Table Of ContentSpin-valley qubit in nanostructures of monolayer semiconductors: Optical control and
hyperfine interaction
Yue Wu,1 Qingjun Tong,1 Gui-Bin Liu,2 Hongyi Yu,1 and Wang Yao1
1Department of Physics and Center of Theoretical and Computational Physics,
The University of Hong Kong, Hong Kong, China
2School of Physics, Beijing Institute of Technology, Beijing 100081, China
We investigate the optical control possibilities of spin-valley qubit carried by single electrons
6
localized in nanostructures of monolayer TMDs, including small quantum dots formed by lateral
1
0 heterojunction and charged impurities. The quantum controls are discussed when the confinement
2 induces valley hybridization and when the valley hybridization is absent. We show that the bulk
valleyandspinopticalselectionrulescanbeinheritedindifferentformsinthetwoscenarios,bothof
n
whichallowthedefinitionofspin-valleyqubitwithdesiredopticalcontrollability. Wealsoinvestigate
a
nuclear spin induced decoherence and quantum control of electron-nuclear spin entanglement via
J
intervalley terms of the hyperfineinteraction. Optically controlled two-qubit operations in a single
7
quantumdot are discussed.
2
PACSnumbers: 78.67.Hc,03.67.Lx,73.61.Le,71.70.Jp
]
l
l
a
h I. INTRODUCTION extended monolayer, e.g. by patterned electrodes, simi-
- larly to the quantum dots in III-V heterostructures,and
s electrically controlled spin qubit has been proposed.9,10
e
Single spins at semiconductor nanostructures have
m Alternatively,quantumdotconfinementcanalsobereal-
been widely explored as information carriers in quan-
izedbythe lateralheterojunctionsona singlecrystalline
. tum computing, quantum spintronics and quantum
t monolayer,butwithdifferentmetalelements inandout-
a metrology.1–3 These solid state qubit systems of inter-
m sidethequantumdotregion(c.f. Fig. 1),wheretheband
est include the spin of single electrons or holes local-
offset between the different TMD compounds forms the
- ized at quantum dots or by impurities formed in various
d potential to confine single electron or hole. Lateral het-
bulksemiconductorssuchastheIII-Vcompounds,silicon
n erostructureswithMoSe islandssurroundedbyWSe on
2 2
o and diamond, and their heterostructures and nanoscry- a crystallinemonolayerhavebeen demonstratedveryre-
c tals. These electron and hole spin qubits have demon- cently using chemical vapor deposition growth,although
[ strated remarkable optical and electrical controllability,
the length scale of the island is µm, still too large for
1 relatively long coherence time at low temperature com- confining single electron.11 The∼TMDs monolayers also
v pared to the unit operation time, and potential integra- hostvariousatomicdefectswhichcanlocalizesingleelec-
8 bilitywithexistingsemiconductortechnologies. Through tronorholeaswell.12–16 Remarkably,recentexperiments
6 the hyperfine interactions with the electron or hole spin
have shown that certain types of defects in monolayer
2 qubits, the lattice nuclear spins also play crucial roles in
WSe are excellentsingle photonsources,emitting atan
7 2
these solid state qubit systems, either as additional in-
0 energy which is a few tens meV below the exciton in
. formation carriers with the advantage of extremely long the 2D bulk.17–21 Such behaviors of the TMDs defects
1 storage time, or as a deleterious noise source that need
resemblethe shallowimpurities inconventionalsemicon-
0
to be suppressed.
6 ductors (e.g. Si donor in III-V compounds) that localize
1 Atomically thin two-dimensional(2D) semiconductors singleelectron(orhole)aswellassingleexciton,implying
: offernewopportunitiesforquantumspintronicsandspin the possibility towards optical control of single electron
v
based quantum information processing. An electrically spin.22
i
X controllable spin qubit system based on 2D materials Optically controlled spin qubit is highly appealing in
r wasfirstproposedingraphene,agaplesssemiconductor.4 monolayerTMDsbecauseoftheinterestingopticalprop-
a
Monolayer group-VIB transition metal dichalcogenides erties of the 2D bulk. The monolayer TMDs have a
(TMDs) have recently emerged as a new class of direct unique band structure with the conduction and valence
gap2Dsemiconductorswithappealingopticalproperties band edges both at the degenerate K and -K valleys at
and rich spin physics, implying their greatpotentials for the corners of the hexagonal Brillouin zone. The direct-
hosting optically controlled spin qubits.5,6 These com- gapopticaltransitionshaveaselectionrule: left-(right-)
pounds are of the chemical composition of MX (M = handed circular polarized photon couples to the inter-
2
Mo,W;X=S,Se). ThemonolayerisaX-M-Xcovalently bandtransitionsintheK(-K)valleyonly.23,24Thisvalley
bonded hexagonal 2D lattice, with a direct bandgap in dependentopticalselectionrulehasmadepossibleinthe
the visible frequency range which is ideal for optoelec- 2D bulk the optical pumping of valley polarization,25–27
tronicapplicationsandfortheexplorationofopticalcon- and optical generation of valley coherence.28 Moreover,
trol of spin.7,8 Single electrons can be confined in quan- the spin-orbit coupling from the metal atoms gives rise
tumdots definedbylateralconfinementpotentialsonan to a pronounced coupling between the valley pseudospin
2
and spin,6,29,30 through which the optical selection rule and present the optical selection rules for the quantum
becomes a spin dependent one, allowing the optical con- confined states. Coherent rotations between valley hy-
trol of spin as well. This suggests that the valley pseu- bridized states by optical control will be discussed. In
dospin and spin of a single electron can be a promis- Sec. IV, we discuss the electron states in the absence of
ing qubit carrier with optical controllabilities, as long as valleyhybridizationwhenthe confinementhas the afore-
these bulk properties can be inherited when the electron mentioned rotational symmetry. The optical control is
is localized in the monolayers. achieved with the help of external magnetic fields. In
A major difference in the spin and valley pseudospin Sec. V, we discuss the hyperfine interactions of the con-
physics expected between the bulk electron and the lo- finedelectronsandholeswithlattice nuclearspinsinthe
calizedelectronis the intervalley coupling andvalley hy- envelope function approximation. We propose to opti-
bridizationbytheconfinement. Thisissuehasbeenstud- cally control the electron-nuclear spin entanglement via
ied for quantum dot confinement potentials on extended intervalley terms of the hyperfine interaction. The de-
monolayers,31 where the intervalley coupling is found to coherence time of the localized electron spin caused by
be weakfor quantum dotwith lateralsize largerthan20 interactingwithlatticenuclearspinsisdiscussed. InSec.
nm ( meV or orders smaller, depending on the shape VI, we discuss the possibility to realize two-qubit logic
ands∼izeofthe dot). In suchacase,the valley hybridiza- operations between the spin qubit and the valley qubit
tion is well quenched by the much stronger spin-valley carried by a single electron in a quantum dot. Finally,
coupling,andthequantumdotcanwellinheritthevalley conclusions are given in Sec. VII. Appendix A uses a
and spin physics of the 2D bulk. Interestingly, a sensi- three-band tight-binding model to estimate the interval-
tive dependence of intervalley coupling strength on the ley coupling strength in the confinement by charged im-
central position of the confinement potentials is discov- purity and small quantum dot. In Appendix B, we an-
ered. ItisfoundthatwhenthepotentialhasC orhigher alyze the terms in the electron-nuclear and hole-nuclear
3
rotationalsymmetry,the intervalleycoupling completely hyperfine interactions based on symmetries of the rel-
vanishes if the potential center is at a chalcogen atom evant atomic orbitals, and estimate the bulk hyperfine
site or the hollowcenter ofthe hexagonformedby metal constants.
and chalcogen atoms, which is due to the dependence of
the eigenvalue of C rotationoperator on the locationof
3
the rotation center.31,32 II. CONFINEMENT OF SINGLE ELECTRON IN
In this work, we investigate the optical control possi- THE NANOSTRUCTURES
bilities of spin-valley qubit carriedby single electrons lo-
calized in nanostructures of monolayer TMDs, including If the length scale of the confinement potential is still
chargedimpurities andsmallquantum dots (with length much larger than the lattice constants, the bound states
scaleof10nm orsmaller). We discuss the quantumcon- are formed predominantly from the band-edge Bloch
trols under two different scenarios: (i) in the presence of statesintheKand-Kvalleysofthe2Dbulk. Ingeneral,
valley hybridization due to the strong confinement and each eigenstate in the confinement is a hybridization of
(ii)intheabsenceofvalleyhybridization. Thelattersce- Bloch states from the K and -K valleys due to the inter-
nario is considered for the confinements that has C3 or valley coupling introduced by the confinement potential,
higher rotationalsymmetry about a chalcogen atom site except when the potential has a C rotational symme-
3
orthehollowcenterofthe hexagonformedbymetaland tryabouteither achalcogenatomsite orahollowcenter
chalcogen atoms, or when this symmetry is only weakly of the hexagon formed by metal and chalcogen atoms.31
broken so that the residue intervalley coupling can be In the absence of valley hybridization, the bound-state
wellquenchedbythespin-valleycoupling. We showthat eigenfunctionscanbeconstructedfromBlochstatesfrom
the bulk valley and spin optical selection rules can be the K or -K valley only, denoted as Ψ where τ =
τ,s
±
inherited in different forms in the two scenarios, both is the valley index for the K valley, and s = ( ) de-
± ↑ ↓
of which allow the coherent rotation between electron notes spin up (down) state. This is a convenient basis
states controlled by optical pulses. The hyperfine inter- for our discussion, even when intervalley coupling and
action between lattice nuclear spins and the electron or valley hybridization are present. Intervalley coupling is
hole spin is also formulated within the envelop function thentheoff-diagonalmatrixelementsbetweenthesebasis
approximation, and the nuclear spin induced decoher- states due to the confinement potential. If confinement
ence of the spin-valley qubit is analyzed. We find that potential is spin-independent, intervalley coupling van-
the short-range nature of the hyperfine interaction gives ishes between states with opposite spin index. With the
risetointervalleyterms,whichcanbeutilizedforoptical large quantization energy in the confinement potential
control of the electron-nuclear spin entanglement. (seeAppendixA),wecanfocusonlyonthegroundstates
The rest of the paper is organized as follows. In Sec. foreachspinandvalleyindex,whiletheexcitedstatesare
II, we give a brief account of the nanostructures being far off resonance concerning either the valley hybridiza-
consideredhereforopticallycontrolledspin-valleyqubit. tion effect or the optical controlof the spin-valley qubit.
In Sec. III, we discuss the electron states in presence Below we consider two types of confinements.
of valley hybridization expected in strong confinement, The firstis the lateralheterostructuresbetweendiffer-
3
Figure 1: Schematics of nanostructures of monolayer TMDs for optically controlled spin-valley qubit: (a) small quantum dots
in lateral heterojunction and (b) charged impurity system. In (a), theheterojuction is formed of aMoSe island in monolayer
2
WSe . An electron is confined in the MoSe quantum dot (bottom panel) and is excited to the trion state (top panel). The
2 2
schematics of the confinement potential is shown on the left. In (b), the impurity system is formed in monolayer WSe when
2
a W atom is replaced by a Reone. The D0 system is shown at thebottom and D0X is shown at thetop.
ent TMDs, for example, a MoSe island surrounded by meV. Valley hybridization for the hole component is al-
2
WSe on a crystalline monolayer. With a type-II band ways negligible due to the giant spin-valley coupling of
2
alignmentbetweenthetwoTMDs,suchaheterostructure hundred of meV in all TMDs.
forms a confinement potential of a vertical wall, which
localizesasingleelectroninthe MoSe regionwitha po-
2
tential depth of few hundred meV. The valley and spin
degrees of this electron can then define a qubit. In the
optical control, an optical field can couple the states of
the single electron to the optical excited states of trion
(i.e. two electrons plus a hole) through the interband The second type of nanostructure is a neutral donor
transition. These trion states are utilized as intermedi- systemD0, for example a Re replacing a W in the WSe
2
ate states for the optical control of the single electron monolayer, where the positively charged impurity binds
states. Althoughthe heterostructureitselfdoesnotform the extra electron and forms a hydrogenic state. Simi-
aconfinementforasinglehole(c.f. Fig. 1),thequantum larly to the quantum dot case, the single electron states
confinementoftheelectronconstituentswillnevertheless canbeopticallycoupledtothedonorboundexcitonD0X
localize the trion at the heterostructure. states. In GaAs, D0-D0X system has been extensively
The intervalley coupling strength grows with the de- exploredfor opticallycontrolledsinglespin.22 Compared
crease in size of the quantum dot. At a lateral size of withthequantumdot,theD0-D0X systeminmonolayer
5 nm, the coupling matrix element reaches 0.1 1 meV TMDs is expected to be a much tighter confinement due
−
depending on the quantum dot shape. The valley hy- tothe enhancedCoulombinteraction. Consequently,the
bridization will then be determined by the competition intervalleycouplingstrengthismuchstronger(unlessthe
of this off-diagonal matrix elements in the basis Ψ , impurity is centered at a chalcogen atom site or a hol-
τ,s
and the diagonal energy differences between Ψ and low center of the hexagon formed by metal and chalco-
+,s
Ψ due to the spin-valley coupling in the band of the genatoms). ForseveralexamplaryelectrostaticCoulomb
,s
2D−bulk. For monolayer MoS , the spin-valley coupling potential as shown in Appendix A, we find the interval-
2
strengthin the conduction band is 3 meV,33 comparable ley coupling strength can be comparable to the electron
to the achievable intervalley coupling in small quantum spin-valleycouplingstrengthinMoSe ,MoS andWSe .
2 2 2
dots. For other three TMDs (MoSe , MoS and WSe ), Therefore,thevalleyhybridizationofelectronisexpected
2 2 2
the spin-valley coupling strength is in the range of 20-40 to be significant in the D0-D0X system.
4
Since intervalley coupling conserves spin, we re-write
the Hamiltonian in a compact form,
cosθ sinθ
H = d~ ~τ =d (2)
0 · (cid:18) sinθ cosθ (cid:19)
−
with d~= (h,0,λs)= d(sinθ,0,cosθ). The eigenenergies
2
are ǫ = sd, with eigenvectors,
1(2),s
±
cosθ
u ,s = 2 , (3)
| 1 i (cid:18) sinθ (cid:19)
2
sinθ
u ,s = 2 . (4)
| 2 i (cid:18) cosθ (cid:19)
− 2
These four spin-valley configurations of the single elec-
tron can be used to construct the qubit.
Figure 2: Valley and spin dependent optical transition selec-
Ourproposedopticallycontrolledqubitoperationsrely
tion rules at the band edge of the monolayer WX (a) and
2
MoX (b). (c) Optical transition selection rules in quan- on the optical selection rules in monolayer TMDs.24 In
2
tum dots with valley hybridization of the localized single 2D bulk of monolayer TMDs, the conduction (valence)
electron. Solid (dashed) horizontal lines denote spin-up (- band edge states mainly consist of transition metal dz2
dvaolwleny). Rsteadteds.ouDblaerdk-agrrreoewned(blilnaceks)decnolootredσe−n-optoelsar+izKed(li−ghKts) 0(d(xm2−y2=±id2x)y.)AortbtithaelswKithptohinetms,atghneetBicloqcuhansttautmesmhcav=e
v
satnrdenbgltuheionndesasdheendotoeneσ+∝-psoinlaθriz(ei.de.liegnhatbs,lewditbhytahefintritaensinittieorn- C3 rotatio±nsymmetry±C3|τ,si=e−i2m3π |τ,si,whichim-
2 pliesanazimuthalselectionrulefortheallowedinterband
valley coupling, see text), and solid one ∝cosθ.
2 opticaltransition(m m 1modulo3)=0. Becauseof
c v
− ∓
inversion symmetry breaking, this optical selection rules
is valley-contrasted. The spin-valley locking of the holes
III. OPTICAL CONTROL OF ELECTRON further makes these selection rules spin-dependent: σ+
STATES IN PRESENCE OF VALLEY
circular polarization optical field can generate spin-up
HYBRIDIZATION
electrons and spin-down holes in valley K, while the ex-
citation in the K valley is simply the time-reversal of
In this section, we consider the scenario where the the above, as sh−own in Fig. 2 (a) for WX systems and
2
confinement potential introduces pronounced valley hy- Fig. 2 (b) for MoX systems. Since these two kinds of
2
bridization of the localized electron. This applies to the systems only differed by the sign of spin splitting in the
confinementpotentialsofsmalllengthscalewhichdonot conduction band33, we illustrate our results with WX
2
have the C3 rotational symmetry about either a chalco- system in all of the following figures.
gen atom site or a hollow center of the hexagon formed In the context of a quantum dot chargedwith a single
bymetalandchalcogenatoms(seeSec. IIandAppendix electron, an optical field can couple the different spin-
A). valley states of the single electron to an charged exci-
We note that valley hybridization is present for elec- tonstate(trion)ofthevariousspin-valleyconfigurations.
trons only. For holes, the band edges of the 2D bulk are Thesetransitionshaveopticalpolarizationselectionrules
spin-valley locked because of the giant spin-orbit cou- inherited from the 2D bulk. With the valley hybridiza-
pling, i.e. valley K (-K) has spin down (up) holes only. tion of electrons, the optical transitions in fact become
Astheconfinementpotentialdoesnotflipspin,valleyhy- more intricate in the quantum dot. As shown in Fig.
bridizationbytheconfinementiscompletelyquenchedfor 2 (c), there are six bright trion states that can be cou-
the spin-valley locked holes. For electrons with a much pled to the four spin-valley states of the single electron.
smaller spin-valley coupling in the 2D bulk band edges, The dashed arrows denote the transitions with strength
we take into account both spin species in each valley, sinθ (i.e. enabled by a finite intervalley coupling
∝ 2
andthequantumdotHamiltonianintheaforementioned h), while the solid arrows denote the transitions with
basis is, strength cosθ.
∝ 2
Amongallthepossibleopticaltransitions,wenotethat
λ u , and u , can both be coupled to the same trion
H0 =hτx+ τzsz, (1) | 1 ↑i | 2 ↑i
2 state |X−,↑i = e†+, e†, h†+, |Gi with a σ+ circular po-
↑ −↑ ⇓
larizedlight. Heree createsanelectronstatewithspin
where h is the intervalley coupling strength, τ and s de- †τ,s
note the pauli matrices operating at valley and real spin s and valley τ and similarly h†τ,s′ creates a hole state
degrees of freedom, and λ is spin-valley coupling of con- with spin s = , with G denoting an empty conduc-
′
⇑ ⇓ | i
duction band. tionbandandfullvalenceband. Therefore u , , u ,
1 2
| ↑i | ↑i
5
is transformed to
0 0 E D sinθe iα
1 0 2 −
H = 0 0 E2D0cosθ2 ,
E D sinθeiα E D cosθ ∆
1 0 2 2 0 2
(6)
where the fast oscillating terms e 2idt have been ne-
−
∝
glected. For large detuning, the trion state is eliminated
via using the adiabatic approximation. The dynamics of
Figure3: Opticallycontrolledrotationbetweenthevalleyhy- the qubit is then described by
bridizedstatesviatheRamantypeprocessesmediatedbythe
trionstate. TherearetwoΛ-typethree-levelsystemsthatcan D2 E2sin2 θ E1E2 sinθeiα
H = − 0 1 2 2 (7)
btiecaclopnturlosellsedwritehspσe+ctipvoellyarviziaattiohneR(aa)moarnσt−yppeoplarroiczeastsiobny(obp)-. eff ∆ (cid:18) E12E2 sinθe−iα E22cos2 θ2 (cid:19)
The quantum dot can have four states for encoding infor- which can be rewritten as
mation that are distinguished by the spin index and can be
H =n I +~n ζ~, (8)
selectively accessed using circularly polarized light. eff 0
·
with
3an(da)|)X. −S,i↑miilfaorrlmy,aaΛσ-−typcierctuhlraere-pleovlaerlizsyedstelimgh(tc.cfo.uFpliegs. n0 = −D02(E12sin222θ∆+E22cos2 2θ),
|u1,↓iand|u2,↓iwith|X−,↓i=e†+, e†, h†, |Gi,form- n = E1E2D02sinθcosα,
ing another Λ-type three-level syste↓m−(c↓.f.−F⇑ig. 3 (b)). x − 2∆
We note that a single quantum dot can now have four E E D2sinθsinα
n = 1 2 0 ,
states for encoding information: u , , u , , u , , y
1 2 1 2∆
t|uiv2e,↓aic}c.esσs+ofotrhσis−Hpiloblaerrtizespdalcieghft{o|rmeaitk↑heiesr|pionsist↑iiabilliez|asteiloe↓nci-, n = D02(E12sin2 θ2 −E22cos2 2θ), (9)
z
− 2∆
readout, or quantum control, where optical control sce-
nariosutilizing the Λ levelscheme canbe borrowedfrom where ζ~ operates on our defined qubit, which precedes
optically controllable III-V quantum dots.34,35 under this pseudo-magnetic field ~n.
For example, coherent rotation between the pair of The effectofintervalleycouplingis involvedin the an-
states u1, , u2, (or u1, , u2, ) can be re- gle θ. Without intervalley coupling, θ = 0, ~n only lies
{| ↑i | ↑i} {| ↓i | ↓i}
alized through an optical Raman process via the inter- in z direction. Therefore, intervalley coupling plays an
mediate trion states X−, (or X−, ) by σ+ (or σ−) crucial role in the optically controlled single-qubit oper-
| ↑i | ↓i
polarizedlightintheΛ-typethree-levelsystem.36 Apply- ation. In general cases with finite intervalley coupling,
ingtwophase-lockedopticalpulseswithσ+ polarization, arbitrary pseudo-magnetic field orientation can be ob-
the light-matter interaction Hamiltonian in the rotating tained by changing the control parameters E , α and
1,2
wave approximation is ∆. Forexample,whenoneofthetwopulsesisturnedoff,
i.e. E or E being set to zero, ~n is in z direction. This
1 2
HI =jX=1,2Ωj(t)(cid:12)X−,↑(cid:11)huj,↑|+H.c., (5) realizes a single-qubit phase-shift gate USφ = (cid:18)10 e0iφ (cid:19)
(cid:12)
if we set E = 0, where φ = D02E22cos2 θ2t. On the other
where the Rabi frequencies are of the forms Ω (t) = 1 ∆
αwDE1i10tDh−=0Eαsuij2njb,2θ≡eeiinDωαg1t−btXheiαe−in1,agmatnphisdleitthuΩreed2lea(ottpo)itvfiect=ahplehtErapas2oneDlas0ibrticeizootewnsdeθ2meleinagiω1th2rtttihx−aeimneαld2-. bshwhyaiatnahpdne,ndowxpMh=teiocn−SalEEEp12q12uDu=la02s∆nestitanwu2nmiθ2tθ2h.daTdonuthdrewaαqtitiu=ohbn0iltat,fts~net=raiatsle2isn|wnπizxote|hu.eoldFfxob3rdenisrflmqeiucp,taptiroehedne-
h ↑| | ↑i 2
ement between the localized electron state and the trion
intervalley coupling is calculated as 1 meV if the lateral
τ,sDτ,s′
state,whichisapproximatelyproportionalto h | | i, confinement potential is set as 0.2 eV (see Appendix A).
aH
where τ,s D τ,s is the optical transition matrix ele- When a light with E D = 0.5 meV is applied, n 1
′ 1 0 x
h | | i | |∼
ment between the bulk conduction and valence states at µeV and t is about 0.8 ns, if we set the detuning ∆=5
f
τK points and a is the Bohr radius of the trion state. meV.
H
Becauseofthe strongCoulombinteraction,D isseveral
0
times larger than the one in III-V semiconductor quan-
tumdots. Thefrequenciesω =E ∆ ǫ arechosento IV. ELECTRON STATES IN ABSENCE OF
j t j
satisfy the Raman conditions with−E −and ∆ being the VALLEY HYBRIDIZATION
t
trion energy and Raman detuning respectively. In the
rotating frame defined by e iǫ1t u , , e iǫ2t u , and If the confinement has C symmetry, intervalley cou-
− 1 − 2 3
e−i(ET−∆)t X−, , the total Ham| ilto↑niian H =| H0↑i+HI pling vanishes when the confinement center is chosen
| ↑i
6
at the chalcogen atom site or the hollow center of the
hexagon lattice.31,32 In this case, valley is a good quan-
tum number, and the quantum dot states are formed
from the Bloch states in a single valley of the 2D bulk.
Theopticaltransitionsofthespin-valleystatesofthesin-
gle electrons to trions in Fig. 2 (c) then reduces to those
in Fig. 4 (a).
Opticalcontrolofthespinstatesisstillpossibleinthe
presenceofamagneticfieldwithanin-planecomponent,
which can couple the spin up and down states from the
samevalley. With externalmagnetic fields,the Hamilto-
nian for the single electron at each valley becomes
λ
H = τ s +B s +B s =d~ ~s, (10)
0′ 2 z z x x z z ′·
whered~ =(B ,0,λτ+B )=d(sinθ ,0,cosθ )istheef-
′ x 2 z ′ ′ ′
fectivefieldonthe spindoubletateachvalley,asplotted Figure 4: (a) Optical transition selection rules in quantum
schematically in Fig. 4 (b), which is valley-dependent in dotswithoutvalleyhybridization,andinabsenceofmagnetic
general. The eigenstates of this Hamiltonian are field. (b) Configurations of effective magnetic fields in the
two valleys. The dark blue, light blue, green and yellow ar-
cosθ′ rows indicate Bz, Bx, λ and total field d~′ respectively. In
|u′1,τi=(cid:18) sinθ2′ (cid:19), (11) each valley, theeigenstates of theeffective magnetic field are
2 indicated (black spots).
sinθ′
|u′2,τi=(cid:18) cos2θ′ (cid:19). (12)
− 2
These spin-coupled states can be used to construct the
qubits. IncontrasttothescenarioinSec. III inpresence
ofthevalleyhybridization,thevalleyindexisnowagood
quantum number while the spin is now quantized along
a direction tilted from z. Coherent rotation between the
pairofstates u ,+ , u ,+ (or u , , u , )can
{| ′1 i | ′2 i} {| ′1 −i | ′2 −i}
be realized through an optical Raman process via the
intermediate trion states (c.f. Fig. 5).
Inanappliedmagneticfieldwithafinitein-planecom-
ponent,therearesixbrighttrionstates,asshowninFig.
5(a). Onecanseethatthetwostates u ,+ and u ,+
| ′1 i | ′2 i
abryeacoσu+plpeodlatroiztehdeltirgihont.sStaimteil|aXrl−y,,+ui,=e†+a,n↓ed†+,u↑h,†+,⇓|aGrei
| ′1 −i | ′2 −i
coupledtothetrionstate|X−,−i=e†, e†, h†, |Giby Figure 5: (a) Optical transition selection rules in quantum
−↑ −↓ −⇑
a σ polarized light (c.f. Fig. 5 (b) and (c)). By virtual dotswithoutintervalleycouplingwhilewithappliedmagnetic
−
excitation of these trion states, single qubit operations fields. Thecouplingstrengthindashedline∝sinθ′ andsolid
2
including the spin initialization and spin rotations can one ∝ cosθ′. There are two Λ-type three-level systems that
be controlled via optical Raman process.35,37 The effec- canbecont2rolledrespectivelyviatheRamantypeprocessby
tive Rabi frequencies for the qubits are the same as the opticalpulseswithσ+polarization(b)orσ−polarization(c).
ones in Eq. (9) while with θ being replaced by θ here. The quantum dot can therefore provide four states that are
′
B playsthe same roleasintervalley couplingh inthe distinguished by the valley index and can be accessed using
x
formercasediscussedinsec. IIIanditscompetitionwith circularly polarized light.
spin-valley coupling λ determines the operation speed.
For MoS with λ being a few meV, B can be the same
2 x
order of magnitude in a magnetic field of a few Tesla.
For the other three group-VIBTMDs, λ 20 40 meV,
∼ − optical control of the states. Note that because of the
which is much larger than B in most conditions. The
x difference in the effective field d~, the effective Rabi fre-
coupling strength of the optical transition from u ,τ ′
| ′1 i quency and hence the operation speed differ by a factor
to X ,τ in the Λ-level scheme is a weak one, propor-
tion|al−to iBλx. This will limit the operation speed for the qBBxx22++((λλ//22+−BBzz))22 for the two valleys.
7
V. INTERPLAY OF LATTICE NUCLEAR SPINS of the wavefunction. For electrons, we consider the pro-
WITH CONFINED ELECTRON AND HOLE jectedformofthehyperfineinteractionbetweenthebasis
states Ψc (~r),Ψc (~r),Ψc (~r),Ψc (~r) . For holes,
+, +, , ,
Electrons and holes localized in semiconductors can n ↑ ↓ −↑ −↓ o
with the giant spin splitting at the valence band top,
be coupled to the environment consists of phonons and
we only need to consider the two-fold spin-valley locked
lattice nuclear spins. At a temperature low for the elec-
basis: Ψv (~r),Ψv (~r) .
trons but high for the nuclear spins (i.e. 10 mK - K), +, ,
n ↑ −↓ o
the effects of phonon can be well suppressed, leaving the We haveusedtwoapproachesto obtainthe band edge
lattice nuclear spins as the ultimate environmental de- Blochfunctions. Inthefirstapproach,weextracttheor-
grees of freedom.35 In MX nanostructures, the stable bital compositions of the band edge Bloch states from
2
isotopes of the relevant elements with nonzero nuclear first principle calculations, and then write the Bloch
spin include: (95Mo, 5/2, 15.92%), (97Mo, 5/2, 9.55%), functions by using the Roothaan-Hartree-Fock atomic
(183W, 1/2, 14.31%), (33S, 3/2, 0.76%) and (77Se, 1/2, orbitals.41,42 In the second approach, we use the numer-
7.63%), where the second number in the bracket gives ically calculated Bloch functions from Abinit.43–45 The
the nuclear spin quantum number and the third gives two approaches give consistent results on the form and
the natural abundance.38 magnitude of hyperfine interactions. Details aregivenin
We derive here the forms of hyperfine interaction be- Appendix B, and the forms are summarized below.
tween the localized electron and hole with these lattice
nuclear spins in the envelope function approximation. MoS MoSe WS WSe
2 2 2 2
This is applicablefor the localizedelectronwavefunction ̺ 0.23 0.23 0.37 0.40
formedlargelyfromthebandedgeBlochfunctionsatthe AcM -0.50 -0.51 0.76 0.79
K points. These band edge Bloch functions are mainly AvM -1.52 -1.53 1.78 1.82
±contributed from the metal d-orbitals and a small but AcX 0.05 0.46 0.08 0.33
finite componentofthe chalcogenp-orbitals. The hyper- AvX -0.16 -1.33 -0.37 -1.63
fine coupling strength with the metal nuclear spins are
Table I: Hyperfine constants evaluated based on Bloch func-
therefore stronger. The chalcogen nuclear spins are ex-
tions constructed using Roothaan-Hartree-Fock atomic or-
pectedtoplaylessimportantroles,forboththeweakness
bitals (see text). Ac(v) is in unit of µeV and ̺ is dimen-
ofthehyperfineinteractionstrengthandthesmallernat- M(X)
sionless.
ural abundance of the stable isotopes with finite nuclear
spins.32,38
(i) Electron hyperfine interaction with M atom:
We find that, similarly to both the electron and hole
hyperfine interactions in III-V semiconductors,39,40 the
hyperfineinteractionhereisoftheshort-rangenature: it HMc = AcM Ω|Fc(R~k)|2 IzkSz +̺(IxkSx+IykSy)
needs to be counted only for nuclear spins in direct con- Xk (cid:2) (cid:3)
tact with the electron or hole, with a coupling strength 1+e−2iK~·R~kτ++e2iK~·R~kτ , (14)
proportional to the electron/hole density at the nuclear ×(cid:16) −(cid:17)
site. As the electron now has the valley pseudospin in
addition to the spin, the hyperfine interaction has in- where Ik and R~k are the spin operatorand position vec-
travalley terms as well as the intervalley terms. The lat- tor of the k-th nuclei of M atom, τ are the rasing and
ter arises from the short-range nature of the hyperfine lowing operators for valley index an±d Ω is the volume of
interaction, which makes possible the coupling between the unit cell. ̺ denotes the ratio between the transverse
the single electron states from different valleys. and the longitudinal interactions.
(ii) Hole hyperfine interaction with M atom:
A. Intravalley and intervalley hyperfine interaction Hv =Av ΩFv(R~ )2IkS . (15)
M M | k | z z
Xk
The hyperfine interaction in the quantum dot is for-
mulated by projecting the complete electron nuclear hy- (iii) Electron hyperfine interaction with X atom:
perfineinteractionintothe basisofthe localizedelectron
1
and hole wavefunctions in the envelope function approx- Hc = Ac ΩFc(R~ )2[I kS + (I kS +I kS )
imation, which are given by, X XXk | k′ | z′ z 8 x′ x y′ y
Ψcτ(,sv)(~r)=Fc(v)(~r)Φcτ(v)(~r)χs (13) +38(e−2iK~·R~′kτ+I−′kS−+H.c.)], (16)
where Φcτ(v)(~r) = eiτK~·~rucτ(v)(~r) is the Bloch wave func- where I′k and R~k′ are the spin operator and position
tion at τK point in the conduction (c) and valence (v) vector of the k-th nuclei of X atom and we have used
bands with uc(v)(~r) being its periodic part, Fc(v)(~r) is theassociatedrasingandlowingoperatorsfornucleiand
τ
the localized envelop function and χ is the spin part electron spins.
s
8
(iv) Hole hyperfine interaction with X atom: Optical Raman processes using these trion states realize
an optical quantum pathway to control these electron-
HXv = AvXXk Ω|Fv(R~k′)|2[Iz′kSz − 14(e−2iK~·R~′kτ+I+′kS+ nthuecleenareregnytasnpgliltetdinsgta(t2ehs.t)Hboewtewveeern,w|ve1nioatnedth|va2t,ibisectayupsie-
+H.c.)]. (17) cally less then 1µeV, the oscillating terms ∝ e−i2htt are
slow ones and can not be neglected in this case as we
All of the hyperfine constants Ac(v) in different MX did in Eq. (6). To realize a coherent rotation of the two
M(X) 2 level system spanned by v and v , we use a single
are listed in Table I. | 1i | 2i
opticalpulsetocouplebothstatestothetrionstate,35,46
Fromtheaboveresults,onecanfindthatthehyperfine
as shown in Fig. 6 (b) and (c). Applying an optical
interactionrelatedto the Mnucleiismuchstrongerthen
pulse with σ+ polarization to virtually excite the trion
the oneto the Xnuclei. Moreimportantly,the hyperfine
interaction may contain both intravalley and intervalley state X2− , the dynamics is governed by the following
Hamil(cid:12)tonia(cid:11)n,
terms. (cid:12)
H′ =htσx−∆ X2− X2− −[Ω(t)|−,↓ie|↑in X2− +H.c.],
B. Optical control of electron-nuclear spin (cid:12) (cid:11)(cid:10) (cid:12) (cid:10) (cid:12)
(cid:12) (cid:12) (cid:12)
entanglement
where Ω(t) is the Rabi frequency in the rotating frame
The intervalley part in the hyperfine interaction sug- and ∆ is the detuning of the laser relative to X2− . For
large detuning, we can use the adiabatic appr(cid:12)oxim(cid:11)ation
gests a possibility for optical control of the electron- (cid:12)
to eliminate the trion state. The dynamics is described
nuclearspinentanglement. Thehyperfineinteractionbe-
by
tween the confined electron and M nucleus is
̺
HHF = AcM(1Ω+|Fτc(0+)|2τhI)z.Sz + 2(I+S−+I−S+)i(18) He′ff = |2Ω∆|2 (cid:18)e−1ihtt ei1htt (cid:19)=ǫ0I+~n(t)·~σ, (20)
+
× −
wherewehaveassumedthattheMnucleiislocatedatpo- describing the qubit state precessing under a time-
sition R~ = 0, where the hyperfine interaction is strong. dependentmagneticfield~n(t)withthestrengthΩ2/(2∆)
0
The term I S shift upwards (downwards) the energy rotating in the x-y plane with the angular frequency h .
z z t
levelswith∝h = 1Ac ΩFc(0)2whenelectronandnuclear Because h is typically less then 1 µeV and Ω2/(2∆)
l 4 M | | t
spinspointinthesame(opposite)direction. Theterm is several hundreds of µeV, the optical pulse in the pi-
∝
[I S (τ +τ )+H.c.] couplesthe differentvalley states cosecond scale can be regarded as an instantaneous one.
+ +
wher−e electro−n and nuclear spins point in the opposite To complete an arbitrary rotation, two subsequent ro-
direction, which can be rewritten as tations along x- and y-directions, which constitute two
SU(2) generators, are needed. Explicitly, at t = 2nπ
H = h σ , (19) ht
1 t x (n is an integer), ~n(t) is in the x-direction. Whereas
where h = Ac ΩF(0)2̺ and σ denotes the Pauli ma- at t = (2n+1/2)π, ~n(t) is in the y-direction. For a
trices detfinedMin t|he tw|o-2dimensional space spanned by 183W nucleiihnttriangular-shapeWS quantumdots with
2
+, , , or +, , , N = 100, the shortest time interval for these two subse-
w{|ith↑tihee|↓siunbsc|r−ipt↓ieea|↑nidn}n deno{t|ing↓eiele|c↑tirnon|a−nd↑ineu|c↓liena}r quent operations is 287 ns. Because h 1, this oper-
t ∝ N
states respectively. The other terms can be neglected, ation time can be shortened by using smaller quantum
because they couple those states separated by the spin- dots.
valley coupling, which is much larger compared to the We note that the possibility to control the coherent
Nhyp=erSfinies tinhteernaucmtiboenr.oTfhuenimtacgelnliitnudtheeofquhatn∝tumN1,dwothseroef rot,ation in, ctohmebsiunbedspwacitehstphaenRneFdcobnytr|o+l,t↑hiaet|↓fliinpsatnhde
Ω |− ↓ie|↑in
areaS. Fora183Wnucleiintriangular-shapeWS quan- nuclear state, can potentially realize the intervalley ro-
2
tum dots with N =100, we estimate h 0.0036µeV. tation in the electron subspace spanned by +, and
The eigenstates of Eq. (19), v =t ∼1 (+, , . We also note that, in our scheme, a si|ngl↑eineuclei
| 1,2i √2 | ↑ie|↓in± |− ↓ie
|−,↓ie|↑in) or |v3,4i = √12(|+,↓ie|↑in ± |−,↑ie|↓in), wenitthvastllaetyes.|siTnhiesiunsteedratcoticoonupstlreeenlgetchtrohn stat1e.s oAfltdeirffnear--
which are electron-nuclear entangled states. These en- t ∝ N
tively, if a connectionof nucleiwith a fully polarizedini-
tangled states contain electron spin state from both val-
tial state , , are used, the interaction strength
leys and can be connected via certain intervalley trion |↓ ↓ ↓···in
state, as shown in Fig. 6 (a). For example, v √ν, so that the operation speed can be increased by
| 1,2i ∝ √N
can be coupled with equal strength to the trion state √νN time,whereν istheabundanceofMnuclei. Inthis
X1− = e†, e†+, h†, |Gi|↓in by a σ− polarized light or case, the electron-nuclear entangled states are v1′,2 =
(cid:12)(cid:12)X2−(cid:11) = e−†↓, e†+↑, h−†+⇑, |Gi|↑in by a σ+ polarized light. √12(|+,↑ie|↓,↓,↓···in±|−,↓ie kck|↓,↓,↑k ··(cid:12)(cid:12)·in).(cid:11)
(cid:12) (cid:11) −↓ ↑ ⇓ P
(cid:12)
9
encebetweentheelectronstateswithoppositespins,and
hence results in pure (inhomogeneous) dephasing. We
assume thatthere is no correlationbetweendifferent nu-
clearspinsandthatthenuclearspinsaredistributeduni-
formly within the quantum dot, the variance of the field
is
h2 =(Ac )2 νΩ2 Fc(R~ )4 Ik2 , (21)
eff M | Q k | z
(cid:10) (cid:11) Xk (cid:10) (cid:11)
where denotes the averageovernuclear spin states.
The cohh·e·r·eince time for electron state is T 2π .
2∗ ∼ √ h2
h effi
The intervalley electron-nuclear flip-flop term
[IkS (τ +τ )+H.c.] causes the population relaxatio∝n
+ +
betw−eenthed−egenerateelectronstates +, >and , >
| ↑ |− ↓
and between +, > and , >. The relaxation time for
| ↓ |− ↑
this process is T 2π ,47 where h2 is the vari-
Feliegcutrreon6:sta(tae)sOinptpicreaslenancedohfyapenruficnleeacrosuppilnin.gTshbeetbwroewenntahre- 1 ∼ qhh′e2ffi D ′effE
ance of the in-plane nuclear field,
rows denote the nuclear spin state, e.g. from a 183W nuclei.
TheΛ-typethree-levelsystemsformedbytheelectron-nuclear
h2 =̺2(Ac )2 νΩ2 Fc(R~ )4 (Ik2+Ik2) .
entangledstatesandtrionstatesareconnectedviaanoptical ′eff M | Q k | x y
pulse with σ+ polarization (b) or σ− polarization (c). (cid:10) (cid:11) Xk (cid:10) (cid:11)
(22)
N 100 500 1000
C. Nuclear spin induced decoherence
ν 25.5% 100% 25.5% 100% 25.5% 100%
MoS T (ns) 298 150 668 337 944 477
2 1
The interaction with lattice nuclear spins causes the T∗(ns) 97 49 217 110 307 155
2
decoherence of localized electron spin.47–52 Because the ν 14.3% 100% 14.3% 100% 14.3% 100%
hyperfine interactionwith the M nuclei is much stronger WS T (ns) 557 210 1246 471 1762 666
2 1
thanthe one with the X nuclei, we considerthe decoher- T∗(ns) 292 110 652 246 922 349
2
ence effect arising from interaction with the former one.
From Eq. (18), we know that there are four decoherence Table II: Decoherence time in MoS2 and WS2 quantum dots
with different numbers of nuclear spin. ν = 25.5% (14.3%)
channels in the basis of +, , , , +, , , .
The first one arises from th{e|ter↑mi |−Ik↓Si ,| wh↓iich|−cau↑sie}s corresponds to the natural abundance in MoS2 (WS2), and
∝ z z ν = 100% corresponds to the case that each metal atom
dephasing between electron states with different spin.
within thequantumdot has a nuclear spin.
Thesecondonearisesfromtheterm [IkS (τ +τ )+
H.c.], which causes relaxation betw∝een+ele−ctro+n st−ates
For an infinite-temperature state, we have Ik2 =
with different spin and valley. The third one arises from x,y,z
athtieontebrmetw∝ee[nIzkeSlezc(tτr+on+sτt−at)e+s wHi.tch.],diwffhericehntcavaulsleesyrwelhaixle- eInkc(Iekt+im1e)/in3c.reSainsecse wPitkhΩt2h|FeQcin(Rc~rke)a|s4e∼ofN1√,Nth.e(cid:10)IdnecToah(cid:11)berle-
the same spin. The last one arises from the term II,welistT andT forquantumdotswithdifferentsize
∝ 2∗ 1
(IkS +H.c.), which causes relaxationbetween electron (represented by N) and the abundance of the nuclear
+
state−s with different spin in each valley. The last two spins. The decoherence time is several hundreds of ns,
relaxation channels are much weaker compared with the which is serval orders larger then the operation time in
former two, because the hyperfine interaction is small the optical quantum control of spin-valley qubit.
compared to the spin-valley coupling so that the energy
cost associated with the transitions (a few to a few tens
meV) is much larger than the hyperfine induced transi- VI. OPTICALLY CONTROLLED TWO-QUBIT
tion matrix element. So relaxation between the initial OPERATIONS IN A SINGLE QUANTUM DOT
and final states are suppressed by the large energy cost.
Therefore,inthefollowing,wemakeanestimationofthe With the extra valley degree of freedom in TMDs, a
decoherence time arising from the first two channels. singleelectroninthegroundstateoftheQDconfinement
For the dephasing between electron states with differ- has four spin-valley configurations that one can exploit
ent spin induced by the term ∝ IzkSz, the effective nu- to encode two qubits. Here we discuss the possibility of
clear field experienced by the localized electron in each utilizing both qubits in a single dot and realizing two-
singlevalleyish =Ac ΩFc(R~ )2Ik. Thestatis- qubit logic controls.
eff M k | Q k | z
ticalfluctuation in the nucPlear spinconfigurationsthere- We consider first the logic operations in the presence
fore corresponds to an uncertainty in the energy differ- of valley hybridization, but in the absence of in-plane
10
couplethe 00 and 01 statesviaaRaman-typeprocess
| i | i
(c.f. Fig. 7 (b)). By controlling the amplitudes and
phasesofthetwopulsessothatthepseudomagneticfield
~n defined in Eq. (9) is in the x direction, a controlled
NOT gate can be realized
1 0 0 0
0 1 0 0
U = . (24)
N 0 0 0 1
0 0 1 0
For MoS quantum dot with an intervalley coupling
2
strengthof1meV(seeAppendix A),ifwesetthedetun-
ing ∆=0.3 meV and E D =0.05 meV, this two-qubit
1 0
gate can be realized in 5 ns.
∼
To realize a SWAP gate, we consider nanostructures
with valley hybridization and in applied external mag-
Figure 7: Optically controlled two-qubit operations between netic field. In this scenario, the electron eigenstates are
the spin qubit and valley qubit carried by a single electron both spin and valley hybridized, which are used as the
in a quantum dot. (a) Definition of the two-qubit states. basis states for the qubits,
The trion states with higher energy (highlighted in red and
blue background) are used as intermediate states for optical 11 = (cosθ + +sinθ )(cosθ +sinθ ),
′ ′
control in two-qubit operations. (b) Energy level scheme for | i | i |−i |↑i |↓i
realizing controlled phase gate and controlled NOT gate in |00i = (cosθ|+i+sinθ|−i)(sinθ′|↑i−cosθ′|↓i),
the presence of valley hybridization. (c) Energy level scheme 01 = (sinθ + cosθ )(sinθ′ cosθ′ ),
| i | i− |−i |↑i− |↓i
for realizing SWAP gate in the presence of valley hybridiza- 10 = (sinθ + cosθ )(cosθ′ +sinθ′ ).
tion and applied magnetic fields. The unwanted transitions | i | i− |−i |↑i |↓i
(dashed doubled-arrowed lines) are detuned from the two- In this definition, the two qubits do not have interaction
photon resonant condition by 2Bz and suppressed. at rest. Each of these states is now optically coupled
tothe trionstatese†, e†+, h†, |Giande†, e†+, h†+, |Gi
(c.f. Fig. 7 (c)).−↓For↑ex−am⇑ple, apply−in↓g σ↑+ p⇓olar-
magnetic field. We define the computational basis as
ized lights, these states can couple to the trion state
mn = m n where m,n = 1,0 , and the sub-
|scripitss|andisv⊗de|niovtespinandvalley{deg}reesoffreedom. e†, e†+, h†+, |Gi with strengths ∝ sinθsinθ′, sinθcosθ′,
Explicitly, 11 = u1, , 10 = u2, , 01 = u1, , co−s↓θco↑sθ′ a⇓nd cosθsinθ′ respectively. In order to se-
| i | ↑i | i | ↑i | i | ↓i
and 00 = u , (see Fig. 7 (a)). Under this defi- lectively control these states, we apply a magnetic field
2
| i | ↓i
nition, the two qubits do not have interaction at rest. in z direction to make the unwanted optical transi-
Theopticalcontrolfortwo-qubitoperationsusesthetwo tions detuned from the two-photon resonant condition
htraivoenestnaetregsiees†−h,↓ieg†+h,e↑rh†−th,⇑an|Gtihaenodthee†−r,↓ter†+io,↑nh†+st,a⇓t|eGsibwyhtihche biny Fthige.Ze7em(ca)n. spVliitatinvgirt2uBazllyanedxcsiutipnpgretshseedt,raiosnshsotawtne
conduction band spin-orbit splitting λ. As highlighted e†, e†+, h†+, |Gi with σ+ light, one can realize coherent
in Fig. 7 (a), these two trion states couple to the four ro−ta↓tion↑s se⇓lectively between 01 and 10 to realize a
| i | i
states of the two qubits by light of different circular po- SWAP gate,
larizations. These two trion states then can be used as
the intermediate states in our control scheme, where the 1 0 0 0
lower energy trion states can be neglected with λ in the 0 0 1 0
U = . (25)
range of a few meV to a few tens of meV32,33 (c.f. Fig. W 0 1 0 0
7 (b) and (c)). Via virtually exciting these trion states 0 0 0 1
with different circularly polarized lights, one can obtain
For MoS quantum dot with an intervalley coupling
the controlled two-qubit gates. For example, applying a 2
single pulse of σ+ light (Ω =0), only the state 00 will strength of 1 meV (see Appendix A) and applied mag-
1
| i netic fields B =B =1 meV, if we set the detuning ∆ =
pick up a phaseshift due to the AC-stark shift (Fig. 7 x z
0.3 meV and E D = 0.05 meV, the SWAP gate can be
(b)), so we have a controlled phase-shift gate 1 0
realized in 2 ns.
∼
1 0 0 0
0 1 0 0
UPφ =0 0 1 0 . (23) VII. DISCUSSION AND CONCLUSIONS
0 0 0 eiφ
In conclusion, we have studied the optical controlla-
One can also use two σ+ polarized pulses to selectively bility of the spin-valley qubit carried by single electrons
