The energy spectrum of spin-orbit coupled states of individual sub-surface boron acceptor dopants in silicon have been investigated using scanning tunneling spectroscopy at cryogenic temperatures. The spatially resolved tunnel spectra show two resonances, which we ascribe to the heavy- and light-hole Kramers doublets. This type of broken degeneracy has recently been argued to be advantageous for the lifetime of acceptor-based qubits [R. Ruskov and C. Tahan, Phys. Rev. B 88, 064308 (2013)]. The depth dependent energy splitting between the heavy- and light-hole Kramers doublets is consistent with tight binding calculations, and is in excess of 1 meV for all acceptors within the experimentally 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.

Dopant atoms in silicon are attractive candidates for spin-based quantum computation. Recent studies have demonstrated long coherence-times for both ensembles of bulk donors1 and individual donors2 in silicon. Meanwhile, rapid progress in scanning tunneling microscopy (STM) based lithography has paved the way towards atomically precise placement of dopant atoms.3 While phosphorous donors in silicon remains among the most compelling candidates for dopant-based quantum computing to date, other impurity systems have recently drawn considerable attention. In particular, boron acceptors could provide a pathway towards electrically addressable spin-qubits via spin-orbit coupling4 analogues to electrically driven spin manipulation in gate defined electron5,6 and hole7 quantum dots in III–V materials. Compared with other spin-orbit qubits, acceptors in silicon have several advantages: gates are not required for hole confinement and each qubit experiences the same confinement potential; furthermore, the hole-spin decoherence, due to the nuclear spin bath, can be effectively eliminated by isotope purification of the silicon host.

Unlike the ground state of donor-bound electrons in silicon, the acceptor-bound hole ground state is four-fold degenerate, reflecting the heavy-hole/light-hole degeneracy of the silicon valence band. Recent theoretical work has suggested the regime of long lifetimes for acceptors with a four-fold degenerate ground state is only accessible for small magnetic fields.4 Interestingly, this work also suggests that symmetry breaking due to strain or electric fields could yield longer-lived qubits based on acceptor-bound holes at higher magnetic fields. The symmetry breaking perturbation of biaxial strain4 renders the lowest two (qubit) levels within a Kramers degenerate pair, such that they do not directly couple to electric fields. Quantum confinement could provide a similar form of protection. In this letter, we demonstrate that the symmetry-reduction of a potential boundary renders the lowest two levels Kramers degenerate and heavy-hole like. Recent transport spectroscopy studies of an individual acceptor embedded in nano-scale transistors have shown that for acceptors ∼10 nm away from an interface the bulk-like four-fold degeneracy is maintained.8 Here, we demonstrate that the presence of a nearby interface (<2 nm) lifts the four-fold degeneracy of the acceptor-bound hole ground state by investigating the energy spectrum and wavefunctions of individual sub-surface boron acceptors using scanning tunneling spectroscopy (STS).

Recent low-temperature STM/STS studies have probed the energy spectrum and wavefunctions of individual impurity atoms in Si9–11 and GaAs.12,13 These studies have revealed the influence of the semiconductor/vacuum interface on the ionization energy of sub-surface dopants,9,13 the spatially resolved structure of the dopant wavefunction10–12 and the mechanisms for charge-transport through these dopants.9,11,14 Here, we use STS to measure the energy difference between the heavy-hole and light-hole states of individual B acceptors less than 2 nm away from the surface. The sample is prepared by repeated flash annealing a 8 × 1018 cm−3 boron doped Si(100) substrate in ultra-high vacuum. The flash annealing not only provides an atomically flat surface but also yields a region of low doping at the interface due to out-diffusion of the B impurities. As a result, the near-interface acceptors are weakly coupled to the bulk acceptors, which form a valence impurity band (VIB). Together with the STM tip, the sub-surface acceptors can be considered as a double-barrier system for single-hole tunneling9 (see Fig. 1(a)). As illustrated in the inset of Figure 1(b), individual acceptors are identified as protrusions at Vb = −1.5 V due to enhancement of the valence band density of states.9 

FIG. 1.

(a) Schematic energy diagram of the tunneling process. Applying a positive sample bias (eVb) results in an upward TIBB. When the energy of a state of an isolated acceptor near the interface is pulled above the Fermi level (EF), there is a peak in the conductance due to resonant tunneling from the VIB through the acceptor state into the tip. In this schematic description, the top layer of silicon is depleted of dopants. The measured acceptors are well-isolated from each other, as well as the underlying nominally 8 × 1018 cm−3 doped region, denoted by VIB. (b) Current and (c) differential conductance measured directly above the isolated acceptor (open circles) as a function of bias voltage. The differential conductance within the band gap is fitted to the sum (solid line) of two thermally broadened Lorentzian line-shapes corresponding to resonant tunnelling through the lowest two acceptor states.

FIG. 1.

(a) Schematic energy diagram of the tunneling process. Applying a positive sample bias (eVb) results in an upward TIBB. When the energy of a state of an isolated acceptor near the interface is pulled above the Fermi level (EF), there is a peak in the conductance due to resonant tunneling from the VIB through the acceptor state into the tip. In this schematic description, the top layer of silicon is depleted of dopants. The measured acceptors are well-isolated from each other, as well as the underlying nominally 8 × 1018 cm−3 doped region, denoted by VIB. (b) Current and (c) differential conductance measured directly above the isolated acceptor (open circles) as a function of bias voltage. The differential conductance within the band gap is fitted to the sum (solid line) of two thermally broadened Lorentzian line-shapes corresponding to resonant tunnelling through the lowest two acceptor states.

Close modal

Figures 1(b) and 1(c) show the current and differential conductance measured over a single sub-surface acceptor. For the bias voltage Vb < 0 V, the charge transport is dominated by holes tunnelling directly from the Si valence band to the tip. Similarly, for Vb > 1.1 V, transport is dominated by holes tunnelling from the tip to the Si conduction band. The observed features in the band-gap, i.e., for 0 V < Vb < 1.1 V, can be attributed to holes tunnelling from the VIB through the localized acceptor states to the tip. Upon increasing the bias voltage from 0 V to 1.1 V, the energy levels of the subsurface acceptor states shift due to tip induced band bending (TIBB). Each time an acceptor level enters the bias window, which is defined by the Fermi level of the tip and the Si substrate, this opens an additional channel for transport resulting in a stepwise increase in the current (Fig. 1(b)) and a corresponding peak in the differential conductance 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 peaks in the band-gap, which we attribute to the lowest two acceptor states entering the bias window.

A spatially resolved differential conductance map is shown in Fig. 2(a). The amplitude of the differential conductance measured over the acceptor is a measure of the probability density of the impurity wavefunction (Fig. 2(b)). The probability density maps of the lowest two acceptor states have the same spatial extent and symmetry, i.e., neither state shows anti-nodes in the surface plane. The similarity between the probability density maps of the two conductance peaks is consistent with what is expected for ±3/2 and ±1/2 states. Both are expected to have the observed predominant s-like envelopes, with similar 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 between these two states can only arise from the difference in pseudo-spin, i.e., from non-degenerate heavy-hole and light-hole states. In the remainder of this letter, we will discuss how the perturbation of the subsurface acceptor wavefunctions by an interface lifts the degeneracy between the heavy- and light-hole Kramers doublets that make up the four-fold degenerate ground state of an unperturbed (bulk) acceptor.

FIG. 2.

(a) Spatially resolved differential conductance map as a function of tip position and bias voltage measured over a sub-surface acceptor (different from Fig. 1). (b) Normalized conductance maps of the lowest two acceptor states. (c) Normalized probability density obtained from tight-binding simulations.

FIG. 2.

(a) Spatially resolved differential conductance map as a function of tip position and bias voltage measured over a sub-surface acceptor (different from Fig. 1). (b) Normalized conductance maps of the lowest two acceptor states. (c) Normalized probability density obtained from tight-binding simulations.

Close modal

Based on the tunneling spectra of the single acceptor we can now extract the energy splittings of the states, and show that they are in reasonable agreement with tight binding (TB) predictions for the s-like 3/2 and 1/2 states, and too small to be associated with charging processes where a second hole is bound to the acceptor. If the lever arm α = dEi/edVi that couples the chemical potential to the bias voltage is known, the voltage Vi for which a localized state i is brought into resonance is a direct measure for the eigenenergy Ei of this state. We determine α by fitting the conductance to a thermally broadened Lorentzian16 (see Fig. 1(b))

g(V)+cosh2(E/2kBT)×12Γ(12Γ)2+(αe[VVi]E)2dE,
(1)

where kB is the Boltzmann factor, T is the temperature (we assume T = 4.2 K for the fit), Vi is the voltage corresponding to the center of the ith conductance peak, is the Planck's constant, and Γ is the sum of the tunnel-in and tunnel-out rates. The interface induced splitting is given by δE = α(V±1∕2 − V±3∕2).

The distance of the subsurface acceptors to the interface is measured from the spectral shift of the valence band edge9 

ΔEVeeQ4πϵ0ϵSi1s2+d2,
(2)

where d is the acceptor depth and Q = e(ϵv − ϵSi)/(ϵv + ϵSi) is the image charge due to the mismatch between the dielectric constants ϵv and ϵSi of the vacuum and silicon, respectively. The onset voltage for tunneling from the valence band VV is determined by finding the voltage axis intercept of the linear extrapolation of the normalised conductance curve at its maximum slope point.17 The spectral shift of the valence-band edge is fitted to Eq. (2) using the depth d of individual acceptors and the modified dielectric constant Q as two, independent, fitting parameters. The obtained values for Q for all measured acceptors agree within experimental error with the expected value Q = e(ϵv − ϵSi)/(ϵv + ϵSi) following the classical half-space approach and experimental values that have previously been reported from STM experiments.18,19

We have calculated the wavefunctions of interface-perturbed acceptor dopants using tight binding, in order to compare their spatial symmetry (and later, energetics) with experiments. The tight-binding Hamiltonian of 1.4 × 106 silicon atoms with a boron acceptor was represented with a 20-orbital sp3d5s* basis per atom including nearest-neighbor and spin-orbit interactions. An acceptor was represented by a Coulomb potential of a negative charge screened by the dielectric constant of Si and subjected to an onsite cutoff potential U0.20 The model provides an accurate solution for the single-hole eigenstates of a bulk acceptor, and an acceptor near an interface. The magnetic field is represented by a vector potential in a symmetric gauge and entered through a Peierls substitution. The full TB Hamiltonian is solved in NEMO3D21 by a parallel Block Lanczos algorithm, and the relevant low energy acceptor wavefunctions are obtained. We do not observe ionization parabolas such as in, for example,18 from which we infer that the states are probed in a condition close to flat-band. Moreover, the lever arm is small, so the tip-induced electric field is strongly screened. We therefore neglect tip-induced electric fields in our TB simulations.

The probability density for the two lowest doublets (Fig. 2(c)) are comparable to experiments (Fig. 2(b)), with an asymmetry between [110] and [1¯10] directions that reflects the difference between these two directions in the surface plane of atoms. From the calculated ratios of the Zeeman splittings for the two lowest energy doublets (ΔE±3∕2E±1∕2 ∼ 3 for B||[001] and ΔE±3∕2E±1∕2 ≪ 3 for B||[100] at low magnetic field), we infer that the lowest Kramers doublet has a predominant heavy-hole (±3/2) character whereas the second Kramers doublet has a predominant light-hole (±1/2) character, and that the quantization axis is along the surface-normal.

Figure 3 shows the measured energy splitting δE between the lowest two acceptor states as a function of acceptor depth for six different acceptors demarcated by the solid diamonds. These results are compared with a fully atomistic TB simulation marked by the grey circles. From the combination of the spatial and spectral measurements, we can conclude that the observed states belong to the s-like manifold, which contains a ±3/2 Kramers doublet and a ±1/2 Kramers doublet. Not only are both of the envelopes predominantly s-like but also the few meV energy scale of the splitting is too small to be associated with higher excited states (at 25 meV)22 or charging transitions (47 meV)23 of an acceptors. We note that the charging energy near the surface could fall as low as 20 meV,11 but this is still too large for our results.

FIG. 3.

Experimental and theoretical depth dependence of the interface induced energy splitting. The vertical error bars on the energy-level splitting are proportional to the confidence of the fit of α. Errors in Vi are negligible. The horizontal error bars are proportional to the confidence of the fit of d, which decreases for acceptors further from the interface as the spectral shift of the valence-band maximum becomes smaller.

FIG. 3.

Experimental and theoretical depth dependence of the interface induced energy splitting. The vertical error bars on the energy-level splitting are proportional to the confidence of the fit of α. Errors in Vi are negligible. The horizontal error bars are proportional to the confidence of the fit of d, which decreases for acceptors further from the interface as the spectral shift of the valence-band maximum becomes smaller.

Close modal

The energy difference between the ±3/2 and ±1/2 doublets for an acceptor near an interface can be understood in terms of different envelope wavefunctions. While the envelopes of both doublets are predominantly s-like, they contain higher spherical harmonics, in particular, d-like states.15 Whereas for the 3/2 doublets these d-like components are orthogonal to the surface-normal, the 1/2 doublets have d-like components parallel to the surface-normal. Near the interface, the parallel components get perturbed more strongly than the orthogonal components, and as a result the ±1/2 is higher in energy than the ±3/2 doublet.

We observe an increase of δE for acceptors closer to the interface, as predicted by the TB calculations. Importantly, all acceptors studied showed an energy splitting in excess of 1 meV, which is a significant perturbation compared to the binding energy of 46 meV.24 For comparison, the electric field required to obtain a splitting of 0.5 meV would be 40 MV/m,25 much greater than the electric field required for field ionization 5 MV/m.26 Our results therefore demonstrate that the presence of an interface provides an effective way to energetically isolate a single Kramers doublet that could serve as the working levels of a spin-qubit. Such a qubit could benefit from elimination of the nuclear spin bath by isotope purification, and could provide a route towards an electrically controllable spin qubit.

This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001027) and the U.S. National Security Agency and the U.S. Army Research Office under Contract No. W911NF-08-1-0527. J.A.M. received funding from the Royal Society Newton International Fellowship scheme. M.Y.S. acknowledges an ARC Federation Fellowship. S.R. acknowledges an ARC Future Fellowship and FP7 MULTI. The authors are grateful to D. Culcer for helpful discussions.

1.
K.
Saeedi
,
S.
Simmons
,
J. Z.
Salvail
,
P.
Dluhy
,
H.
Riemann
,
N. V.
Abrosimov
,
P.
Becker
,
H.-J.
Pohl
,
J. J. L.
Morton
, and
M. L. W.
Thewalt
,
Science
342
,
830
(
2013
).
2.
J. T.
Muhonen
,
J. P.
Dehollain
,
A.
Laucht
,
F. E.
Hudson
,
R.
Kalra
,
T.
Sekiguchi
,
K. M.
Itoh
,
D. N.
Jamieson
,
J. C.
McCallum
,
A. S.
Dzurak
, and
A.
Morello
,
Nat. Nanotechnol.
9
,
986
(
2014
).
3.
M.
Fuechsle
,
J. A.
Miwa
,
S.
Mahapatra
,
H.
Ryu
,
S.
Lee
,
O.
Warschkow
,
L. C. L.
Hollenberg
,
G.
Klimeck
, and
M. Y.
Simmons
,
Nat. Nanotechnol.
7
,
242
(
2012
).
4.
R.
Ruskov
and
C.
Tahan
,
Phys. Rev. B
88
,
064308
(
2013
).
5.
K. C.
Nowack
,
F. H. L.
Koppens
,
Y. V.
Nazarov
, and
L. M. K.
Vandersypen
,
Science
318
,
1430
(
2007
).
6.
S.
Nadj-Perge
,
S. M.
Frolov
,
E. P. A. M.
Bakkers
, and
L. P.
Kouwenhoven
,
Nature
468
,
1084
(
2010
).
7.
V. S.
Pribiag
,
S.
Nadj-Perge
,
S. M.
Frolov
,
J. W. G.
van den Berg
,
I.
van Weperen
,
S. R.
Plissard
,
E. P. A. M.
Bakkers
, and
L. P.
Kouwenhoven
,
Nat. Nanotechnol.
8
,
170
(
2013
).
8.
J.
van der Heijden
,
J.
Salfi
,
J. A.
Mol
,
J.
Verduijn
,
G. C.
Tettamanzi
,
A. R.
Hamilton
,
N.
Collaert
, and
S.
Rogge
,
Nano Lett.
14
,
1492
(
2014
).
9.
J. A.
Mol
,
J.
Salfi
,
J. A.
Miwa
,
M. Y.
Simmons
, and
S.
Rogge
,
Phys. Rev. B
87
,
245417
(
2013
).
10.
K.
Sinthiptharakoon
,
S. R.
Schofield
,
P.
Studer
,
V.
Brázdová
,
C. F.
Hirjibehedin
,
D. R.
Bowler
, and
N. J.
Curson
,
J. Phys.: Condens. Matter
26
,
012001
(
2013
).
11.
J.
Salfi
,
J. A.
Mol
,
R.
Rahman
,
G.
Klimeck
,
M. Y.
Simmons
,
L. C. L.
Hollenberg
, and
S.
Rogge
,
Nat. Mater.
13
,
605
(
2014
).
12.
A. M.
Yakunin
,
A. Y.
Silov
,
P. M.
Koenraad
,
J.-M.
Tang
,
M. E.
Flatte
,
J. L.
Primus
,
W.
van Roy
,
J.
de Boeck
,
A. M.
Monakhov
,
K. S.
Romanov
,
I. E.
Panaiotti
, and
N. S.
Averkiev
,
Nat. Mater.
6
,
512
(
2007
).
13.
A. P.
Wijnheijmer
,
J. K.
Garleff
,
K.
Teichmann
,
M.
Wenderoth
,
S.
Loth
,
R. G.
Ulbrich
,
P. A.
Maksym
,
M.
Roy
, and
P. M.
Koenraad
,
Phys. Rev. Lett.
102
,
166101
(
2009
).
14.
J. A.
Miwa
,
J. A.
Mol
,
J.
Salfi
,
S.
Rogge
, and
M. Y.
Simmons
,
Appl. Phys. Lett.
103
,
043106
(
2013
).
15.
D.
Schechter
,
J. Phys. Chem. Solids
23
,
237
(
1962
).
16.
E.
Foxman
,
P.
Mceuen
,
U.
Meirav
,
N.
Wingreen
,
Y.
Meir
,
P.
Belk
,
N.
Belk
,
M.
Kastner
, and
S.
Wind
,
Phys. Rev. B
47
,
10020
(
1993
).
17.
R.
Feenstra
,
Phys. Rev. B
50
,
4561
(
1994
).
18.
K.
Teichmann
,
M.
Wenderoth
,
S.
Loth
,
R. G.
Ulbrich
,
J. K.
Garleff
,
A. P.
Wijnheijmer
, and
P. M.
Koenraad
,
Phys. Rev. Lett.
101
,
076103
(
2008
).
19.
D. H.
Lee
and
J. A.
Gupta
,
Science
330
,
1807
(
2010
).
20.
R.
Rahman
,
C.
Wellard
,
F.
Bradbury
,
M.
Prada
,
J.
Cole
,
G.
Klimeck
, and
L.
Hollenberg
,
Phys. Rev. Lett.
99
,
036403
(
2007
).
21.
G.
Klimeck
,
S. S.
Ahmed
,
H.
Bae
,
N.
Kharche
,
S.
Clark
,
B.
Haley
,
S.
Lee
,
M.
Naumov
,
H.
Ryu
,
F.
Saied
,
M.
Prada
,
M.
Korkusinski
,
T. B.
Boykin
, and
R.
Rahman
,
IEEE Trans. Electron Devices
54
,
2079
(
2007
).
22.
G.
Wright
and
A.
Mooradian
,
Phys. Rev. Lett.
18
,
608
(
1967
).
23.
W.
Burger
and
K.
Lassmann
,
Phys. Rev. Lett.
53
,
2035
(
1984
).
24.
A.
Ramdas
and
S.
Rodriguez
,
Rep. Prog. Phys.
44
,
1297
(
1981
).
25.
A.
Köpf
and
K.
Lassmann
,
Phys. Rev. Lett.
69
,
1580
(
1992
).
26.
G.
Smit
,
S.
Rogge
,
J.
Caro
, and
T.
Klapwijk
,
Phys. Rev. B
70
,
035206
(
2004
).