# Depth dependence of the radiative lifetime of shallow color centers in
single crystalline diamond

Maryam Zahedian1,∗, Jietian Liu1,∗, Ricardo Vidrio1, Shimon Kolkowitz2,
Jennifer T. Choy1,∗∗ ∗These authors contributed equally to this work.
1Department of Engineering Physics, University of Wisconsin - Madison,
Madison, WI 53706  2Department of Physics, University of Wisconsin - Madison,
Madison, WI 53706  ∗∗Email<EMAIL_ADDRESS>
###### Abstract

Optically active defects in diamond are widely used as bright single-photon
sources for quantum sensing, computing, and communication. For many
applications, it is useful to place the emitter close to the diamond surface,
where the radiative properties of the emitter are strongly modified by its
dielectric environment. It is well-known that the radiative power from an
electric dipole decreases as the emitter approaches an interface with a lower-
index dielectric, leading to an increase in the radiative lifetime. For
emitters in crystalline solids, modeling of this effect needs to take into
account the crystal orientation and direction of the surface cut, which can
greatly impact the emission characteristics. In this paper, we provide a
framework for analyzing the emission rates of shallow ($<100$ nm) defects, in
which optical transitions are derived from electric dipoles in a plane
perpendicular to their spin axis. We present our calculations for the depth-
dependent radiative lifetime for color centers in (100)-, (110)-, and
(111)-cut diamond, which can be extended to other vacancy defects in diamond.

diamond color centers, fluorescence lifetime, quantum sensing

##  I Introduction

Color centers in diamond, such as the negatively charged nitrogen-vacancy (NV)
centers, are solid-state defects whose spin properties enable sensitive
measurements of the local magnetic fields Hong  _et al._ (2013); Barry  _et
al._ (2016); Rondin  _et al._ (2014), strain Doherty  _et al._ (2014);
Ovartchaiyapong  _et al._ (2014); Knauer  _et al._ (2020), and temperature
Plakhotnik  _et al._ (2014); Acosta  _et al._ (2010); Chen  _et al._ (2011)
through spin-dependent photoluminescence. For sensing applications, the
measurement sensitivity and spatial resolution benefit from placing the spin
defect in close proximity (within few to tens of nanometers) to the physical
quantity being measured. Typical sensing measurements thus involve placing
color-center-containing nanodiamonds close to the sensing target (via
attachment to scanning probes Schell  _et al._ (2011) or spin-casting) or
using implanted Orwa  _et al._ (2011) or delta-doped Ohno  _et al._ (2012)
shallow color centers in single crystal diamond.

For color centers close to a planar surface, it is important to consider the
effect of their dielectric environment on the emission characteristics, in
particular the rate of energy emission. There is a host of theoretical
Drexhage (1970); Lukosz and Kunz (1977); Barnes (1998) and experimental Danz
_et al._ (2002); Schell  _et al._ (2014) work on the dependence of the emitted
power of optical dipoles on their distance to an interface. For emitters in
single-crystalline solids, this effect is complicated by the crystal
orientation and direction of the surface cut which constrain the relative
orientation of the spin and thereby electric dipoles to the surface.
Understanding the characteristics of shallow-color-center emission, including
how the radiative lifetime is affected by proximity to the surface, is
especially crucial for implementing near-field sensing measurements such as
Förster resonance energy transfer (FRET) Tisler  _et al._ (2013); Nelz  _et
al._ (2020); Radtke  _et al._ (2019) and for designing photonic structures to
control light-matter interactions with color centers Faraon  _et al._ (2011);
Hausmann  _et al._ (2013); Wambold  _et al._ (2021).

This paper aims to provide a framework for analyzing the emission rates of
near-surface color center in diamond. We primarily focus on the negatively
charged NV center (hereby abbreviated as NV) and calculate the radiative
lifetimes of NVs close to the air-diamond interface, by integrating their
total radiative power. We first summarize and review the relevant literature
on the photo-physics of NVs, which inform the treatment of the electric
dipoles in our model. We then explain our approach to calculating radiative
lifetimes of optical dipoles near surfaces. Calculations of NV lifetimes as a
function of depth for diamond with (100)-, (110)-, and (111)-surface cuts are
then presented, showing the depth-dependence of the NV radiative lifetime
within 100 nm from the interface and the effects of dipole orientation and
diamond surface cut. We conclude with a discussion on extending our approach
to other color centers and the feasibility of using lifetime measurements to
determine emitter depth.

##  II Model of optical dipole transitions in vacancy defects

Vacancy defects in diamond, including the NV center and other split-vacancy
defects (such as most Group IV color centers) are formed from missing carbon
atoms next to substitutional or interstitial atoms. The electron spin, whose
interactions with external electromagnetic and strain fields provide the
physical basis for most quantum sensing schemes, is oriented along one of the
crystallographic axes ($i.e.$, $[111]$, $[1\overline{1}\overline{1}]$,
$[\overline{1}1\overline{1}]$, or $[\overline{1}\overline{1}1]$). Optical
emission from the spin defect arises from pairs of orthogonal electric dipoles
(conventionally denoted as ${X}$ and ${Y}$) that are orthogonal to the spin
axis Hughes and Runciman (1967); Epstein  _et al._ (2005) (Figure 1a). The
particular orientation for each set of electric dipoles for a given NV center
is set by the non-axial local strain, such that for a large ensemble of
emitters, the dipole orientations are homogeneously distributed on the plane
perpendicular to the spin axis Radko  _et al._ (2016). This physical picture
of the electric dipoles can be most clearly illustrated by polarization
studies of NV excitation and photoluminescence in (111)-cut diamond at room
temperature Alegre  _et al._ (2007); Abe  _et al._ (2017); Rohner  _et al._
(2019) and near 4 K Batalov  _et al._ (2009); Kaiser  _et al._ (2009); Fu  _et
al._ (2009). For an [111]-oriented NV in (111)-cut diamond, the spin axis is
normal to the diamond surface while the electric dipoles are in the diamond
surface plane. At low temperatures, resonant excitation of each of the ${X}$
and ${Y}$ dipoles with polarized light follows Malus’s law, with the
sinusoidal modulations from the two orbital branches exhibiting opposite
phases Kaiser  _et al._ (2009). The ability to distinguish between the
polarized emissions from the $X$ and $Y$ transitions has a strong temperature
dependence. At temperatures above 40 K, the emission from the [111]-oriented
NV appears unpolarized Abe  _et al._ (2017); Fu  _et al._ (2009).

For the NV center, the optical dipole transitions correspond to energies
around 1.945 eV (or 637 nm in wavelength) Davies and Hamer (1976). In
fluorescence spectroscopy, the optical dipole transitions are observed as a
zero phonon line followed by a broad phonon sideband that extend to $\sim 800$
nm Jelezko and Wrachtrup (2006). The branching ratio of the zero phonon line
is typically $\sim 3\%$ Faraon  _et al._ (2011). Local strain lifts the
degeneracy of the two dipole moments, so that they are typically separated by
a few GHz which is considerably below the thermal energy in most measurement
scenarios (4 K to room temperature). Therefore, excitation of either dipole
moment is followed by phonon-assisted population exchange between the excited
states and eventually radiative decay to the ground states via one of the
dipole transitions (Figure 1b). If the population exchange within the excited
states occurs more quickly than radiative decay, then the excited-state
populations achieve thermal equilibrium prior to decay leading to both dipole
transitions occurring with equal probability Kaiser  _et al._ (2009) (see
Supplementary Section 1). Emission from the electric dipoles incoherently sums
to yield radiation of photons that spectrally overlap at room temperature,
with the overall rate of single-photon emission being the average between the
radiative rate of each dipole moment ($R_{X}$ and $R_{Y}$):
$R=(R_{X}+R_{Y})/2$. This emission rate is proportional to the total radiative
power, $L$, and is the inverse of the the mean optical lifetime
$\tau_{mean}=1/R$.

In homogeneous dielectric environments (such as the case of an emitter deeply
embedded in a bulk diamond crystal or an emitter-containing nanocrystal
suspended in a uniform medium), $R_{X}=R_{Y}$ and therefore the mean lifetime
is identical to the lifetime associated with each dipole transition (defined
as $\tau_{X}$ and $\tau_{Y}$). In bulk diamond, this mean lifetime is
typically $\sim 13$ ns Martin (1999). For an emitter in a non-isotropic medium
($e.g.$, near a surface or in nanostructures), the radiative rates from the
$X$ and $Y$ dipoles can be affected unequally and thus contribute differently
to the mean lifetime. The mean lifetime can then be calculated as a weighted
average of the lifetimes (see Supplementary Section 1):

$\tau_{mean}=\frac{1}{R}=\frac{2}{R_{X}+R_{Y}}=\frac{R_{X}}{R_{X}+R_{Y}}\frac{1}{R_{X}}+\frac{R_{Y}}{R_{X}+R_{Y}}\frac{1}{R_{Y}}=\frac{R_{X}}{R_{X}+R_{Y}}\tau_{X}+\frac{R_{Y}}{R_{X}+R_{Y}}\tau_{Y}$
(1)

Finally, the total decay rate of the emitter is a sum of the rates of all
possible radiative and non-radiative channels. While only radiative
transitions are shown in the simplified energy scheme in Figure 1b, the
internal quantum efficiency for the NV is around 0.7 Radko  _et al._ (2016).
Excited NVs can also decay via non-radiative channels which include an
intersystem crossing mechanism Goldman  _et al._ (2015) as well as
recombination with electron traps in the diamond crystal and at the interface.
For the NV center, the spin-dependent intersystem crossing decay is the basis
of optical initialization and readout of the NV spin and occurs on a timescale
much longer (on the order of hundreds of ns) than the radiative lifetime
Jelezko and Wrachtrup (2006). Therefore this pathway contributes negligibly to
the measured fluorescence decay. Meanwhile, other non-radiative channels
related to the environment tend to be fast (few ns) and thus can be discerned
from radiative emission by multi-exponential fitting and by correlating
fluorescence lifetime with emission intensity Tisler  _et al._ (2013); Inam
_et al._ (2013); Mohtashami and Koenderink (2013).

##  III Calculation of the radiative lifetime for an NV close to an interface

We use a numerical implementation of the angular spectrum method to calculate
the radiative lifetimes of the $X$ and $Y$ dipoles near the surface. Figure 2a
illustrates an electric dipole inside diamond (with refractive index
$n_{1}\sim 2.4$) interfacing with air ($n_{2}\sim 1$). The $z$ axis (which is
also the surface cut of the diamond) is defined as the normal vector of the
diamond-air interface while the depth $z_{0}$ is defined as the distance
between the color center and the interface. The angle between the surface
normal and the electric dipole is defined as $\theta_{e}$. Here, we use the
subscript $e$ to denote electric dipole emission.

Since the lifetime of any electric dipole is inversely proportional to its
radiative power Lukosz and Kunz (1977), we can calculate the depth-dependent
lifetime ($\tau(z_{0})$) of the dipole relative to its lifetime in homogeneous
medium ($\tau(z_{0})$) as follows:

$\tau(z_{0})/\tau_{\infty}=[L(z_{0})/L_{\infty}]_{e}^{-1}$ (2)

where $[L({z_{0}})]_{e}$ and $[L_{\infty}]_{e}$ are the powers radiated by the
electric dipole at $z_{0}$ and $z_{0}\rightarrow\infty$.

For any electric dipole at $\theta_{e}$, the radiative power ratio is
comprised of angle-dependent contributions from the parallel
($[L(z_{0})/L_{\infty}]_{e\parallel}$) and perpendicular
($[L(z_{0})/L_{\infty}]_{e\perp}$) components of the radiative power Lukosz
and Kunz (1977):

$[L(z_{0})/L_{\infty}]_{e}=\cos^{2}\theta_{e}[L(z_{0})/L_{\infty}]_{e\perp}+\sin^{2}\theta_{e}[L(z_{0})/L_{\infty}]_{e\parallel}.$
(3)

We adapt the equations introduced by Lukosz and Kunz Lukosz and Kunz (1977) to
find the radiative power for an electric dipole, perpendicular and parallel to
the surface as follows:

$[L(z_{0})/L_{\infty}]_{e\perp}=1+\frac{3}{2}\mathrm{Re}[(\int_{0}^{1}+\int_{i\infty}^{0})r_{1,2}^{(p)}(1-\nu^{2})\mathrm{exp}{(2ik^{(1)}z_{0}\nu)}\
d\nu]$ (4)

$[L(z_{0})/L_{\infty}]_{e\parallel}=\frac{1}{2}\\{1+[L(z_{0})/L_{\infty}]_{e,\perp}\\}+\frac{3}{4}\mathrm{Re}\\{(\int_{0}^{1}+\int_{i\infty}^{0})[r_{1,2}^{(s)}-r_{1,2}^{(p)}]\mathrm{exp}{(2ik^{(1)}z_{0}\nu)}\
d\nu\\}$ (5)

where $k^{(1)}$ and $k^{(2)}$ are respectively wave vectors in medium 1 and 2;
$\nu={k_{z}^{(1)}}/{k^{(1)}}$ is defined as the normalized z-component of the
wave vector in medium 1. $r_{1,2}^{(s)}$ and $r_{1,2}^{(p)}$ are reflection
coefficients for s and p polarized light (see Supplementary Section 3):

$r_{1,2}^{(s)}(\nu)=\frac{k^{(1)}\nu-k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\nu^{2}}}{k^{(1)}\nu+k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\nu^{2}}}$
(6)

$r_{1,2}^{(p)}(\nu)=\frac{\frac{\epsilon_{2}}{\epsilon_{1}}k^{(1)}\nu-k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\nu^{2}}}{\frac{\epsilon_{2}}{\epsilon_{1}}k^{(1)}\nu+k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\nu^{2}}}$
(7)

where $\epsilon_{1,2}$ are the dielectric constants in medium 1 and 2 and
$n={n_{2}}/{n_{1}}$. To acquire the lifetime shown in Equation 2, the
radiative power ratio was calculated by numerically solving the integrals of
Equations 4 and 5. We note that the terms involving integration over the
imaginary plane (Re$\int_{i\infty}^{0}$) are zero except when $n>1$ ($e.g.$
diamond is in contact with a higher index material), due to the presence of
evanescent waves at the interface Lukosz and Kunz (1977).

The depth-dependent lifetimes are shown in Figure 2b and exhibit monotonic
increase (up to $>16$-fold at the interface) for the perpendicular electric
dipole at $z_{0}<200$ nm, while the radiative lifetime of the parallel
electric dipole modulates slightly (with $<15\%$ variation) with depth. These
results are consistent with previous studies of classical dipoles near
interfaces Drexhage (1970); Cui  _et al._ (2015) and have their physical
origins in the constructive and destructive interference between the dipole
emissions and the reflected waves from the interface. In addition, we
calculated the changes in lifetime of the electric dipole close to an
interface with $n>1$ and observed a decrease in lifetime as expected (see
Supplementary Section 5).

To account for the strong presence of the phonon sideband in the NV emission
described in Section II, the spectrally averaged powers (represented by
$\langle[L(z_{0})/L_{\infty}]_{e\perp}\rangle$ and
$\langle[L(z_{0})/L_{\infty}]_{e\parallel}\rangle$ for electric dipoles
perpendicular and parallel to the interface) are calculated as follows:

$\langle[L(z_{0})/L_{\infty}]_{e\perp/\parallel}\rangle=\frac{\int[L(z_{0})/L_{\infty}]_{e\perp/\parallel}I_{NV}(\lambda)d\lambda}{\int
I_{NV}(\lambda)d\lambda}$ (8)

where $I_{NV}(\lambda)$ is the emission spectrum of NV acquired at room
temperature for an NV in bulk diamond.

Using Equations 2 and 3, we can now calculate the lifetime of each of the $X$
and $Y$ dipoles of the NV:

$[\tau(z_{0},\alpha,\theta_{m})/\tau_{0}]_{X/Y}=[\cos^{2}\theta_{eX/Y}\langle[L(z_{0})/L_{\infty}]_{e\perp}\rangle+\sin^{2}\theta_{eX/Y}\langle[L(z_{0})/L_{\infty}]_{e\parallel}\rangle]^{-1}$
(9)

where $\theta_{eX}$ and $\theta_{eY}$ are the angles between the surface
normal and the $X$ and $Y$ electric dipoles, with

$\displaystyle X:\begin{cases}\sin\theta_{eX}=\sin(\alpha)\sin(\theta_{m})\\\
\cos\theta_{eX}=\sqrt{1-\sin^{2}(\alpha)\sin^{2}(\theta_{m})}\end{cases}$
$\displaystyle Y:\begin{cases}\sin\theta_{eY}=\cos(\alpha)\sin(\theta_{m})\\\
\cos\theta_{eY}=\sqrt{1-\cos^{2}(\alpha)\sin^{2}(\theta_{m})}\end{cases}$ (10)

Equations 9 and 10 demonstrate the dependence of $\tau_{X}$ and $\tau_{Y}$ on
the spin axis orientation ($\theta_{m}$) and the orientation of the dipoles on
the plane perpendicular to the spin axis (defined through angle $\alpha$ as
shown in Figure 1a). These relationships can be combined with Equation 1 to
yield the mean radiative lifetime for a single NV (see Supplementary Section
4):

$\begin{split}[\tau(z_{0},\theta_{m})/\tau_{\infty}]_{mean}=[\frac{1}{2}\sin^{2}\theta_{m}\langle[L(z_{0})/L_{\infty}]_{e\perp}\rangle+\frac{1}{2}(1+\cos^{2}\theta_{m})\langle[L(z_{0})/L_{\infty}]_{e\parallel}\rangle]^{-1}\end{split}$
(11)

It can be inferred from Equation 11 that the mean radiative lifetime for any
NV is a function of $\theta_{m}$ and $z_{0}$ but independent of $\alpha$.
Therefore, for any arbitrary pairs of orthogonal dipoles $X$ and $Y$ on the
plane normal to the defined spin axis, the depth dependence of the mean
radiative lifetime is the same.

##  IV Results and Discussion

We now explicitly calculate the NV radiative lifetimes associated with near-
surface spins along the four spin quantization axes in (100)-, (110)-, and
(111)-diamond. Table 1 shows the possible $\theta_{m}$ values for any NV and
the corresponding spectrally averaged weights of
$\langle[L(z_{0})/L_{\infty}]_{e\perp}\rangle$ and
$\langle[L(z_{0})/L_{\infty}]_{e\parallel}\rangle$ in the determination of the
total radiative power $\langle[L(z_{0})/L_{\infty}]_{e}\rangle$. Here, we use
the notation $NV_{i}$ ($i=1,2,3,4$) to respectively indicate the four possible
spin axes in diamond $[111]$, $[1\overline{1}\overline{1}]$,
$[\overline{1}1\overline{1}]$ and $[\overline{1}\overline{1}1]$ which can be
determined by magnetometry and/or polarization-resolved measurements Alegre
_et al._ (2007); Schloss  _et al._ (2018); McCullian  _et al._ (2022). Due to
trigonal symmetry, the normalized power of NVs in (100)-oriented diamond is
the sum of the perpendicular and parallel components with respective weights
$\frac{1}{3}$ and $\frac{2}{3}$, regardless of the NV spin orientation. This
result is consistent with calculations shown in Radtke  _et al._ (2019); Radko
_et al._ (2016). For (110)-diamond, two unique power distributions are derived
for NVs with $\theta_{m}=35.3^{\circ},144.7^{\circ}$ and
$\theta_{m}=90^{\circ}$. Since the radiative power of the perpendicular dipole
component experiences more quenching as the emitter approaches the surface
(Figure 2b), we expect the NVs with $\theta_{m}=90^{\circ}$ which have a
larger contribution from $[{L_{\infty}}]_{e\perp}$, to be more sensitive to
their proximity to the surface. Finally, NVs in (111)-diamond with the spin
axis perpendicular to the surface radiate similarly to a parallel dipole and
be minimally affected by depth, while the other three orientations (all with
$\theta_{m}=109.5^{\circ}$) show stronger depth dependence.

NV direction | $NV_{1}:[111]$ | $NV_{2}:[1\overline{1}\overline{1}]$ | $NV_{3}:[\overline{1}1\overline{1}]$ | $NV_{4}:[\overline{1}\overline{1}1]$
---|---|---|---|---
(100) surface
$\theta_{m}$ | $54.7^{\circ}$ | $54.7^{\circ}$ | $125.3^{\circ}$ | $125.3^{\circ}$
$\langle[\frac{L(z_{0})}{L_{\infty}}]_{e}\rangle$ | $\frac{1}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{2}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{2}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{2}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{2}{3}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$
(110) surface
$\theta_{m}$ | $35.3^{\circ}$ | $90.0^{\circ}$ | $90.0^{\circ}$ | $144.7^{\circ}$
$\langle[\frac{L(z_{0})}{L_{\infty}}]_{e}\rangle$ | $\frac{1}{6}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{5}{6}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{2}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{1}{2}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{2}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{1}{2}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{1}{6}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{5}{6}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$
(111) surface
$\theta_{m}$ | $0.0^{\circ}$ | $109.5^{\circ}$ | $109.5^{\circ}$ | $109.5^{\circ}$
$\langle[\frac{L(z_{0})}{L_{\infty}}]_{e}\rangle$ | $\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{4}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{5}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{4}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{5}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$ | $\frac{4}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\perp}\rangle+\frac{5}{9}\langle[\frac{L(z_{0})}{L_{\infty}}]_{e\parallel}\rangle$

Table 1: $\theta_{m}$ and power ratio for different NV directions in different
surface cuts.

The radiative lifetime in each case is then calculated as an inverse of the
radiated power. Our depth-dependent lifetimes for all possible NV orientations
are plotted in Figures 3, 4, and 5 for (100)-, (110)-, and (111)-diamond. For
each possible NV orientation, we also show the variation in the lifetimes of
individual dipole transitions $X$ and $Y$ as a function of $\alpha$ at
$z_{0}=1$ nm using Equation 9, along with the mean lifetime of the NV emission
which is invariant to $\alpha$ (Figures 3c, 4c, and 5c).

Figures 3d, 4d, and 5d show that proximity to the surface can significantly
modify the lifetime at depths shallower than $\sim$ 100 nm, but the effect
greatly depends on the NV orientation and surface cut. In (100)-diamond, there
is a monotonic increase in the mean radiative lifetime as NVs are closer to
the surface, leading to a $\sim$1.5-fold increase in the lifetime at the
surface. For defects in (110)-diamond, the mean radiative lifetime for NVs
with $\theta_{m}=90^{\circ}$ increases by a factor of $>1.8$ within a few
nanometers from the surface. Meanwhile, NVs with $\theta_{m}=35.3^{\circ}$ and
$144.7^{\circ}$ experience small increase as well as sinusoidal modulation in
the lifetime with decreasing depth, due to self-interference between the
emitter and reflected light. Similarly for $NV_{1}$ with
$\theta_{m}=0^{\circ}$ in (111)-diamond, the mean lifetime is slightly
modulated by depth (Figure 5d). The other three NV centers with identical
$\theta_{m}=109.5^{\circ}$ in (111)-oriented diamond, exhibit $\sim$1.8-fold
increase in their mean radiative lifetime (Figure 5d).

We also performed the lifetime calculation for an NV in diamond interfacing
with different dielectrics (see Supplementary Section 5). As expected, a
smaller mismatch in the dielectric constants reduces the suppression in
radiative power, resulting in a reduction of $7\%$ in the lifetime increase
between a diamond-air and diamond-oil interface. Meanwhile NVs close to an
interface with higher refractive index will exhibit enhanced emission rate and
corresponding shortening in the lifetime, with $>35\%$ lifetime reduction in
comparison to bulk for a silicon-like material.

Our numerical results for (100)-surface cut diamond are consistent with
experimental data by Radtke et al and Ofori-Okai et al Radtke  _et al._
(2019); Ofori-Okai  _et al._ (2012), which demonstrated prolonged lifetimes
for shallow NVs in comparison to lifetimes in the bulk. Even longer lifetimes
(up to $\sim 48$ ns) were observed for NVs embedded in nanoposts Choy  _et
al._ (2011). In the work by Radtke and colleagues, it was observed that for
NVs $\sim 7$ nm below the surface, the radiative lifetime is prolonged by 1.2
times. While our calculation predicts a longer lifetime by a factor of 1.4 at
that depth, the discrepancy may arise from the underestimation of NV depth due
to the channeling effect of ion-implanted color centers Toyli  _et al._
(2010); Pham  _et al._ (2016).

The calculations presented here are applicable for other color centers in
diamond such as Group-IV-split-vacancy centers Bradac  _et al._ (2019), where
polarization-selective photoluminescence studies revealed the presence of sets
of electric dipoles similar to NV centers Hepp  _et al._ (2014).

Finally, accurate determination of the color center depth is important for
nuclear magnetic resonance (NMR) spectroscopy using spin defects DeVience  _et
al._ (2015) as well as correlation studies between spin coherence and
proximity to sources of electric and magnetic noise Sangtawesin  _et al._
(2019). Current methods for depth determination tend to be destructive ($e.g.$
Secondary ion mass spectroscopy Toyli  _et al._ (2010); Fiori  _et al._
(2014)) or rely on interfacing diamond with another material that provides
sources of electron acceptors (in the case of FRET Tisler  _et al._ (2013)) or
proton spins (for NMR techniques Pham  _et al._ (2016)). We propose that
radiative lifetime measurement of shallow color centers in diamond has the
potential to be employed as a simple technique for emitter depth estimation
given that the orientation of the color center and crystal cut have been
determined. However, the resolution of the technique may be limited due to the
relatively small dynamic range of the lifetime increase as well as
experimental uncertainties Laube  _et al._ (2019) in optical lifetime
measurements.

Supporting Information

Supporting Information can be found towards the end of the document.

Acknowledgements

This work is supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences under Award $\\#$DE-SC0020313. We are grateful for
insightful discussions with Mikhail Kats on modeling of the NV dipoles.

Figure 1: (a) Atomic structure for the NV center. $X$ and $Y$ are electric
dipoles in the pink-colored plane perpendicular to the spin axis along [111]
in (100)-oriented diamond. The intersection of the surface plane and plane
perpendicular to the spin axis is shown as a dashed line. $\alpha$ is defined
as an angle rotating clockwise from the intersection line to $X$, when
observing from the opposite direction of the NV spin. Various angles are
defined relative to the surface normal, to denote the orientations of the spin
axis $\theta_{m}$ and the $X$ ($\theta_{eX}$) and Y ($\theta_{eY}$) dipoles.
(b) Simplified energy level diagram for vacancy defects indicating the X and Y
optical transitions. The transition energies for $X$ and $Y$ are shown to be
offset by a small amount (much smaller than the thermal energy at room
temperature) due to local strain.

Figure 2: (a) Schematic showing an electric dipole at any angle ($\theta_{e}$)
at distance $z_{0}$ from the interface. (b) Normalized lifetime of an electric
dipole perpendicular (blue; $\theta_{e}=0^{\circ}$) and parallel (red;
$\theta_{e}=90^{\circ}$) to the interface as a function of $z_{0}$ at
$\lambda=637$ nm (zero phonon line of $NV^{-}$ in diamond), with $n_{1}=2.4$
(refractive index of diamond), and $n_{2}=1$ (refractive index of air).

Figure 3: (a) 3D plot of NVs along four possible orientations of
crystallographic axes in a (100)-oriented diamond and its coordinate axes. (b)
Atomic structure of the NV centers and arbitrary set of $X$ (blue) and $Y$
(red) electric dipoles associated with each NV. (c) The radiative lifetime
ratios for the $X$ (blue) and $Y$ (red) electric dipoles at $z_{0}$=1 nm,
along with the mean lifetime (black), as a function of $\alpha$. (d) The mean
radiative lifetime ratio for $NV_{1}$, $NV_{2}$, $NV_{3}$, and $NV_{4}$ as a
function of $z_{0}$.

Figure 4: (a) 3D plot of NVs along four possible orientations of
crystallographic axes in a (110)-oriented diamond and its coordinate axes. (b)
Atomic structure of the NV centers and arbitrary set of $X$ (blue) and $Y$
(red) electric dipoles associated with each NV. (c) The radiative lifetime
ratios for the $X$ (blue) and $Y$ (red) electric dipoles at $z_{0}$=1 nm,
along with the mean lifetime (black), as a function of $\alpha$. (d) The mean
radiative lifetime ratio for $NV_{1}$, $NV_{2}$, $NV_{3}$, and $NV_{4}$ as a
function of $z_{0}$.

Figure 5: (a) 3D plot of NVs along four possible orientations of
crystallographic axes in a (111)-oriented diamond and its coordinate axes. (b)
Atomic structure of the NV centers and arbitrary set of $X$ (blue) and $Y$
(red) electric dipoles associated with each NV. (c) The radiative lifetime
ratios for the $X$ (blue) and $Y$ (red) electric dipoles at $z_{0}$=1 nm,
along with the mean lifetime (black), as a function of $\alpha$. (d) The mean
radiative lifetime ratio for $NV_{1}$, $NV_{2}$, $NV_{3}$, and $NV_{4}$ as a
function of $z_{0}$.

## Supplementary information

###  IV.1 Rate equations for single-photon emission from an NV

As described in the main text, optical emission from the NV arises from two
orthogonal electric dipoles denoted as $X$ and $Y$. While each of the $X$ and
$Y$ dipole moments can be independently excited, phonon processes lead to
population exchange between the excited states. Eventually, radiative decay
can occur via either transition. The rate equations describing the radiative
decay of an excited NV can be written as:

$\begin{cases}\frac{dP_{X}}{dt}=-R_{X}P_{X}-\gamma(P_{X}-P_{Y})\\\
\frac{dP_{Y}}{dt}=-R_{Y}P_{Y}-\gamma(P_{Y}-P_{X})\end{cases}$ (S.12)

where $R_{X}$ and $R_{Y}$ are the decay rates from states $X$ and $Y$;
$\gamma$ is the rate of phonon-assisted population exchange between the $X$
and $Y$ excited states; $P_{X}$ and $P_{Y}$ are the normalized excited state
populations at $X$ and $Y$. We note that non-radiative decay processes of the
NV through the singlet states can be neglected since they occur on much longer
timescales. Solving the set of rate equations yields:

$\displaystyle\Rightarrow\frac{d^{2}P_{X}}{dt^{2}}+(R_{X}+R_{Y}+2\gamma)\frac{dP_{X}}{dt}+(R_{X}R_{Y}+R_{X}\gamma+R_{Y}\gamma)P_{X}=0$
$\displaystyle\Rightarrow
P_{X}=Ae^{\beta_{1}t}+Be^{\beta_{2}t},\mathrm{where}$
$\begin{cases}\beta_{1}=\frac{-(R_{X}+R_{Y}+2\gamma)+\sqrt{(R_{X}-R_{Y})^{2}+4\gamma^{2}}}{2}\\\
\beta_{2}=\frac{-(R_{X}+R_{Y}+2\gamma)-\sqrt{(R_{X}-R_{Y})^{2}+4\gamma^{2}}}{2}\end{cases}$
(S.13)

We set the initial populations to be $P_{X}(0)=P_{Y}(0)=\frac{1}{2}$,
corresponding to the case in which both $X$ and $Y$ dipole moments have equal
probabilities of being excited. The time-dependent excited state populations
are then:

$P_{X}=\frac{\beta_{2}+R_{X}}{2(\beta_{2}-\beta_{1})}e^{\beta_{1}t}+\frac{\beta_{1}+R_{X}}{2(\beta_{1}-\beta_{2})}e^{\beta_{2}t}\\\
$ (S.14)

$P_{Y}=\frac{\beta_{2}+R_{Y}}{2(\beta_{2}-\beta_{1})}e^{\beta_{1}t}+\frac{\beta_{1}+R_{Y}}{2(\beta_{1}-\beta_{2})}e^{\beta_{2}t}$
(S.15)

The total excited state population, $P$, is the sum of $P_{X}$ and $P_{Y}$ and
is proportional to the number of emitted photons after excitation in a
fluorescence measurement. Using $\tau_{X}=\tau_{0}$ and $\tau_{Y}=2.7\tau_{0}$
(a valid set of orthogonal dipoles for (100)-diamond as shown in Figure 3c in
the main text), we plot the time trace of excited state population for
different values of $\gamma$ in Figure S.1. Our results show that for for
$\gamma^{-1}\ll\tau_{0}$, the mean lifetime corresponds to the inverse of the
average emission rates between $R_{X}$ and $R_{Y}$, as follows:

Figure S.1: Time trace of excited state population for an NV in (100)-diamond,
for different $\gamma$ values.

$\displaystyle\Rightarrow\begin{cases}\beta_{1}\approx-\frac{R_{X}+R_{Y}}{2}\\\
\beta_{2}\approx-\frac{R_{X}+R_{Y}+4\gamma}{2}\end{cases}$
$\displaystyle\Rightarrow\begin{cases}P_{X}\approx\frac{1}{2}e^{-(R_{X}+R_{Y})t/2}\\\
P_{Y}\approx\frac{1}{2}e^{-(R_{X}+R_{Y})t/2}\end{cases}$

So the total radiated rate $R$ for the whole system becomes:

$R=\frac{R_{X}+R_{Y}}{2}.$ (S.16)

$\gamma$ is dependent on the temperature and is fast ($\gg\tau_{0}^{-1}$) at
room temperature Martin (1999); Fu  _et al._ (2009), leading to the appearance
of unpolarized emission Kaiser  _et al._ (2009). As $\gamma$ decreases with
temperature, polarized emissions from the $X$ and $Y$ transitions can be
distinguished. In the regime that $\gamma\lesssim\tau_{0}^{-1}$, the excited
state decay tends towards a bi-exponential behavior with longer time
constants.

###  IV.2 Determination of orthogonal dipole orientations

In this section, we describe our approach to generating valid sets of dipole
orientations for our lifetime calculations. As noted in the main text, a set
of orthogonal electric dipoles for a given NV center can be oriented along any
direction on the plane perpendicular to the spin axis and thus the solutions
provided here are not unique.

If we define $X$ and $Y$ dipole directions for $NV_{1}$ to be
$X\parallel[\overline{1}\overline{1}2]$ and $Y\parallel[1\overline{1}0]$, then
we use rotation matrix rotating from $NV_{1}$ to other three different
direction to find different $X$ and $Y$ for different $NV$ direction.

If we want to find the rotation matrix $A$ that rotates $\hat{NV_{1}}$ to
$\hat{NV_{2}}$, we need to know

$\displaystyle\vec{w}=\hat{NV_{1}}\times\hat{NV_{2}}=\frac{2}{3}[01\overline{1}]$
(S.17) $\displaystyle s=\lVert w\lVert\textrm{ (sine of angle)}$ (S.18)
$\displaystyle c=\hat{NV_{1}}\cdot\hat{NV_{2}}\textrm{ (cosine of angle)}$
(S.19)

Then the rotation matrix $A$ is

$A=I+[w]_{\times}+\frac{1-c}{s^{2}}[w]_{\times}^{2}$ (S.20)

where $[w]_{\times}$ is the skew-symmetric cross-product matrix of $w$

$[w]_{\times}\stackrel{{\scriptstyle\text{def}}}{{=}}\begin{bmatrix}0&-w_{3}&w_{2}\\\
w_{3}&0&-w_{1}\\\ -w_{2}&w_{1}&0\\\
\end{bmatrix}=\frac{2}{3}\begin{bmatrix}0&1&1\\\ -1&0&0\\\ -1&0&0\\\
\end{bmatrix}$ (S.21)

Then we can get rotation matrix $A=\frac{1}{3}\begin{bmatrix}-1&2&2\\\
-2&1&-2\\\ -2&-2&1\\\ \end{bmatrix}$

We apply this rotation matrix to $X$ and $Y$, then we can get $X^{*}$ and
$Y^{*}$ for $NV_{2}$

$\displaystyle X^{*}$ $\displaystyle=AX=\frac{1}{3}\begin{bmatrix}-1&2&2\\\
-2&1&-2\\\ -2&-2&1\\\ \end{bmatrix}\begin{bmatrix}-1\\\ -1\\\ 2\\\
\end{bmatrix}=\begin{bmatrix}1\\\ -1\\\ 2\\\ \end{bmatrix}$ (S.22)
$\displaystyle Y^{*}$ $\displaystyle=AY=\frac{1}{3}\begin{bmatrix}-1&2&2\\\
-2&1&-2\\\ -2&-2&1\\\ \end{bmatrix}\begin{bmatrix}1\\\ -1\\\ 0\\\
\end{bmatrix}=\begin{bmatrix}-1\\\ -1\\\ 0\\\ \end{bmatrix}$ (S.23)

Using the same method, we can find out the direction of $X$ and $Y$ for three
other $NV$ directions shown in table 2.

NV direction | $NV_{1}:[111]$ | $NV_{2}:[1\overline{1}\overline{1}]$ | $NV_{3}:[\overline{1}1\overline{1}]$ | $NV_{4}:[\overline{1}\overline{1}1]$
---|---|---|---|---
X direction | $[\overline{1}\overline{1}2]$ | $[1\overline{1}2]$ | $[\overline{1}12]$ | $[\overline{1}\overline{1}\overline{2}]$
Y direction | $[1\overline{1}0]$ | $[\overline{1}\overline{1}0]$ | $[110]$ | $[1\overline{1}0]$
$\theta_{e}$ angle for (100) surface cut
$\theta_{eX}$ | $114.1^{\circ}$ | $65.9^{\circ}$ | $114.1^{\circ}$ | $114.1^{\circ}$
$\theta_{eY}$ | $45.0^{\circ}$ | $135.0^{\circ}$ | $45.0^{\circ}$ | $45.0^{\circ}$
$\theta_{e}$ angle for (110) surface cut
$\theta_{eX}$ | $125.3^{\circ}$ | $90.0^{\circ}$ | $90.0^{\circ}$ | $125.3^{\circ}$
$\theta_{eY}$ | $90.0^{\circ}$ | $180.0^{\circ}$ | $0.0^{\circ}$ | $90.0^{\circ}$
$\theta_{e}$ angle for (111) surface cut
$\theta_{eX}$ | $90.0^{\circ}$ | $61.9^{\circ}$ | $61.9^{\circ}$ | $160.5^{\circ}$
$\theta_{eY}$ | $90.0^{\circ}$ | $144.7^{\circ}$ | $35.3^{\circ}$ | $90.0^{\circ}$
Table 2: X and Y dipole directions for four different NV directions and the
angle between the normal vector of surface cut and X or Y

###  IV.3 Comparison of numerical integration with full-wave simulation and
Taylor expansion approaches

To calculate $[L(z_{0})/L_{\infty}]_{e,\perp}$ and
$[L(z_{0})/L_{\infty}]_{e,\parallel}$, we explored using the Finite-Difference
Time-Domain (FDTD) method Taflove  _et al._ (2005) (using commercial software
Lumerical) in addition to numerically integrating Equations. 4 and 5.

To solve the integrals, we need to first find $r_{1,2}^{(s)}(\nu)$ and
$r_{1,2}^{(p)}(\nu)$, where reflection coefficients are defined as follows
Lukosz and Kunz (1977):

$r_{1,2}^{(s)}=\frac{k_{z}^{(1)}-k_{z}^{(2)}}{k_{z}^{(1)}+k_{z}^{(2)}}$ (S.24)

$r_{1,2}^{(p)}=\frac{\frac{\epsilon_{2}}{\epsilon_{1}}k_{z}^{(1)}-k_{z}^{(2)}}{\frac{\epsilon_{2}}{\epsilon_{1}}k_{z}^{(1)}+k_{z}^{(2)}}$
(S.25)

and $k_{z}^{(1)}$, and $k_{z}^{(2)}$ are z-components of the the plane wave
vectors in medium 1 and 2.

Then, from Equations. S.24 and S.25, and substituting $k_{z}^{(1)}$,
$k_{z}^{(2)}$, and $\frac{n_{2}}{n_{1}}$ with $k^{(1)}\cos\theta_{e1}$,
$k^{(1)}\cos\theta_{e2}$, and $n$, respectively we get:

$r_{1,2}^{(s)}=\frac{k_{z}^{(1)}-k_{z}^{(2)}}{k_{z}^{(1)}+k_{z}^{(2)}}=\frac{k^{(1)}\cos\theta_{e1}-k^{(2)}\cos\theta_{e2}}{k^{(1)}\cos\theta_{e1}+k^{(2)}\cos\theta_{e2}}=\frac{k^{(1)}\cos\theta_{e1}-k^{(2)}\sqrt{1-\sin^{2}\theta_{e2}}}{k^{(1)}\cos\theta_{e1}+k^{(2)}\sqrt{1-\sin^{2}\theta_{e2}}}$

$=\frac{k^{(1)}\cos\theta_{e1}-k^{(2)}\sqrt{1-(\frac{n_{1}}{n_{2}})^{2}\sin^{2}\theta_{e1}}}{k^{(1)}\cos\theta_{e1}+k^{(2)}\sqrt{1-(\frac{n_{1}}{n_{2}})^{2}\sin^{2}\theta_{e1}}}=\frac{k^{(1)}\cos\theta_{e1}-k^{(2)}\sqrt{1-\frac{1}{n^{2}}\sin^{2}\theta_{e1}}}{k^{(1)}\cos\theta_{e1}+k^{(2)}\sqrt{1-\frac{1}{n^{2}}\sin^{2}\theta_{e1}}}$

$r_{1,2}^{(s)}=\frac{k^{(1)}\cos\theta_{e1}-k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\cos^{2}\theta_{e1}}}{k^{(1)}\cos\theta_{e1}+k^{(2)}\sqrt{1-\frac{1}{n^{2}}+\frac{1}{n^{2}}\cos^{2}\theta_{e1}}}$
(S.26)

where $\theta_{e1}$ and $\theta_{e2}$ are the angles between normal vector of
surface plane and $k^{(1)}$ and $k^{(2)}$ wave vectors. Finally we have
$r_{1,2}^{(s)}(\nu)$ and $r_{1,2}^{(p)}(\nu)$ by replacing
${k_{z}^{(1)}}/{k^{(1)}}$ with $\cos\theta_{e1}$, where $\nu=\cos\theta_{e1}$
Drexhage (1970).

We also explored a third approach to calculate the radiative power, using a
first-order Taylor series Lukosz and Kunz (1977) of total radiative power in
the regime that $z_{0}\ll\lambda_{1}$.

Figure S.2 plots the computed lifetimes by the three approaches (FDTD, full
numerical integration, and first-order Taylor expansion). The depth-dependent
lifetimes are in great agreement between FDTD and numerical integration
methods. However, our results indicate that first order Taylor expansion only
works well for a limited range of $z_{0}\leq 6$ nm (for $\lambda$=$637$ nm and
$n_{1}=2.4$). The results in the main text are thus obtained using numerical
integration.

Figure S.2: Normalized lifetime of an electric dipole, a) perpendicular
($\theta_{e}=0^{\circ}$), and b) parallel ($\theta_{e}=90^{\circ}$) to the
interface.

###  IV.4 Calculation of radiative powers

The spectrally averaged power ratios for the $X$ and $Y$ dipole transitions
are calculated from Eqs. 9 and 10 as follows:

$\langle[L(z_{0},\alpha,\theta_{m})/L_{\infty}]\rangle_{X}=\sin^{2}(\alpha)\sin^{2}(\theta_{m})\langle[L(z_{0})/L_{\infty}]_{\perp}\rangle+[1-\sin^{2}(\alpha)\sin^{2}(\theta_{m})]\langle[L(z_{0})/L_{\infty}]_{\parallel}\rangle$
(S.27)
$\begin{split}\langle[L(z_{0},\alpha,\theta_{m})/L_{\infty}]\rangle_{Y}&=\sin^{2}(\alpha+\frac{\pi}{2})\sin^{2}(\theta_{m})\langle[L(z_{0})/L_{\infty}]_{\perp}\rangle+[1-\sin^{2}(\alpha+\frac{\pi}{2})\sin^{2}(\theta_{m})]\langle[L(z_{0})/L_{\infty}]_{\parallel}\rangle\\\
&=\cos^{2}(\alpha)\sin^{2}(\theta_{m})\langle[L(z_{0})/L_{\infty}]_{\perp}\rangle+[1-\cos^{2}(\alpha)\sin^{2}(\theta_{m})]\langle[L(z_{0})/L_{\infty}]_{\parallel}\rangle\end{split}$
(S.28)
$\begin{split}\langle[L(z_{0},\alpha,\theta_{m})/L_{\infty}]\rangle_{\frac{X+Y}{2}}=\frac{1}{2}\sin^{2}\theta_{m}\langle[L(z_{0})/L_{\infty}]_{\perp}\rangle+\frac{1}{2}(1+\cos^{2}\theta_{m})\langle[L(z_{0})/L_{\infty}]_{\parallel}\rangle\end{split}$
(S.29)

###  IV.5 Shallow NV lifetimes in different media

Figure S.2: Normalized lifetime of an electric dipole, for $n>1$, with
$n_{1}=2.4$, and $n_{2}=3.4$ at $\lambda=637$ nm a) perpendicular, and b)
parallel to the interface. c) Spectrally averaged normalized lifetime of NV in
a (100) surface cut diamond versus $z_{0}$. Figure S.2: Comparison of NV
lifetime in different $n_{2}$ medium in a) (100), (b) (110), and (c) (111)
diamond surface cuts

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# High degree of current rectification at nanoscale level

Madhumita Saha  Physics and Applied Mathematics Unit, Indian Statistical
Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India  Santanu K.
Maiti<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India

###### Abstract

We address an unexpectedly large rectification using a simple quantum wire
with correlated site potentials. The external electric field, associated with
voltage bias, leads to unequal charge currents for two different polarities of
external bias and this effect is further enhanced by incorporating the
asymmetry in wire-to-electrode coupling. Our calculations suggest that in some
cases almost cent percent rectification is obtained for a wide bias window.
This performance is valid against disorder configurations and thus we can
expect an experimental verification of our theoretical analysis in near
future.

##  I Introduction

Designing of an efficient rectifier at nanoscale level has been the subject of
intense research after the prediction of molecular rectifier by Aviram and
Ratner r1  in $1974$. Following this pioneering work, interest in this area
has rapidly picked up with several theoretical propositions and experimental
verifications and most of these works involve small organic molecules with
donor-acceptor pair between metallic electrodes r2 ; r3 ; r4 ; r5 ; r55 ; r6 ;
r7 ; r8 ; r9 ; r10 ; r11 ; r12 . Recently a DNA-based rectifier has also been
established r13  which exhibits a large rectification ratio of about $15$ at
$1.1\,$V.

To achieve rectification (viz, $I(V)\neq I(-V)$), energy levels of the
bridging material have to be aligned differently for the positive and

Figure 1: (Color online). Quasi-periodic (Fibonacci, $5$th generation i.e.,
$8$ atomic sites) chain attached to $1$D source (S) and drain (D) electrodes.
The filled red and blue circles correspond to two different lattices (say) $A$
and $B$, respectively.

negative biases. This can be done in two different ways: (i) placing an
asymmetric conductor between source and drain electrodes r9 ; r11 , keeping
identical conductor-to-electrode couplings, and (ii) considering a symmetric
conductor with unequal conductor-electrode couplings r12 . The understanding
is that for both these two cases resonant energy levels, in presence of finite
bias, are arranged distinctly for two different biased conditions which
results finite rectification. Therefore, when both these two conditions are
satisfied one can expect maximum rectification.

So far mostly molecular systems r6 ; r7 ; r8 ; r9 ; r10 ; r11 ; r12  have been
used to design rectifying diodes, but constructing them using single molecules
is a major challenge janes  and still many open questions remain which
certainly demand further study. Therefore, designing of a rectifier using
simple geometric structure which provides high rectification ratio is a matter
of great interest. In the present work we essentially focus on it and make an
attempt to establish that a simple $1$D chain with correlated site potentials
can exhibit a very high degree of rectification, and sometimes it becomes
nearly close to $\pm 100\%$ for a wide bias window. This performance is valid
against disordered configurations which we confirm by comparing the results of
different $1$D quasi-periodic chains like Fibonacci (Fibo), Thou-Morse (TM),
Copper-mean (CM) and Bronze-mean (BM) and all these systems are constructed by
using two primary lattices, namely $A$ and $B$, following the specific
inflation rules quasiall ; tm ; skmfib . For the case of Fibonacci chain the
rule is: $A\rightarrow AB$ and $B\rightarrow A$. Therefore, applying
successively this substitutional rule, staring with $A$ or $B$ lattice we can
construct the full lattice chain for any particular generation, say $p$-th
generation, obeying the prescription $F_{p}=F_{p-1}\otimes F_{p-2}$. So, if we
start with $A$ lattice then the first few generations of the Fibonacci series
are $A$, $AB$, $ABA$, $ABAAB$, $ABAABABA$, $\dots$, etc. The inflation rules
for the other three quasiperiodic chains which we consider here i.e., TM, CM
and BM are: $A\rightarrow AB$, $B\rightarrow BA$; $A\rightarrow ABB$,
$B\rightarrow A$, and $A\rightarrow AAAB$, $B\rightarrow A$, respectively.
Using these rules we construct the quasiperiodic chains for any desired
generation starting with any lattice site $A$ or $B$.

The rest of the paper is arranged as follows. In Sec. II we present the model
and theoretical framework for calculations. The results are described in Sec.
III, and finally, in Sec. IV we conclude our findings.

##  II Model and theoretical framework

The calculations are worked out using wave-guide theory based on tight-binding
(TB) framework. In this framework the Hamiltonian of the full system,
schematically shown in Fig. 1, can be written as $H=H_{\mbox{\tiny
el}}+H_{\mbox{\tiny ch}}+H_{\mbox{\tiny tn}}$, where $H_{\mbox{\tiny el}}$,
$H_{\mbox{\tiny ch}}$ and $H_{\mbox{\tiny tn}}$ correspond to the Hamiltonians
of the electrodes (source and drain), quasi-periodic chain and chain-to-
electrode tunneling coupling, respectively. In terms of on-site potential
$\epsilon_{i}$ and nearest-neighbor hopping integral $t$, the TB Hamiltonian
of the chain can be written as:

$H_{\mbox{\tiny
ch}}=\sum\limits_{i=1}\epsilon_{i}c_{i}^{\dagger}c_{i}+\sum\limits_{i=1}t\left(c_{i+1}^{\dagger}c_{i}+c_{i}^{\dagger}c_{i+1}\right)$
(1)

where $c_{i}^{\dagger}$ ($c_{i}$) represents the electronic creation

Figure 2: (Color online). Voltage dependent site energies of a $200$-site
chain for three different electrostatic potential profiles, shown by three
distinct colored curves, at the bias voltage $V=0.2\,$V.

(annihilation) operator. In a similar way we can write the TB Hamiltonian
$H_{\mbox{\tiny el}}$ of the two side-attached perfect $1$D electrodes
parameterized by $\epsilon_{0}$ and $t_{0}$. These electrodes are coupled to
the sites $1$ and $N$ of the conductor (described by $H_{\mbox{\tiny tn}}$)
through the coupling parameters $\tau_{S}$ and $\tau_{D}$ (see Fig. 1), where
$N$ being the total number of lattice sites of the bridging conductor.

In presence of a finite bias $V$ between the source and drain, an electric
field is developed across the chain and hence its site energies are voltage
dependent el1 ; el2 . It gives
$\epsilon_{i}=\epsilon_{i}^{0}+\epsilon_{i}(V)$, where $\epsilon_{i}^{0}$ is
voltage independent and it becomes $\epsilon_{A}$ or $\epsilon_{B}$ depending
on the lattice sites $A$ or $B$. The dependence of $\epsilon_{i}(V)$ is
associated with electron screening as well as bare electric field at the
junction. In the absence of any screening electric field is uniform across the
junction el1 ; el2  which makes $\epsilon_{i}(V)=V/2-iV/(N+1)$ (linear
variation, red line of Fig. 2) for a $N$-site chain. Whereas, long-range
electron screening makes the profile non-linear as shown by the green and blue
curves of Fig. 2. In our calculations we consider these three different
potential profiles to have a complete idea about the bare and the screened
electric field profiles, and their effects on rectification as in realistic
case different materials possess different electron screening which will yield
different field variations. For a slight variation from these potential
profiles no significant change is observed in the physical properties, and
thus our findings may be implemented in realistic cases.

To evaluate transmission probability across the conducting junction we solve a
set of coupled linear equations containing wave amplitudes of distinct lattice
sites of the chain wg ; skm . Assuming a plane wave incidence, we can write
the wave amplitude at any site $n$ of the source as $A_{n}=e^{ikn}+re^{-ikn}$,
where $k$ is the wave-vector and $r$ being the reflection coefficient. While,
for the drain electrode we get $B_{n}=\tau e^{ikn}$, where $\tau$ represents
the transmission coefficient. For each $k$, associated with the injecting
electron energy, we find the transmission probability from the expression
$T(E)=|B_{1}|^{2}=|\tau|^{2}$. Once it is determined, the net junction current
for a particular bias voltage $V$ at absolute zero temperature is obtained
from the relation datta

$I(V)=\frac{e}{\pi\hbar}\int\limits_{E_{F}-\frac{eV}{2}}^{E_{F}+\frac{eV}{2}}T(E)\,dE$
(2)

where $E_{F}$ represents the Fermi energy. Finally, we define the
rectification ratio as r13  $RR=|I(V)|/|I(-V)|$. $RR=1$ suggests no
rectification. In all calculations we set the common parameter values as:
$t=1\,$eV, $\epsilon_{0}=0$, $t_{0}=3\,$eV and, unless otherwise stated
$E_{F}=0$.

##  III Results and discussion

Now we present our results. In Fig. 3 we show the variations of

Figure 3: (Color online). $|I|$ and $RR$ as a function of voltage for a $8$th
generation ($N=34$) Fibonacci chain with $-\epsilon_{A}=\epsilon_{B}=0.5\,$eV
considering both linear (red curve) and two non-linear (green and blue curves)
potential profiles. In the first column the results are shown for the
symmetric coupling ($\tau_{S}=\tau_{D}=1\,$eV), while in the second column
they are presented for the asymmetric coupling ($\tau_{S}=0.1\,$eV and
$\tau_{D}=1\,$eV). Dashed line corresponds to $RR=1$.

$|I|$ both for the forward and reverse biased conditions along with the
rectification ratio $RR$ considering three different electrostatic potential
profiles where the first and second columns correspond to the symmetric and
asymmetric wire-to-electrode couplings, respectively. Two observations are
noteworthy. First, the currents for the positive and negative biases are quite
close to each other in the limit of symmetric coupling (Fig. 3(a)), whereas
they differ distinctively in the case of asymmetric coupling (Fig. 3(c))
though the currents in this case are much less than the previous one. Second,
for a wide bias window ($\sim 1$-$2\,$V) the rectification ratio is
significantly large, in the limit of asymmetric coupling, reaching a maximum
of $\sim 11$. This is a reasonably large value compared to the reported
results for different conducting junctions considering both symmetric
molecular structures as well as asymmetric molecule-to-lead couplings where RR
varies between $2$ to $10$ r9 ; r12 .

To illustrate the mechanism of rectification let us focus on the spectra given
in Fig. 4. For the fully perfect chain ($\epsilon_{i}^{0}=0\,\forall\,i$),
$T$-$E$ spectrum in presence of positive bias (red curve of Fig. 4(a)) exactly
matches with what we get in the case of negative

Figure 4: (Color online). Transmission probability $T(E)$ (red color) as a
function of energy $E$ for ordered ($\epsilon_{i}=0\,\forall\,i$) and
Fibonacci chains ($-\epsilon_{A}=\epsilon_{B}=0.5\,$eV) considering a linear
bias drop. The average density of states $\rho(E)$ (black color) is
superimposed in each spectrum. Here we set $\tau_{S}=\tau_{D}=1\,$eV and
$N=8$.

bias (red curve of Fig. 4(b)). Therefore, for this junction identical currents
are obtained upon the integration of the transmission function $T$, resulting
a vanishing rectification. While, comparing the spectra given in Figs. 4(c)
and (d), worked out for the positive and negative biases considering a
Fibonacci chain, it is clearly seen that the transmission spectra differ
sharply which results a finite rectification. To achieve rectification the
essential thing is that, as stated earlier, the energy levels of the bridging
material have to be aligned differently for the positive and negative biases.
The alignments of energy levels for the two different wires are clearly
reflected from the $\rho$-$E$ spectra (black curves of Fig. 4). Thus
increasing the misalignment higher $RR$ is expected and it can be done further
by including additional asymmetric factors like asymmetric environmental
effects env , inconsistent gating gat , etc.

Figure 5 shows the percentage of rectification (defined as
$(|I(V)|-|I(-V)|)/(|I(V)|+|I(-V)|)=(RR-1)/(RR+1)$) as a function of voltage
for three different sizes of the Fibonacci chain. Quite interestingly we see
that for wide voltage regions nearly cent percent rectification is obtained,
and thus the present system can be utilized as a perfect rectifier.

In order to justify the robustness of rectification against disorderness, in
Fig. 6 we present $RR$-$V$ characteristics for some typical quasi-periodic
lattices. All these junctions provide finite rectification where $RR$ varies
in a wide range, and from these curves it can be emphasized that any one of
such lattices can be used to achieve

Figure 5: (Color online). Percentage of $RR$ as a function of bias voltage $V$
for three different sizes of the Fibonacci chain considering a linear bias
drop, where (a), (b) and (c) correspond to $N=55$, $89$, $233$, respectively.
Here we choose $\tau_{S}=0.1\,$eV and $\tau_{D}=1\,$eV, and
$-\epsilon_{A}=\epsilon_{B}=0.5\,$eV.

the goal of rectification action. Looking carefully into the spectrum (Fig. 6)
it is observed that $RR$ reaches to zero ($\sim-100\%$)

Figure 6: (Color online). $RR$-$V$ characteristics for four different quasi-
periodic chains with inconsistent wire-to-electrode coupling
($\tau_{S}=0.1\,$eV and $\tau_{D}=1\,$eV) in presence of linear potential
profile. Here we set $N=34$, $32$, $43$ and $43$ for Fibo, TM, CU and BM
chains, respectively, and choose $-\epsilon_{A}=\epsilon_{B}=0.5\,$eV.

for a reasonable voltage window ($\sim 0.23$-$0.38\,$V) for the junction
containing BM wire. One could also get opposite scenario i.e., $\sim+100\%$
rectification through any one of these junctions. This is solely associated
with the interplay between the arrangements of lattice sites and electrostatic
potential profile.

Finally, in Fig. 7 we discuss the possibilities of regulating the
rectification ratio externally for a fixed bias voltage. This can be

Figure 7: (Color online). Percentage of $RR$ as a function of Fermi energy
$E_{F}$ for a $5$th generation Fibonacci chain
($-\epsilon_{A}=\epsilon_{B}=0.5\,$eV) considering a linear bias drop at
$V=2\,$V, where (a) and (b) correspond to the symmetric
($\tau_{S}=\tau_{D}=1\,$eV) and asymmetric ($\tau_{S}=0.1\,$eV and
$\tau_{D}=1\,$eV) wire-to-electrode couplings, respectively.

achieved by tuning the Fermi energy, which on the other hand is controlled by
external gate voltage. It is worthy to note that upon changing $E_{F}$ one can
get a wide variation of $RR$ (viz, $-100\%$ to $+100\%$), and thus it can be
emphasized that the present model can be utilized to get externally controlled
rectifier at nano-scale level.

Before the end, we would like to point out that, apart from rectifying action
all these quantum wires characterized by quasi-periodic lattices show another
uncommon property of junction current where an increase in the bias voltage
results a reduction of net current (see Figs. 3 and 6). This is the so-called
negative differential conductance (NDC) effect ndc , and its detailed analysis
will be given in our forthcoming paper.

##  IV Concluding remarks

To conclude, in the present work we have attempted to establish a model
quantum system that can exhibit a high degree of current rectification at
nanoscale level. An unexpectedly large rectification ratio has been obtained,
and most importantly, we see that in some cases nearly cent percent
rectification can be achieved for a wide bias window. Our results are also
valid against disordered configurations which we have confirmed by considering
different kinds of disordered systems. Though the proposition given here is
based on purely theoretical arguments, we hope that its experimental
verification can be done in near future.

Finally, we want to note that although the results have been worked out at
zero temperature, all the physical features remain invariant at finite
temperature ($\sim 300$ K) as thermal broadening of energy levels is much
weaker than the broadening caused as a result of wire-to-electrode couplings
datta .

##  V Acknowledgment

MS would like to acknowledge University Grants Commission (UGC) of India for
her research fellowship.

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