# Position-controlled functionalization of vacancies in silicon by single-ion
implanted germanium atoms

###### Abstract

Special point defects in semiconductors have been envisioned as suitable
components for quantum-information technology.

The identification of new deep centers in silicon that can be easily activated
and controlled is a main target of the research in the field. Vacancy-related
complexes are suitable to provide deep electronic levels but they are hard to
control spatially.

With the spirit of investigating solid state devices with intentional vacancy-
related defects at controlled position, here we report on the
functionalization of silicon vacancies by implanting Ge atoms through single-
ion implantation, producing Ge-vacancy (Ge$V$) complexes. We investigate the
quantum transport through an array of Ge$V$ complexes in a silicon-based
transistor.

By exploiting a model based on an extended Hubbard Hamiltonian derived from
ab-initio results we find anomalous activation energy values of the thermally
activated conductance of both quasi-localized and delocalized many-body
states, compared to conventional dopants. We identify such states, forming the
upper Hubbard band, as responsible of the experimental sub-threshold transport
across the transistor.

The combination of our model with the single-ion implantation method enables
future research for the engineering of Ge$V$ complexes towards the creation of
spatially controllable individual defects in silicon for applications in
quantum information technologies.

###### keywords:

Quantum transport, single-ion implantation, point defects, Ge-vacancy complex,
Hubbard model

Simona Achilli* Nguyen H. Le Guido Fratesi Nicola Manini Giovanni Onida Marco
Turchetti Giorgio Ferrari Takahiro Shinada Takashi Tanii Enrico Prati

S. Achilli, G. Fratesi, N. Manini, G. Onida
Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
20133 Milano, Italy.
Email Address<EMAIL_ADDRESS>
Nguyen H. Le
Advanced Technology Institute and Department of Physics, University of Surrey,
Guildford GU2 7XH, United Kingdom.

Marco Turchetti
Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche,
Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
Current affiliation: Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA.

Giorgio Ferrari
Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di
Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

Takahiro Shinada
Center for Innovative Integrated Electronic Systems (CIES), Tohoku University,
468-1 Aramaki Aza Aoba, Aoba-ku, Sendai, Miyagi, 980-8572, Japan

Takashi Tanii
Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku,
169-8555 Tokyo, Japan.

Enrico Prati
Istituto di Fotonica e Nanotecnologie, Consiglio Nazionale delle Ricerche,
Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Deep-level impurities in group IV semiconductors have been envisioned as a
possible route toward room-temperature quantum information processing [1, 2,
3, 4]. Vacancies in silicon could in principle provide deep electrically
relevant individual centers suitable for high temperature quantum information
manipulation [5, 6], but their positioning is rarely controllable [7]. Indeed,
the miniaturization of nanoelectronic devices for quantum information
processing [8] faces many technological and fundamental challenges, such as
the control of effects induced by structural disorder [9] and those related to
the localization of the dopant wavefunction in nanometer-size channels [10].
While the former can be partly addressed by exploiting high-precision
implantation techniques [11, 12] or scanning-tunneling microscopy [13], the
wavefunction localization could be a major issue, due to the crucial role it
plays in phenomena related to electronic correlation, transport, and
eventually in setting up controlled electronic properties suitable for
exploitation in quantum information devices.

Quantum transport in silicon field-effect transistor with As, B, P dopants has
been probed both at cryogenic and room temperature [14, 15, 16, 17]. By
controlling the dopant density with atomic precision it is possible to explore
different quantum conduction regimes [18, 19] ranging from single-electron
Coulomb blockade to Hubbard-band transport in the Anderson-Mott transition
region [20, 14]. Nevertheless the application of conventional dopants as
qubits requires operating temperature below 1.5 K [21], preventing the
exploitation of this technology between soft cryogenic and room temperatures.

Such drawback can be mitigated by exploiting vacancy-related complexes that,
by virtue of their deep electronic levels, could be used for quantum transport
at room temperature. The functionalization through foreign species would also
allow a spatial control of these defects and could be exploited to induce a
spatial anisotropy of the electronic wavefunctions without altering the deep
electronic spectrum, leading to a reduction of the interference effects [16].
Fast optical manipulation like NV centers and divacancies in SiC enable in
principle up to room temperature operations [4]. Moreover, tailoring
correlation effects in nanostructure via a precise control of both the
positioning and the composition of point defects would open the way to quantum
metamaterials. This challenge benefits of the interplay between advanced
experimental techniques and theoretical models able to predict the properties
of interest [22].

Based on ab-initio calculations [3], some of us have recently demonstrated
that Ge$V$ complexes in silicon are stable defects, characterized by excited
states deep in the bandgap, at about $\simeq-0.5$ eV and $\simeq-0.35$ eV from
the conduction band, consistent with experiment [23]. Such impurity states
have large on-site electron-electron repulsion ($\simeq 150$ meV) due to their
highly localized wavefunctions (the decay length of the lowest charged state
is $\simeq 0.45$ nm). This localization far exceeds that of conventional P and
B states [24]. Here we exploit this theoretical prediction for the fabrication
of the first transistor with Ge$V$ complexes in the silicon channel, and we
interpret the electric conductance measurements by means of a Hubbard-like
model.

In view of addressing the challenge of functionalizing vacancies in silicon as
deep-level spatially-controlled centers, we generate Ge$V$ complexes by
single-ion implantation (SII) of germanium atoms. The overall strategy
consists in investigating the trans-conductance of silicon transistors of
length of both 200 nm and 500 nm, operated from cryogenic to room temperature.
The transistor’s channel consists of a linear array of up to 100 implanted Ge
atoms, which have been activated at the moderate annealing temperature of
$550^{\circ}$C.

The quantum transport across an array of such defects reveals the formation of
impurity bands inside the silicon band gap. Differently from similar
experiments carried out with phosphorous impurities, in which activation
energies for thermally activated quantum transport are of the order of few meV
[20], Ge$V$ complexes show different kinds of thermally-activated processes of
the Hubbard band, that are characterized by activation energies ranging from
hundreds of $\mu$eV to tens of meV.

For the modeling of a one-dimensional array of few Ge$V$ centers with linear
density and disorder comparable to experiments, we exploit an extended Hubbard
approach. The model exploits Hamiltonian parameters derived from the ab-initio
wavefunctions instead of a Kohn-Luttinger description [25, 26, 27], going
beyond an effective-mass theory approach [28]. Through this formalism we shed
light on the mechanisms of quantum transport in such array of Ge$V$ complexes,
in comparison to conventional dopants.

A side-by-side analysis of the experimental and model data allows us to
explain the observed impurity band as due to the upper Hubbard band. By
raising temperature from $4.2$ K up to room temperature, we demonstrate that
the conductance is driven by weakly localized states at low temperature, while
at high temperature delocalized states start to contribute.

The spatially deterministic implantation of up to 100 Ge$V$ centers and
electrical readout in our experiment may be further developed in the future to
achieve, - thanks to lower implantation energies [11], the sufficient spatial
control to address for instance large scale analog quantum simulation of
strongly-correlated models with deep impurities in silicon. This system size
is far beyond the capability of conventional-computer exact simulations and is
thus the regime of interests for future semiconductor-based quantum simulators
[29, 30]. Although it is not granted that a nanowire of 100 implanted Ge$V$
sites may eventually realize a quantum simulator with 100 qubits, impurities
in silicon complement cold-atom platforms [31, 32], as they enable us to
access interaction strengths and hopping amplitudes that are orders of
magnitude higher, easing the need for extreme cooling [29]. Ge$V$ is also
deeper than shallow dopants, with the binding energy of the double occupancy
state more than 200 times the $1.7$ meV value in Si:P [26], allowing in
principle a much better protection against ionization to the conduction band.
Our transport measurements allow us to confirm starting fabrication parameters
which could be further developed in the future to realize a quantum device
based on Ge$V$ impurities.

##  2 Results

###  2.1 Optimization of the linear implantation density

As our approach relies on measuring the conductance provided by the electrons
hopping from one localized state to the next, the device must attain a large
enough linear density of Ge$V$ sites for “impurity” bands to form. Thus,
compared to the wide impurity states of As and P [14], the implantation pitch
for Ge must be adapted to the size of the Ge$V$ localized states. To assess
this size, we resort to ab-initio calculations.

The Ge$V$ complex is a neutral defect, being Ge isoelectronic to silicon, and,
differently from conventional dopants, does not show hydrogen-like shallow
states, but deep energy levels related to the dangling bonds in the vacancy.
In the Ge$V$ ground state a deep level within the valence band and a defect
state in the gap are filled by the four unpaired electrons from the dangling
bonds. Further electrons, injected into the channel by applying a small bias
voltage between the source and the drain electrodes, occupy strongly-bound
defect states in the energy gap, experiencing a relatively large on-site
Coulomb repulsion ($U\simeq 150$ meV). Depending on the amplitude of the
inter-site hopping, the channel may behave as a more-or-less strongly-
correlated system, where the Mott-Hubbard physics applies [33, 34].

We model the one-dimensional Ge$V$ array with an extended Hubbard Hamiltonian
[35, 20, 36]

$\displaystyle H=$ $\displaystyle\sum_{i}\epsilon_{i}n_{i}-\sum_{\langle
i,j\rangle}\sum_{\sigma=\uparrow,\downarrow}t_{i,j}(c^{\dagger}_{i,\sigma}c_{j,\sigma}+c^{\dagger}_{j,\sigma}c_{i,\sigma})$
(1) $\displaystyle+U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}+\sum_{i\neq
j}W_{i,j}n_{i}n_{j}\,.$ (2)

which includes the on-site energy level $\epsilon_{i}$ for each defect, the
coupling $t_{i,j}$ between nearest-neighboring sites $\langle i,j\rangle$, the
on-site Hubbard correlation $U$ energy, plus inter-site long-range electron-
electron repulsion $W_{i,j}$. The $c_{i,\sigma}$ is a second-quantization
fermionic operator which destroys a spin-$\sigma$ electron at site $i$;
$n_{i\sigma}=c^{\dagger}_{i,\sigma}c_{i,\sigma}$, and
$n_{i}=\sum_{\sigma}n_{i\sigma}$ are the number operators. The screening by
the host silicon matrix is accounted for, scaling $W_{i,j}$ by the dielectric
constant of silicon ($\epsilon_{\rm Si}=11.7)$. For further details, see
Section 5.

For charged defects such as donor or acceptor impurities, the on-site energy
$\epsilon_{i}$ must include the long-range Coulomb interaction $V$ from the
ion core of nearby sites [35], giving rise to position-dependent on-site
energies $\epsilon_{i}$ (see bottom of Figure 1b). Since Ge$V$ is a neutral
object, here the on-site energies are the same for all the sites in the array,
($\epsilon_{i}=\epsilon$, see Figure 1b). Such difference between Ge$V$ and
conventional dopants leads to a different shape of the conductance spectrum,
as discussed below.

Figure 1:  (a) Wavefunction of the lowest charged state bound to the Ge$V$
complex, $-0.5$ eV below the conduction band. Two complexes are aligned in the
$[110]$ direction through the vacancy center (black crosses), and the panel
reports the cut through a $[\overline{1}01]$ plane. Si ions in the outermost
crystal layer are indicated by black dots; gray dots mark ions in the second
layer. The Ge atom is out of plane. (b) The conduction mechanism through the
array of Ge$V$ defects in the low-filling (top) and half-filling (middle)
regime. The comparison with the case of P impurities in silicon is reported
(bottom) showing the localization of the single electron wavefunction due to
the different on-site energies of the impurities. For Ge$V$, the single-
electron and half-filling configurations are reported illustrating the effect
of the on-site Hubbard correlation. (c) Computed hopping parameter $t_{i,j}$
(dots) as a function of the inter-site distance. The dotted line is the fitted
exponential behavior. (d) The electron-electron interaction obtained from the
ab-initio wavefunction (dots). The dotted line indicates the large-distance
asymptotic decay (see text).

We evaluate the hopping and inter-site repulsion parameters using the
wavefunction of the electronic state occupied by an additional electron on the
defect, namely the charged excitation sitting $-0.5$ eV below the conduction-
band bottom [3]. The use of a wavefunction obtained from ab-initio calculation
allows a more accurate description of deep energy levels of Ge$V$ with respect
to the usually adopted Kohn-Luttinger wavefunction which is suitable for
shallow levels of conventional dopants but not appropriate for Ge$V$. Figure
1a shows a section of such ab-initio-computed wave function across a high-
symmetry crystal plane through two vacancy centers. In the model, the Ge atoms
bound to the vacancies lie along the [111] crystal direction (outside the
plane reported in Figure 1a). The wavefunction in the proximity of the defect
is asymmetric, with a larger amplitude at the Ge side of the vacancy, and
exhibits a fast decay along the [110] crystal direction that we chose as
direction for implantation. In real samples one can expect the Ge ions to
randomly occupy one of the four substitutional positions around the vacancy.
Nevertheless, due to their fast decay, defect wavefunctions at the typical
implantation distances are essentially isotropic. We can hence expect that the
overlap between wavefunctions at different sites is weakly affected by the
orientations of the Ge$V$ complexes and by the Ge$V$-Ge$V$ joining line.

Figures 1c and 1d report the hopping and inter-site repulsion parameters
computed with the ab-initio wavefunction of Figure 1a. The hopping $t_{ij}$
follows approximately an exponential decay as a function of distance, with
smaller oscillations than typically observed for P in silicon. Such
oscillations are usually more marked at small distances, while they are
quenched for large inter-site separations. Accordingly, for $t_{ij}$ we take
the exponential function of distance (black line in Figure 1c), fitted on the
ab-initio overlaps evaluated for $d<3$ nm.

At all distances the hopping parameter is significantly smaller than those
relevant for conventional dopants [35] such as P, due to the stronger
localization of the Ge$V$ wavefunction. Guided by Figure 1c, the Ge
implantation pitch needs to be of order $d\simeq 5$ nm, for $t_{i,j}$ to reach
an amplitude that supports measurable conductance in conventional P-doped
single-impurity transistors, namely in the region around $t_{i,j}\simeq 1$
meV. In our setup, we have implanted the ions at a $10$ nm spacing, compared
to the $\sim 100$ nm spacing used for P [14].

The inter-site electron-electron repulsion $W_{ij}$ decays as $V_{0}/d_{ij}$
(with $V_{0}=e^{2}/4\pi\epsilon_{0}\epsilon_{\rm Si}$) for $d_{ij}\gtrsim 2$
nm (black curve in Figure 1d). Given the adopted spacing in the SII
fabrication process, for our calculations we stick to the
$W_{ij}=V_{0}/d_{ij}$ asymptotic formula, namely the same expression used for
other kinds of dopants.

###  2.2 Thermally-activated transport in a disordered 1D array of Ge$V$
defects

We fabricated a set of nanotransistors doped by single-ion implantation [6]
whose channel charge is controlled by a backgate, so the implantation is
carried out from the top. The channel of the transistor is either $200$ nm or
$500$ nm long, depending on the sample. Eight and fifty pairs of Ge ions are
implanted in the former and the latter, respectively (Figure 2b) to form a
1-dimensional (1D) array connecting the source and the drain electrodes, as
detailed in Section 6 below.

Figure 2:  (a) A scanning microscopy image and (b) a scheme of the Ge-
implanted transistor. Two different lengths of the silicon channel are
considered: 500 nm or 200 nm with 50 or 8 pairs of Ge ions, respectively,
implanted by single-ion implantation. The silicon-on-insulator (SOI) thickness
is 90 nm and the underlying BOX is of 125 nm. (c) $I$–$V$ trans-
characteristics, measured from 67 K to room temperature. The array shows
impurity band transport up to room temperature. The yellow hexagon marks the
bias voltage corresponding to low-filling regime at which the activation
energies are extracted (see Figure 5c). (d) Low-temperature $I$–$V$ trans-
characteristics. A thermally activated band is formed below the conduction-
band edge at +3.5 V. The cyan square marks the bias voltage corresponding to
low-filling regime at which the activation energies are extracted (see Figure
5d).

The $I$–$V$ trans-characteristics (reported in Figure 2c-d and Figure 4b for
the long and short channel transistor, respectively) shows that below the
threshold of the transistor additional transport is observed after the
implantation at sufficiently high density, due to impurity-band formation. We
have explored temperatures ranging between 4.2 K and 90 K for the short-
channel sample (see also Supporting Information) while for the long one we
extend up to room temperature.

A thermally activated impurity band is observed up to room temperature, whose
peculiar activation energy values are discussed below.

In order to characterize the impurity-band transport, we simulate the
conductance by exploiting the rate-equation formalism [37, 38, 35]. This
method focuses on the tunneling of electrons from the leads into the array,
leading to a change of the many-body state from $(n_{e}-1,\alpha)$ to
$(n_{e},\beta)$, where $n_{e}$ is the number of electrons in the array and
$\alpha,\beta$ label the states. A transition rate equation describes the
tunneling of one electron from the left electrode into the array and from the
array into the right electrode and a temperature dependent linear response
conductance is obtained, see Section 5.

Figure 3a reports the zero-bias conductance for an ideal array with equally-
spaced Ge$V$ sites placed at a mutual distance of 4 nm, illustrating the
effect of a variable number of sites.

Figure 3:  (a) Computed conductance at $T$=$20$ K, as a function of the
chemical potential for Ge$V$ arrays consisting of $N_{S}$=$4$, 6 and 8 sites
placed at a regular spacing of 4 nm. The chemical potential is referred to the
conduction band bottom. The positions of the peaks match the addition energies
recorded in the horizontal axis of panel (b), obtained at $T$=$0$ K. The half-
filling (HF) configuration for each array is marked. (c) Computed bandwidth
and low-filling conductance $G_{\rm 1\,e}$=$G(\mu_{n=1}$) of an array of 8
Ge$V$ sites as a function of the spacing between the sites. (d) The
conductance of an array of $N_{S}=4$ P sites in silicon at the same 4 nm
spacing and $T$=$20$ K as panel (a).

The conductance peaks correspond to the progressive filling of the $N_{S}$
sites in the array with up to $2N_{S}$ electrons, forming many-body states.

For comparison, the horizontal axis of Figure 3b reports the addition energies
defined by $E_{ad}(n_{e})=E_{0}(n_{e})-E_{0}(n_{e}-1)$, in terms of the
ground-state energy $E_{0}(n_{e})$ for $n_{e}$ electrons. At $T=0$, the shift
in the chemical potential that matches the addition energy leads to the
injection of one extra electron into the channel.

The model exhibits the formation of two Hubbard bands separated by an energy
gap which is a result of the on-site electronic correlation. By increasing the
number $N_{S}$ of sites in the array, the Hubbard gap between these bands
decreases due to the long-range Coulomb interaction $W_{ij}$.

Indeed the electron-electron repulsion term $W_{ij}$ increases the width of
the Hubbard bands that would otherwise be of order $t_{i,j}$, namely very
narrow compared to the Hubbard gap. This bandwidth increases for increasing
number of sites $N_{S}$ (Figure 3b), while it decreases by increasing the
distances between the defects (Figure 3c), of course due to the decrease in
both $W_{i,j}$ and $t_{i,j}$.

On the basis of the theoretical description of the Hubbard system we can infer
that only the upper Hubbard band is accessible in the experimental
measurements, being the lower one very deep in energy. Although it was not
possible to determine the lever arm factor because the transport regime does
not involve a Coulomb blockade, the observation of the only upper Hubbard band
is compatible with previous reports on the same batch of devices, for which
the chemical potential could be tuned by approximately 20 meV per V of applied
back-gate voltage [20]. However, the applied voltage may not exceed -10 V,
since larger values would lead to dielectric breakdown of the oxide. This
implies that the accessible energy range is limited to about 200 meV below the
conduction band, not sufficient to reach the lower Hubbard band (500 meV below
the conduction band).

In the energy gap of silicon the electric current in the device is driven by
electronic states of the electrons injected in the channel. In the ideal array
coherent resonant tunneling [39] is carried out by delocalized states in the
array, characterized by a sizable wavefunction amplitude overlapping the left
and right electrodes. In contrast, localized states generated by either
Anderson or Mott localization (or both) contribute poorly to the conductance.
These states are expected to contribute to thermally-activated nearest-
neighbor hopping or variable-range hopping [40]. On this basis, and inspecting
the main contributions to the conductance in Figure 3a, we can infer that
many-body configurations at low-filling of both the lower and upper Hubbard
band are delocalized across the whole array. The large value of the
conductance for single-electron filling is a consequence of the degeneracy of
the electronic level at the individual sites in the array, due to the neutral
character of the Ge$V$ complex. Consequently, the resonant transport of a
single electron is favored because no localization effects are present.

As the Hubbard band fills up, Coulomb correlations prevail, causing a drop in
conductivity, as can be observed above -450 meV and above -250 meV for the
lower and upper Hubbard band, respectively.

At half-filling the system undergoes Mott localization leading to an
insulating state with one electron per site.

The quantum transport process in Ge$V$ shows therefore significant differences
compared to the conventional donors. Being the latter charged impurities, the
on-site energies depend on the interaction with the nearby ionic centers,
leading to deeper levels at the middle of the array, as shown in the bottom
part of Figure 1b. As a consequence, in such kind of defects the one-electron
transport is disfavored, due to the spatial localization of its ground-state
wavefunction at the central sites, with small overlap with the electrodes.

Indeed, the conductance of a one-dimensional array of P atoms in silicon,
obtained with a similar approach [35] (see Figure 3d), is maximum near half-
filling, where the impurities are neutral, with the result that the electron-
electron repulsion ($W_{i,j}$ term) compensates the unequal on-site energies
$\epsilon_{i}$ due to the electron-ion core long-range attraction.

The fingerprint of Mott localization in Ge$V$ is more evident in longer arrays
that show a contribution to the conductance only from a small percentage of
the associated many-body states (see Figure 3a). Differently, by keeping
constant the number of sites in the regularly-spaced array and increasing
their mutual distance, the conductance increases (see Figure 3c) due to the
reduction of the inter-site Coulomb interaction.

In order to account for the main experimental features, the ideal assumption
of a perfect regularly-spaced array has to be relaxed.

Figure 4:  (a) Calculated temperature-dependent conductance (upper Hubbard
band contribution) of an array of $N_{S}$=$8$ Ge$V$ complexes at distances
4$\pm$2 nm. The chemical potential is referred to the conduction band bottom.
(b) Trans-characteristics of the short-channel sample with eight pairs of Ge
ions implanted by SII. Blue, red and green vertical dotted lines mark the
three sub-bands identified by fitting the data with Fermi Dirac functions (see
the text). These three lines and the purple vertical one also mark the gate
potential at which the activation energies in Figure 5d and c are extracted.

For a more realistic model we have included the effect of structural disorder,
by introducing a random variability of the inter-site distance, up to $50\%$.
Furthermore, the improved model includes an energy-dependent coupling between
the array and the electrodes that takes into account the slower decay of the
wavefunctions of the leads near the top of the tunneling barrier (see Section
5).

Figure 5:  (a) High-temperature dependence of the measured conductance of the
short-channel sample at very low filling. The corresponding $V_{g}$ is marked
in Figure 4b by a purple vertical line at $\sim$3.8 V. b) Low-temperature
dependence of the measured conductance of the short-channel sample at middle-
high filling. The activation energies are extracted at the $V_{g}$ marked by
vertical lines with corresponding color in Figure 4b. (c) High-temperature
dependence of the measured conductance of the long-channel sample in the low-
filling condition (see corresponding $V_{g}$ in Figure 2c). The activation
energies extracted through a fit with the Arrhenius law are reported. (d) The
same as panel (e) for high-filling (see corresponding $V_{g}$ Figure 2d) and
lower-temperature regime. (e) Arrhenius fits of the high-temperature
dependence of the theoretical conductivity. The individual peaks corresponding
to less than a quarter filling of the upper Hubbard band are reported, labeled
by their characteristic values of $n_{e}$. (f) The same as panel a) for the
low-temperature regime and high-filling of the upper Hubbard band.

Figure 4a reports the computed conductivity for the upper Hubbard band of a
1D-array with 8 Ge$V$ sites, spaced randomly in the 2–6 nm range, and
temperature from 4.2 K to 87 K. We focus on the upper Hubbard band because the
interpretation of the experimental measurements based on the theoretical
results suggest that the lower Hubbard band is too deep in energy to be
measured. Comparison with Figure 3a indicates a sharp reduction in
conductivity, which we shall discuss in the next section.

In the disordered array the contribution of states at the middle of the upper
Hubbard band ($n_{e}=12$–14, $\mu\in[-187,-140]$ meV) prevails over the low-
filling ones ($n_{e}=9$–11, $\mu\in[-267,-212]$ meV) as an effect of the
energy dependent coupling to the electrodes that hinders the conductivity of
deep energy states. The calculated conductance can be compared to the trans-
characteristics of the short-channel transistor reported in Figure 4b for
temperatures between 10 K and 35 K and in the Supporting Information for lower
and higher temperatures. For the short-channel sample three sub-bands can be
identified and can be related to three different filling regimes, i.e. low,
middle and high filling. By fitting the trans-characteristic with three Fermi-
Dirac functions we have extracted the $V_{G}$ corresponding to the onset of
the three sub-bands, which is reported in Figure 4 by blue, red and green
lines (see also the Supporting Information). An overall agreement between
theory and experiments can be observed, although the tiny number of defects in
the model gives rise to peaks instead of the measured plateau, which is
related to the denser spectrum that characterizes channels with many sites.
For what concerns the temperature dependence we observe a general increase of
the conductance with the temperature, although in correspondence of some peaks
this behavior is not satisfied due to the absence, at these specific energies,
of excited states that can be populated by increasing temperature, a condition
which is mitigated in long arrays.

By fitting the temperature dependence of the conductance derived from the
trans-characteristics in Figures 2c,d and 4b with an Arrhenius law:

$\ln G(T)=\text{const}-E_{a}/(k_{\text{B}}T)$

we can extract the activation energies $E_{a}$. We consider two different
regimes, i.e. those corresponding to very low (less-than-a-quarter) filling
(Figure 5a and c) and to middle-high (more-than-a-quarter) filling (see Figure
5b and d) We find that at middle-high filling and low temperature (4 K–20 K)
the the measured activation energies of Ge$V$ defects are of the order of 1
meV or less (red and green line of Figure 5b and Figure 5d), i.e. smaller than
those of typical donors such as arsenic and phosphorous at low temperature
($\sim 1.2$–$2.5$ meV), as observed in few-dopants arrays.

Also the activation energy at high temperature, i.e. above 20 K (Figure 5d),
is smaller than that reported for conventional dopants (4.6 meV versus 7 meV).

In the low-filling regime (see Figure 5a and c) the measured conductance at
high temperature is characterized by a very large activation energy, of the
order of 30 meV. In particular for the short-channel transistor we found an
activation energy of 22.2 meV while for the long-channel one it reaches the
value of 26.7 meV.

The theoretical model is able to reproduce the experimental behavior for both
the regimes as observed in Figure 5e and f for an array with eight Ge$V$ sites
and in the Supporting Information for longer systems including up to twelve
sites. In particular the conductance peaks in the middle of the band
($n_{e}=12$, $n_{e}=13$ in Figure 5f) are characterized by activation energies
of the order of meV in the temperature range $4$–$20$ K. Differently, the
peaks $n_{e}=9$ and $n_{e}=10$ (Figure 5e) display large activation energies
at high temperature, reaching 30 meV at the bottom of the upper Hubbard band,
in agreement with the experiments.

It is worth noting that despite the theoretical model can be directly compared
to the measurements performed on the short-channel sample, which counts
approximately the same number of sites in the channel, the peculiar
temperature dependence observed for different filling and temperature regimes
is reproduced also by increasing the number of sites up to twelve and
increasing the distance between the defects (see Supporting Information).

Moreover the statistical analysis, performed on 10 different random spatial
disorder realizations for each theoretical sample with fixed number of sites,
confirms the observed behavior (see Supporting Information).

##  3 Discussion

The sharp reduction of the conductivity computed for the realistic model
compared to the perfect array is due to the strong localization of the
electrons induced by the structural disorder enhanced by the large sensitivity
of $t_{i,j}$ on the inter-site spacing, a result of the fast decay of the
wavefunctions of deep levels. Disorder is especially effective in reducing the
intensity of the conductance at low-filling of the upper Hubbard band,
relative to the ideal array. Consequently, the contribution of extended states
is quenched, with a residual conductance supported by rather localized states,
explaining the very low conductance in the experiments and the need for a very
dense array of implanted complexes to measure a current signal.

Calculations exhibit the largest conductance at the middle of the upper
Hubbard band, corresponding to a filling $n_{e}/N_{S}\simeq 1.5$. In this
configuration the many-body wavefunction is sufficiently spread over multiple
sites to support electronic transport. Indeed this intermediate filling takes
the best advantage of the on-site Coulomb repulsion, which tends to favor a
leveling of the on-site average occupations, contrasting disorder and long-
range Coulomb effects. As a result, at the middle of the upper Hubbard band a
partial balancing between the localization induced by disorder and the strong
Coulomb repulsion that characterizes states at complete filling supports
transport.

The model based on a relatively short array does not grasp all experimental
features. We can expect that simulations of a longer array would provide a
larger bandwidth and a denser electronic spectrum, with a fast-growing number
of excited states. These excitations are expected to play a significant role
in the thermally-activated conductivity. By increasing $N_{S}$, the peaks
observed in the conductivity at finite $T$ would turn into a continuous band
and the conductivity in the Hubbard band would appear as a plateau, as in the
experiment.

By raising temperature, the peak conductivity increases due to the thermally-
activated contributions of excited states delocalized across the entire 1D
array. Despite mismatched array lengths, measurements and calculations agree
on indicating that at low temperature the conductance behaves approximately as
the parallel of two thermally-activated currents, associated to small
activation energies near 1 meV and 5 meV (see Figure 5b,d and f). This mild
temperature dependence of the conductance is likely the result of few
delocalized states intersparsed in a dense excited-state energy spectrum of
mainly localized strongly-correlated states.

Also the large activation energy observed at large negative gate voltage
(Figure 5a and c) is reproduced by the model at the beginning of the upper
Hubbard band, $n_{e}/N_{S}\gtrsim 1$ (Figure 5e). In this small-filling regime
measurements and calculations agree on an activation energy of the order of 30
meV. It is worth noting that the large activation energy extracted from the
calculation for temperatures between 20 and 87 K suggests that at room
temperature the small-filling contributions ($n_{e}=9,10$ in Figure 5e) are
indeed expected to reach a value comparable to that of the main peaks, leading
to a measurable signal.

This large activation energy observed in the Ge$V$ 1D array compared to other
dopants can be related to the delocalized nature of the states supporting
electronic transport. At $n_{e}/N_{S}\gtrsim 1$, an essentially single
electron is pushed across the Ge$V$ array charged by the other $n_{e}=N_{S}$
electrons. The long-range Coulomb repulsion and disorder tend to localize all
or almost all its low-energy states. Delocalized states start to become
available beyond an energy of the order of the difference in Coulomb repulsion
near the centre and at the ends of the array, i.e. for $k_{\text{B}}T\simeq
30$ meV. This condition affects less the $n_{e}/N_{S}\simeq 1.5$ filling,
where the on-site Coulomb repulsion fabricates a few relatively delocalized
current-carrying many-body states even at much lower energy.

The model analysis allows us to formulate two different pictures for the two
kind of impurities, namely conventional P dopants and Ge$V$ complexes. The
former are characterized by a larger intersite hopping, a smaller on-site
repulsion, and neutral energy-aligned sites in the upper Hubbard band. For
these reasons, P or As arrays are less affected by disorder and tend to
support larger values of conductance than Ge$V$ arrays.

On the other hand, in Ge$V$ arrays both the disorder and effective Coulomb-
repulsion site-dependent profile are very effective in acting against
transport. Here localized states prevail by far and conductance increases
weakly by raising temperature. As an exception, at low-filling in excess of
half-filling a few highly-excited delocalized states can provide a sizable
contribution to conduction at room temperature.

##  4 Conclusions

Ge$V$ complexes provide a viable tool to exploit deep levels typical of the
vacancy defect in silicon, by means of the precise positioning of single-ion
implantation. By investigating the quantum transport through an array of Ge$V$
complexes in silicon-based transistors, we demonstrate the activation of the
Ge$V$ complexes at the low annealing temperature of $550^{\circ}$C and the
formation of an impurity band below the conduction-band edge, representing the
upper Hubbard band. The peculiar activation energies of the Ge$V$ complexes
induce effects which differ significantly from those of conventional dopants,
such as P and As.

By exploiting an extended Hubbard model based on ab-initio-derived parameters,
we explain the thermally-activated contribution to the conductance of weakly-
localized states at high filling and low temperature, for which an activation
energy of less than 1 meV is observed. In contrast, at low-filling we
associate fairly delocalized states to a high activation energy, in the 30 meV
region.

We have shown that the single-ion implantation method enables the engineering
of Ge$V$ complexes, a step toward future construction of spatially-
controllable single defects in silicon, with potential exploitation in the
control of quantum information encoded in electron states, with additional
interest for the analog quantum simulation of strongly-correlated models.

##  5 Theoretical model

We employ an extended Hubbard description of the 1D array of defects in
silicon, Eq. (1). Its parameters are obtained by ab-initio calculations [3].
The on-site correlation energy is evaluated as the energy separation between
the Ge$V^{-}$ and Ge$V^{2-}$ charge states, amounting to $U=150$ meV. The
other parameters are calculated through the matrix elements, whose expression
is provided in Ref. [35]. The involved integrals are evaluated exploiting ab-
initio wavefunctions computed previously [3], centered on different defects.
The wavefunction of the in-gap state localized around the defect is available
on a real-space grid, whose extension coincides with the size of the ab-initio
simulation cell.

In order to account for the long-distance decay of the wavefunction we have
extended the size of the simulation cell compared to our previous work [3], to
a $4\times 4\times 4$ silicon bulk supercell. This enlargement of the
computational cell has been made possible by adopting a theoretical approach
based on a localized basis set and pseudopotential description of the
electron-ion interaction, as implemented in the SIESTA package [41]. For the
exchange-correlation part, rather than an hybrid functional we adopted a
GGA-1/2 approximation [42] , which is able to reproduce both the silicon
energy gap and the excited states of the defect, in very good agreement with
the results we obtained with the more accurate (but computationally more
expensive) hybrid functional [43]. Being the GGA-1/2 approach not suitable to
compute forces, we kept the defect geometry as previously determined in the
relaxation done with the hybrid functional [3]. For the calculations of the
two-centers integrals following Eq. 3-7 of Ref. [35], we extended the
wavefunction to larger distances, by extrapolating its natural decay.

The conductance related to resonant transport is obtained by exploiting the
rate equation derived from the Fermi golden rule [37, 38, 35] that yields the
tunneling probability of electrons from the left to the right lead, accounting
for the transition from a $\Psi_{\beta}^{n_{\sigma}-1,n_{\sigma^{\prime}}}$ to
a $\Psi_{\alpha}^{n_{\sigma},n_{\sigma^{\prime}}}$ many-body state in the
array:

$\displaystyle G=$ $\displaystyle\frac{2\pi\Gamma e^{2}}{\hbar
k_{\text{B}}T}\sum_{n_{\sigma},n_{\sigma}^{\prime}}\sum_{\alpha,\beta}\frac{M_{\alpha,\beta,\sigma}^{L,n_{\sigma},n_{\sigma}^{\prime}}M_{\alpha,\beta,\sigma}^{R,n_{\sigma},n_{\sigma}^{\prime}}}{M_{\alpha,\beta,\sigma}^{L,n_{\sigma},n_{\sigma}^{\prime}}+M_{\alpha,\beta,\sigma}^{R,n_{\sigma},n_{\sigma}^{\prime}}}$
(3) $\displaystyle\times
P_{i}^{n_{\sigma},n_{\sigma}^{\prime}}[1-f_{\text{FD}}(E_{i}^{n_{\sigma},n_{\sigma}^{\prime}}-E_{j}^{n_{\sigma}-1,n_{\sigma}^{\prime}}-\mu)]\,.$

Here $E_{\alpha}^{n_{\sigma},n_{\sigma}^{\prime}}$ is the energy of the
$\alpha$-th many-body state with $n_{e}=n_{\sigma}+n_{\sigma}^{\prime}$
electrons; $M_{\alpha,\beta,\sigma}^{L,n_{\sigma},n_{\sigma}^{\prime}}$
($M_{\alpha,\beta,\sigma}^{R,n_{\sigma},n_{\sigma}^{\prime}}$) are the
transition rates between state $(\alpha,n_{\sigma},n_{\sigma}^{\prime})$ and
$(\beta,n_{\sigma}\pm 1,n_{\sigma}^{\prime})$ that involve the injection of
one electron from (to) the left (right) lead; $P$ is the the grand canonical
ensemble probability that controls the filling of a many-body level at a given
temperature; $f_{\text{FD}}$ is the Fermi-Dirac distribution; finally,
$n_{\sigma}$ is the number of the involved up- or down-spin electrons in the
array, with $n_{\sigma}^{\prime}=n_{e}-n_{\sigma}$ those of the other spin
kind. In the evaluations of $G$, at very low temperature ($T\leq 3$ K) in the
sums we include only the 2 lowest-energy many-body states for each
$(n_{\sigma},n_{\sigma}^{\prime})$; at higher temperature we include the 10
lowest excited states.

The pre-factor $\Gamma$ includes the coupling to the electrodes, which
accounts for the hopping between the leads and the end sites in the array. For
conventional shallow dopants $\Gamma$ is taken as a constant energy of the
order of a fraction of meV. In the calculations for the ideal system (Figure
3a,d) we fix $\Gamma=0.1$ meV.

The assumption of a constant coupling to the leads should be released for
complexes with deep electronic levels, because the wavefunctions of the Fermi-
gas electrons in the lead decays at a significantly different rate as a
function of the energy at which they execute the transition, thus affecting
the tunneling overlap with the electrodes. Accordingly, we modulate the
coupling $\Gamma$ with the transmission probability through the potential
barrier separating the lead from the channel. Assuming for simplicity a
rectangular potential barrier of height $V_{B}$ and width $a$, the
transmission of an electron at energy $E<V_{B}$ is given by [44]:

$\vartheta=\left[1+\frac{V_{B}^{2}\sinh^{2}(ka)}{4E(V_{B}-E)}\right]^{-1}$

where $k=\sqrt{2m(V_{B}-E)/\hbar)}$.

In our device the n-doped silicon forming the leads is smoothly connected to
the channel due to the diffusion of P dopants, thus the barrier can be
considered extremely thin. We take $a=a_{Si}~{}0.34$ nm; the height of the
barrier for the lowest $n_{e}=1$ state coincides with the excitation energy
$0.5$ eV of the ground state of a Ge$V$. The coupling to the electrodes
$\Gamma=\nu\theta$ is set equal to $0.001$ meV for the deepest electronic
level of the Ge$V$ complex in the lower Hubbard band. This choice of
parameters leads to a computed conductance which is comparable to the
experimental one and the thermally activated behavior is mildly dependent on
the choice of $\Gamma$.

##  6 Experiment

Transistors were fabricated on a (100)-oriented n-type (phosphorus $\sim
10^{15}$ cm-3) silicon-on-insulator (SOI) wafer by the standard complementary
metal-oxide-silicon (CMOS) process (Figure 1). The source and drain regions
are doped with phosphorus to form the n+ diffusion layer. The channel region
(500 nm $\times$ 100 nm in the long-channel sample and 500 nm $\times$ 100 nm
in the short-channel one) is exposed to the implantation that can be performed
as the last step of the transistor fabrication process. Using the single-ion
implantation technique [45, 46], germanium ions were implanted in pairs into
the channel region to form a one-dimensional array with 10 nm intervals
between neighboring sites. Higher spacing of 40 nm pitch between the
implantation sites returned no sub-threshold band. The implantation of pairs
of ions increases indeed the probability to have at least one activated
complex per site. The residual randomness of the position of the atoms
implanted by SII may cause difficulties for a straightforward implementation
of single atom-based devices. This effect could be mitigated in those
application where a small cluster of impurities is sufficient instead of a
single atom. Combination of masks with SII can be also considered.

Different annealing temperatures have been applied after the implantation
process and we observe that the Ge-related complexes are activated by
annealing at 550∘C for 1 min in N2 gas, as previously reported [23, 47].

Figure 6:  The trans-characteristics of an implanted nanotransistor at three
different annealing temperature of $350^{\circ}$C (green), $450^{\circ}$C
(orange) and $550^{\circ}$C (red). Transport is restored at $450^{\circ}$C (to
be compared with the inset, which shows a device as before implantation),
while the Ge$V$ defects are activated only above $550^{\circ}$C as from
previous reports. All conductance measurements have been carried out at room
temperature.

In order to demonstrate the activation of the defects at 550∘C, a set of
devices have been prepared to be annealed at different temperature. Figure 6
shows the trans-characteristics of the transistor both before the implantation
(in the inset) and after annealing at three different temperatures. After the
implantation, damages in the crystal significantly reduce the mobility of the
electrons. The annealing at $350^{\circ}$C is insufficient to restore the
lattice. By the annealing at 450∘C, the transport of the transistor
conductance is almost restored, even though with a lower mobility than the
pre-implantation one. As known from the literature [23, 47] such temperature
is insufficient for the activation of the Ge$V$ defect, thus here Ge acts by
just affecting the mobility. Finally, by applying an annealing temperature of
550∘C the Ge$V$ defects are activated, as demonstrated by the formation of an
impurity band.

The Ge$V$ generation in silicon, according to transport data, looks at least
of the same order than formation of Ge$V$ in diamond for which the probability
to have a single emitter over 200 implanted Ge ions per spot was estimated
equal to 53% [48].

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements

The Authors acknowledge financial support of the NFFA infrastructure under
Project ID 517. Computational resources were provided by the INDACO Platform
for High Performance Computing at the Università degli Studi di Milano
http://www.indaco.unimi.it, and by CINECA through the supercomputing grant
project HP10C3S9Z0. EP, TS and TT acknowledges JSPS, Ministero Affare Esteri,
MEXT and CNR Short Mobility program for funding.

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Table of Contents

Spatially controlled functionalized vacancy are created in a silicon channel
via high-precision single-ion implantation. Upon annealing they form point
defects with deep states in the silicon gap operating up to room temperature.
Quantum transport through an array of Ge$V$ complexes displays contributions
from weakly localized and delocalized states in the upper Hubbard band,
characterized by very different activation energy ($E_{A}$).