TQDs havebeen performed only in the many-electron regime [14–17]. Thus, many of the electronic structures and asso-ciated spin dynamics remain unexplored, including cou-pling with nuclear spins (NSs), which could be investi-gated in the few-electron regime by using an SB [36, 37].In this Letter, we report transport measurementsthrough a three-terminal GaAs lateral TQD in the few-electron regime. We study two-path transport, whereelectrons enter the center QD and leave from either theleft or right one. Thus, the TQD system can be viewed as a splitter comprised of two sets of DQDs sharing thecenter QD. When only one of the two paths is allowed toenter the transport window, finite-bias transport througheach path exhibits a bias triangle with an SB, character-istic of a DQD [38]. In contrast, when both paths areinside the transport window and simultaneously spin-blockaded, we find a correlative enhancement of trans-port for both DQDs. We explain this correlation by amechanism in which the two DQDs cooperate to lift theSB. Detailed transport spectra show that different kindsof cooperative mechanisms operate, reflecting the size ofthe exchange splitting with respect to NS fluctuations,which modifies details of the spin configuration of thethree-electron states. Our results indicate that a vari-ety of correlation phenomena can be explored in three-terminal TQDs.The TQD used in this study was defined with Ti/Augates in a GaAs/Al . Ga . As heterostructure contain-ing a two-dimensional electron gas (density 2 × m − )95 nm below the wafer surface [Fig. 1(a)]. The three QDsare serially arranged with finite tunnel coupling only be-tween adjacent pairs, t ℓc ( t cr ), for the left and center(center and right) QDs. Each QD has a separate electri-cal lead. The electrochemical potentials of the left and V C =-1.69 V ( , , )( , , )( , , ) ( , , )( , , ) (b) V L ( V ) - . - . V R (V) -0.72 -0.68 V R (V) -0.72 -0.68300 pA 0.3 pA (Log) I C
500 nm (a) V R V C V L I C I R (c) (2,0,1) (1,0,2) FIG. 1: (color online) (a) Scanning electron micrograph ofthe TQD sample. (b) I C as a function of V R and V L in thefew-electron regime at V C = − .
69 V. (c) Charge detectionsignal measured simultaneously with (b). E i genene r g y ( µ e V ) - (c) ε >0 - ε ε ε - ε - (d) - (e) ε <0 Overlap ε lr ( µ eV)
20 pA0.2 pA (Log) Le ft D Q D R i gh t D Q D - . - . V L ( V ) -0.70 -0.68 V C =-1.734 V (a) - . - . -0.68 -0.66 V L ( V ) V R (V) ∆ V R,2 ∆ V R,1 ∆ V ∆ V V C =-1.756 V (b) I C I C ε =0 ε l ε l ε r ε r ε c ε l ε r ε c ε c FIG. 2: (color online) (a), (b) I C vs V R and V L around thebias triangles of left and right DQDs, taken at V C = − . − .
756 V, respectively. (c)-(e) Energy level diagrams of ε ℓ , ε c , and ε r as a function of ε ℓr for (c) ε >
0, (d) ε = 0,and (e) ε < right leads are fixed at − µ eV, while that of the centerlead is kept grounded. We measure the influx currents( I R and I C ) through the right and center leads as a func-tion of gate biases V L , V C , and V R . Current conservationallows us to deduce the influx current through the leftlead as I L = − I R − I C . The occupancies ( k, m, n ) of theleft, center, and right QDs are determined using a side-coupled quantum point contact charge sensor. Through-out this Letter, k and n are the actual electron numbers,while m represents the effective number, which is smallerthan the actual number by two. All measurements wereperformed with the sample mounted in a dilution refrig-erator with a base temperature of 20 mK.Figure 1(b) shows I C measured as a function of V L and V R at V C = − .
69 V. The simultaneous chargedetection reveals charge addition lines with three differ-ent slopes [Fig. 1(c)], indicating the formation of a TQD.DQD-like bias triangles [38] are clearly observed in I C around the charge degeneracy points between (2 , ,
1) and(1 , ,
1) and between (1 , ,
1) and (1 , , , , → (2 , , , , → (1 , , I C near these bias triangles, taken at two slightly differ-ent V C values. For V C = − .
734 V [Fig. 2(a)], the two (d)
Spinrelaxation (e) (f) (g) (h)
20 pA (Log scale) V C =-1.756 V - . - . V L ( V ) -0.68 -0.67 V R (V) -0.68 -0.67 V R (V) -0.68 -0.67 V R (V) - I L I C - I R (a) (c)(b) FIG. 3: (color online) (a) I C , (b) I R , and (c) I L vs V R and V L at V C = − .
756 V. (d)-(h) Schematic illustration of processesinvolved in the cooperative lifting of the SB. bias triangles are separated. In the bottom part of eachtriangle, current is strongly suppressed by the DQD-likeSB [36]. A strikingly different behavior emerges when V C is slightly decreased ( V C = − .
756 V) in such a way thatthe two bias triangles overlap each other [Fig. 2(b)]. In-terestingly, I C in the overlapped region is larger than thesimple sum of the currents in the non-overlapped regions.The basic behavior of the bias triangles can be under-stood in terms of the relative alignment of the energylevels (2 , , , , , , ε ℓ , ε c and ε r , respectively. The important parametersare the detuning ε ℓr ≡ ε ℓ − ε r between ε ℓ and ε r andthe relative alignment of their average with respect to ε c , given by ε ≡ ( ε ℓ + ε r ) − ε c . Experimentally, ε and ε ℓr are independently tunable by gate sweeps alongthe ∆ V and ∆ V axes defined in Fig. 2(b). The energydiagrams in Figs. 2(c)-(e) plot ε ℓ , ε c , and ε r as a func-tion of ε ℓr for (c) positive, (d) zero, and (e) negative ε .Transport through the left (right) DQD is governed bythe (2 , , , ,
1) [(1 , , , , ε ℓr = − (+)2 ε , which we hereafter referto as the left (right) DQD resonance. Energy conser-vation restricts transport through the left (right) DQDto the region ε ℓr ≤ − ε ( ε ℓr ≥ ε ), where ε ℓ ( r ) ≤ ε c is met. For ε >
0, the transport regions of the twoDQDs are separated by a region where both DQDs arein the Coulomb blockade [Fig. 2(c)]. This corresponds tothe situation at V C = − .
734 V, where the bias trianglesare separated [Fig. 2(a)]. In contrast, for ε <
0, thetransport regions of the two DQDs overlap [Fig. 2(e)],which leads to the overlapping bias triangles observed at V C = − .
756 V [Fig. 2(b)].The enhanced leak transport in the overlapped re-gion is demonstrated more vividly by plotting I R and I L recorded simultaneously with I C [Figs. 3(a)-(c)]. Both I R and I L are clearly enhanced in the same gate-voltage re-gion corresponding to the overlap. This clearly indicatesthat, although I R flows only through the right DQD, it isinfluenced by current flowing through the left DQD andvice versa.Before discussing the mechanism for the current en-hancement, we introduce notation for specifying TQDstates. We use σ ℓ , σ c , and σ r (= ↑ , ↓ ) to specify thespin of the singly occupied state in the left, center,and right QDs, respectively. For a doubly occupiedstate, we only consider a singlet 2 S , as the triplet liesmuch higher in energy. Thus, the states for the (1 , , , ,
1) charge configurations can be denoted by | σ ℓ , σ c , σ r i and | S, , σ r i , respectively. The SB in thenon-overlapped region can be understood in the sameway as that in a single DQD [10]. For example, trans-port via the (1 , , → (2 , ,
1) transition is blocked if theinitial state | σ ℓ , σ c , σ r i has no overlap with | (1 , S, σ r i ≡ √ ( |↑ , ↓ , σ r i − |↓ , ↑ , σ r i ).We propose a mechanism for the observed current en-hancement, which we refer to as cooperative lifting ofthe SB. As an example, we consider the situation whereboth DQDs are spin-blockaded by the occupation of |↑ , ↑ , ↑i [Fig. 3(d)]. Suppose that the SB is lifted by aspin flip in the right QD, resulting in |↑ , ↑ , ↓i [Fig. 3(e)].Since |↑ , ↑ , ↓i is orthogonal to | (1 , S, σ r i but not to | σ ℓ , (1 , S i , a sequential tunneling through the rightDQD |↑ , ↑ , ↓i → |↑ , , S i → |↑ , , σ r i → |↑ , σ c , σ r i fol-lows. The spins of the electrons tunneling off from |↑ , , S i [Fig. 3(f)] or on to |↑ , , σ r i are random. Thus,the resultant state |↑ , σ c , σ r i can take the four spin con-figurations shown in Figs. 3(d), (e), (g), and (h). It is seenthat the SB is restored in (d), whereas the other threeconfigurations can lead to successive sequential tunnelingthrough the (e) right, (g) left, and (h) either DQD. It isnoteworthy that the TQD state, which was orthogonal to | (1 , S, σ r i after the initial spin flip in the right QD [(e)],now has a finite overlap with | (1 , S, σ r i in (g) and (h)as a result of the SB’s lifting and resultant transport inthe right DQD. Such a sequence can occur independentlyof the initial SB state or in which QD the initial spin flipoccurs. Thus, even if the SB in the right DQD is restoredas in (g), the non-blockaded spin configuration in the leftDQD leads to reloading into a (1 , ,
1) state in which theSB in the right DQD is lifted. As a consequence of suchcooperative effects, the lifting of the SB in either DQDinduces a larger number of electrons to flow through bothDQDs than in the case of independent DQDs.Deeper insights into the cooperative effects are pro-vided by more detailed measurements around the SB-SB overlap [Fig. 4(a)-(c)]. As shown schematically inFig. 4(d), the axes of Figs. 4(a)-(c) are taken nearly par-allel to the ∆ V and ∆ V axes, with their scales projectedonto the V R axis. The dashed lines labeled “L-res” and“R-res” mark the left- and right-DQD resonances, respec-tively. The data reveal that the enhanced leak transportis comprised of several fine structures. This is also seen inFigs. 4(e)-(h), where I R and I L along the five horizontallines in Fig. 4(b) and (c) are displayed. First, we discuss (g) (h)(f)(e) . . ∆ V R , ( m V ) R -r e s R - bp L -r e s L - bp - I R , - I L ( p A ) (a) (b) (c) I C - I R - I L ∆ V R ,1 (mV) (Log scale)0.1 pA -6.0 0.0-6.0 0.0 ∆ V R ,1 (mV) ∆ V R ,1 (mV) f A f A . . . . . L-bpL-bpR-bpR-bp R -r e s R -r e s L -r e s L -r e s L - bpL - bp R - bp R - bp ∆ V R ,2 =2.4 mV ∆ V R ,2 =2.4 mV1.8 mV1.8 mV1.2 mV1.2 mV ∆ V R ,2 =5.0 mV ∆ V R ,2 =5.0 mV4.5 mV4.5 mV R -r e s R -r e s L -r e s L -r e s ∆ V R , ∆ V R ,1 (d) R -r e s L -r e s FIG. 4: (color online) (a)-(c) Detailed measurements of (a) I C , (b) I R , and (c) I L around the SB-SB overlap as a functionof ∆ V R, and ∆ V R, . (d) Schematic showing the correspon-dence between the region measured in (a)-(c) and the biastriangles. (e), (g) [(f), (h)] Slices of data in (b) [(c)] takenat ∆ V R, values indicated by the horizontal lines in (b) and(c). (e) and (f) correspond to the non-overlapped region, and(g) and (h) to the overlapped region. The traces are offsetvertically by 500 fA for clarity. the broad peaks running parallel to the right- and left-DQD resonance, indicated by the solid lines labeled “L-bp” and “R-bp”. These peaks appear in both the over-lapped and non-overlapped SB regions, but with differentbehavior. In the non-overlapped SB regions [Figs. 4(e)and (f)], the height of the peaks does not depend on ε (i.e., on ∆ V R, ), as expected for independent DQD trans-port. The leak current of ∼
300 fA is consistent with thatreported for a single DQD, which is known to arise fromspin relaxation induced by an inhomogeneous NS field[36, 39]. In contrast, inside the overlapped SB region[Figs. 4(g) and (h)], the peak height varies with ε by asmuch as a factor of ∼
2, being enhanced up to ∼ . ε in the over-lapped region is a manifestation of an interdependentrelationship between the two DQDs in the cooperativetransport. It is conceivable from the simplified picture inFigs. 3(d)-(h) that the transport through one DQD fa-cilitates that through the other. Thus, one would expectthe height of the peak in I R ( I L ) to depend on the valueof I L ( I R ) at that gate voltage. Indeed, in Figs. 4(g)and (h), the broad peak in I R ( I L ) grows as ∆ V R, (andhence ε ) increases and, accordingly, I L ( I R ) at the cor-responding gate voltage grows, reflecting the changes in D (c)|| ∆ B → N || || ∆ B → N || E i genene r g y ( µ e V ) (b) A α , -2 ε lr ( µ eV) (a) . - . Q QD D D D - ε ε A l, A r, ε lr ( µ eV) . - . E i genene r g y ( µ e V ) FIG. 5: (color online) (a) Energy levels of (1 , , t ℓc = t cr = 1 µ eV for the same value of ε ( <
0) as in Fig. 2(e). (b) A ℓ, and A r, for the same param-eters as in (a). (c) Energies of eigenstates near the quadruplet( Q ) calculated for ε = 0. the tuning parameters of the left (right) DQD.Another noticeable feature in Fig. 4 is the sharp peaksin I R and I L that appear along the R-res and L-res lines,respectively. These are distinct from the broad peaks dis-cussed above in that transport occurs only with the helpof the other DQD. Indeed, outside the overlap region,there is no transport feature visible along the R-res or L-res lines. In conventional DQDs, SB leak transport at res-onance is suppressed by interdot tunnel coupling, whichopens an anticrossing gap and pushes the singlet statesout of the energy window accessible from the triplet viastatistical fluctuations of the NS field k ∆ ~B N k [36]. In thepresent case, the absence of transport features along theR-res or L-res lines outside the overlap region indicatesthat both DQDs have large enough interdot tunnel cou-pling for such a situation to occur. Thus, the emergenceof resonant leak transport in the overlap region suggestsa non-trivial transport mechanism assisted by the othernon-resonant DQD.To see this, we examine the nature of three-electronstates in the presence of both t ℓc and t cr . | σ ℓ , σ c , σ r i , | S, , σ r i , and | σ ℓ , , S i are hybridized to form four dou-blets (total spin 1 /
2) and one quadruplet (total spin 3 / | D i i ( i = 1 , · · · ,
4) and | Q i , respec-tively. Figure 5(a) shows their energy levels for ε < | D i and | D i , which exhibit anticrossing, have finite | S, , σ r i ( | σ l , , S i ) components near the left-DQD (right-DQD)resonance. On the other hand, | D i as well as | Q i areconstructed from | σ ℓ , σ c , σ r i only and thus have purely(1 , , | D i , which is nearly degenerate with | Q i . Inthis state, the two spins in the left DQD form a tripletnear the left-DQD resonance, while those in the rightDQD form a triplet near the right-DQD resonance. This is seen in the squared overlap integrals with the statescomprising a (1 ,
1) singlet in the left or right DQD: A ℓ, ≡ |h (1 , S, σ r | D i| and A r, ≡ |h σ ℓ , (1 , S | D i| [Fig. 5(b)]. A ℓ, ( A r, ) sharply drop near the left-DQD(right-DQD) resonances at ε ℓr = − ε ( ε ℓr = 2 ε ). Theimportant observation is that A r, remains finite when A ℓ, vanishes and vice versa. Therefore, although relax-ation from | Q i to | D i does not directly contribute toSB leak current through the DQD on resonance, the oc-cupation of | D i leads to sequential tunneling throughthe other (off-resonance) DQD, by which the system canbe reloaded into | D i or | D i . Note that these are res-onant transport states of the on-resonance DQD, whichaccounts for the observed sharp peak along the resonance.Finally, we note that current is suppressed at the cross-point of the R-res and L-res lines [Figs. 4(a)-(c)]. Nearsuch a double-resonance point, the effects of t ℓc and t cr are no longer separable. As shown in the energy diagramfor ε = 0 [Fig. 5(c)], near the resonance all the dou-blets | D i i are split off from the quadruplets | Q i , leavingno doublets available within a window of ±k ∆ ~B N k / | Q i ’s, theSBs in both DQDs are protected from NS fluctuations,and all three spins remain locked parallel to one another.Such an SB mechanism is distinct from that of the con-ventional SB in a DQD and is of genuine TQD nature.In summary, we have demonstrated transport mea-surements through a three-terminal TQD in the few-electron regime. The SB-SB overlap brings out a cor-relation transport through the cooperation of two DQDsin the SB’s lifting. Competition between exchange inter-action and NS fluctuations leads to distinct cooperativemechanisms manifested by multiple peaks in the trans-port spectra. Our results show the potential of TQDs asa platform hosting a variety of correlation physics.We thank Y. Tokura and T. Fujisawa for fruitful dis-cussions and H. 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Takashi Kobayashi, Takeshi Ota, Satoshi Sasaki, and Koji Muraki
NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan (Dated: February 8, 2018)
HAMILTONIAN
Eigenstates | D i i ( i = 1 , · · · ,
4) and | Q i without magnetic field are calculated with 5 × H S z =+1 / ofthe subspace where z -projection of the total spin is +1 / H S z =+1 / = ( ε ℓ − ε c ) | S, , ↑ih S, , ↑ | + ( ε r − ε c ) | ↑ , , S ih↑ , , S |− t ℓc ( | ↓ , ↑ , ↑ih S, , ↑ | + | S, , ↑ih↓ , ↑ , ↑ | ) + t ℓc ( | ↑ , ↓ , ↑ih S, , ↑ | + | S, , ↑ih↑ , ↓ , ↑ | )+ t cr ( | ↑ , ↓ , ↑ih↑ , , S | + | ↑ , , S ih↑ , ↓ , ↑ | ) − t cr ( | ↑ , ↑ , ↓ih↑ , , S | + | ↑ , , S ih↑ , ↑ , ↓ | )= ε ℓr / ε − t ℓc t ℓc − t ℓc t ℓc t cr − t cr t cr − t cr − ε ℓr / ε , on a basis set of | S, , ↑i , | ↓ , ↑ , ↑i , | ↑ , ↓ , ↑i , | ↑ , ↑ , ↓i , and | ↑ , , S ii