Voltage tunability of single-spin states in a quantum dot

Abstract

Single spins in the solid state offer a unique opportunity to store and manipulate quantum information, and to perform quantum-enhanced sensing of local fields and charges. Optical control of these systems using techniques developed in atomic physics has yet to exploit all the advantages of the solid state. Here we demonstrate voltage tunability of the spin energy-levels in a single quantum dot by modifying how spins sense magnetic field. We find that the in-plane g-factor varies discontinuously for electrons, as more holes are loaded onto the dot. In contrast, the in-plane hole g-factor varies continuously. The device can change the sign of the in-plane g-factor of a single hole, at which point an avoided crossing is observed in the two spin eigenstates. This is exactly what is required for universal control of a single spin with a single electrical gate.

ARTICLE

Received 5 Sep 2012 | Accepted 17 Jan 2013 | Published 26 Feb 2013

Anthony J. Bennett1, Matthew A. Pooley1,2, Yameng Cao1,3, Niklas Skld1, Ian Farrer2, David A. Ritchie2 & Andrew J. Shields1

Single spins in the solid state offer a unique opportunity to store and manipulate quantum information, and to perform quantum-enhanced sensing of local elds and charges. Optical control of these systems using techniques developed in atomic physics has yet to exploit all the advantages of the solid state. Here we demonstrate voltage tunability of the spin energy-levels in a single quantum dot by modifying how spins sense magnetic eld. We nd that the in-plane g-factor varies discontinuously for electrons, as more holes are loaded onto the dot. In contrast, the in-plane hole g-factor varies continuously. The device can change the sign of the in-plane g-factor of a single hole, at which point an avoided crossing is observed in the two spin eigenstates. This is exactly what is required for universal control of a single spin with a single electrical gate.

DOI: 10.1038/ncomms2519

Voltage tunability of single-spin states in a quantum dot

1 Toshiba Research Europe Limited, Cambridge Research Laboratory, 208 Science Park, Milton Road, Cambridge CB4 0GZ, UK. 2 Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK. 3 Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK. Correspondence and requests for materials should be addressed to A.J.B. (email: mailto:[email protected]

Web End [email protected] ).

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The spin of charges in quantum dots (QDs) has long been considered a suitable qubit for quantum operations1. The three-dimensional connement offered by a single

semiconductor QD reduces many decoherence mechanisms, allowing impressively long coherence times to be observed in coherent population trapping2 or using spin-echo techniques3,4. In the latter, control of the spins was achieved with resonant, ultrafast optical pulses. An alternative mechanism for controlling single spins is for an electric eld to vary their coupling to a xed magnetic eld (B), described by the g-tensor (g)5. This method allows multiple closely spaced spin qubits to be individually addressed via nano-electrodes, without resonant lasers or localized magnetic elds. Critical to this concept is the ability to change the sign of one component of the g-tensor5,6. Then through careful alignment of the magnetic eld direction it is possible to switch between two electric elds where the precession directions of the spin (given by g.B) are orthogonal on the Bloch sphere. In such a system, universal control can map any point on the Bloch sphere onto any other point. Although experimental studies have been made of the g-tensor in QDs712, the change of sign of one component with electrical eld has yet to reported.

Early work used semiconductor quantum wells to electronically tune the g-tensor of multiple spins by shifting their carrier wavefunctions into areas of different material composition13. Extending this work to single charges trapped in zero-dimensional structures has not been straightforward as the carriers tunnel out of the structure when electric eld is applied.

One approach was demonstrated using electronically coupled pairs of QDs14. Carriers displayed the g-tensor of the material in which they were located, so when a voltage was applied to localize the charge in one dot, the g-factor measured was that of the indium-rich QD. However, when the wavefunction was delocalized between the dots, there was a much greater spatial overlap with aluminium arsenide semiconductor in the barrier, and a change in g was observed.

Recently, experiments showed that vertical electric elds can change the g-factor relevant for out-of-plane magnetic elds (g>)9 in dots that are engineered to have increased height and reduced indium composition. The in-plane g-factor of an s-shell hole (gkh;s) was also modied by vertical electric eld10 over a modest eld range of 20 kV cm 1. However, both of these measurements were made in the photo-current regime, where carriers rapidly tunnel from the dot greatly limiting the spin lifetime. Conversely, an in-plane electric eld can change the gtensor, but there the tunnelling problem is even more severe11,12.

We solve these problems by locating single dots in the centre of a p-i-n diode where barriers that hinder tunnelling allow us to apply electric elds, F, up to 500 kV cm 1, while still

observing photoluminescence15. We study changes in the gtensor as a function of electric eld and show that a high degree of control can be achieved for both electrons and holes. We observe that continuous variation in the g-factor of holes in a parallel magnetic eld can be obtained. Different behaviour is observed depending on whether the hole is in the s- or p-shell. When gkh;s has a low value at zero electric eld these devices are capable of tuning it through an avoided crossing at nite eld, and changing its sign, without carriers escaping.

ResultsCharged exciton transitions in magnetic and electric eld. When the magnetic eld is orthogonal to the plane of the sample (Faraday geometry, B>), we see that the g-factors are barely affected by electric eld (see Supplementary Fig. S1 and Supplementary Note 1). However, when the magnetic eld is aligned in the plane of the sample (Voigt geometry, B||), strikingly

different behaviour is observed. The separate in-plane g-factors of s-shell electrons (gke;s) and holes (gkh;s) may be determined from the decay energies of the positively (X ) and negatively (X )

charged excitons. The magnetic eld splits both the upper (gkh;smBBk) and lower states (gke;smBBk) of X , where mB is the Bohr Magneton. Four transitions (E1E4) are observed as shown in Fig. 1a,b. The highest and lowest energy transitions of this quadruplet (E1 and E4) emit photons with electric eld orthogonal to B and the intermediate transitions (E2 and E3) parallel to B. Fitting the energies of each transition resulting from the X state, one can determine gke;sX and gkh;sX using gke;sX mBBk E1 E3 E2 E4 and gkh;sX mBBk E1 E2 E3 E4. Similar arguments can be made to determine gke;sX and gkh;sX from the X transi

tions. There is not enough information in this measurement alone to determine the sign of these g-factors. However, for nearly all dots, we see an increase in the ne-structure splitting of the neutral exciton state with magnetic eld, which is a signature that both have the same sign16, which we take to be negative14,17.

gke;s appears to be constant for a given exciton complex, but on switching between the X and X transitions an abrupt step is always observed. The reason for this is that gke;sX is

determined by the initial state (when there are two holes also present in the dot). These holes are better conned than the electron and provide a coulomb attraction that reduces the extent of the electron wavefunction, pushing gke;s closer to

2(ref. 18). However, gke;s when no holes are present is determined from the nal state of the X transition. For the sample of 15 dots studied j gke;sX j 0:266 0:012 and j gke;sX j 0:178 0:033, where the numbers quoted are

the means.d.

In contrast to the behaviour of the electron, gkh;s varies linearly with electric eld for both X and X in Fig. 1e. We estimate that the discontinuity in gkh;s on switching between X and X is on average an order of magnitude smaller than the similar effect for gke;s, as expected, given the greater spatial extent of the electron wavefunction. There is remarkable homogeneity in the rate at which gkh;s can be tuned with electric eld for different dots, x j dgkh;s=dF j 5:7 1:5 10 4 cm kV 1, where j gkh;s j at zero electric eld is 0.4690.110. The scatter in the value of gkh;s

at F 0 is greater than the comparable value for the electron, as

this is affected more strongly by variations in dot height and lateral size18. The rate x compares well with the recent publication of Godden et al.10, which determines j gkh;s j from the energy

splitting of the X in a time-resolved photo-current measurement. This paper reports a linear variation in gkh;s at a rate of3.5 10 4 cm kV 1 over a range of only 20 kV cm 1. The rate

of shift is also of the same order of magnitude as that predicted theoretically, for dots of greater height and uniform composition6. It will be interesting to see whether further theoretical work can fully explain the variations in the g-tensor we observe.

Minimizing the g-factor of the s-shell hole. With the range of elds accessible in these samples, any dot with j gkh;s j 0.285 at

F 0 can be tuned to a minimum j gkh;s j at some eld, F0. We now discuss data from a dot with j gkh;s j 0.174 at F 0. This

same QD also displays a minimum in the ne-structure splitting

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2

1

1

+

X X +

2

2

+ 1

2

2

1

gh||,s (X )

ge||,s (X +)

E3

E3

E1

E4

E1

E4

E2

E2

ge||,s (X )

1

gh||,s (X +)

1

2

1

2

+

1

x2F F02 gkh;sF02

q

1

+ 2

Energy (meV)

1,330.9

1,330.8

1,330.7

1,330.6 1,331.1

1,331.2

1,331.4

y tan 1

gkh;sF0xF F0 gkh;sF

" #

2

In Fig. 2g,h, we show data summarising the behaviour of the two hole eigenstates at 4T (the lowest eld at which we can spectrally resolve all four transitions) and 5T (the highest eld available with our magnet) tted with this model. We observe that the magnitude of the anti-crossing in energy appears to scale linearly with B||, thus

j gkh;sF0 j is constant, at least in the range of elds we can probe.

The resulting variation of y with F is the same for both magnetic elds (Fig. 2h), in accordance with equations (1) and (2). It will be interesting to further probe the behaviour of this effect in higher magnetic elds.

g-factor of the p-shell hole. Finally, we study the decay of the positively changed biexciton, XX , which consists of a lled s-shell and a excess p-shell hole. These transitions are observed on the low-energy side of the X transition22,23. We determine the g-factors of the p-shell hole which, to our knowledge, has not been possible before (although the Voigt-geometry electron p-shell g-factor has been probed24). We nd that in the Faraday geometry, there is no variation in the p-shell hole g-factor as a function of electric eld. In a Voigt geometry, the brightest radiative decays from XX involve recombination of an s-shell electron and hole. The resulting photons are linearly polarized as shown in Fig. 3a (blue and red arrows have orthogonal linear polarization), with the initial state XX split by gkh;pmBBk, where gkh;p is the Voigt p-shell hole g-factor. However, the nal states can either have spin S 1/2 or 5/2: their splittings are partly

determined by the electron-hole exchange between the s-shell electron and p-shell hole, which has not been well studied. Empirically, we see that the spin splitting of the S 1/2 nal state is

below the system resolution at B|| 0, but increases with magnetic

eld. In contrast, the S 5/2 nal state has a spin splitting of

several hundred meV at B|| 0 but is reduced with B||. Nevertheless,

it is possible to measure the initial-state splitting XX using either the S 1/2 or S 5/2 nal state quadruplets, and thus infer j gkh;p j . When this is done, both quadruplets lead to the same value of j gkh;p j (Fig. 3e), as expected. We nd that j gkh;p j has a greater magnitude than j gkh;s j and varies non-linearly with

electric eld. The greater extent of the p-shell hole wavefunction outside the dot is likely to bring gkh;p closer to the value determined by the wetting layer and surrounding Gallium arsenide (GaAs).

DiscussionSeveral proposals exist for universal control of a single spin in a QD1,5,6,25. The ability of the device reported here to change

E1

E3

E4

1,331.3

E3

0 30 60 90 HWP angle (o)

0 30 60 90 HWP angle (o)

0.5

0.4

0.3

0.2

0.1

0 100 200 300 400

gh||,s (X +)

gh||,s (X )

g-factor

ge||,s (X +)

ge||,s (X )

Electric field (kV cm1)

Figure 1 | Electric eld tuning of s-shell electron and hole g-factors. (a,b) The energy levels of the negatively and positively charged excitons (X and

X, respectively) in a Voigt-geometry magnetic eld is showed. Transitions E1 and E4 (red) result in linearly polarized emission orthogonal to the magnetic eld, and E2 and E3 (blue) are parallel to the magnetic eld.

(c,d) Polarization-dependent spectra from the X at 78.5 kVcm 1 and

X at 385.7 kVcm 1, respectively, at a eld of 4T, as a half-wave plate

(HWP) is rotated. (e) The extracted s-shell electron g-factor, j gke;s j , and s-shell hole g-factor, j gkh;s j , as a function of electric eld is showed.

of the neutral exciton of 1.8 meV at 57.1 kV cm 1. We nd that j dgkh;s=dF j 7:74 10 4 cm kV 1, and thus we are able to tune the hole eigenstate splitting D j gkh;s j mBB towards a

minimum value at an electric eld of F0 225.0 kV cm 1. For

elds above F0 (such as shown in Fig. 2a), we observe that the sign of gkh;s is the same as for all other dots in the ensemble. For electric elds below F0 (such as in Fig. 2e), gkh;s has the opposite sign, which manifests itself as a clear difference in the orientation angle at which the largest difference in transition energies is observed. Figure 2f plots the X transition energies as a function of electric eld, F, to clearly show the form of the anti-crossing in the hole states (the mean value of all four transition energies has been subtracted for clarity, to remove the Stark shift). The minimum hole-state splitting corresponds to j gkh;sF0 j 0.042, but we stress that on either side of this minimum value, the gkh;s has different sign.

The behaviour of D is reminiscent of the anti-crossing in neutral exciton states that has been observed with electric eld15,19,20, however, in this case the states that are coupled together contain only a single hole. Indeed, in the analysis of Plumhof et al.21 who studied the anti-crossing of the neutral exciton states under externally applied strain, it was the hole wavefunction that dominated the orientation of the eigenstates relative to the laboratory (y) and anti-crossing of the eigenenergies. As with the neutral exciton, we t the avoided crossing with a coupling parameter gkh;sF0mBBk, where the splitting between the hole states

varies linearly away from F0 at a rate xmBB||.

D gkh;sFmBBk mBBk

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a

Angle to B, [afii9835](o ) Hole-state splitting, ( eV)

1,345.8

60

f

g

5T

50

1,345.7

E1

E2

E4

40

40

30

4T

b

c

d

e

1,343.9

3T

2T

1T

gh||,s[afii9839]BB

1,343.8

20

20

1,342.5

10

Energy (meV)

0

ge||,s[afii9839]BB

0

1,342.4

E3

80

h

1,340.5

60

1,340.4

40

1,336.2

1,336.1

1

0

20

1,336.0

0

100 200 300 400 500

Electric field, F (kV cm1)

500 Electric field, F (kV cm1)

0 30 60 90

100 200 300 400

Half wave plate angle (degrees)

Offest energy (eV)

20

40

60

Figure 2 | Changing the sign of the Voigt-geometry s-shell hole g-factor with electric eld. Polarized spectra of the positively charged exciton, X , for an in-plane magnetic eld of 5T and electric elds of (a) 107.1 kVcm 1, (b) 185.7 kVcm 1, (c) 221.4 kVcm 1, (d) 264.3 kVcm 1 and

(e) 335.7 kVcm 1. (f) The energies of the four transitions of the X offset by their mean value at each electric eld is showed. (g) The magnitude of the

hole g-factor, j gkh;s j , and the (h) orientation (y) of the states relative to the magnetic eld. Both g,h show ts for 15T based on equations (1) and (2).

a

B = 0T

B > 0T

1,350

XX +

=

0.1

0.0

0.1

0.1

0.0

0.1

b

d

X +

Energy (meV) p- shell hole g- factor

1

1,345

XX +

=

2

=

1,340

S = 5/2

1

1

,

XX ++

2

2

1

+

1,335

2

S = 1/2 1,330

2

Offset energy (meV)

S = 5/2 S = 1/2

1

c

e

S= 5/2S= 1/2

1

1

,

2

2

+

0.65

1

2

1

0.60

2

1

1

,

2

2

1

+

0.55

2

100 300 400

Spin flip into X +

200 100 300 400

200 Electric field (kV cm1)

Figure 3 | Determination of Voigt-geometry p-shell hole g-factor. (a) The allowed transitions for recombination of an s-shell electron-hole pair of the positively charge biexciton, XX , is showed. Red and blue arrows indicate photon emission with opposite linear polarization. (b) The energies of the quadruplet with S 5/2 nal state, offset by their mean at 4T, and (c) the energies of the quadruplet with S 1/2 nal state, offset by their mean at 4T, as

a function of electric eld. (d) The absolute energy of the transitions shown in a versus electric eld, at 4T. From b,c, we independently extract the magnitude of the p-shell hole g-factor (e) for the S 1/2 (red) and S 5/2 (blue) transitions.

the sign of j gkh;s j , combined with the reduced holehyperne

interaction and greater hole-spin lifetime open up the possibility of all-electrical 4p manipulation of the hole spin. Alternatively, controlled phase shifts may be achieved on a qubit encoded on the spin of the electron by addition of two holes for a

predetermined time, which could be achieved by controlled charging.

The timescale of any electrical control sequence is limited by the resistance and capacitance of the diode to tens of picoseconds26, which is signicantly greater than that achieved

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with coherent optical pulses. However, the ability to achieve full Bloch-sphere control with only a single electrical gate is a promising avenue of investigation. Such a device could nd applications in a spin-based quantum memory27, spin-echo techniques3,4, spin-based quantum computing1,28 and generation of photonic cluster-states29.

Methods

Sample design. The sample consists of a single layer of self-assembled QDs grown in the centre of a 10-nm wide GaAs quantum well, clad with a 75% AlGaAs superlattice, which suppresses the tunnelling of carriers. These dots are grown in a single deposition of InAs at a substrate temperature of 470 C and with a transition to self-assembled 3D growth at 60 s. The resulting dots are 23 nm in height, and are capped in 5 nm of GaAs at 470 C before raising the substrate temperature for growth of the superlattice. p- and n-doping regions are arranged symmetrically above and below the QD layer, with a total intrinsic region thickness of 140 nm. The diode is encased in a weak planar microcavity, with micron-sized apertures in a metallic layer on the surface to allow optical addressing of single dots.

Experimental arrangement. The sample is mounted inside the bore of a super-conducting magnet applying elds of up to 5T. When the sample growth direction is aligned with the magnetic eld, a single on-axis microscope objective is used to excite and collect the emission from the sample. When the magnetic eld is in the plane of the sample (Voigt geometry), an additional 45 mirror is mounted to allow optical access to the sample. Photoluminescence is excited from the sample with a continuous wave 850 nm laser diode, and passed through a rotatable half-wave plate and polariser before detection. For the data in Fig. 2, spectral measurements conrm that the sample was orientated within 0.1 of the magnetic eld direction.

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Acknowledgements

This work was partly supported by the EU through the Integrated Project QESSENSE (project reference 248095), the Marie Curie Initial Training Network (ITN) Spin-Optronics (project number 237252) and EPSRC.

Author contributions

The samples were grown by I.F. and D.A.R. and processed by M.A.P. The optical measurements were made by A.J.B., M.A.P., Y.C. and N.S. A.J.S. guided the work. All authors discussed the results and their interpretation. A.J.B. wrote the manuscript, with contributions from the other authors.

Additional information

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

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How to cite this article: Bennett, A. J. et al. Voltage tunability of single-spin states in a quantum dot. Nat. Commun. 4:1522 doi: 10.1038/ncomms2519 (2013).

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