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Received 11 Jan 2013 | Accepted 27 May 2013 | Published 27 Jun 2013

C.H. Yang1, A. Rossi1, R. Ruskov2, N.S. Lai1, F.A. Mohiyaddin1, S. Lee3, C. Tahan2, G. Klimeck3,A. Morello1 & A.S. Dzurak1

Although silicon is a promising material for quantum computation, the degeneracy of the conduction band minima (valleys) must be lifted with a splitting sufcient to ensure the formation of well-dened and long-lived spin qubits. Here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metaloxide semiconductor quantum dot, providing splittings spanning 0.30.8 meV. The splitting varies linearly with applied electric eld, with a ratio in agreement with atomistic tight-binding predictions. We demonstrate single-shot spin read-out and measure the spin relaxation for different valley congurations and dot occupancies, nding one-electron lifetimes exceeding 2 s. Spin relaxation occurs via phonon emission due to spinorbit coupling between the valley states, a process not previously anticipated for silicon quantum dots. An analytical theory describes the magnetic eld dependence of the relaxation rate, including the presence of a dramatic rate enhancement (or hot-spot) when Zeeman and valley splittings coincide.

DOI: 10.1038/ncomms3069

Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting

1 Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology, School of Electrical Engineering & Telecommunications, The University of New South Wales, Sydney 2052, Australia. 2 Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, MD 20740, USA. 3 Network for Computational Nanotechnology, Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA. Correspondence and requests for materials should be addressed to A.R. (email: mailto:[email protected]

Web End [email protected] ).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3069

Silicon is at the heart of all modern microelectronics. Its properties have allowed the semiconductor industry to follow Moores law for nearly half a century, delivering

nowadays billions of nanometre-scale transistors per chip. Remarkably, silicon is also an ideal material to manipulate quantum information encoded in individual electron spins13. This is a consequence of the weak spinorbit coupling and the existence of an abundant spin-zero isotope, which can be further enriched to obtain a semiconductor vacuum in which an electron spin can preserve a coherent quantum superposition state for exceptionally long times4.

In order to dene a robust spin-1/2 qubit Hilbert space, it is necessary that the energy scale of the two-level system is well separated from higher excitations. In this respect, a major challenge for the use of silicon is represented by the multi-valley nature of its conduction band. In a bulk silicon crystal the conduction band minima are six-fold degenerate, but in a two-dimensional electron gas (2DEG), the degeneracy is broken5 into a two-fold degenerate ground state (G valleys) and a four-fold degenerate excited state (D valleys), owing to vertical connement of electrons with different effective mass along the longitudinal and transverse directions, respectively. Furthermore, the G valley degeneracy is generally lifted by a sharp perpendicular potential69 and the relevant energy separation is termed valley splitting (VS).

The VS depends on physics at the atomic scale1012 (for example, roughness, alloy and interface disorder), and so it is not surprising that experiments have revealed a large variability of splittings among devices, ranging from hundreds of meV5,1315 up to tens of meV in exceptional cases16. At present, the lack of a reliable experimental strategy to achieve control over the VS is driving an intense research effort for the development of devices that can assure robust electron spin qubits by minimizing multi-valley detrimental effects17,18, or even exploit the valley degree of freedom19,20 for new types of qubits.

Another crucial parameter to assess the suitability of a physical system to encode spin-based qubits is the relaxation time of spin excited states (T1). Spin lifetimes have been measured for gate-dened Si quantum dots (QDs)21, Si/SiGe QDs22,23 and donors in Si (ref. 24), reporting values that span from a few milliseconds to a few seconds. Furthermore, the dependence of the spin relaxation rate (T 11) on an externally applied magnetic eld (B)

has been investigated. Different mechanisms apply to donors and QDs, accounting for the observed T 11pB5 and B7 dependencies25, respectively. In principle, T 11(B) depends on the valley conguration and the details of the excited states above the spin ground state. However, until now, no experimental observation of the effects of a variable VS on the relaxation rate has been reported.

Here, we demonstrate for the rst time that the VS in a silicon QD can be nely tuned by direct control of an electrostatic gate potential. We nd that the dependence of the VS on vertical electric eld at the Si/SiO2 interface is strikingly linear, and show that its tunability is in excellent agreement with atomistic tight-binding predictions. We demonstrate accurate control of the VS over a range of about 500 meV and use it to explore the physics of spin relaxation for different QD occupancies (N 1, 2, 3). We

probe both the regime where the VS is much larger than the Zeeman splitting at all magnetic elds and that where the valley and spin splittings are comparable. We observe a dramatic enhancement of the spin decay rate (relaxation hot-spot) when spin and valley splittings coincide. To our knowledge, such hot-spots have been predicted for relaxation involving orbital states26,27 (not valley states), but these are yet to be observed. We develop an analytic theory that explains the B-eld dependence of the relaxation rates and the details of the relaxation hot-spot in

terms of admixing of spin-valley states. This mechanism is seen to be signicantly more prominent than the conventional spinorbit hybridization28.

ResultsQD addition spectrum. Our device is fabricated using a multilevel gated metaloxidesemiconductor (MOS) technology29, and its architecture is depicted in Fig. 1ad. A quantum dot is formed under gate P by applying a positive bias voltage to induce an electron accumulation layer. Strong planar connement for the dots potential well is achieved by negatively biasing gates B, C1 and C2. A 2DEG reservoir is also induced by positively biasing gates R1 and R2, and the QD occupancy can be modied by inducing electrons to tunnel between this reservoir and the dot. The remaining gates, namely, SB1, SB2 and ST, are employed to dene a single-electron transistor (SET), capacitively coupled to the QD and used as a read-out device. The high exibility of our design would allow us to use the same device also as a (single-lead) double-dot structure by rearranging the gate bias (for example, dots can be formed under gates B and C2). However, in this work, we only present results relevant to the single-dot conguration.

In order to characterize the addition spectrum of the QD, we make use of a technique previously developed for GaAs-based systems, which combines charge detection and gate pulsing30. There is no direct transport through the single-lead QD and, therefore, addition/removal of charge is only detected via modications in the SET current. In particular, charge transitions are detected as current peaks in the SET signal whenever the QD energy eigenstates come into resonance with the reservoirs Fermi level. Note that the SETQD coupling is merely capacitive (via Ccpl) and electrons do not tunnel between them. In order to maximize charge sensitivity in the detector, we employ dynamic voltage compensation31 on different gates, which makes our read-out signal virtually unaffected by slow charge drifts and random charge rearrangements. A comprehensive discussion of the charge stability measurements can be found in Supplementary Note 1.

Figure 1e illustrates the addition energy spectrum for the rst 14 electron additions to the QD. There is very little variation of charging energy (EC) for high occupancies (ECE11 meV for

N49). However, by decreasing the electron number, the charging energy steadily increases, as expected when the dot size is signicantly affected by the electron number. This evidently indicates that the few-electron regime has been achieved. Most interestingly, the energy spectrum shows peaks for the addition of the fth and thirteenth electrons. The extra addition energy needed for those transitions can be attributed to complete lling of the rst and second orbital shells. As illustrated in Fig. 1f, this is consistent with the energy spectrum of two-valley 2D Fock Darwin states32, where the rst and second orbital shells hold four and eight electrons, respectively. This conrms that we can probe the occupancy until the last electron. To our knowledge, such a clear manifestation of two-dimensional shell structure has been observed before only in InGaAs dots33 and in Si/SiGe dots34.

Spin-valley lifetimes. In order to measure the spin state of individual electrons in the QD, we use an energy-selective readout technique35. The read-out protocol consists of three phases clocked by a three-level pulsed voltage applied to gate P, which directly controls the dots electrochemical potential (see Fig. 2a). First, an electron of unknown spin is loaded into the dot, causing a sudden decrease in the sensor current. Next, the potential of the dot is lifted so that the Fermi level of the reservoir lies between the spin-up and spin-down states of the dot, meaning that a

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a b

SB2

2DEG QD

SET

0 2 4 6 8 10 12 14

ISET

SB1

Ccpl

30 nm

c d

25 nm

2DEG 8 nm

N=1 2

N=4 5

2nd Orbit

N=12 13

3rd Orbit

2D Fock-Darwin states

N=4

N=12

(0, 1, +)

(0, 1, )

(0, 1, +)

(0, 1, )

(0, 0, +)

(n, l, v) = (0, 0, )

E C(meV)

Figure 1 | Device architecture and addition energy spectrum. (a) Schematic (top view) of the devices gate layout. Different colours represent different layers within the gate stack. (b) Schematic diagram of the single-lead QD (left) and SET detector (right). Regions where an electron layer is formed are coloured in orange. The read-out signal (ISET) is sensitive to the QD charge state due to the QD/SET capacitive coupling (Ccpl). (c) Device cross-sectional schematic. An electron layer is formed underneath the positively biased gates: R1 and R2 dene the QD reservoir; P controls the QD population, and ST the sensors island. The SiO2 layer (in purple) thickness and plunger gate width are indicated. (d) Energy diagram showing qualitatively the conduction band prole in the device. Electrons accumulate wherever the gate bias lowers the conduction band below the Fermi level, EF. (e) Charging energy as a function of electron number. Spikes corresponding to complete 2D shell lling are observed. (f) Schematic of electron lling for two-valley 2D FockDarwin states. Each state can hold two electrons of antiparallel spin and is identied by a pair of quantum numbers (n,l) and its valley occupancy (v).

spin-up electron can tunnel off the dot while a spin-down electron is blocked. This is the read phase, during which the presence of a spin-up state would be signalled by a current transient (spin-up tunnels out and then spin-down tunnels in) whereas a spin-down electron would lead to no current modication. Finally, the dots potential is further lifted to allow the electron to tunnel off, regardless of its spin orientation. In Fig. 2b, single-shot traces for both spin-up (in blue) and spin-down (in green) detection are plotted. The longer the system is held in the load phase before performing a read operation, the more likely it is for the spin-up excited state to decay to the spin-down ground state. Thus, by varying the length of the load phase and monitoring the probability of detecting a spin-up electron, we can determine24,35 the spin lifetime, T1. In our experiments the B eld is directed along the [110] crystallographic axis. A comprehensive discussion of both the spin-up fraction measurements and the tting procedure to evaluate T1 is

included in the Supplementary Note 2. As shown in Fig. 2d, we observe a wide range of spin lifetimes as a function of magnetic eld, with lifetimes as long as 2.6 s at the lowest elds studied, B 1.25 T. These are some of the longest lifetimes observed to

date in silicon quantum dots25.

A key focus of our experiment was to electrostatically tune the valley energy separation and measure relaxation rates in different valley congurations and QD electron occupancies. As we show below, our data denitively indicate that excited valley states have

a critical role in the spin relaxation processes. We develop a theory to explain how changes in the VS affect the spin-valley state mixing and leads to the observed relaxation times.

As we detail in the following section, we have attained accurate gate control of the VS, allowing us to tune it over a range of hundreds of meV. This permits us to conduct experiments in regimes where the VS (EVS) is either larger or smaller than the Zeeman spin splitting (EZ), depending on the magnitude of the magnetic eld (see Fig. 2c).

Figure 2d presents measurements of spin relaxation rates as a function of magnetic eld for two VS values at a xed dot population of N 1. We start by examining a conguration

where the valley separation is larger than the spin splitting at all elds (green data set). In other words, we operate in a regime for which

EVS 4 EZ gmBB 1 where g is the electron gyromagnetic ratio, mB is the Bohr magneton and B is the applied in-plane magnetic eld.

For EVS 0.75 meV (green data in Fig. 2d), we observe a

monotonic increase in the rate with respect to B that becomes increasingly fast as EZ approaches EVS. In our experimental conditions (B eld parallel to [110]), the T 11pB5 dependences for known bulk-like mechanisms in silicon36,37 should not apply, while predicted23,25,27,38 rates pB7 do not explain the experimental data.

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Load

[afii9840]+ EF

[afii9840]

Read

Load Read

Empty

V Pulse

(mV)

Empty

Read

0 2 4 6 8 10

Time (ms)

103

N = 1

103

Energy

102

101

104

T 11 (s 1 )

[afii9840]+

100

103

E VS

[afii9829]

101

102

101

100

10 1 2 3 4 5 6

101

B (T)

Figure 2 | Spin read-out and relaxation rates for single-electron occupancy. (a) Schematic diagram showing the effect of the three-level pulse sequence on the electro-chemical potential of the dot. Energy levels in the QD are Zeeman split according to spin polarization and valley degeneracy is lifted. For clarity, only lower valley states are shown to be loaded/unloaded. (b) Pulsing sequence (top) for the single-shot spin read-out and normalised SETsignal for spin-up (middle) and spin-down (bottom). (c) Energy diagram of the one-electron spin-valley states as a function of B eld. Maximum mixing of spin and valley degrees of freedom occurs at the anticrossing point where Zeeman and valley splittings coincide. Relevant relaxation processes are sketched. (d) Relaxation rates as a function of magnetic eld for different valley splittings. Data points for EVS 0.75 meV, EVS 0.33 meV are shown as

green and red circles, respectively. Dashed lines are the calculated relaxation rates tted with r 1.7 nm (green), r 1.1 nm (red).

By decreasing the valley separation to EVS 0.33 meV, we can

achieve the condition where the Zeeman splitting matches or exceeds the VS. The red data in Fig. 2d illustrate the situation where inequality (1) only holds for Bo2.8 T. When EZ EVS

(that is, for B 2.8 T), a spike in the relaxation rate occurs.

Relaxation hot-spots have been previously predicted to occur for spin relaxation involving orbital states in single and coupled QDs26,27,39,40. To our knowledge, this is the rst experimental observation of such a phenomenon.

In order to understand the relaxation mechanisms, we have developed a model that takes into account the perturbations in pure spin states due to spinorbit coupling (SOC), yielding eigenstates that are admixtures of spin and valley states. The four lowest spin-valley states (see Fig. 2c) are dened as 1

j i v ; #

j i,

j i v ; "

2 j i, 3

j i v ; #

j i, 4

j i v ; "

g2 3

j i 2

j i. These states are con

sidered to be only very weakly affected by higher excitations, such as orbital levels that are at least 8 meV above the ground state in our device41. In Supplementary Note 3 we detail how mixing to a 2p-like orbital state leads to a B7 dependence in T 11 and is, therefore, important mainly for high B elds (above the anticrossing point). At lower elds, the prominent mechanism is the spin-valley admixing, which we now discuss in detail.

The relaxation between pure spin states is forbidden because the electronphonon interaction does not involve spin ipping. However, in the presence of interface disorder, SOC can mix states that contain both the valley and spin degrees of freedom, thus permitting phonon-induced relaxation. Indeed, in the non-ideal case of QDs with a disordered interface, roughness

can perturb the envelope function of both valleys (otherwise identical for ideal interfaces) and allows one to assume non-zero dipole matrix elements connecting the valley states (see Supplementary Note 3), such as r hv j r j v i,

r hv j r j v i, r hv j r j v i (for ideal

interfaces these are non-zero only due to a strongly suppressed Umklapp process). By means of perturbation theory, we dene renormalized excited states 2

j i and 3

j i that can relax to the

ground state 1

j i, as they have an admixture of the state 3

j i of the

same spin projection (see Fig. 2c). The details of the SOC Hamiltonian, HSO, and perturbation matrix are reported in

Supplementary Note 3. The leading-order wavefunctions are given by:

j i0 sin

g2 2

j i cos

j i0 cos

g2 2

j i sin

g2 3

j i 3

q is an expression involving the detuning from the anticrossing point, d EVS EZ, and the energy splitting

at the anticrossing:

Da 2 j hv ; " j HSO j v ; #i j

mtEVS2

p "h bD aR

where cosg=2 1 a=2

1=2, sing=2 1 a=2

1=2, and

a d=

d2 D2a

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Energy

B ST = 5.0 T

103

+ N = 2

102

[afii9840]+

T+

|32e

1 (s 1 )

B (T)

N = 1 N = 3

E VS

101

|[afii9829]|/g[afii9839]B (T)

T 1

1 2 3 4 5

1 2 3 4

[afii9829]

103

|22e

[afii9840]

103 102

100

1 (s1 )

B (T)

101

100

1 (s1 ) T 1

102

T 1

101

|12e

BST

2 3 4 5 6 7 8

Figure 3 | Spin-valley relaxation for multi-electron occupancy. (a) Energy diagrams of the two-electron spin-valley states in the dots potential well (left) and as a function of B eld (right). Dashed line indicates that T is not accessible (see text). (b) Relaxation rate as a function of B eld for N 2 and

EVS 0.58 meV. Red (blue) crosses represent data points at elds smaller (larger) than the anticrossing point. Dashed lines are the calculated rates tted

with r2e 4.76 nm. Right inset: Data from the main graph re-plotted as a function of the modulus of the detuning energy, d. Points of equal absolute

detuning have nearly the same decay rates. d5 dependence (grey line) is a guide for the eye. Left inset: Relaxation rate as a function of B eld for N 3 and

EVS 0.58 meV. Results for N 1 are also shown for comparison.

where bD (aR) is the Dresselhaus (Rashba) SOC parameter, : is the reduced Plancks constant and mt 0.198me is the transverse

effective electron mass.

By evaluating the relaxation rate via the electronphonon deformation potentials (proportional to the deformation potential constants, d;u), we obtain the rate below the anticrossing as:

G"21 cos2

g2 Gv

d2 D2a

q d

d2 D2a

Gv0v 5

where the pure-valley relaxation rates are (for longitudinal and transverse phonons):

Gsv0v DEv0v; r

DE5v0v

4pr"h6

r2 v7s

Is 6

where r is the silicon mass density, vs is the speed of sound in silicon, Il 4 2u 35 2 u d=15 2d 3 , It 16105 2u are the

angular integrals, and DEv0v and r are the energy difference and the dipole matrix element relevant to the transition, respectively (see also Supplementary Note 3). The experimental condition for which the hot-spot occurs (that is, EVS EZ) is modelled as an

anticrossing point for the mixed states 2

j i and 3

j i. At that point,

spin relaxation is maximized and G"21 approaches the valley relaxation rate, as d-0 in equation (5).

Above the anticrossing (that is, EVSoEZ), the relevant

relaxation transitions are 3

j i ! 1

j i and 3

j i ! 2

j i (the subse

quent decay 2

j i ! 1

j i is in the form of a fast inter-valley

transition, and is therefore neglected). The analytical formulations of these contributions read:

G"31 sin2

g2 Gv

d2 D2a

q d

d2 D2a

Gv0v 7

G"3"2 sin2

g2 cos2

g2 Gv

DEv

0v; r r 8

The dashed lines in Fig. 2d show the calculated relaxation rates relevant to the two experimental values of EVS discussed, also including B7 contribution from SOC mixing with the higher

orbital state (see Supplementary Note 3). We use dipole matrix elements as a single free parameter by assuming

j r j j r r j r. A least-square t to the

experimental data is performed by xing the SOC strength to (bD aR)E4560 m s 1 (justied by the high electric

eld E3 107 V m 1, see Wilamowski et al.42 and Nestoklon

et al.43). The t then extracts a dipole size rE12 nm for both values of EVS.

The good agreement between the calculations and the experiment, as well as the presence of a hot-spot at the point of degeneracy between Zeeman and VS, provide strong evidence of our ansatz that the spin relaxation is predominantly due to a new mechanism: that of mixing with the excited valley states via Rashba/Dresselhaus-like SOC in the presence of interface disorder.

Both the splitting at the anticrossing, equation (4), and the inter-valley relaxation, equation (6), depend crucially on the size of the dipole matrix element, r, predicting a fast phonon relaxation of E107108 s 1 for r 13 nm, at the hot-spot of

Fig. 2d. This conrms our core ndings that when spin-valley states anticross, the inter-valley rates are fast for these samples, with the only available relaxation mechanism being the inter-valley decay. We point out that these relaxation rates are expected to be sample/material-dependent, given the effect of interface disorder on valley mixing.

We now examine the case where N 2 electrons, and

investigate the dependence of the relaxation rate on the magnetic eld at a xed VS (EVS 0.58 meV). We note that the energy

levels accessible for loading the second electron in the dot, when the N 1 spin-down ground state is already occupied, are either

the singlet (S) or the two lower triplets (T , T0), whereas the

higher triplet (T ) would require a spin ip and is, therefore, not

readily accessible (see Fig. 3a). In general, for triplet states, the antisymmetry of the two-electron wavefunction requires one electron to occupy a higher energy state. For our multi-valley QD (Fig. 1f), this requirement is fullled when the two electrons occupy different valley states (see Fig. 3a). For low elds, the ground state is S and the triplets have higher energies. This results in excited states (triplets) that extend over two valleys and relax to a single-valley ground state (singlet).

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As the magnetic eld is increased, S and T undergo an

avoided crossing (B BST), and then T becomes the ground

state. We adjust the levels of our pulsed read-out protocol so that during the load phase only S and T are below the reservoirs

Fermi energy, whereas, during the read phase, the Fermi energy is positioned within the singlettriplet (ST) energy gap. As a consequence, for BoBST (B4BST) a T (S) state would be signalled with a current transient, and relaxation rates can be extracted as for the N 1 occupancy. The experimental relaxation

rates in Fig. 3b show a strongly non-monotonic behaviour, approaching an absolute minimum at the anticrossing point (BST 5 T). The trend is strikingly symmetric, as can be

appreciated when T 11 is plotted against the detuning energy, as shown in the right inset. This symmetry is reected in the QD energy spectrum (Fig. 3a), as far as the detuning d is concerned.

For BoBST, the ST energy gap decreases with increasing B, resulting in slower relaxation rates. By contrast, for B4BST, the

ST energy gap increases with increasing eld, and so does the relaxation rate.

As opposed to the one-electron case, we note that the two-electron eigenstates anticrossing leads to a minimum in the relaxation rate (cold-spot), dened by a splitting at the anticrossing, D2ea, of the same order as that of equation (4) (see

Fig. 3a and Supplementary Note 3). The occurrence of this minimum does not strictly depend on the nature of the states involved in the decay (spin-like, valley-like, orbital-like or admixtures). It is due to the fact that the avoided crossing takes place between the ground and the rst excited state, whereas for the case N 1 it involves the rst and the second excited states

without affecting the ground state.

To model the two-electron case, we build the wavefunctions for S and T from the single-particle states by considering

the Coulomb interaction as a perturbing averaged eld. The corresponding states are dened as 1

j i2e v ; v ; S

j i,

j i2e v ; v ; T

j i. Next, the additional perturbation given by

SOC leads to renormalized eigenstates that are admixtures of singlet and triplet:

j i02e sin

g2 1

j i2e cos

g2 2

j i2e 9

j i02e cos

g2 1

j i2e sin

g2 2

j i2e 10

being similar forms to equations (2) and (3). As we show in Supplementary Note 3, by evaluating the electronphonon Hamiltonian matrix element for the transition between these states, one nds that it coincides in its form with its one-electron counterpart for 3

j i1e! 2

j i1e. Therefore, we can conclude that the

corresponding relaxation rate, G2e"21, has the same functional form as those derived in equation (8), although the matrix elements for the two cases will be different (see Supplementary Note 3). The dashed lines in Fig. 3b represent the calculated rates that are tted to the experimental data similarly to the case where N 1. Once

again, the model convincingly reproduces the main features of the experimental trend, in particular the rates for elds away from the anticrossing, together with the symmetry of the characteristics with respect to BST. Further work may be needed to improve the t in the vicinity of the anticrossing point.

We also measured the relaxation rates for N 3 electrons.

When the QD occupancy is set at N 2, the lower valley is fully

occupied and for low B elds the ground state is a singlet. In this condition, the read-out protocol is adjusted to probe spin relaxation within the upper valley upon loading/unloading of the third electron. By keeping EVS 0.58 meV and using the same

methodology described before, we measure relaxation rates for the third electron spin state. We nd that there is no signicant difference between the spin relaxation rates for N 3 and N 1,

as shown in the left inset of Fig. 3b. Two main conclusions can be drawn from this. First, we can infer that the effect of electron electron interactions on the multi-valley spectrum may be negligible44, which is plausible. Indeed, for valley three-electron states, two electrons are just spectators, so that the remaining electron establishes the same energy level structure as in Fig. 2c, and the Coulomb corrections do not affect the VS. Second, as we report in Yang et al.41, in small QDs for higher occupancies a signicantly reduced energy separation between the ground state and the rst excited orbital state is observed. This would introduce a non-negligible perturbation on the relaxation if this were affected by the orbital degree of freedom. Hence, the similarities in behaviour in terms of decay rates are a further indication that for our QD the dominant relaxation mechanism resides in the degree of spin-valley admixing, as opposed to the spinorbit admixing relevant for other semiconductor systems28.

Valley splitting control. We now turn to the experimental demonstration of accurate control of the VS, EVS, via electrostatic gating. To determine EVS, we use two different experimental approaches. One utilizes the rapid increase in spin relaxation at the hot-spot, and is applicable in the low-magnetic-eld regime. The other is based on magnetospectroscopy, and is relevant for high elds.

The rst technique stems from the fact that the hot-spot can be reliably detected by monitoring the spin-up probability as a function of magnetic eld. In Fig. 4a, we show measurements of the spin-up probability performed with the same method as the one used to evaluate spin lifetimes (see Supplementary Note 2). We see that the probability of detecting a spin-up electron decreases signicantly at some magnetic elds. A sudden drop of the spin-up fraction in a narrow range of eld identies the increase in relaxation rate associated with the hot-spot. Given that valley and Zeeman splittings coincide at the hot-spot, one can extract the valley separation as EVS gmBBHS, where BHS is dened as the eld at which the hot-spot

is observed. For varying gate-voltage congurations, we scan B in the range 2.8 ToBo5 T, and identify BHS by setting an arbitrary probability threshold (green-shaded area in Fig. 4a) below which the hot-spot is assumed to occur. The use of this technique is limited to Bo5 T because the lifetime drop at the hot-spot can be therein condently assessed. At higher elds the relaxation becomes increasingly fast and its enhancement at the hot-spot is indistinguishable within our measurement bandwidth (E10 kHz).

In order to evaluate EVS at higher magnetic elds, we use a more conventional magneto-spectroscopic approach, as shown in Fig. 4b. By employing the same gate-pulsed technique used for the charge stability experiments (see Supplementary Note 1), we focus on the singlettriplet ground-state transition as we load the second electron into the dot (that is, N 1-2 transition) in the

range 5 ToBo6.5 T. This is clearly identied as the point where the S (light grey feature) and T (dark grey feature) states cross.

Here, EVS gmBBST, as seen in Fig. 3a.

The data points in Fig. 4c represent the measured valley separation as a function of VP, obtained by means of the aforementioned techniques. The solid line t shows remarkable consistency between the two sets of data and reveals that EVS

depends linearly on the gate voltage over a range of nearly 500 meV, with a slope of 640 meV V 1. In order to keep constant the dots occupancy and tunnelling rates for different VP, a

voltage compensation is carried out by tuning gates C1 and B accordingly. We note that we previously reported VSs of comparable magnitude (few hundreds of meV) in devices realized with the same technology41,45. However, to our knowledge, this is the rst demonstration of the ability to accurately tune the VS electrostatically in a silicon device.

6 NATURE COMMUNICATIONS | 4:2069 | DOI: 10.1038/ncomms3069 | http://www.nature.com/naturecommunications

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3069 ARTICLE

N=2

100

Bave [afii9846]B =

3.78 0.03T

| Probability, %

B ST = 5.1 T

B (T)

V P (mV)

N = 1

3.6 3.8 4

0 2 4 6 8

B (T)

|Fz| (MV m1)

0.8

30 2.1

B C2

0 2

z (nm)

45 0

x (nm)

0 200

400 ECB (meV)

Atomistic

EVS/VP

(meV V1)

Effective mass

Exp.

0.640

Atom.

0.597

Eff. m.

0.541

E VS (meV)

exp

0.6

E VS (meV)

1.9

mod

0.4

1.7

1.4 1.6 1.8 2

VP (V)

Figure 4 | Tunability of the VS via gate-voltage control. (a) Spin-up probability as a function of magnetic eld for VP 1.59 V. The occurrence of

a hot-spot minimizes the spin-up fraction and allows one to extract BHS

(shaded area) and, in turn, EVS. (b) Pulsed-voltage magnetospectroscopy showing dISET/dVP at N 1-2 transition. A square pulse of amplitude

16 mV at 287 Hz is applied to gate P. The evolution of the energy difference between the singlet state (light grey) and the triplet state (dark grey) allows one to extract BST (dashed line) and, in turn, EVS. (c) Valley splitting as a function of plunger gate voltage (bottom axis) and modulus of interface vertical electric eld in the QD (top axis). Blue and red dots show the valley separation (left axis) measured with hot-spot and magneto-spectroscopic techniques, respectively. Error bars indicate standard deviations for the measured values. The linear t (solid line) indicates a valley tunability of0.640 meV V 1. Dashed lines are calculations performed via atomistic (purple) and effective mass (green) methods (right axis). Bottom inset:

Table comparing calculated and experimental tunabilities. Top inset: TCAD simulation of the conduction band prole at VP 1.4 V in the (x,z) plane.

Negative energy region in red reveals where the dot is formed.

A linear dependence of the VS with respect to the vertical electric eld has been predicted for 2DEG systems via effective mass theory7,9,10. A similar dependence for MOS-based QDs46 has also been reported by employing atomistic tight-binding calculations47. In order to compare our experimental nding with the theoretical predictions, we simulate the vertical electric eld (FZ) in the vicinity of the dot for the range of gate voltages used in the experiments. We employ the commercial semiconductor software ISE-TCAD48 to model the devices electrostatic potential, and thereby the electric elds in the nanostructure. For this purpose, TCAD solves the Poisson equation with an approximation of Newtons iterative method49 to obtain convergence at low temperatures.

The spatial extent of the dot is identied by regions where the calculated conduction band energy drops below the Fermi level

(red area in the top inset of Fig. 4c). Note that our calculations are performed on a three-dimensional geometry identical to the real device with the only free parameter being the amount of offset interface charge. This is adjusted to match the experimental threshold voltage of the device (Vth 0.625 V), as explained in

the Supplementary Note 4.

The computed variation of interface electric eld with gate voltage VP is used to determine the VS according to both the atomistic46 and effective mass7 predictions. Dashed lines in Fig. 4c depict the trends for both approaches, with both exceeding by more than 1 meV of the measured values. Despite this offset, the atomistic calculations give a tunability of the VS with gate voltage, DEVS/DVP, in good agreement with the experiments. The calculated value of 597 meV V 1 agrees with the measured value to within less than 7%. The value of 541 meVV 1 calculated using the effective mass approach reveals a larger deviation (E15%) from the experiments. The presence of an offset in the computed VS may be due to the contribution of surface roughness that is not accounted for in the models, and is thought to be responsible for a global reduction of EVS8,1012,50,51. We emphasize, however, that the gate tunability would remain robust against this effect, which is not dependent on electric eld.

DiscussionIn this work, we have shown that the VS in a silicon device can be electrostatically controlled by simple tuning of the gate bias. We used this VS control, together with spin relaxation measurements, to explore the interplay between spin and valley levels in a few-electron quantum dot.

The relaxation rates for a one-electron system exhibit a dramatic hot-spot enhancement when the spin Zeeman energy equals the VS, whereas for a two-electron system the rates reach a minimum at this condition. We found that the known mechanisms for spin relaxation, such as the admixing of spin and p orbital states, were unable to explain the key features of the experimental lifetime data, and so introduced a novel approach based on admixing of valley and spin eigenstates. Our theory, which showed good agreement with experiment, implies that spin relaxation via phonon emission due to spin-orbit coupling can occur in realistic quantum dot systems, most likely due to interface disorder.

Our results show that by electrical tuning of the VS in silicon quantum dots, it is possible to ensure the long lifetimes (T141 s)

required for robust spin qubit operation. Despite this, the excited valley state will generally be lower than orbital states in small quantum dots, placing an ultimate limit on the lifetimes accessible in very small dots, due to the spin-valley mixing described above.

Electrical manipulation of the valley states is also a fundamental requirement to perform coherent valley operations. However, the experimental relaxation rate at the observed hot-spot was found to be fast (T 1141 kHz) for our devices, implying a fast inter-valley relaxation rate.

Finally, in the context of realizing scalable quantum computers, these results allow us to address questions of device uniformity and reproducibility with greater optimism. Indeed, our work suggests that issues related to the wide variability of the VS observed in silicon nanostructures to date can shift from the elusive atomic level (surface roughness, strain, interface disorder) to the more accessible device level, where gate geometry and electrostatic connement can be engineered to ensure robust qubit systems.

Methods

Device fabrication. The samples fabricated for these experiments are silicon MOS planar structures. The high purity, near intrinsic, natural isotope silicon substrate has n ohmic regions for source/drain contacts dened via phosphorous

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3069

diffusion. High quality SiO2 gate oxide is 8 nm thick and is grown by dry oxidation at 800 C. The gates are dened by electron-beam lithography, Al thermal evaporation and oxidation. Three layers of Al/Al2O3 are stacked and used to selectively form a 2DEG at the Si/SiO2 interface and provide quantum connement in all the three dimensions.

Measurement system. Measurements are carried out in a dilution refrigerator with a base temperature TbE40 mK. Flexible coaxial lines tted with low-temperature low-pass lters connect the device with the room-temperature electronics. In order to reduce pick-up noise, the gates are biased via battery-powered and opto-isolated voltage sources. The SET current is amplied by a room-temperature transimpedance amplier and measured via a fast-digitizing oscilloscope and a lock-in amplier for the single-shot and energy spectrum experiments, respectively. Gate voltage pulses are produced by an arbitrary wave-function generator and combined with a DC offset via a room-temperature bias-tee.

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Acknowledgements

The authors thank D. Culcer for insightful discussions, and F. Hudson and D. Barber for technical support. This work was supported by the Australian National Fabrication Facility, the Australian Research Council (under contract CE110001027) and by the U.S. Army Research Ofce (under contract W911NF-13-1-0024). The use of nanoHUB.org computational resources operated by the Network for Computational Nanotechnology, funded by the US National Science Foundation under grant EEC-0228390, is gratefully acknowledged.

Author contributions

C.H.Y. carried out the measurements. N.S.L. designed and fabricated the device. C.H.Y. and A.R. analysed the data. C.H.Y., A.R., S.L., G.K., A.M. and A.S.D. discussed the results. R.R. and C.T. modelled the relaxation rates. F.A.M. simulated the electric eld proles. A.S.D. conceived and planned the project. A.R. wrote the manuscript with input from all coauthors.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting. Nat. Commun. 4:2069 doi: 10.1038/ncomms3069 (2013).

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Abstract

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Although silicon is a promising material for quantum computation, the degeneracy of the conduction band minima (valleys) must be lifted with a splitting sufficient to ensure the formation of well-defined and long-lived spin qubits. Here we demonstrate that valley separation can be accurately tuned via electrostatic gate control in a metal-oxide-semiconductor quantum dot, providing splittings spanning 0.3-0.8 meV. The splitting varies linearly with applied electric field, with a ratio in agreement with atomistic tight-binding predictions. We demonstrate single-shot spin read-out and measure the spin relaxation for different valley configurations and dot occupancies, finding one-electron lifetimes exceeding 2 s. Spin relaxation occurs via phonon emission due to spin-orbit coupling between the valley states, a process not previously anticipated for silicon quantum dots. An analytical theory describes the magnetic field dependence of the relaxation rate, including the presence of a dramatic rate enhancement (or hot-spot) when Zeeman and valley splittings coincide.

Details

Title

Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting

Author

Yang, C H; Rossi, A; Ruskov, R; Lai, N S; Mohiyaddin, F A; Lee, S; Tahan, C; Klimeck, G; Morello, A; Dzurak, A S

Pages

2069

Publication year

2013

Publication date

Jun 2013

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms3069

ProQuest document ID

1371821811

Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting

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