Bob Daniels delights in the West Tisbury farmer's market year after year.
I like my vegetables as much as the next guy, but lately I've been feeling out of place at the West Tisbury Farmers' Market. Admittedly, I've never felt reverential about chard and arugula and still haven't jumped on the kale bandwagon. It's not just the exotic vegetables, but the oat soap and green olive gluten-free tapenade, the Maitake mushrooms, squash blossoms, smoked peppercorn pesto and kombucha.
My first inkling of feeling out of place occurred on July 9, 2017, the day when dogs were banned from the market. No more outings for Milo. A more hygienic, less Labradoodly venue. Lots of hurt feelings and squeals.
It's hardly how I remember growing up in Ohio, watching Lassie on Sunday nights: Jeff plowing the North 40, chickens squawking by the clothesline. Now that's a farm.
At the farmers' market, I'm finding myself bewildered. West Chop Inner Tea. What makes it inner? And when did they start growing tea in West Chop? Then there's Katama Chamomile, "an especially calming blend." What if my insurance doesn't cover it?
When it's 93 degrees outside, I find myself longing for a supermarket. It's always Antarctica in the Cronig's produce section. And while their employees don't feel obliged to lecture you on the life cycle of a parsnip, they will point out the potato counter. (They call them potatoes, not fingerlings. Potatoes as in salad, baked, mashed.) None of the staff seem interested in sharing tips of how to rid your garden of slugs, but there are shelves of admittedly the most un-organic pesticides.
I stop at another farm stand and read the poster for "Pork Belly, Pork Chops, Loin Roasts and Country Ribs." A man mentions there's a new litter of piglets at the Farm Institute. The cutest, an orange one, is called Sweet Potato. I ponder the name, then it hits me: Sweet potato will one day be loin roast and then it is all becomes too much. How can I ever reconcile my love for pork with my fondness for cute? I reconsider my carnivore ways, reflect on Ghandi, Babe and Tofu, then head next door for a popover.
She's right. And I've never realized it before.
I end up at Island Bees, where bee-ziness is booming. The varroa mites plaguing bees around the world is not a problem on the Vineyard. The bees are thriving here. And so, in its own way, is the West Tisbury Farmers' Market.
Ted Sutton lives in Newton and Aquinnah.


text:
Six Members of IAAS Indonesia Learned the Dynamics of Agricultural and Livestock Commodities Production Cycle at The 60th IAAS World Cogress in Mexico
The Power of Qurban
Let's Make a Change with Agriventor
Ciudad de México, Mexico – International Association of Students in Agricultural and Related Sciences (IAAS) is the largest agricultural student organization in the world based in Leuven, Belgium. IAAS has been established since 1957 and has over 10,000 members spread across four continents, Africa, America, Asia-Pacific and Europe. In order to commemorate the 60th anniversary, IAAS in July 2017 was held a World Congress with the theme "Mexico 360" in Mexico which attended by 60 participants from 19 countries, including Indonesia. The theme "Mexico 360" was appointed with the intention to invite participants in a comparative study of various cycles of production of major agricultural commodities in Mexico, such as wheat, corn, and agave. Not only that, the participants also studied the supply chain of animal feed and livestock products, such as eggs, meat, milk, cheese and others.
In 2016, IAAS Indonesia was honored to host The 59th IAAS World Congress. This year, IAAS Mexico in cooperation with the Tecnologico de Estudios Superiores de Monterrey (ITESM) college, is entrusted to organize The 60th IAAS World Congress which runs for 21 days from July 9 to July 30, 2017 in Mexico. Various series of main events and activities including the General Assembly (GA); Workshop; Industry, Farm and Campus Visit; Trade Fair; Development Fund; Excursion; And City Tour was held in 7 cities, namely Ciudad de México; Pachuca; Querétaro; La Piedad; Guadalajara; Aguascalientes; and Oaxaca.
Six Indonesian students consisting of three Bogor Agricultural University (IPB) students, Azwar A. Sujaswara, Levina Caroline and M. Irfan Fathoni, then Syuga Invicta from Sebelas Maret University (UNS), and two students from Mataram University (UNRAM), Meita Khastin and Rima Avianti was chosen to represent IAAS Indonesia in this World Congress. They took the mission to conduct annual reports on exchange programs, social movements and village concept projects of IAAS Indonesia and Asia Pacific, promoting The 9th IAAS Olympic and The 2nd IYSCA event that will be held in IPB on the next October 2017, supporting the Indonesian candidate in nominating as the next Executive Board of IAAS World, and promoting the Indonesian culture and International Student Mobility programs in Indonesia.
The results of this World Congress are IAAS Indonesia was awarded as The Most Outgoing Exchange Performance and one of the Indonesian delegates, Levina, was elected as Vice President of Communication of IAAS World for the period of 2017/2018. Not only that, other Indonesian delegations also played an active role during the congress. Azwar was selected as the project leader in one of the IAAS New Worldwide Project for this year, the Climate Change Challenge, and two other delegates, Meita, was elected as Vice-Chairman and Rima as Secretary of General Assembly. Syuga is involved and actively contributes in Exchange Program development workshops and Irfan involved in finance topics. It shows the existence of IAAS Indonesia members who are not only able to contribute actively in student organization and youth movements at the national level, but also able to contribute actively at the international level as well.
According to Irfan as the head of the delegation, IAAS World Congress in Mexico is very interesting and the delegates get a lot of insight and applicative knowledges from this series of programs. Besides knowing the social-culture of Mexico, the delegates also have the opportunity to become an audience at The 14th International Congress Holstein de las Americas and interact with the experts in the cattle breeding and dairy industry in Latin America. The delegates also had the opportunity to have visits and comparative studies to International Maize and Wheat Improvement Center (CIMMYT), both national and multinational leading companies that engaged in agriculture in Mexico, and have a direct dialogue with some local government officials at the region level as well and representatives of Mexican agriculture ministries that known as SAGARPA.
From this World Congress it known that Mexico has applied better agricultural technology and sciences than Indonesia. So, as youth let's contribute for the better Indonesia especially in agricultural sector!
*Here is some other information related to the implementation of The 60th IAAS World Cogress in Mexico
New Executive Board of IAAS World 2017/2018
Anne-Marie Dzinoreva from IAAS Zimbabwe as new President of IAAS World,
Celine Feoux-milan from IAAS France as Vice President of Finance,
Levina Caroline from IAAS Indonesia as Vice President of Communication,
Paulina Elena-Manthey from IAAS Germany as Vice President of Exchange,
Luis Gandarillas from IAAS Mexico as Vice President of External Relations.
New Control Board of IAAS World 2017/2018
Siti Nisrina Hasna from IAAS Indonesia,
Natalie Bidner from IAAS USA,
Madan Poudel from IAAS Nepal
IAAS World New Project
IAAS Training Committee
Alumni Guideline
Expro Catalogue
Development Funds (DF) Day
Climate Change Challenge (Triple C)
Social Project (VCP)
IAAS "How to" Video
The 61th IAAS World Cogress will be held in Croatia (2018)
The 62th IAAS World Cogress will be held in Cote de Ivoire (2019)
IAAS Goes to School – Youth Agricareture 2019
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redpajama_set_name: RedPajamaCommonCrawl


"You have to say this is a special, all-time defense," said Broncos defensive coordinator Wade Phillips, who was The Associated Press Assistant Coach of the Year.
Trevathan didn't hesitate to call the Broncos' defense one of the best ever.
NFL World Champions... I'll take it!!!!!
On Sunday, Denver's defense kept Newton, the league's MVP, jittery all day. Despite wearing gold shoes before the golden Super Bowl, Newton couldn't finish off a dynamic season.
Von Miller, who was named Super Bowl MVP, twice stripped the ball from Newton, once for a touchdown, the second time setting up a clinching TD.
Denver's top-ranked defense, the one that ran roughshod over Tom Brady in the AFC championship, simply wouldn't let Newton get comfortable.
"It's every one of these guys who got me to this," Miller said.
Newton was sacked six times — receiver Ted Ginn Jr., went down once on an aborted trick play — and if Miller wasn't torturing him, DeMarcus Ware was. Ware had two of the seven sacks, equaling the most ever by one team in the Super Bowl.
"He was stressed," cornerback Bradley Roby said of Newton.


The modern women do consider something different than the traditional gold band on her wedding ring along with the most popular wedding ring could be the white gold ring for ladies. It is shiny but won't have the regular yellow gleam from the regular wedding ring. This is a new metal produced by the jewelers to create something out from the ordinary. This metal provides classy look but differs from the others from your general. You can set stones into it being a ruby or emerald that may catch the gleam with the light on it. You can engrave modern designs into it too which will set it apart and provide your individual memory in the day.
A good wedding checklist includes your choice of dates and different church options. You might even choose to have your wedding day outdoors, which may entail an entirely different kind of wedding checklist. Instead of looking for venues where you can hold your party following the ceremony, you will end up considering businesses that erect large marquee tents. Unless you rent a hall that delivers everything for the wedding, you will also ought to consider renting tables and chairs, getting flower bouquets done, finding catering and selecting sometimes a band or DJ to liven up the festivities.
(In Chile the tradition is opposite – while using couple wearing their engagement rings on the right hands, then transferring them to their left hands after the marriage is created official in the marriage ceremony.) Even other societies – including among Russians, Nordics, Colombians, and Polish – that display the ring on the fourth finger from the right hand instead of the left nevertheless credit the standard Romans' belief within the vena amoris for their culture's basis for the chosen finger. Outside of the cultures when the right hand houses the original ring finger, wedding rings and wedding bands on the right hand can have several unique meanings. For instance, widows often switch their rings on the other hand to symbolize their widowhood, or left-handed people may decide to wear their rings on their own right hands that it is less of an obstruction on their everyday activities that demand more utilization of their dominant hand.
They require considerable planning, skill and expertise, finding yourself in an overseas location, couples may encounter a couple of difficulties with regards to the foreign culture and language. A wedding planner is important to translate and negotiate expenses using the local staff. Many resorts and hotels offer discounts, complimentary gifts to guests communicate with all the event planner and soon to be couple to prepare wedding and reception.


text:
CSS dynamique, ou DCSS, est un terme générique pour un ensemble de technologies utilisées pour créer des feuilles de style dynamiques, en utilisant une combinaison de n'importe quel langage de programmation côté serveur (comme PHP/ASP/Perl/JSP) et des feuilles de style en cascade (CSS). La première idée de DCSS a été proposée en juillet 2002 par Jori Koolstra, un programmeur néerlandais.

DCSS permet de travailler avec des variables en CSS et la réécriture dynamique de code source en CSS. De nombreux systèmes de gestion de contenu ont créé des modules supplémentaires, par exemple Drupal, pour gérer le DCSS. Sass et Moins peuvent être employés comme langages de feuilles de style dynamiques.

Exemple en PHP
Généralement, une page web utilisant les feuilles de style en cascade dynamiques est configurée via trois fichiers. Un premier fichier contenant les variables CSS, un fichier .php qui reprend le contenu CSS et la page web pour laquelle le CSS est requis.

Un fichier .dcss ressemble normalement à ceci :
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1//EN" "http://www.w3.org/TR/xhtml11/DTD/xhtml11.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
  <head>
    <title>Exemple DCSS</title>

    <style type="text/css">
      <?php
         /* Inclure la feuille de style */
         require_once("sheet.dcss.php");
      ?>
    </style>

  </head>
  <body>
  </body>
</html>

Ci-après, le fichier .dcss.php inclus. Remarquez qu'un fichier dcss a toujours une extension .php :

<?php
  /* Inclusion du fichier variables */
  require_once("vars.php");

  echo "p";
  echo "{";
  echo "font-family:    \"$FONT_TYPE\";";
  echo "}";
?>

Et le fichier qui contient les variables pour le fichier dcss.php (souvent nommé vars.php) :

<?php
  $FONT_TYPE = "Courier New";
?>

Voir aussi
 Page web dynamique
 HTML dynamique

Références

Liens externes
 Codewalkers.com - tutoriel de CSS Dynamique avec PHP

CSS
Conception de sites web

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

Electron spins carry much promise as state variables for scaled
classical logic\cite{zutic04} and as quantum bits (qubits) in quantum
computer architectures.\cite{kane98} A key challenge of current
research in spin-based electronics (``spintronics'') and quantum
computation is to devise methods to probe spin polarization in
semiconductors with weak spin-orbit coupling,\cite{zutic06} like
silicon and silicon-germanium alloys.  Silicon combines many special
features that make its electronic spin a promising basis for classical
and quantum logic devices: Band structure properties such as weak
spin-orbit coupling and indirect band gap, and the possibility of
preparing a nuclear spin free environment lead to extremely long spin
relaxation times in comparison to other
semiconductors.\cite{tyryshkin03}

However, the same features that lead to long spin relaxation times
also make spin detection more challenging.  Optical methods for spin
detection such as Faraday and Kerr rotation have been used successfully
in III-V semiconductors,\cite{kikkawa99,crooker05} but unfortunately, these
methods are inapplicable to silicon and related materials, whose
spin-selective optical transitions are extremely weak and ineffective.

\begin{figure}
\includegraphics[width=3in]{device}
\caption{(Color online) Silicon transistor with a ferromagnetic
  source, inspired by the devices in [\onlinecite{jonker07,appelbaum07}]. Our
  theory shows that doping with different types of group V donor
  species along the transistor channel enables the determination of
  the conduction electron spin polarization distribution along the
  channel. Each donor species has a distinct set of hyperfine-split
  electron spin resonance frequencies, whose amplitude is directly
  proportional to the local carrier spin and donor electron spin
  polarizations.\label{device}}
\end{figure}

Recently, injection and detection of conduction electron spin
polarization was demonstrated in silicon.  In Ref.~[\onlinecite{jonker07}], a Si
device with a ferromagnetic (FM) source was
 interfaced with GaAs to enable optical spin detection,
while in [\onlinecite{appelbaum07}] a spin valve based on a
 FM-Si interface was used.

Both experiments achieve spin detection at the interface, but no local
or ``in-situ'' spin detection. The interface limits spatial resolution
and introduces additional scattering effects.  In studies with
accumulation channel silicon transistors, spin-dependent
scattering was demonstrated by electrical detection of magnetic
resonance (EDMR).\cite{ghosh92,lo07}

In this article we present a theory of spin-dependent transport in the
scattering of spin polarized conduction electrons in the two
dimensional electron gas of field effect transistors.  Based on the
scattering theory, we propose a donor-based approach to the problem of
\emph{in-situ carrier spin detection in silicon}.  At the limit of a
single donor impurity, this mechanism also provides a path for the
readout of single donor spin states that does not require proximity to
(or presence of) charge traps.\cite{kane98,xiao04,elzerman04,kaplan78}

Consider a silicon transistor with a series of different group V donor
impurities implanted along its conduction channel, Fig.~\ref{device}.
Spatially resolved characterization of carrier spin polarization is
obtained by measuring the source-drain current when each donor species
is in resonance with a microwave field.

This is possible due to spin-dependent scattering (SDS), which relies
solely on the symmetry of impurity scattering
events.\cite{honig66,schmidt66,ghosh92} While this effect was believed to be
observed through EDMR in [\onlinecite{ghosh92}] amidst competing
heating effects, it was only recently that SDS was confirmed
experimentally using a small ensemble ($6 \times 10^6$) of implanted
antimony (Sb) donors.\cite{lo07} In the following, we present a general
microscopic theory of SDS, its efficiency and tunability in a
two dimensional electron gas (2DEG).  We show that the core physical
mechanisms underlying SDS are virtual transitions mediated by
negatively ionized donor levels.  Our results apply to all
semiconductors, but are particularly useful as a probe of spin
polarization in group IV and related materials with weak spin-orbit
coupling, whose spin detection is intrinsically challenging.  Our
physical optimization of this effect has exciting consequences for
spintronics and single donor electron or nuclear spin readout.\cite{sarovar07}

We remark that SDS is quite different from other EDMR experiments
based on spin-dependent recombination of electron-hole
pairs.\cite{lepine72,kaplan78,christmann95,brandt04,stegner06,mccamey06,morley08}
As showed by Kaplan, Solomon, and Mott,\cite{kaplan78} spin-dependent
recombination has important contributions that \emph{do not} depend on
carrier spin polarization $p_c$. Hence it is difficult to use
recombination based EDMR as a probe of carrier spin polarization
$p_c$.

This article is organized as follows. Section~\ref{sdsedmr} presents a
general theory of SDS and its detection by EDMR in the case of thermal
equilibrium in an external magnetic field (no spin injection).
Section~\ref{numerics} describes our explicit numerical calculations,
together with a comparison between theory and two
experiments,\cite{ghosh92,lo07} and a discussion on how SDS can be
optimized with respect to donor impurity location.
Section~\ref{kondosection} discusses the validity of our perturbation
theory approach, by showing explicit calculations of the Kondo
temperature as a function of donor impurity location.
Section~\ref{spininjec} generalizes our theory to the case of
non-equilibrium spin injection, and discusses how EDMR scales with
carrier spin polarization. Section~\ref{conclusions} presents our
conclusions.

\begin{figure}
\includegraphics[width=3in]{band_diagram}
\caption{(Color online) Band diagram along the $z$ direction
  perpendicular to the 2DEG plane, as shown in Fig.~\ref{device}.
  Here $E_c$ is the conduction band edge, $\epsilon_F$ is the 2DEG
  Fermi energy, $\epsilon_{\rm{sb},l}$ is the ground state energy for
  the $l$-th subband, and $\epsilon_1$ is the ground state energy of a
  neutral donor (singly occupied).  We also show the donor impurity
  ground state wavefunction $\Phi_1$, and the conduction electron
  wavefunction $\Psi_{k,l}$ with its characteristic length $z_0$.  As
  we show below, only donors located at $R\gg z_0$ can be detected by
  EDMR.\label{band_diagram}}
\end{figure}

\section{Theory of spin-dependent impurity scattering and electrically
  detected magnetic resonance\label{sdsedmr}}

Impurity scattering depends on whether the two-particle
states formed by a conduction and an impurity electron are in a
singlet (S) or triplet (T) configuration,
\begin{equation}
\Psi^{S/T}_{kl}=\frac{1}{\sqrt{2}}
\left[\Psi_{kl}(\bm{r}_{1})\Phi_1(\bm{r}_{2})
\pm \Psi_{kl}(\bm{r}_{2})\Phi_1(\bm{r}_{1})\right].
\label{st}
\end{equation}
Here $\Psi_{kl}$ and $\Phi_1$ are orbital wavefunctions, respectively
of a conduction electron with momentum $k$ in the $l$-th subband, and
a localized donor impurity electron in the ground state.
Fig.~\ref{band_diagram} shows the band diagram and potential profile
for conduction and impurity wavefunctions in the 2DEG.

Access to additional channels for virtual scattering in the singlet
state (Fig.~\ref{diagram}) translates into distinct neutral impurity
scattering times for triplets vs. singlets, $\tau_{T}\neq \tau_{S}$.
With spin polarizations
$p=(p_{\uparrow}-p_{\downarrow})/(p_{\uparrow}+p_{\downarrow})$, for
conduction ($p_c$) and impurity ($p_i$) electrons, the occupation
fraction for singlets is given by $p_S=(1-p_i p_c)/4$, and the
fraction for triplets $p_T=1-p_S=(3 + p_i p_c)/4$.  Due to this
difference in singlet vs. triplet scattering times, the device current
is directly related to the spin polarizations according to
\begin{equation}
I \propto \langle \tau\rangle = \left(p_S \tau_{S}+p_T \tau_{T}\right)
\propto I_0 \left(1+\alpha  p_c p_i\right).
\label{mu}
\end{equation}
Here we introduced the parameter $\alpha$ as a figure of merit for the
SDS effect.  Eq.~(\ref{mu}) forms the basis of the SDS mechanism of
EDMR detection, which only occurs when the carrier spin polarization
$p_c$ is non-zero. At thermal equilibrium, and in the low temperature ``degenerate limit''
($k_BT\ll \rm{Min}_l\{\epsilon_F-\epsilon_{\rm{sb},l}\}$), $p_c$
arises due to Pauli paramagnetism,
\begin{equation}
  p_c=\frac{n_\uparrow-n_\downarrow}{n_\uparrow+n_\downarrow}=
  \frac{g\mu B}{2\left[\epsilon_F-\frac{1}{2}(\epsilon_{\rm{sb,1}}
          +\epsilon_{\rm{sb,2}})\right]}.
\end{equation}
Here $n_{\sigma}$, $\sigma=\uparrow,\downarrow$ is the electron
density for spin species $\sigma$, $B$ is the external magnetic field,
$g\mu/\hbar=\gamma_e=1.8\times 10^{7}$~(sG)$^{-1}$ is the free
electron spin gyromagnetic ratio, and
$\epsilon_{\rm{sb},1}$, $\epsilon_{\rm{sb},2}$ are the 2DEG subband
ground state energies.

\begin{figure}
\includegraphics[width=3in]{diagram}
\caption{(Color online) Contributions to neutral impurity scattering.
  (a) The largest individual contribution to singlet scattering is a
  virtual transition to a negatively ionized donor level, where two
  electrons occupy the same (ground) orbital donor level $\epsilon_1$.
  Such a transition is symmetry forbidden for triplets due to Pauli
  exclusion. (b) Triplet scattering occurs due to virtual transitions
  into negatively ionized donor levels where one electron occupies the
  level $\epsilon_1$, and the other electron occupies an excited
  orbital state $\epsilon_i>\epsilon_1$. These same transitions into
  excited orbital states are also allowed for singlets.\label{diagram}}
\end{figure}

Under microwave irradiation, the donor spin transition rates are given
by $W\equiv W_{\uparrow}=W_\downarrow$ (the subscript refers to the
initial state of the transition),
\begin{equation}
  W(\omega)= 2\pi \left|\gamma_e B_{\perp}\right|^2
\frac{1}{\pi T_{2}^{*}}\frac{1}{(\omega-\omega_m)^{2}+\left(\frac{1}{T_{2}^{*}}\right)^{2}},
\label{weqn}
\end{equation}
where $B_{\perp}$ is the amplitude of the microwave
field, $\omega_m= \gamma_e (B+h_m)$ is the resonance frequency of
the donor with nuclear spin state $m=-I,-I+1,\ldots, +I$ ($h_m$ is a
hyperfine shift), and $\frac{1}{T_{2}^{*}}$ is the inhomogeneous linewidth.
The microwave rate $W$ competes with the donor spin-flip rate
$\frac{1}{T_1}=\Gamma_{\uparrow}+\Gamma_{\downarrow}$ in order to determine a
steady state impurity spin polarization,
\begin{equation}
p_i=\frac{p_{i0}}{1+2T_1W},
\label{pi}
\end{equation}
where $p_{i0}=\tanh{\left(\frac{g\mu B}{2k_BT}\right)}$ is
the equilibrium donor spin polarization.

When donor atoms are at resonance ($\omega=\omega_m$), and satisfy the
saturation condition $(\gamma_e B_{\perp})^{2}\gtrsim 1/(T_1
T^{*}_{2})$, their spin polarization vanishes, $p_i \approx 0$,
effectively \emph{reducing the device current by $\Delta I/I_0 \approx
  \alpha p_{c}p_{i0}$}. In the ``clean limit'', when no other
scattering sources are competing with donor impurity scattering, the
figure of merit $\alpha$ is given by
\begin{equation}
\alpha=\frac{\sum_l \left(\tau_{T,l}-\tau_{S,l}\right)}
{\sum_l \left(3\tau_{T,l}+\tau_{S,l}\right)},
\label{alpha}
\end{equation}
where we added a subscript $l$ to account for different subbands.
Note that the saturation condition depends
critically on the microwave amplitude $B_{\perp}$, donor spin-flip
rate $\frac{1}{T_1}$ and
resonance linewidth $\frac{1}{T^{*}_{2}}$.

In transport measurements, SDS is always in competition with other
scattering sources, such as surface roughness and lattice defects. We
describe these phenomenologically by adding a rate $\frac{1}{\tau_0}$:
$\frac{1}{\tau_{S/T}}\rightarrow
\frac{1}{\tau_0}+\frac{1}{\tau_{S/T}}$.
Usually, transport is
dominated by scattering sources other than neutral donors
($\frac{1}{\tau_{0}}\gg\frac{1}{\tau_{S/T}}$), leading to a reduced
figure of merit $\alpha \approx \frac{\tau_0}{4}
\sum_l\left(\frac{1}{\tau_{S,l}}-\frac{1}{\tau_{T,l}}\right)\ll 1$.

The scattering amplitudes to second order in the Born approximation are given
by
\begin{eqnarray}
{\cal A}^{S/T}_{kl,k'l'}&=&\langle \Psi^{S/T}_{kl}| {\cal H}| \Psi^{S/T}_{k'l'}\rangle
\nonumber\\&&+\sum_{i} \frac{\langle \Psi^{S/T}_{kl}| {\cal H}| i^{S/T}\rangle
\langle i^{S/T}| {\cal H}| \Psi^{S/T}_{k'l'}\rangle}{(\epsilon_1+\epsilon_{kl})-
(\epsilon_1+\epsilon_i+U)},
\label{ast}
\end{eqnarray}
and the scattering rates due to a single donor impurity are
\begin{eqnarray}
\frac{1}{\tau_{S/T,l}}&=&\frac{2\pi}{\hbar}\sum_{k',l'}\left|
{\cal A}^{S/T}_{kl,k'l'}\right|^{2}\left(1-\cos{\theta_{kk'}}\right)\delta(\epsilon_{kl}-\epsilon_{k'l'})
\nonumber\\
&=& \frac{2\pi}{\hbar}\rho\sum_{l'}\left\langle|{\cal
  A}^{S/T}_{kl,k'l'}|^{2}(1-\cos{\theta_{kk'}})\right\rangle,
\label{taust}
\end{eqnarray}
where $\theta_{kk'}$ is the angle between incoming ($\bm{k}$) and
outgoing ($\bm{k}'$) conduction electron wavevectors, with
$|\bm{k}'|=k$ averaged over the Fermi distribution.  In the degenerate
limit we may set $k$ equal to the Fermi wavevector $k_F$, but a more
general averaging procedure will be considered below.  Here
$\rho=\frac{m^{*}A}{2\pi \hbar^{2}}$ is the 2DEG energy density, with
$A$ the 2DEG area. The energy of the conduction electrons is assumed
to be $\epsilon_{kl}=\epsilon_{\rm{sb},l}+\frac{\hbar^2
  k_{\perp}^{2}}{2m^{*}}$, with $\bm{k}_{\perp}=(k_x,k_y)$, $m^{*}$
the effective mass, and $\epsilon_{\rm{sb},l}$ the ground state energy
of the $l$-th subband.  Eqs.~(\ref{ast})~and~(\ref{taust}) do not
depend explicitly on electron Zeeman energy because we assumed that
the $g$-factor of the donor is the same as the $g$-factor of the
electrons [while it is essential to consider different $g$-factors for
EDMR detection, e.g.  in Eq.~(\ref{weqn}), the $g$-factor differences
give only a negligible effect in the calculation of transport
properties, provided the external field $B$ is not too large].

The second order contribution in Eq.~(\ref{ast}) is a sum over virtual
(intermediate) states, and is generally much larger than the 1st order
contribution.  Fig.~\ref{diagram} illustrates the most important
channels for neutral impurity scattering. These depend on the
two-particle Hamiltonian ${\cal H}$ only through the donor
single-particle orbital energies $\epsilon_i$, donor ``on-site''
Coulomb repulsion $U$, and the overlap between conduction and impurity
wavefunctions.  The largest individual contribution is a virtual
transition to $(\epsilon_1 +U)$, which is a negatively ionized donor
level with two electrons in the ground orbital state
[Fig.~\ref{diagram}(a)]. This transition is only allowed for singlets
due to Pauli exclusion.  Since singlet scattering also has the
channels for excited orbital states available, it is always stronger
than triplet scattering (which has access only to the excited states)
and in general $\alpha > 0$.

In these neutral donor scattering events, the total spin of conduction
and donor electrons is always conserved, but their respective spin
states may be exchanged.\cite{bastard90,mahan08} In our notation,
$\Gamma_{\sigma}$ is the rate for a donor with spin $\sigma$ to
exchange spin with a conduction electron having spin $-\sigma$;  using
$\sigma=+,-$ to denote the spin state, the rates can be written as
\begin{eqnarray}
\Gamma_\sigma &=& \frac{2\pi}{\hbar}\sum_{kl,k'l'} \left|J_{kl,k'l'}\right|^{2}
\delta(\epsilon_{kl}-\epsilon_{k'l'})\nonumber\\
&&\times f\left(\epsilon_{kl}+\sigma \frac{g\mu B}{2}\right)
\left[1-f\left(\epsilon_{k'l'}-\sigma\frac{g\mu B}{2}\right)\right].
\quad\phantom{.}
\label{gammasigma}
\end{eqnarray}
Here $J_{kl,k'l'}={\cal A}^{T}_{kl,k'l'}-{\cal A}^{S}_{kl,k'l'}$ are exchange
scattering amplitudes, and
\begin{equation}
f(\epsilon)=\frac{1}{\textrm{e}^{\frac{\epsilon-\epsilon_F}{k_BT}}+1}
\end{equation}
are Fermi functions. After converting the sums into integrals over $k$ space,
the donor spin-flip scattering rate
becomes
\begin{equation}
\frac{1}{T_1}=\Gamma_{\uparrow}+\Gamma_{\downarrow}= \frac{2\pi}{\hbar}\rho^{2}
\left\langle|J_{k,k'}|^2\right\rangle g\mu B \coth{\left(\frac{g\mu B}{2k_BT}\right)}.
\label{t1}
\end{equation}
As a check, note that the equilibrium donor spin polarization is given
by $p_{i0}=(\Gamma_{\downarrow}-\Gamma_{\uparrow})
/(\Gamma_{\uparrow}+\Gamma_{\downarrow})= \tanh{\left(\frac{g\mu
      B}{2k_BT}\right)}$.  When $k_BT\gg g\mu B$,
$\frac{1}{T_1}$ scales linearly with the temperature, while in the
opposite regime it scales linearly with $B$.

When other donor spin-flip mechanisms are active, we must add their
rate to Eq.~(\ref{gammasigma}).  Nevertheless, we will see below that
exchange scattering is the dominant contribution for donors in silicon
in a wide parameter range.

\section{Model calculations and comparison to experiment\label{numerics}}

\subsection{Model Hamiltonian and virtual two-particle donor states}

The two-particle Hamiltonian ${\cal H}$ that models the coupling
between a conduction plus a donor impurity electron can be written
explicitly as
\begin{equation}
{\cal H}= {\cal H}_{0}(1)+{\cal H}_{0}(2)+ {\cal C}_{R}(1)+
{\cal C}_{R}(2) + {\cal C}_{ee}(1,2)-2\epsilon_F,
\label{htotal}
\end{equation}
where for notational simplicity $(1)$ denotes coordinate $\bm{r}_1$,
and $(2)$ denotes coordinate $\bm{r}_{2}$.  Here ${\cal
  H}_{0}=\hat{T}+V_{z}(z)$ is the translational invariant
single-particle Hamiltonian, with $\hat{T}$ the kinetic energy and
$V_z(z)$ the 2DEG confinement (triangular at low $z$ but flattens out
at high $z$, see Fig.~\ref{band_diagram}). ${\cal C}_R$ is the
attractive Coulomb potential of the impurity, and ${\cal C}_{ee}$ is
the Coulomb electron-electron repulsion,
\begin{subequations}
\begin{eqnarray}
{\cal C}_{R}(i)&=& -\frac{e^{2}}{\kappa}\frac{\textrm{e}^{-q_{\rm{TF}}|\bm{r}_i-\bm{R}|}}
{\left|\bm{r}_i-\bm{R}\right|},\label{cn}\\
{\cal C}_{ee}(1,2)&=&\frac{e^{2}}{\kappa}\frac{\textrm{e}^{-q_{\rm{TF}}|\bm{r}_1-\bm{r}_2|}}
{\left|\bm{r}_{1}-\bm{r}_{2}\right|},\label{cee}
\end{eqnarray}
\end{subequations}
where $\bm{R}=R\hat{z}$ is the location of the donor impurity, and
$\kappa\approx 12$ the dielectric constant for silicon.
Since we are in the degenerate limit, it is important to account for screening;
we use Thomas-Fermi screening with wavevector
\begin{equation}
q_{\rm{TF}}=\sqrt{\frac{6\pi e^{2}n_{\rm{2D}}|\phi(z)|^{2}}
{\epsilon_F-\frac{1}{2}(\epsilon_{\rm{sb,1}}+\epsilon_{\rm{sb,2}})}},
\end{equation}
where $n_{\rm{2D}}$ is the 2DEG density and $\phi(z)$ the subband
wavefunction defined below. Note that the last term in
Eq.~(\ref{htotal}) is the chemical potential times the number of
particles.

In order to compute the scattering amplitudes [Eq.~(\ref{ast})], we need
to choose a set of virtual states $|i^{S/T} \rangle$ that forms a
complete basis for the two-particle Hilbert space.  An important
insight is that the conduction electron may hop into the impurity
site, form a negatively ionized virtual donor state, and then hop back
into the Fermi sea.  This motivates the choice of a molecular orbital
basis of negatively ionized donor states:
\begin{equation}
| i^{S/T} \rangle = \frac{1}{\sqrt{2}}
\left[ | \Phi_1 \Phi_i\rangle \pm | \Phi_i \Phi_1\rangle\right], \;i=1,2,3,\ldots,
\label{twoparti}
\end{equation}
where $\Phi_i(\bm{r})$ is a donor orbital state with single particle
energy $\epsilon_i$, satisfying $\left({\cal H}_{0}+{\cal
    C}_{R}\right)|\Phi_i\rangle = \epsilon_i |\Phi_i\rangle$. Note
that $i=1$ refers to the ground orbital state, and $i=2,3,\ldots$
refers to other excited donor single-electron orbitals.  From
Eq.~(\ref{twoparti}) we see that the state $|1\rangle$ only exists as
a singlet, $|1^{T}\rangle=0$.

The simplest molecular orbital approximation is to assume
\begin{equation}
{\cal H}|i^{S/T}\rangle\approx
(\epsilon_1+\epsilon_i+U-2\epsilon_F)|i^{S/T}\rangle.
\label{molorb}
\end{equation}
Using this relation, we see that the 2nd order contribution
in Eq.~(\ref{ast}) depends only on the energies $\epsilon_i$, $U$,
and on the overlap integral $S_{kl,i}=\langle \Psi_{kl}|\Phi_i\rangle$. Note
also that states of the type $|\Phi_i\Phi_j\rangle\pm
|\Phi_j\Phi_i\rangle$ with $i,j\neq 1$ do not contribute to
Eq.~(\ref{ast}) [they are orthogonal to Eq.~(\ref{st})].

\subsection{Microscopic model for single particle states and overlap integral}

Bulk silicon has a six-fold degenerate conduction band, with energy
minima located at $k_0=0.85\times 2\pi/(5.43~\rm{\AA})$ along the set
of $\langle 100\rangle$ directions in the Brillouin zone. Its
effective mass is anisotropic. Each valley has a heavy mass equal to
$0.98 m_e$ along the valley direction, and a much lighter mass $m^{*}
=0.19 m_e$ in the perpendicular plane ($m_e$ is the free electron
mass).  In the presence of a (001) interface, electrons in the valleys
along $\pm (001)$ will have considerably lower energy, a consequence
of the heavy longitudinal mass.\cite{ando82} This leads to the
following two-subband model for the conduction electron wavefunctions:
$\Psi_{kl}(\bm{r})=\psi_l (z)\textrm{e}^{i\bm{k}\cdot
  \bm{r}_{\perp}}/\sqrt{A}$, with $A$ the 2DEG area and $l=1,2$
a subband label.
The subband wavefunctions are given by
$\psi_{1}(z)=\phi(z)\sqrt{2}\cos{(k_0 z)}$ and
$\psi_{2}(z)=\phi(z)\sqrt{2}\sin{(k_0 z)}$.
For $z>0$, we use the Takada-Uemura envelope function,
\begin{equation}
\phi(z)=\sqrt{\frac{3}{2 z_0}}\left(\frac{z}{z_0}\right)\textrm{e}^{-\frac{1}{2}\left(\frac{z}{z_0}\right)^{3/2}},
\label{env}
\end{equation}
that is known to be an excellent analytic approximation to the
self-consistent 2DEG ground state (see p. 468 of
Ref.~[\onlinecite{ando82}]). We assume $\phi(z)=0$ for $z\leq 0$.  The
characteristic length scale $z_0$ models the 2DEG width, that can be
controlled electrically by the gate voltage.  Our choice of $\psi_l$
corresponds to pure imaginary intervalley scattering at the interface
(other choices of intervalley scattering phase simply give a phase
shift to the cosine and sine functions, that is equivalent to changing
the position of the interface).\cite{sham79} The conduction electron
states are assumed to satisfy ${\cal
  H}_0|\Psi_{k,l}\rangle=\epsilon_{kl}|\Psi_{k,l}\rangle$, where
$\epsilon_{kl}=\frac{\hbar^2}{2m^{*}}k_{\perp}^{2}+\epsilon_{\rm{sb},l}$.
Each subband has energy $\epsilon_{\rm{sb},l}$ at $k=0$, with
valley-splitting
$(\epsilon_{\rm{sb},2}-\epsilon_{\rm{sb},1})/k_B=0-100$~K depending on
interface quality and device geometry.\cite{takashina06,goswami07}

We now discuss model wavefunctions for the donor impurity.  We expect
that donors located too close to the interface will have quite high
$\frac{1}{T_1}$, and the EDMR saturation condition $(\gamma_e
B_{\perp})^2\gtrsim 1/(T_1 T_{2}^{*})$ will not be
satisfied.  For this reason, we expect that
only donors located far enough from the interface (at depths $R\gg z_0$) can
contribute to EDMR. Therefore it is sufficient to use donor wave
functions that are good approximations in the bulk. Those are the
Kohn-Luttinger wavefunctions,\cite{kohn57}
\begin{equation}
\Phi_i(\bm{r-R})= F_i (\bm{r-R})\sum_{j=1}^{6} \eta_{ij}
\textrm{e}^{i\bm{k}_j\cdot (\bm{r-R})}.
\label{kohnlutt}
\end{equation}
For simplicity, we used an isotropic envelope $F_i(\bm{r-R})\approx
\textrm{e}^{-r'/a^{*}_i}/\sqrt{\pi a^{*3}_{i}}$, with
$r'^{2}=x^2+y^2+(z-R)^{2}$. The Bohr radius depends on the hydrogenic
principal quantum number $n$ according to $a^{*}_{i}\approx n^2
a^{*}_{1}$ (we used $a^{*}_{1}=18.6$~\AA). The principal quantum
number $n$ relates to the subscript $i$ according to
$n=(i~\rm{div}~6)+1$. Below we will list the donor energy levels with
the zero of energy chosen at flat band (the bulk conduction band
edge).

The valley vectors $\bm{k}_{j}$ are given by $\bm{k}_{1}=+k_0\hat{x}$,
$\bm{k}_{2}=-k_0\hat{x}$, $\bm{k}_{3}=+k_0\hat{y}$,
$\bm{k}_{4}=-k_0\hat{y}$, $\bm{k}_{5}=+k_0\hat{z}$, and
$\bm{k}_{6}=-k_0\hat{z}$. The corresponding $\eta_{ij}$ are conveniently
written as 6-dimensional vectors. Each hydrogenic envelope has six
different valley-split states, classified by symmetry.  The lowest
energy state for Sb has $\epsilon_1= -45$~meV with A1 symmetry,
$\bm{\eta}_{A1}=\frac{1}{\sqrt{6}}(1,1,1,1,1,1)$.\cite{grimeiss82}
This is followed by a three-fold degenerate T1 symmetry level, with
energy $\epsilon_2=-33$~meV and
$\bm{\eta}_{T1}=\frac{1}{\sqrt{2}}(1,-1,0,0,0,0)$,
$\frac{1}{\sqrt{2}}(0,0,1,-1,0,0)$,
$\frac{1}{\sqrt{2}}(0,0,0,0,1,-1)$. Note that only the last state
couples to the (001) subbands. The highest energy level has E symmetry
with energy $\epsilon_3=-31$~meV and is two-fold degenerate,
$\bm{\eta}_{E}=\frac{1}{2}(1,1,-1,-1,0,0)$, or
$\frac{1}{2}(1,1,0,0,-1,-1)$. Again, only one of these couples to
electrons in the lowest energy subband. For principal quantum number
$n\geq 2$ we assumed $\epsilon_n=\epsilon_1/n^{2}$ with
degenerecence $6n^2$. Finally, we used $U=43$~meV for the on-site
Coulomb repulsion.  This corresponds to a binding energy of
$E_{D-}=-(\epsilon_1+U)=2$~meV; this is quite similar to the binding
energy of $1.6$~meV measured for phosphorous (P) in silicon,\cite{narita85}
and calculated by Oliveira and Falicov.\cite{oliveira86} We are not
aware of measurements of $E_{D-}$ for Sb impurities.

One important point is that the overlap integral between conduction electrons
in the $l$-th valley and the impurity
electron $S_{kl,i}=\langle \Psi_{kl}| \Phi_i\rangle$ is strongly
oscillatory on R. For notational convenience, define $s_1= \cos{(k_0 R)}$, and
$s_2=\sin{(k_0 R)}$, and set
$s_3\equiv s_1$. After a simple calculation we see that
$S_{kl,i}\propto s_{l}$ apart from a smooth envelope whenever
the valley symmetry of donor orbital $i$ is ``even''
($\propto\cos{[k_0(z-R)]}$), and $S_{kl,i}\propto s_{l+1}$ when the
valley symmetry of orbital $i$ is ``odd'' ($\propto\sin{[k_0(z-R)]}$).
These oscillations occur due to valley interference,
in a similar fashion as the exchange oscillations between two donor
impurities in silicon.\cite{cullis70,koiller01,wellard05}

\subsection{Contributions to the scattering amplitude}

We start by evaluating explicitly the first order contributions to
Eq.~(\ref{ast}). To get an idea of the order of magnitude of each
contribution, we will quote numerical values for a single donor
located at $R=6 z_0$, with parameters $A=(1\mu \rm{m})^{2}$,
$\epsilon_{F}=-9.32$~meV,
$\epsilon_{\rm{sb},1}=\epsilon_{\rm{sb},2}=-10$~meV, and $z_0=40$~\AA.
In this case the overlap integral between conduction electron at
$\epsilon_F$ and donor ground state was $S_{k,1}s_1$, with
$S_{k,1}=1.31\times 10^{-4}$.

Let's consider the first order contribution to Eq.~(\ref{ast}).
Using ${\cal H}_{0}|\Psi_{kl}\rangle = \epsilon_{kl} |\Psi_{kl}\rangle$ and $[{\cal
H}_{0}+{\cal C}_{R}]|\Phi\rangle = \epsilon_1 |\Phi\rangle$, we
get
\begin{eqnarray}
\langle \Psi^{S/T}_{kl} | {\cal H}|\Psi^{S/T}_{k'l'}\rangle &=&
\langle \Psi_{kl}| {\cal C}_{R}|\Psi_{k'l'}\rangle +\langle \Psi_{kl}\Phi| {\cal C}_{ee}|\Psi_{k'l'}\Phi\rangle
\nonumber\\
&\pm& \left[
(\epsilon_1 -\epsilon_{F})
S_{k,1}S^{*}_{k',1}s_ls_{l'}\right.\nonumber\\
&& + \langle \Phi |{\cal C}_{R}| \Psi_{k'l'}\rangle
S_{k,1}s_{l'}\nonumber\\&& \left.
+ \langle \Phi \Psi_{kl}|{\cal C}_{12} |\Psi_{k'l'}\Phi\rangle
\right].
\end{eqnarray}
The first contribution on the right hand side is attractive Coulomb
potential scattering (always negative), and the second contribution is
repulsive scattering from the donor electron cloud. This has the same
order of magnitude as the first term, but the opposite sign.  Assuming
$s_l=s_{l'}=1$ (maximum value) these terms are respectively
$-8.1$~neV, and $+8.5$~neV.  The first term inside the brackets equals
$-0.61$~neV, the second $-0.3$~neV, and the third $+0.2$~neV.  Hence
we have $\langle \Psi^{S}_{kl} | {\cal
  H}|\Psi^{S}_{k'l'}\rangle=-0.3$~neV, and $\langle \Psi^{T}_{kl} |
{\cal H}|\Psi^{T}_{k'l'}\rangle=+1.1$~neV. \emph{Interestingly, in
  this particular case the first order contribution to Eq.~(\ref{ast})
  favors triplet scattering}. Actually, computations for different
donor positions $R$ shows that the first order contribution changes
sign several times as the donor position $R$ is varied.

We now turn to second order contributions to Eq.~(\ref{ast}). We
remark that these are always negative for singlets and for triplets
because $\epsilon_1<\epsilon_F < \epsilon_1 +U < \epsilon_i+U$ for
$i=2,3,\ldots$.  The largest individual contribution is a virtual
transition to a negatively ionized donor state with two electrons
occupying the donor ground state. This contributes exclusively to
singlet scattering. Using Eq.~(\ref{molorb}) we get
\begin{equation}
\frac{\langle \Psi^{S}_{kl}| {\cal H}| 1^{S}\rangle
\langle 1^{S}| {\cal H}| \Psi^{S}_{k'l'}\rangle}{\epsilon_{kl}-\epsilon_1-U}=
\frac{-2(2\epsilon_1 +U -2\epsilon_F)^2\left|S_{k,1}\right|^2 s_l s_{l'}
}{\epsilon_1+U-\epsilon_F},\qquad\phantom{.}
\label{large2nd}
\end{equation}
which equals $-3.7$~neV for the particular case considered. This is
more than $3$ times larger than the 1st order contribution mentioned
above. We checked several other parameter regimes, and found
Eq.~(\ref{large2nd}) to be 2-10 times larger than the 1st order
contribution (singlet or triplet).

All the other second order terms contribute equally to singlet and
triplet. The largest of these involve virtual excited states to the
$n=1$ state with T1 or E symmetry. The T1
state contribution is
\begin{equation}
-3\frac{(\epsilon_1+\epsilon_4+U-2\epsilon_F)^2\left|S_{k,1}\right|^2 s_{l+1} s_{l'+1}}
{\epsilon_4+U-\epsilon_F}=-0.95~\rm{neV},
\end{equation}
and the E contribution is
\begin{equation}
-\frac{3}{2}\frac{(\epsilon_1+\epsilon_6+U-2\epsilon_F)^2\left|S_{k,1}\right|^2 s_{l} s_{l'}}
{\epsilon_6+U-\epsilon_F}=-0.47~\rm{neV}.
\end{equation}

The contributions with principal quantum number $n\geq 2$ may be lumped together in a sum:
\begin{eqnarray}
-\sum_{n=2}^{\infty}\Lambda_n &=& -\sum_{n=2}^{\infty} n^2
\frac{(\epsilon_1 +\epsilon_n +U -2 \epsilon_F)^2}
{\epsilon_n+U-\epsilon_F}\left|S_{k,n}\right|^2 \nonumber\\
&&\times\left(\frac{5}{2}s_l s_{l'}+3s_{l+1}s_{l'+1}\right)
\label{nmax}\\
&\approx& -1.3 \times 10^4 \;\rm{neV},\nonumber
\end{eqnarray}
since there are $3n^2$ orbitals coupling to the 2DEG valleys, each
with energy $\epsilon_n =
\epsilon_1/n^2$. The envelope of the overlap integrals is defined as
\begin{equation}
S_{k,n} = \frac{1}{\sqrt{3}}\int d^{3}r \phi(z)
\frac{\textrm{e}^{i\bm{k}\cdot \bm{r}_{\perp}}}{\sqrt{A}}F_{n}(\bm{r}-\bm{R}).
\end{equation}
We get $\Lambda_2 = 390$~neV, $\Lambda_3 =
6170$~neV, $\Lambda_4 = 4164$~neV, $\Lambda_5 = 1367$~neV, $\Lambda_6
= 425$~neV, $\Lambda_8 = 50$~neV. The $\Lambda_n$ reach a maximum for
$n=3$, and then decrease appreciably with increasing $n$ because the
$S_{k,n}$'s become exponentially small.  As a result, we get a good
approximation by evaluating the sum up to $n_{\rm{max}}=4$. We found
that the contributions for $n\geq 5$ produce negligible changes to our
final result.

Hence the sum of the second order contributions in Eq.~(\ref{ast}) can
be as much as $10^4$ times larger than the first order contributions.
It is interesting to note that the difference between singlet and
triplet rates is determined by the \emph{difference between squared
  amplitudes}: $\frac{1}{\tau_S}-\frac{1}{\tau_T}\propto ({\cal
  A}^{S}_{kl,k'l'})^{2}-({\cal A}^{T}_{kl,k'l'})^{2}$, which is of the
order of $10^4$~neV$^{2}$.  This is several orders of magnitude larger
than the square of the exchange scattering amplitude, that determines
exchange scattering: $\frac{1}{T_1}\propto({\cal
  A}^{T}_{kl,k'l'}-{\cal A}^{S}_{kl,k'l'})^{2}$ is only about $\sim
10$~neV$^{2}$. Therefore, we see that exchange scattering is quite
different than SDS, in the sense that the latter benefits from a large
number of additional electronic transitions.

\subsection{Dependence of EDMR parameters on donor depth}

We now show explicit numerical calculations of the parameters
determining EDMR detection, and discuss their dependence with donor
depth $R$. The figure of merit $\alpha$ is shown in
Fig.~\ref{alphafig}, and the donor spin flip rate $\frac{1}{T_1}$ due
to the exchange scattering mechanism is shown in Fig.~\ref{t1rate}.
Both $\alpha$ and $\frac{1}{T_1}$ decrease appreciably as $R$
increases relative to the 2DEG thickness $z_0$, and are quite
sensitive to $\epsilon_F-\epsilon_{\rm{sb},l}$ (or, equivalently, the
2DEG area density). We used
$\epsilon_F-\epsilon_{\rm{sb,1}}=0.68$~meV, and considered two
different cases: Valley degenerate with
$\epsilon_{\rm{sb,1}}=\epsilon_{\rm{sb},2}=-25$~meV, and valley-split
with $\epsilon_{\rm{sb},2}=-15$~meV and
$\epsilon_{\rm{sb},1}=-25$~meV.  Both $\alpha$ and $\frac{1}{T_1}$ are
independent of the 2DEG area $A$; $\alpha$ depends sensitively on the
scattering time due to other sources; we used $\tau_0=0.4$~ps which is
equivalent to a typical transistor mobility of $4\times
10^{3}$~cm$^{2}/(\rm{Vs})$ ($\frac{1}{T_1}$ is independent of
$\tau_0$). We assumed $T=5$~K and $B=0.36$~T ($\alpha$ does not depend
on $B$, and is nearly independent of $T$ in the valley degenerate
case).

Fig.~\ref{t1rate} shows that exchange scattering is several orders of
magnitude stronger than conventional spin-phonon coupling of isolated
donor impurities. For example, Feher and Gere\cite{feher59} measured
$\frac{1}{T_1}\sim 10^{-1}$~s$^{-1}$ for phosphorous donors in bulk
silicon at $T=5$~K and $B=0.36$~T.

Interestingly, the EDMR amplitude depends on whether the device
temperature is larger than the valley-splitting energy or not.  At
large temperatures (or small valley splittings), $\alpha$ and
$\frac{1}{T_1}$ are smooth functions of $R$ (apart from tiny
oscillations), because contributions from the two subbands complement
each other (dark curves). For the opposite regime of temperature lower
than valley-splitting, the parameters are strongly oscillatory on $R$
(grey curves). In this case $\frac{1}{T_1}$ becomes quite small
for some donor positions close to the interface, suggesting that a
fraction of the donors located at ``lucky sites'' might be detectable
by EDMR. In Figs.~\ref{alphafig}~and~\ref{t1rate} we used filled
circles to denote the actual silicon lattice sites that may be
occupied by substitutional donors.

\begin{figure}
\includegraphics[angle=-90,width=3in]{alpha}
\caption{Figure of merit $\alpha$ as a function of donor depth $R$
  divided by the 2DEG thickness $z_0$. Dark curve: Valley degenerate
  case. Grey curve: Valley split case, with
  $\epsilon_{\rm{sb},2}-\epsilon_{\rm{sb},1}\gg k_B T$. Filled circles
  denote actual sites of the diamond lattice, which may be occupied by
  donors.\label{alphafig}}
\end{figure}

\begin{figure}
\includegraphics[angle=-90,width=3in]{t1rate}
\caption{Donor impurity exchange scattering rate $\frac{1}{T_1}$ as a
  function of donor depth $R/z_0$ (same parameters as in
  Fig.~\ref{alphafig}).\label{t1rate}}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{c |c |c | c }
 &    & Experiment & Theory \\
$n_{\rm{2DEG}}$ ($10^{11} \rm{cm}^{-2}$) &  $B_{\perp}$ (G) &  $\frac{|\Delta
I|}{I_0}$ ($10^{-8}$)  &  $\frac{|\Delta I|}{I_0}$ ($10^{-8}$) \\
\hline
1.1  &  0.3  & 10 & 5.6 \\
2.2 &  0.3  & 4.0 & 2.2 \\
3.8  & 0.3   & 2.0  & 0.5 \\
\hline
2.2 &  0.07  & 1.0 & 0.5 \\
2.2 &  0.14  & 2.5 & 1.0 \\
2.2 & 0.19  & 3.5  & 1.4 \\
2.2 & 0.28   & 4.0  & 2.1 \\
2.2 & 0.55   & 5.0  & 3.8 \\
\hline
\end{tabular}
\caption{Comparison between theory and experiment\cite{lo07} at
  $T=5$~K and $B=0.36$~T with an inhomogeneously broadened linewidth
  of $\frac{1}{\gamma_e T_{2}^{*}}=2$~G.  Our theoretical calculations of
$\Delta I/I_0$ were obtained by multiplying the donor density by the
scattering rate due to a single donor, and integrating over donor depth.  We
used 2DEG thickness $z_0=50$~\AA, valley-degenerate
  subband energy $\epsilon_{\rm{sb},1}=\epsilon_{\rm{sb},2}=-25$~meV (zero energy is at flat
  band), and transistor mobility $4\times 10^3$~cm$^2$/Vs (corresponds
  to $\tau_0=0.4$~ps).}
\label{table1}
\end{center}
\end{table}

\subsection{Comparison to experiment}

Our theory can be directly compared to experiments. Ghosh and
Silsbee\cite{ghosh92} measured $\Delta
I/I_0=(I_{\rm{off~res.}}-I_{\rm{at~res.}})/I_0 \sim - 10^{-5}$ at high
power (the current increased upon resonance), while at lower power
they found that $\Delta I$ changed sign.  In this study, a silicon
transistor bulk doped with phosphorous impurities was used.  Lo {\it
  et al.}\cite{lo07} measured much lower amplitudes $|\Delta I|/I_0=
10^{-8}-10^{-7}$, with antimony donors \emph{implanted only into the
transistor channel}. Unfortunately the use of derivative detection did
not allow the determination of the sign of $\Delta I$.

Our theory disagrees in sign with the measurements of
[\onlinecite{ghosh92}], suggesting that their high power signal was
not due to the SDS mechanism, instead it was likely due to the 2DEG
heating mechanism.\cite{stein83}

We now compare our explicit numerical results with the experimental
data of Lo {\it et al}.\cite{lo07} Our theory allows the prediction of
the device current for a given donor distribution \emph{without any
  fitting parameters}.  Denote $N_j=A\Delta z \;n_d(z_j)$ the number
of donor impurities located in the interval $\Delta z=z_{j+1}-z_{j}$,
with $n_d(z_j)$ the volume density of donors in the $\Delta z$-thick
layer. The EDMR signal averaged over the donor profile
is simply given by
\begin{equation}
\frac{I-I_0}{I_0} = p_c\sum_j N_j\alpha(z_j)p_i(z_j).
\label{iavg}
\end{equation}
We obtained the depth dependent donor density $n_d(z)$ from Secondary
Ion Mass Spectroscopy measurements (SIMS).  Our explicit model
calculations of the EDMR current are shown in Table~\ref{table1} for
all measurements made in [\onlinecite{lo07}].  The EDMR response
predicted by theory is in reasonable agreement with experiment.

\subsection{Physical optimization of EDMR}

\begin{figure}
\includegraphics[angle=-90,width=3in]{current_z}
\caption{EDMR amplitude at $T=5$~K as a function of donor placement,
  for zero valley-splitting (dark curve) and for
  valley-splitting equal to $100$~K (oscillating gray curve, with
  filled circles denoting actual donor sites of the diamond
  lattice).\label{current_z}}
\end{figure}

Table~\ref{table1} shows that the SDS contribution to EDMR is much
weaker than anticipated on the basis of previous
measurements.\cite{ghosh92} We now describe a physical optimization
that aims at maximizing $|\Delta I|/I_0$ for the minimal number of
donor impurities.

The SDS parameters $\alpha$ and $\frac{1}{T_1}$ decrease rapidly with
increasing donor depth $R$, an effect that has important implications
for the optimization of the EDMR amplitude. Notably, for given
$B_{\perp}$, the EDMR signal will be maximal for donors satisfying
$\frac{1}{T_1}(R^*)\approx T_{2}^{*}(\gamma_e B_{\perp})^{2}$.  The location
$R^*$ is the closest one to the interface that satisfies
the saturation condition $(\gamma_e B_{\perp})^{2}\gtrsim
1/(T_1T^{*}_{2})$.

Fig.~\ref{current_z} shows the EDMR amplitude as a function of donor
location, for a \emph{single donor implanted in the center of a transistor of
area
$0.1(\mu\rm{m})^{2}$}.  We assumed microwave amplitude $B_{\perp}=
0.3$~G, 2DEG density $1.1\times 10^{11}$~cm$^{2}$, with other
parameters as in Table I.

Interestingly, the EDMR amplitude depends on whether the device
temperature is larger than the valley-splitting energy or not.  At
large temperatures, $|\Delta I|/I_0$ is a smooth function of $R$,
because contributions from the two subbands complement each other. In
this case a single donor placed at the sweet spot $R^*\approx 5.5~z_0$
is optimally detected by EDMR (dark curve in Fig.~\ref{current_z}, with
$\epsilon_{\rm{sb1}}=\epsilon_{\rm{sb2}}=-25$~meV).  For the opposite
regime of temperature lower than valley-splitting, $|\Delta I|/I_0$ is
strongly oscillatory on $R$, and donors closer to the interface
can be detected as well (oscillating gray curve in Fig.~\ref{current_z}, with
$\epsilon_{\rm{sb1}}=-25$~meV and $\epsilon_{\rm{sb2}}=-15$~meV).

Fig.~\ref{current_z} shows that EDMR is able to detect a single donor spin
implanted in a transistor of area $0.1 (\mu\rm{m})^2$. For
typical $I_0\sim 5$~$\mu$A, current modulations $\Delta I\sim 0.1$~pA
are detectable with standard techniques, \emph{provided the donor is
  placed at the sweet spot $R^*$}. A single donor resonance may be
identified by the presence of only one hyperfine satellite line out of
the full satellite spectrum with 2I+1 lines (for donor nuclear spin
I), for measurements within the nuclear spin-flip time.\cite{sarovar07}

The EDMR amplitude is directly proportional to the donor area density
per monolayer.  Therefore, if $6\times 10^6$ Sb donors were to be
placed exactly at the sweet spot $R^*$ in a large transistor of area
$\sim 10^3 (\mu\rm{m})^{2}$, the signal would be $|\Delta I|/I_0\sim
10^{-5}$, two orders of magnitude higher than in [\onlinecite{lo07}].

\section{Kondo temperature and validity of perturbation theory\label{kondosection}}

Our approach is based on perturbation theory
[Eq.~(\ref{ast})]. This is known to be a good approximation only when the temperature
is larger than the characteristic Kondo temperature $T_{\rm{Kondo}}$.
For our problem, the Kondo temperature may be written as\cite{haldane78}
\begin{equation}
k_B T_{\rm{Kondo}}=\sqrt{\left(\epsilon_1+U-\epsilon_F\right)
\left(\epsilon_F-\epsilon_{\rm{sb}}\right)\left|J\rho\right|}\textrm{e}^{-\frac{1}{|J\rho|}},
\label{kondo}
\end{equation}
where $J=\langle J_{kl,k'l'}\rangle$ is an average exchange
energy at the Fermi level (averaged over subbands), and
$\epsilon_{\rm{sb}}$ is taken as the lowest subband energy.
Eq.~(\ref{kondo}) applies when $\epsilon_1+U>\epsilon_F$ and
$\epsilon_1<\epsilon_{\rm{sb}}<\epsilon_F$.  The first term in the
square root is a characteristic particle excitation bandwidth for
electrons tunneling into the donor, while the second term is a
hole excitation bandwidth.

Fig.~\ref{kondofig} shows our calculated Kondo temperature as a
function of donor depth. Here we see that for $R>3 z_0$, we have
$T_{\rm{Kondo}}\ll 0.1$~K. Hence, even at the lowest temperatures
achievable experimentally, there is no Kondo effect for donors located
at $R>3 z_0$ ($T\gg T_{\rm{Kondo}}$).

In the valley degenerate regime, only donors around $R\sim 5.5 z_0$
can be detected by EDMR, as shown in Fig.~\ref{current_z} (donors
closer to the interface can be detected at higher power).  Hence we
see that donors located in this region satisfy $T\gg T_{\rm{Kondo}}$
for the lowest temperatures achievable in the laboratory, and our
perturbation theory approach is justified. In the valley degenerate
case, both the EDMR current as well as the Kondo temperature are
strongly oscillatory with donor depth, and again it can be seen that
all donors that are detectable by EDMR have extremely low
$T_{\rm{Kondo}}$. Note that the behavior of $T_{\rm{Kondo}}$ as a
function of donor depth is qualitatively similar to exchange
scattering; as a consequence, all donors with $\frac{1}{T_1}$ low
enough to be detectable by EDMR have a corresponding quite small
$T_{\rm{Kondo}}$.

\begin{figure}
\includegraphics[angle=-90,width=3in]{kondo}
\caption{Kondo temperature as a function of donor depth $R/z_0$ (same
  parameters as in Fig.~\ref{alphafig}). Note that the Kondo temperature
  is exponentially small for $R>3 z_0$; this justifies our
  perturbative approach.\label{kondofig}}
\end{figure}

\section{Spin injection \label{spininjec}}

We now discuss the spin injection regime. Under spin injection, each
spin species have different quasi-Fermi energies, $\epsilon_{F\uparrow}\neq
\epsilon_{F\downarrow}$.\cite{zutic04}
For $k_B T\ll
\rm{Min}\{\epsilon_{F\sigma}-\epsilon_{\rm{sb,l}}\}$, the carrier
polarization is well approximated by
\begin{equation}
p_c =\frac{n_{\uparrow}-n_{\downarrow}}{n_{\uparrow}+n_{\downarrow}}
\approx\frac{2\left(\epsilon_{F\uparrow}-\epsilon_{F\downarrow}+g\mu
B\right)}{\sum_l \left(\epsilon_{F\uparrow}
+\epsilon_{F\downarrow}-2\epsilon_{\rm{sb,l}}\right)}.
\end{equation}
Here $n_{\sigma}=\int \rho d\epsilon f_{\sigma}(\epsilon)$ is the 2DEG
area density for each spin subband.

The impurity scattering times
$\tau_{S/T}$ must now be calculated for each spin subband:
\begin{equation}
\left\langle \tau_{S/T,l}\right\rangle_{\sigma} = \frac{\int \rho
  d\epsilon \left(\epsilon-\epsilon_{\rm{sb},l}+\sigma\frac{g\mu
  B}{2}\right)\left(-\frac{\partial f_{\sigma}}{\partial
  \epsilon}\right)\tau_{S/T,kl}}
{\int \rho d\epsilon
  \left(\epsilon-\epsilon_{\rm{sb},l}
+\sigma\frac{g\mu B}{2}\right)\left(-\frac{\partial f_{\sigma}}
{\partial \epsilon}\right)},
\end{equation}
where we added a spin subscript to the scattering times and the Fermi functions.
The energy integrals are from
$(\epsilon_{\rm{sb},l}-\sigma\frac{g\mu B}{2})$ to $\infty$. The current
is now calculated as in Eq.~(\ref{mu}), using $p_S=(1-p_i)/4$ for the
spin up subband, and $p_S=(1+p_i)/4$ for the spin down subband, with
$p_T=1-p_S$ in each case. In addition, each $\langle
\tau_{S/T}\rangle_{\sigma}$ must be multiplied by its
corresponding density $n_{\sigma}$.
The source-drain current becomes
\begin{equation}
\frac{I-I_0}{I_0}=\frac{\sum_{l}n_{\uparrow}\left(\langle
    \tau_{Tl}\rangle_\uparrow-\langle\tau_{Sl}\rangle_\uparrow\right)
-n_{\downarrow}\left(\langle
    \tau_{Tl}\rangle_\downarrow-\langle\tau_{Sl}\rangle_\downarrow\right)}
{\sum_{l}n_{\uparrow}\left(3\langle
    \tau_{Tl}\rangle_\uparrow+\langle\tau_{Sl}\rangle_\uparrow\right)
+n_{\downarrow}\left(3\langle
    \tau_{Tl}\rangle_\downarrow+\langle\tau_{Sl}\rangle_\downarrow\right)}p_i.
\label{dioiinj}
\end{equation}
Interestingly, under spin injection the scattering times are
intertwined with the densities $n_{\sigma}$, and a figure of merit
$\alpha$ independent of $p_c$ can not be defined.  Note that when
$\langle \tau_{S/T}\rangle_{\uparrow}=\langle
\tau_{S/T}\rangle_{\downarrow}$, Eq.~(\ref{dioiinj}) becomes $\Delta
I/I_0 = \alpha p_c p_i$, as obtained previously.

The donor spin transition rates are also modified. Following from
Eq.~(\ref{gammasigma}), $\Gamma_\sigma\propto \int d\epsilon
f_{-\sigma}(\epsilon)[1-f_{\sigma}(\epsilon -\sigma \frac{g\mu
  B}{2})]$, leading to
\begin{eqnarray}
\frac{1}{T_1}&=& \frac{2\pi}{\hbar}\rho^{2}
\left\langle|J_{kl,k'l'}|^2\right\rangle
\left(\epsilon_{F\uparrow}-\epsilon_{F\downarrow}+g\mu B\right)\nonumber\\
&&\times\coth{\left(\frac{\epsilon_{F\uparrow}-\epsilon_{F\downarrow}+g\mu B}{2k_BT}\right)}.
\label{t1inj}
\end{eqnarray}
The steady state impurity
polarization becomes
\begin{equation}
p_{i0}=\tanh{\left(\frac{\epsilon_{F\uparrow}-\epsilon_{F\downarrow}+g\mu
      B}{2k_B T}\right)}.
\label{pi0inj}
\end{equation}

Fig.~\ref{current_pc} shows EDMR lineshapes under spin injection,
using $T=5$~K, $B_{\perp}=0.3$~G, and doping profile and other
parameters similar to [\onlinecite{lo07}].  The 2DEG density is fixed
at $2\times 10^{11}$~cm$^{-2}$, and the conduction electron spin
polarization is varied between 2\% and 75\%. At low $p_c$, the EDMR
amplitude scales as $p_c p_{i0}\propto p_{c}^{2}$, since $\alpha$
remains unchanged and $p_{i0}$ scales proportional to $p_c$ due to
exchange scattering [Eq.~(\ref{pi0inj})].  In this regime, EDMR can be
used as a local probe of carrier spin polarization.

\begin{figure}
\includegraphics[angle=-90,width=3in]{current_pc}
\caption{EDMR lineshape under spin injection, for carrier spin polarization
  $p_c=2,6,10,20,30$\%.  Inset: EDMR amplitude at resonance, as a
  function of $p_c$.\label{current_pc}}
\end{figure}

However, as $p_c$ is increased beyond the threshold
\begin{equation}
p_T = \frac{\textrm{Max} \{2k_BT,g\mu B\}}
{(\epsilon_{F\uparrow}+\epsilon_{F\downarrow})-(\epsilon_{\rm{sb1}}+\epsilon_{\rm{sb2}})},
\end{equation}
the spin-flip rate starts increasing as $\frac{1}{T_1}\propto p_c$,
because any donor spin antiparallel to the 2DEG spins will relax
rapidly [note that Eq.~(\ref{t1inj})
becomes proportional to $p_c$ when $p_c>p_T$].  By virtue of these
larger $\frac{1}{T_1}$'s, EDMR will excite donors further away from
the interface, tending to decrease $\Delta I/I_0$.  Remarkably,
this effect competes against the $p_{c}^{2}$ scaling, saturating
$\Delta I/I_0$ for $p_c\gg p_T$. While this limits EDMR as a probe of
carrier spin for $p_c\lesssim p_T$, it also demonstrates that EDMR
detection is optimal at $p_c\approx p_T$.

\section{Conclusions \label{conclusions}}

In conclusion, we presented a microscopic theory of spin-dependent
scattering in the interaction of conduction electrons with neutral
donor atoms. Our results are based on an effective mass approximation.
More sophisticated approaches are likely to reduce the valley
oscillations,\cite{wellard05} with no modification to our conclusions.
The considered mechanism requires temperatures lower than the impurity
binding energy ($T<100$~K), but even higher temperatures may be
achieved using deep level magnetic atoms or clusters.

We showed that SDS is determined by virtual transitions into doubly
occupied donor states. As a result, SDS always leads to a reduced
current upon EDMR saturation, since $\alpha>0$.

A recent experiment\cite{beveren08} demonstrates that the EDMR current
due to SDS is indeed reduced upon donor spin saturation (Fig.~2(c) in
[\onlinecite{beveren08}]), in agreement with our proposed virtual
transition mechanism.

Spin-dependent scattering detection is challenging due to competing
heating effects,\cite{ghosh92} which are directly proportional to the
number of donors present.  Our finding that SDS arises solely from
impurity spins located within a narrow depth window with respect to
the 2DEG shows a path to significant optimization of spin signal
intensity for a minimal number of donors placed into this depth
window, and underpins the development of single spin readout devices.

Our theory shows how the EDMR amplitude will scale with carrier spin
polarization in the regime of spin
injection.\cite{zutic06,jonker07,appelbaum07} Therefore, the
monitoring of donor electron spin resonances can be utilized for the
spatially resolved characterization of conduction electron spin
polarization, providing a sensitive probe for optimization of spin
injection and spin transport in semiconductors with indirect band gap
and weak spin-orbit coupling.

We thank M. Friesen, T. Schenkel, A.M. Tyryshkin, and I.
\v{Z}uti\'{c} for a careful reading of the manuscript, and A.L. Efros,
X. Hu, B. Koiller, S.A.  Lyon, I.  Martin, J.E. Moore, and A.G.
Petukhov for useful discussions.  RdS acknowledges support from NSERC
and the UVic Faculty of Sciences; CCL and JB acknowledge support from
WIN and NSA.

redpajama_set_name: RedPajamaArXiv


text:
Well Isn't This Interesting...?
While many of the world's top players packed up their suitcases to head Down Under in preparation for the Australian Open, a couple of the most elite stayed in the Western hemisphere to fight amid the desert in Doha. And with both Rafael Nadal and Roger Federer entered in the ExxonMobil Open in Qatar, us fans started to get excited about the possibility of another match-up between the two greats.
We'll have to wait a bit longer.
For most of the week the seedings in Doha played out as expected. Seven of the eight top players made the quarters and the first four all reached the semis -- Rafa and Roger were joined by Frenchman Jo-Wilfried Tsonga and defending champ Nikolay Davydenko. But despite their rankings, the ascent of the latter two was a bit surprising -- a knee injury took Tsonga out of contention after Monpellier while a nagging wrist fracture derailed what should have been a strong 2010 for the Russian and dropped him out of the top twenty for the first time in almost six years.
A promising run for Tsonga ended Friday when Federer took their semifinal match in less than an hour and a half. But Davydenko delivered the shock of the tournament a few hours later when he took on world #1 Nadal. He capitalized on weak serving by the Spaniard and broke his opponent's serve an amazing five times. He built a 5-0 lead in the second set and, though he allowed an ailing Rafa to get one break back and four more chances to even the set in the ninth game, the Russian eventually closed out the match in under ninety minutes.
It was the second straight time Davydenko had beaten Nadal in Doha -- they played the finals here last year -- and the third win in a row he's notched against the Spaniard. To successfully defend his title, of course, he still has to battle through a renewed Roger, though he's pulled off the one-two punch of beating both champions before -- and if he's healthy, he could very well do it again.
But whatever the result tomorrow, Nikolay has certainly put himself back on the radar for the Australian Open, which kicks off in just over a week. Ranked well below his ability, he could be in a position to pull off quite a few upsets and wreak havoc with the draw.
And I don't think anyone will be comfortable with Nikolay Davydenko in his path.
Labels: Doha, Jo-Wilfried Tsonga, Nikolay Davydenko, Rafael Nadal, Roger Federer
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