Interface-induced heavy-hole/light-hole splitting of acceptors in silicon

1.1 MB·English

by  J. A. Mol

Interface-induced heavy-hole/light-hole splitting of acceptors in silicon
J. A. Mol,1,2 J. Salfi,1 R. Rahman,3 Y. Hsueh,3 J. A. Miwa,1,4 G. Klimeck,3 M. Y. Simmons,1 and S. Rogge∗1
(∗To whom correspondence should be addressed; E-mail: [email protected].)
1Centre for Quantum Computation and Communication Technology,
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
2Department of Materials, University of Oxford, 16 Parks Road, Oxford OX1 3PH, UK
3Purdue University, West Lafayette, IN 47906, USA
4Department of Physics and Astronomy, Interdisciplinary Nanoscience
Center (iNANO), Aarhus University, 8000 Aarhus C, Denmark
(Dated: January 26, 2015)
5
1 Theenergyspectrumofspin-orbitcoupledstatesofindividualsub-surfaceboronacceptordopants
0 insiliconhavebeeninvestigatedusingscanningtunnelingspectroscopy(STS)atcryogenictemper-
2 atures. The spatially resolved tunnel spectra show two resonances which we ascribe to the heavy-

n and light-hole Kramers doublets. This type of broken degeneracy has recently been argued to be
a advantageous for the lifetime of acceptor-based qubits [Phys. Rev. B 88 064308 (2013)]. The
J depthdependentenergysplittingbetweentheheavy-andlight-holeKramersdoubletsisconsistent
2 with tight binding calculations, and is in excess of 1 meV for all acceptors within the experimen-
2 tally accessible depth range (< 2 nm from the surface). These results will aid the development of
tunable acceptor-based qubits in silicon with long coherence times and the possibility for electrical

] manipulation.
l
l
a
h Dopant atoms in silicon are attractive candidates for form of protection. In this Letter we demonstrate that
-
s spin-based quantum computation. Recent studies have the symmetry-reduction of a potential boundary renders
e demonstrated long coherence-times for both ensembles thelowesttwolevelsKramersdegenerateandheavy-hole
m
of bulk donors [1] and individual donors [2] in silicon. like. Recent transport spectroscopy studies of an indi-
. Meanwhile, rapid progress in scanning tunnelling mi- vidual acceptor embedded in nano-scale transistors have
t
a croscopy (STM) based lithography has paved the way shownthatforacceptors∼10 nmawayfromaninterface
m
towards atomically precise placement of dopant atoms thebulk-likefour-folddegeneracyismaintained[8]. Here
- [3]. While phosphorous donors in silicon remains among we demonstrate that the presence of a nearby interface
d
the most compelling candidates for dopant-based quan- (<2 nm) lifts the four-fold degeneracy of the acceptor-
n
tum computing to date, other impurity systems have re- boundholegroundstatebyinvestigatingtheenergyspec-
o
c centlydrawnconsiderableattention. Inparticular,boron trum and wavefunctions of individual sub-surface boron
[ acceptors could provide a pathway towards electrically acceptors using scanning tunnelling spectroscopy (STS).

1 addressable spin-qubits via spin-orbit coupling [4] ana- Recent low-temperature STM/STS studies have suc-
logues to electrically driven spin manipulation in gate
v cessfully probed the energy spectrum and wavefunctions
9 definedelectron[5,6]andhole[7]quantumdotsinIII-V of individual impurity atoms in Si [9–11] and GaAs
6 materials. Compared with other spin-orbit qubits, ac- [12, 13]. These studies have revealed the influence of the
6 ceptors in silicon have several advantages: gates are not
semiconductor/vacuum interface on the ionization en-
5 requiredforholeconfinementandeachqubitexperiences
ergy of sub-surface dopants [9, 13], the spatially resolved
0
the same confinement potential; furthermore the hole-
structure of the dopant wavefunction [10–12] and the
.
1 spin decoherence, due to the nuclear spin bath, can be mechanisms for charge-transport through these dopants
0 effectivelyeliminatedbyisotopepurificationofthesilicon
[9, 11, 14]. Here, we use STS to measure the energy
5 host.
difference between the heavy-hole and light-hole states
1
: Unlike the ground state of donor-bound electrons in of individual B acceptors less than 2 nm away from the
v
silicon, theacceptor-boundholegroundstateisfour-fold surface. The sample is prepared by repeated flash an-
i
X degenerate, reflecting the heavy-hole/light-hole degener- nealing a 8×1018 cm−3 boron doped Si(100) substrate
acy of the silicon valence band. Recent theoretical work in ultra-high vacuum. The flash annealing not only pro-
r
a has suggested the regime of long lifetimes for acceptors videsanatomicallyflatsurfacebutalsoyieldsaregionof
with a four-fold degenerate ground state is only accessi- low doping at the interface due to out-diffusion of the B
ble for small magnetic fields [4]. Interestingly, this work impurities. As a result, the near-interface acceptors are
also suggests that symmetry breaking due to strain or weakly coupled to the bulk acceptors, which form a va-
electric fields could yield longer-lived qubits based on lence impurity band (VIB). Together with the STM tip,
acceptor-boundholesathighermagneticfields. Thesym- the sub-surface acceptors can be considered as a double-
metry breaking perturbation of biaxial strain [4] renders barriersystemforsingle-holetunneling[9](seeFig.1(a)).
the lowest two (qubit) levels within a Kramers degener- As illustrated in the inset of Figure 1(b), individual ac-
ate pair, such that they do not directly couple to elec- ceptors are identified as protrusions at V =−1.5 V due
b
tricfields. Quantumconfinementcouldprovideasimilar to enhancement of the valence band density of states [9].
2
(a) (b)
(a) Tip Vacuum Sample ±1/2 ±3/2
0.8
±3/2
V) 0.6
e ( g (pS)
g 3 nm
eVb Acceptor Volta 0.4 100 (c) ±1/2
VIB as 0.2
Bi 0
B E 0
B F
TI
EV −0.2 −2 0 2
Tip Position (nm)
(b) (c)
10−10 100
VB ±1/2 S) VB
p FIG. 2. (a) Spatially resolved differential conductance map
−11 ±3/2 e ( 80 ±1/2 asafunctionoftippositionandbiasvoltagemeasuredovera
10 c
n sub-surfaceacceptor(differentfromFig. 1). (b,c)Probability
a
A) uct 60 density map of the lowest two acceptor states.
ent (10−12 ond
Curr −13 Vb=−1.5 V ntial C 40 ±3/2 (TIBB).Eachtimeanacceptorlevelentersthebiaswin-
10 ere 20 dow,whichisdefinedbytheFermilevelofthetipandthe
1 nm Diff Si substrate, this opens an additional channel for trans-
10−14 0 port resulting in a stepwise increase in the current (Fig.
0 0.5 1 0 0.5 1 1(b)) and a corresponding peak in the differential con-
Bias Voltage (V) Bias Voltage (V) ductance as shown in Fig. 1(c). The conductance peaks
are only observed in the presence of an acceptor, and
for each measured acceptor we observe two conductance
FIG. 1. (a) Schematic energy diagram of the tunneling pro-
peaks in the band-gap which we attribute to the lowest
cess. Applying a positive sample bias (eV ) results in an up-
b
two acceptor states entering the bias window.
wardtip-inducedbandbending(TIBB).Whentheenergyofa
stateofanisolatedacceptorneartheinterfaceispulledabove A spatially resolved differential conductance map is
the Fermi level (E ) there is a peak in the conductance due shown in Fig. 2(a). The amplitude of the differential
F
to resonant tunneling from the valence impurity band (VIB) conductance measured over the acceptor is a measure
throughtheacceptorstateintothetip. Inthisschematicde- of the probability density of the impurity wavefunction
scription, the top layer of silicon is depleted of dopants. The (Fig. 2(b,c)). The probability density maps of the low-
measured acceptors are well-isolated from each other, as well
est two acceptor states have the same spatial extent and
astheunderlyingnominally8×1018 cm−3 dopedregion,de-
symmetry, i.e. neither state shows anti-nodes in the sur-
noted by VIB. (b) Current and (c) differential conductance
face plane. The similarity between the probability den-
measured directly above the isolated acceptor (open circles)
sitymapofthetwoconductancepeaksisconsistentwith
as a function of bias voltage. The differential conductance
within the band gap is fitted to the sum (solid line) of two whatisexpectedfor±3/2and±1/2states. Bothareex-
thermallybroadenedLorentzianline-shapescorrespondingto pectedtohavetheobserveds-likeenvelopes,withsimilar
resonant tunnelling through the lowest two acceptor states. spatial extents [15]. In contrast, a two-hole state would
have a larger spatial extent. When two states have the
same envelope wavefunctions the energy difference be-
Figure 1(b,c) shows the current and differential con- tween these two states can only arise from the difference
ductance measured over a single sub-surface acceptor. in pseudo-spin, i.e. from non-degenerate heavy-hole and
For the bias voltage V < 0 V the charge transport is light-hole states. In the remainder of this Letter we will
b
dominated by holes tunnelling directly from the Si va- discuss how the perturbation of the subsurface acceptor
lence band to the tip. Similarly, for V > 1.1 V trans- wavefunctions by an interface lifts the degeneracy be-
b
port is dominated by holes tunnelling from the tip to tween the heavy- and light-hole Kramers doublets that
the Si conduction band. The observed features in the make up the four-fold degenerate ground state of an un-
band-gap, i.e. for 0 V< V < 1.1 V, can be attributed perturbed (bulk) acceptor.
b
to holes tunnelling from the VIB through the localized Based on the tunneling spectra of the single acceptor
acceptorstatestothetip. Uponincreasingthebiasvolt- we can now extract the energy splittings of the states,
age from 0 V to 1.1 V the energy levels of the subsurface and show that they are in reasonable agreement with
acceptor states shift due to tip induced band bending tightbindingpredictionsforthes-like3/2and1/2states,
3
F
and too small to be associated with charging processes
8
where a second hole is bound to the acceptor. If the V) experiment
lever arm α = dE /edV that couples the chemical po- e
i i m
tential to the bias voltage is known, the voltage Vi for E ( 6 tight binding
which a localized state i is brought into resonance is a δ
direct measure for the eigenenergy E of this state. We g
i n
determine α by fitting the conductance to a thermally plitti 4
broadened Lorentzian [16] (see Fig. 1(b)): S
e
c
g(V)∝(cid:90) +∞cosh−2(E/2k T) erfa 2
B nt
−∞ (1) I
1(cid:126)Γ
× 2 dE, 0
(1(cid:126)Γ)2+(αe[V −V ]−E)2 0 1 2 3
2 i
Depth (nm)
where k is the Boltzmann factor, T the temperature
B
(we assume T = 4.2 K for the fit), V the voltage cor- FIG. 3. Experimental and theoretical depth dependence of
i
responding to the center of the ith conductance peak, (cid:126) theinterfaceinducedenergysplitting. Theverticalerrorbars
Planck’s constant and Γ the sum of the tunnel-in and ontheenergy-levelsplittingareproportionaltotheconfidence
of the fit of α. Errors in V are negligible. The horizontal
tunnel-out rates. Theinterfaceinduced splitting isgiven i
error bars are proportional to the confidence of the fit of d
by δE =α(V −V ).
±1/2 ±3/2 whichdecreasesforacceptorsfurtherfromtheinterfaceasthe
The distance of the subsurface acceptors to the inter-
spectralshiftofthevalence-bandmaximumbecomessmaller.
face is measured from the spectral shift of the valence
band edge [9]:
e−Q 1
symmetricgaugeandenteredthroughaPeierlssubstitu-
∆E ≈e √ , (2)
V 4π(cid:15)0(cid:15)Si s2+d2 tion. ThefullTBHamiltonianissolvedinNEMO3D[21]
by a parallel Block Lanczos algorithm, and the relevant
where d is the acceptor depth and Q=e((cid:15)v−(cid:15)Si)/((cid:15)v+ low energy acceptor wavefunctions are obtained. From
(cid:15)Si) is the image charge due to the mismatch between the calculated magnetic field dependence of the accep-
thedielectricconstants(cid:15)v and(cid:15)Si ofthevacuumandsil- tor states (not shown) we infer that the lowest Kramers
icon, respectively. The onset voltage for tunneling from doublet has a predominant heavy-hole (±3/2) character
the valence band VV is determined by finding the volt- whereas the second Kramers doublet has a predominant
age axis intercept of the linear extrapolation of the nor- light-hole (±1/2) character.
malised conductance curve at its maximum slope point
Fromthecombinationofthespatialandspectralmea-
[17]. The spectral shift of the valence-band edge is fitted
surements, we can conclude that the observed states
to equation 2 using the depth d of individual acceptors
belong to the s-like manifold which contains a ±3/2
and the modified dielectric constant Q as two, indepen-
Kramersdoubletanda±1/2Kramersdoublet. Notonly
dent, fitting parameters. The obtained values for Q for
are both of the envelopes s-like, but the few meV energy
all measured acceptors agree within experimental error
scale of the splitting is too small to be associated with
with the expected value Q = e((cid:15) −(cid:15) )/((cid:15) +(cid:15) ) fol-
v Si v Si
higher excited states (at 25 meV) [22] or charging tran-
lowing the classical half-space approach and experimen-
sitions (47 meV) [23] of an acceptors. We note that the
tal values that have previously been reported from STM
charging energy near the surface could fall as low as 20
experiments [18, 19].
meV [11], but this is still too large for our results.
Figure 3 shows the measured energy splitting δE be-
tween the lowest two acceptor states as a function of ac- We observe an increase of δE for acceptors closer to
ceptordepthforsixdifferentacceptorsdemarcatedbythe the interface, as predicted by the TB calculations. Im-
solid diamonds. These results are compared with a fully portantly, all acceptors studied showed an energy split-
atomistic tight binding (TB) simulation marked by the tinginexcessof1meV.Forcomparison,theelectricfield
greycircles. Thetight-bindingHamiltonianof1.4million required to obtain a splitting of 0.5 meV would be 40
siliconatomswithaboronacceptorwasrepresentedwith MV/m [24], much greater than the electric field required
a 20-orbital sp3d5s* basis per atom including nearest- for field ionization 5 MV/m [25]. Our results therefore
neighbor and spin-orbit interactions. An acceptor was demonstrate that the presence of an interface provides
represented by a Coulomb potential of a negative charge an effective way to energetically isolate a single Kramers
screened by the dielectric constant of Si and subjected doublet that could serve as the working levels of a spin-
toanonsitecutoffpotentialU [20]. Themodelprovides qubit. Suchaqubitcouldbenefitfromeliminationofthe
0
an accurate solution for the single-hole eigenstates of a nuclear spin bath by isotope purification, and could pro-
bulk acceptor, and an acceptor near an interface. The videanewroutetowardsanelectricallycontrollablespin
magnetic field is represented by a vector potential in a qubit.
4
ACKNOWLEDGMENTS W911NF-08-1-0527. J.A.M. received funding from the
Royal Society Newton International Fellowship scheme.
This research was conducted by the Australian Re- M.Y.S. acknowledges an ARC Federation Fellowship.
search Council Centre of Excellence for Quantum Com- S.R acknowledges an ARC Future Fellowship and FP7
putation and Communication Technology (project num- MULTI. The authors are grateful to D. Culcer for help-
berCE110001027)andtheUSNationalSecurityAgency ful discussions.
andtheUSArmyResearchOfficeundercontractnumber
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