# Towards a measurement theory in QFT: “Impossible” quantum measurements are
possible but not ideal

Nicolas Gisin  Group of Applied Physics, University of Geneva, 1211 Geneva,
Switzerland  Constructor University, Geneva, Switzerland  Flavio Del Santo
Group of Applied Physics, University of Geneva, 1211 Geneva, Switzerland
Constructor University, Geneva, Switzerland

###### Abstract

Naive attempts to put together relativity and quantum measurements lead to
signaling between space-like separated regions. In QFT, these are known as
impossible measurements. We show that the same problem arises in non-
relativistic quantum physics, where joint nonlocal measurements (i.e., between
systems kept spatially separated) in general lead to signaling, while one
would expect no-signaling (based for instance on the principle of no-
nonphysical communication). This raises the question: Which nonlocal quantum
measurements are physically possible? We review and develop further a non-
relativistic quantum information approach developed independently of the
impossible measurements in QFT, and show that these two have been addressing
virtually the same problem. The non-relativistic solution shows that all
nonlocal measurements are localizable (i.e., they can be carried out at a
distance without violating no-signaling) but they (i) may require arbitrarily
large entangled resources and (ii) cannot in general be ideal, i.e., are not
immediately reproducible. These considerations could help guide the
development of a complete theory of measurement in QFT.

##  I Introduction

Like every empirical science, physics owns its success to the possibility of
inquiring nature, i.e., performing measurements. Although we are about to
celebrate the centenary of quantum physics, the most empirically successful
theoretical framework ever, there are still fundamental problems when it comes
to understanding measurements. On the one hand, there is no consensus on the
solution of the measurement problem, i.e., what interactions constitute a
measurement and what physical mechanism (if any at all) determines the
appearance of a single outcome. (Note that producing an outcome in the form of
a piece of classical information is the main feature of a measurement). On the
other hand, and this will be the main focus of this paper, there is little
discussion on whether any theoretically acceptable quantum measurement
(including of nonlocal variables) can be physically implemented, namely, such
that the implementation does not imply the violation of any known physical
principles, like, e.g., the no-signaling principle.

The issue of which quantum measurements are physically possible was firstly
raised in the attempt of putting together quantum physics with relativity.
Intuitively the problems stem from the following tension. On the one hand, the
most fundamental principle of relativity states that signals cannot travel
faster than a maximum speed (incidentally that of light). Quantum
measurements, on the other hand, involve the “collapse” of the state vector at
the time of a measurement, which (when considering entangled states of
multipartite distant systems) seems to impose an instantaneous change of the
state, possibly in a space-like separated region. It so became manifest quite
early that relativity and quantum measurements do not easily coexist [1, 2, 3,
4, 5, 6].

The theory that extends quantum mechanics to a relativistic framework is
quantum field theory (QFT). Therein, all the problems with distant systems
seem solved by the assumption of microcausality, i.e., the algebras of
operators defined on any two space-like separated regions commute. However,
QFT still lacks to date a complete theory of measurement (i.e., one that
yields measurement outcomes and it is therefore able to explicitly model all
known quantum phenomena), an issue that has been called “a major scandal in
the foundations of quantum physics” [7]. One may then try to naively apply the
standard rules of quantum measurements in the context of QFT. However, it was
pointed out by Sorkin already in 1993 [8] that this leads to a violation of
no-signaling between space-like separated regions (provided that measurements
take place on finite regions of space-time). These measurements that violate
no-signaling are commonly referred to as impossible measurements (see section
II).

Figure 1: Spacetime diagram of Sorkin’s impossible measurement scenario in
QFT.

The problem of impossible measurements in QFT has gained interest in recent
years (for a historical review, see [9]) and two main approaches have been
proposed to solve this conundrum: The first one, pursued in Refs. [10, 11,
12], accepts that microcausality is not enough to single out the physical
measurements and puts forward further constraints. The second approach,
advanced in Refs. [13, 14, 15, 16], finally provides QFT with an explicit
measurement scheme (by coupling the quantum fields to a probe, which is in
turn another quantum field) such that Sorkin’s impossible measurements do not
seem possible to arise in the first place.

While these proposed solutions represent a valuable advancement in
understanding relativistic quantum measurements, they are completely rooted in
the formalism of QFT. Yet, it should be noted that impossible measurements are
not a characteristic problem of QFT, but rather have a more general scope. In
fact, also in non-relativistic quantum theory one may ask what nonlocal joint
measurements (i.e., performed on multipartite systems located at a distance
from each other) lead to signaling. Studying this problem in a non-
relativistic quantum framework is at the same time simpler and more general.
It is simpler because standard quantum mechanics is provided with a well-
established theory of measurement, and one can thus look for the limitations
that no-signaling imposes on the theoretically allowed measurements. It is
more general because no-signaling could be studied at a more abstract level,
in principle without resorting to any physical theory – in particular to
relativity – to motivate it. In fact, it is true that the principles of
relativity theory represent the most striking justification to impose no-
signaling between distant regions, but this is in the end only one possible
physical justification of why one would impose no-signaling. One can think of
different physical motivations, e.g., a thick wall (or any other impenetrable
potential barrier), or an interruption in the communication channel. These
scenarios exemplify the general principle of no-nonphysical communication,
which states that communication is possible only when information is encoded
into a physical carrier (a letter, a particle, a radio wave, etc.) which
physically transports the information from the sender to the receiver [17].

Interestingly, the problem of what nonlocal quantum measurements lead to
signaling (using a non-relativistic quantum information formalism) has already
been studied in a series of papers [18, 19, 20, 21, 22, 23, 24, 25, 26].
Therein it is shown that the ideal measurements (for definitions on different
types of measurements, see section II.1) of nonlocal variables (which do not
lead to signaling) are those that completely erase the information from the
local states. We borrow the useful terminology from Ref. [27], although with a
different specific meaning, and we call these measurements localizable.111In
Ref. [27], they originally called those measurements causal and localizable,
but we emphasize that if a measurement is localizable in the sense used here,
this implies that it does not signal. As we shall see (in section IV), these
measurements are almost never ideal, i.e., the updated post-measurement state
is in general not an eigenstate of the measurement operator and cannot
therefore be immediately reproduced.

Note that what Refs. [27] and [13] call a measurement actually yields no
outcomes. In the former, in particular, they merely diagonalize the density
matrix in the measurement eigenbasis, which is equivalent to carry out an
ideal measurement and ignoring the outcomes (this defines what is usually
called a quantum measurement channel). However, if a measurement is not ideal,
it does not in general define a quantum channel. Throughout this paper, we
will assume that a measurement is defined by yielding a single outcome with
the probability given by the Born rule (see section II.1 for specific
definitions).

In doing so, we will build on the series of results in Refs. [18, 19, 20, 21,
22]. These works, however, received relatively little attention and were
developed independently of Sorkin’s impossible measurements. Moreover, these
two theoretical frameworks – despite addressing incredibly similar, if not
exactly the same, problems – have never been connected so far. It is the aim
of this paper to review the main results of this non-relativistic quantum
information approach, developing further some of the proposed protocols, and
explicitly show how this satisfactorily addresses the problem of Sorkin’s
impossible measurements in the non-relativistic limit. This will allow us to
better understand the theory of nonlocal quantum measurements in general, and
hopefully to help guide the development of a complete theory of measurement in
QFT.

##  II Impossible measurements in QFT and in the non-relativistic limit

Let us start by restating Sorkin’s impossible measurement scenarios in QFT
[8]. Let $S$ (for Sender), $M$ (for Middle), and $R$ (for Receiver) be three
bounded regions of (Minkowski) spacetime, arranged as in Fig.1, each provided
with a local algebra $\mathfrak{U}(S)$, $\mathfrak{U}(M)$, and
$\mathfrak{U}(R)$, respectively. Since $S$ and $R$ are space-like separated,
microcausality imposes that all the operators of one region commute with the
ones of the other, i.e., $[\hat{S}_{i},\hat{R}_{j}]=0$,
$\forall\hat{S}_{i}\in\mathfrak{U}(S)$ and
$\forall\hat{R}_{j}\in\mathfrak{U}(R)$. This supposedly ensures that no
operation performed in region $S$ can signal to region $R$.

Figure 2: Non-relativistic limit of a Sorkin’s impossible measurement
scenario.

Let us now consider, however, the third region $M$, which is chosen to be
partly in the causal future of $S$ and partly in the causal past of
$R$.222Note that $M$ needs to be an extende region, i.e., to have a finite
size. For any such region, however, it is always possible to find two regions
$S$ and $R$ arranged in the way just described. Consider now the following
protocol. The Sender, situated in region $S$, applies to the state of the
field, $|{\psi_{0}}\rangle$, a “kick” operator, i.e., a unitary element of the
local algebra $\hat{S}(\gamma)\in\mathfrak{U}(S)$, which depends on the
parameter $\gamma$. The Middle observer in region $M$ performs a nonselective
measurement – i.e., one which also produces outcomes but they are ignored
(i.e., traced over) – of the observable
$\hat{\Omega}^{(M)}=\\{\hat{\Pi}^{(M)},\mathds{1}-\hat{\Pi}^{(M)}\\}\in\mathfrak{U}(M)$,
where $\hat{\Pi}^{(M)}$ is a projector. Finally, the Receiver, in region $R$,
measures the expectation value of an observable $\hat{R}\in\mathfrak{U}(R)$.
The application of the state-update rule after the measurement in $M$, yields
the result:

$\displaystyle\langle\hat{R}\rangle$ $\displaystyle=$
$\displaystyle\langle{\psi_{0}}|\hat{S}^{*}(\gamma)\hat{\Pi}^{(M)}\hat{R}\hat{\Pi}^{(M)}\hat{S}(\gamma)|{\psi_{0}}\rangle$
$\displaystyle+\langle{\psi_{0}}|\hat{S}^{*}(\lambda)(\mathds{1}-\hat{\Pi}^{(M)})\hat{R}(\mathds{1}-\hat{\Pi}^{(M)})\hat{S}(\gamma)|{\psi_{0}}\rangle.$

It is easy to show that, given the state-updated after the projective
measurement in $M$, in general $\langle\hat{R}\rangle$ depends on the “kick”
parameter $\gamma$ and therefore the Sender can signal to the Receiver thanks
to the intervention of the Middle observer’s measurement (see a simple example
in section III).

Let us turn to the non-relativistic scenario, as depicted in Fig. 2. In this
case, $S$, $M$, and $R$ are three separate laboratories, each provided with a
system associated to a suitable Hilbert space, $\mathcal{H}_{S}$,
$\mathcal{H}_{M}$, and $\mathcal{H}_{R}$, respectively. The systems in $S$ and
$R$ are completely arbitrary and each of them can be, in the simplest case, a
qubit (i.e., $\mathcal{H}_{S}=\mathbb{C}^{2}=\mathcal{H}_{R}$). However, we
model the fact that $M$ needs to be an extended region of space by it being
composed of two spatially separated qubits, $A$ and $B$, such that
$\mathcal{H}_{M}=\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Following step by step
the construction of Sorkin’s scenario in QFT, we allow, in this order, $S$ to
communicate with $A$, and $B$ to communicate with $R$, whereas no information
can be exchanged between $A$ and $B$. This resembles the relativistic case in
which $M$ partly overlaps with the future causal cone of $S$ and with the past
causal cone of $R$, allowing therefore the (ordered) communication.

We now want to investigate what operations can be performed without violating
the assumption of no-signaling between $S$ and $R$ (in QFT this is due to
space-like separation, but as explained in section I, one can motivate this
without resorting to spacetime with the principle of no-nonphysical
communication). Exactly in the same fashion as the relativistic scenario,
assume that at time $t_{0}$ the global state is $|{\psi_{0}}\rangle$ and the
Sender performs the local unitary “kick” $U^{(S)}$ on her system in $S$. At
time $t_{1}>t_{0}$, the Middle observer performs a joint nonselective
measurement $\Omega^{(M)}$ on the qubits $A$ and $B$. Finally, at
$t_{2}>t_{1}$, the Receiver measures a local observable $\Lambda^{(R)}$,
yielding the outcome $\lambda$. The no-signaling constraint is respected if
the application of $U^{(S)}$ cannot influence the measurement statistics at
$R$, i.e.,

$\displaystyle\text{Prob}(\Lambda^{(R)}=\lambda\text{ at
}t_{2}||{\psi_{0}}\rangle\text{ at }t_{0};\Omega^{(M)}\text{at }t_{1})=$ (2)
$\displaystyle\text{Prob}(\Lambda^{(R)}=\lambda\text{ at
}t_{2}|U^{(S)}|{\psi_{0}}\rangle\text{ at }t_{0};\Omega^{(M)}\text{at
}t_{1}).$

Like for Sorkin’s impossible measurements in QFT, it is straightforward to
show that the no-signaling condition in Eq. (2) is violated by a standard
measurement (see section III).

As we have already noted, both in the QFT and in the non-relativistic
descriptions, the communication between $S$ and $M$ and between $M$ and $R$
are actually allowed, therefore, nothing problematic can stem from there. So
the source of the problem fully lies in the (nonlocal) measurement performed
at $M$. Hence, before going into the main argument, let us sharpen some useful
definitions about measurements.

###  II.1 Definitions about measurements

To begin with, by measurement on a system prepared in a state $\rho$ we simply
mean a quantum process that (i) produces a piece of classical information
(outcome) corresponding to an eigenstate $a_{i}$ of the operator associated to
the measured observable $A$, and (ii) the probability to obtain $a_{i}$ is
given by the Born rule: ${\text{Prob}}(\Pi_{i})=\text{tr}(\rho\Pi_{i})$, where
$\Pi_{i}$ is the projector onto the $i$-th eigenspace of $A$. Note that a
quantum measurement is completed at the moment when the classical outcome is
produced, even if the readout for a single observer may require to collect
(through classical communication) different pieces of classical information.
We also emphasize that the way the state is updated after the measurement is
not part of the measurement itself.

Figure 3: General scheme on how to implement bipartite localizable
measurements with the aid of shared maximally entangled states (ebits) as a
resource. To reconstruct the measurement result of a nonlocal measurement, an
observer $O$ needs to put together the outcomes which are separately produced
in the local laboratories $A$ and $B$.

A measurement is then said to be ideal if, when carried out again after an
arbitrary small amount of time, it yields the same outcome $a_{i}$ with
probability 1. This happens if the measurement process updates the state as
(Lüders rule):
$\rho\rightarrow\rho^{\prime}=\frac{\Pi_{i}\rho\Pi_{i}}{\text{tr}(\rho\Pi_{i})}$,
i.e., the post-measurement state lies within the eigen-subspace corresponding
to the outcome. For non-degenerate operators, ideal measurements are
projective. Here, for simplicity, we identify ideal and projective
measurements. Note that in textbook quantum mechanics measurements are usually
assumed to be ideal, but we will see that this can be problematic.

It ought to be remarked that quantum measurements as defined by (i)-(ii) above
do not define any measurement channel. In fact, measurement channels also
require to have post-measurement states, in which case one merely ignores
(i.e., traces over) the measurement outcomes. Ideal measurement channels, in
particular, correspond to ideal measurements where one ignores the outcome and
only keeps the post-measurement states (with appropriate probability
amplitudes that satisfy the Born rule). In this way, ideal measurement
channels merely diagonalize the density matrix in the measurement basis but
yield no outcome, i.e, they are not properly measurements.333In Ref. [27], the
authors, by diagonalizing the density matrix to perform the “measurement” are
tacitly assuming an ideal state-update.

Degenerate measurements can be realized by merely grouping outcomes. However,
by doing so, an ideal non-degenerate measurement may turn into a non-ideal
degenerate measurement.

A measurement is said to be joint if it is carried out on two (or more)
subsystems, i.e., if the corresponding operator lives in the Hilbert space
composed of the tensor product of the Hilbert spaces describing each of the
subsystems.444Note that joint measurements formally also include product ones.
Despite the fact that product measurements can trivially be carried out at a
distance on their respective subspaces, for a single observer to know the
result one still needs to classically communicate the local outcomes. Non-
relativistic quantum mechanics on its own does not impose any limitations on
the possibility of measuring any observable, in our specific case, in the
laboratory $M$ – i.e., any Hermitian operator $\Omega$ living in
$\mathcal{H}_{M}$ – since the Hilbert space structure does not encompass the
spatial separation of the two qubits in $S$ and $R$.

In a real life experiment, to perform a joint measurement on both qubit
systems one would bring them together and locally measure both. However, in
order to claim that the region $M$ is an extended region, as it is necessary
for an impossible measurement scenario, we assume that the two qubits in $M$
cannot be brought together. A joint measurement in which the two subsystems
cannot be physically brought together to perform it is said to be nonlocal
(this is a physical constraint that is not encapsulated by the formalism of
quantum mechanics).

A nonlocal measurement is called localizable if it can be carried out by
quantum operations performed on each of the parties without the possibility of
signaling to each other (to reveal the result of the measurement one could
need additional classical communication, but the quantum operations that
determine the results can be locally performed at two arbitrarily close time
instants).555In relativity, this means that it exists a foliation such that
the two events are arbitrarily close in time, possibly simultaneous. We assume
that to perform nonlocal measurement the parties can resort to some preshared
resources, such as entanglement. A general scheme to implement a localizable
measurement is depicted in Fig. 3: The measurement of a nonlocal observable
with initial state $\psi_{i}$ is carried out indirectly by locally performing
quantum operations $\Omega_{A}$ and $\Omega_{B}$ on the qubits located at $A$
and $B$, respectively. This produces classical outcomes that, if subsequently
collected by an observer ($O$), unambiguously discriminate between the
eigenstates of the nonlocal observable, i.e., reveals the result of the
nonlocal measurement. We stress that all the discussions on nonlocal
measurements and their localizability are insensible to the application of
local unitaries on $A$ and $B$. Therefore in the remainder of this paper we
will always tacitly assume that the result apply on equivalent classes of
measurements up to local unitary transformations.

We have now all the ingredients to cast the main question about which
measurements are physically possible in this framework, that is: What nonlocal
joint measurements are localizable (i.e., can be implemented with local
operations on spatially separated systems without violating no-signaling)?

##  III An example of impossible measurements

As already remarked in section II, the whole problematic step of the
impossible measurement scenarios (both non-relativistically and in QFT) is the
measurement carried out in the intermediate extended region $M$, because in
any other step communication is allowed. Therefore, in the remainder of this
paper, we will focus on the measurements at the middle region $M$ only,
modeled, in the non-relativistic scenario, by the two separate qubits $A$ and
$B$.

Let us start by showing that ideal nonlocal measurements lead in general to
signaling. Following [20, 22], consider the observable $T$ whose eigenstates
are the orthogonal product states but twisted, i.e., they form the basis
$\\{|{00}\rangle,|{01}\rangle,|{1+}\rangle,|{1-}\rangle\\}$, where
$|{00}\rangle=|{0}\rangle_{A}\otimes|{0}\rangle_{B}$ etc., and
$|{\pm}\rangle=1/\sqrt{2}(|{0}\rangle\pm|{1}\rangle)$. Assume that the initial
state of the two qubits $A$ and $B$ is $\psi_{i}=|{00}\rangle$. Alice can then
decide whether to do nothing or to flip her qubit (i.e., to apply the “kick”
unitary $\sigma_{x}^{(A)})$, while Bob subsequently always measures in the
computational basis.

If Alice does nothing to her qubit, Bob finds the outcome “0” with probability
$1$. But when she applies $\sigma_{x}^{(A)}$, the state gets updated to either
of $|{1+}\rangle$ or $|{1-}\rangle$, therefore the local measurement of Bob
yields either of the outcomes “0” or “1”, each with probability $1/2$. This is
clearly a violation of Eq. (2), leading therefore to signaling. We emphasize
that what leads to signaling – and therefore to the problem of the impossible
measurement – is the standard state-update rule (or the projection postulate)
which is necessary for ideal measurements.

##  IV Localizable quantum measurements

###  IV.1 Twisted measurements

We now show – following Popescu, Groisman, Reznik, and Vaidman [19, 21, 22] –
that there are ways to carry out a non-local measurement of $T$ (defined in
section III) without violating no-signaling, i.e., a measurement of $T$ is
localizable. To do so, assume that Alice and Bob share the initial state
$\psi_{i}$ and an extra ebit as a resource, say
$\Phi^{+}=1/\sqrt{2}(|{00}\rangle+|{11}\rangle)$ (see Fig. 4). The task is
here to find a scheme that allows one to discriminate any of the eigenstates
of the observable $T$ defined above. As previously pointed out, we are
interested here only in nonlocal measurements, i.e., we require $A$ and $B$ to
be carried out at a distance. However, both Alice and Bob are allowed to
perform local joint measurements on the respective two qubits that they
locally control.

The protocol goes as follows. Bob, performs a BSM on his two local qubits,
obtaining the outcome $b=0,1,2,3$. This updates the state of the two qubits on
Alice’s side as $\mathds{1}\otimes\sigma_{b}\psi_{i}$, where
$\sigma_{0}=\mathds{1}$, and $\sigma_{j}$, with $j=1,2,3$ are the standard
Pauli matrices. Alice, on the other hand, independently and therefore possibly
before or after or at a time arbitrarily close to Bob’s measurement, carries
out a measurement of her first qubit in the Z-basis. If she finds the outcome
$a_{1}=0$ she measures her second qubit in the Z-basis as well; otherwise, if
she finds $a_{1}=1$, she measures the second qubit in the X-basis. The second
measurement of Alice (in either basis), yields a bit $a_{2}$. Let us indicate
by $a=\\{a_{1},a_{2}\\}$ the 2-bit outcome of Alice’s measurements. This ends
the quantum part of the measurement, for all the outcomes are now encoded into
pieces of classical information. To read out the result of this nonlocal
measurement, Alice and Bob need to bring together the classical outcomes they
respectively obtained, by for example sending them to an observer $O$ (see
Fig. 3; henceforth the classical communication part will be omitted in the
figures). The merit of this scheme is that the whole quantum part of the
protocol is carried out while keeping Alice and Bob spatially separated (it is
nonlocal) but does not violate no-signaling (it is localizable).666An
alternative technique to carry out a localizable measurement of this
observable – equivalent in terms of resources (i.e., it also uses an extra
ebit) – was previously proposed in [21]. However, this measurement is not
ideal, for the post-measurement state is not an eigenstate of the measured
observable $T$ (Bob’s qubit ends up in a totally mixed state).

It was shown in [21, 23] that this scheme can be extended to implement
localizable measurements in the general product bases of two qubits of the
form
$T_{\alpha}=\\{|{00}\rangle,|{01}\rangle,|{1\alpha}\rangle,|{1\bar{\alpha}}\rangle\\}$,
where
$|{\alpha}\rangle=\text{cos}(\frac{\alpha}{2})|{0}\rangle+\text{sin}(\frac{\alpha}{2})|{1}\rangle$,
and
$|{\bar{\alpha}}\rangle=\text{sin}(\frac{\alpha}{2})|{0}\rangle-\text{cos}(\frac{\alpha}{2})|{1}\rangle$,
with $0<\alpha<\pi$. This measurement is asymptotically localizable, i.e., it
requires an arbitrarily large number of ebits as resources to implement it.
More precisely, for $n$ additional ebits, the probability of successfully
carrying out a measurement of $T_{\alpha}$ is in general $1-1/2^{n}$. However,
if $\alpha=k\pi/2^{n}$, only $n$ additional ebits are a sufficient resource to
carry out a nonlocal measurement of $T_{\alpha}$. Hence, for any rational
multiple of $\pi$, the number of resources needed to implement a localizable
measurement is finite, and in particular can be very small, e.g., for
$\alpha=\pi/4,\frac{3}{4}\pi$, $n=3$; for
$\alpha=\pi/8,\frac{3}{8}\pi,\frac{5}{8}\pi$, $n=4$, and so on [21].

Figure 4: Scheme to implement a localizable measurement in a basis of twisted
product states using an additional ebit as a preshared resource. $M_{a_{1}}$
is the measurement that Alice carries out on her second qubit, depending on
the outcome $a_{1}$ of the measurement performed on her first qubit.

Finally, based on numerical evidence, we formulate the following:

######  Conjecture 1.

The only localizable measurements on two qubits that can be performed using
one ebit as a resource, and not less, are the twisted measurement, and the
BSM.

###  IV.2 Bell State Measurements

The most well known joint (entangled) measurement is surely the BSM, i.e., the
observable whose eigenvectors are the four Bell states
$\\{\Psi^{\pm},\Phi^{\pm}\\}$. BSMs are localizable with the aid of only one
extra ebit (Fig. 5). The pair of qubits on which they want to carry the
nonlocal BSM is prepared in an initial state $\psi_{i}$. The protocol goes as
follows. Both Alice and Bob perform BSM on their respective local qubits, each
of them obtaining an outcome, $a=0,1,2,3$ and $b=0,1,2,3$, respectively. These
correspond to the four possible outcomes of the measurements, associated,
respectively, to the eigenstates of the BSMs $\Phi^{+}$, $\Psi^{+}$,
$\Psi^{-}$, and $\Phi^{-}$.

It is well-known that for BSMs performed on a pairs of qubits distributed
along each of the edges of a polygon, here a rectangle (the black one in Fig.
5), the amount of “$\Phi$”’s (alternatively “$\Psi$”’s) and the amount “+”’s
(alternatively “-”’s) should be even. Therefore, if the resource ebit is
prepared in a fixed Bell state, say again $\Phi^{+}$, the outcomes $a$ and $b$
fully determine the result of the BSM. For example, if Alice finds $a=2$,
corresponding to the eigenstate $\Psi^{-}$, whereas Bob finds $b=3$,
corresponding to $\Phi^{-}$, then the result corresponds to the eigenvalue
associated to $\Psi^{+}$, with probability
$\text{Prob}(a=2,b=3)=|\langle{\psi_{i}}\vphantom{\Psi^{+}}|{\Psi^{+}}\vphantom{\psi_{i}}\rangle|^{2}$.
An observer collecting the classical pieces of information $a$ and $b$ can
therefore figure out the result of the nonlocal measurement.

Figure 5: Scheme to implement a localizable BSM using an additional ebit as a
resource. If a further ebit (represented as dashed, in red) is preshared, the
measurement can be ideal. BSM is the only (non-product) measurement of two
qubits that is localizable and ideal.

This scheme shows that BSM is therefore localizable with the use of an
additional ebit, but it is not ideal, i.e., the post-measurement state is not
prepared in an eigenstate of the measurement operator.777Note that also
partial BSM can be performed with this scheme, although not ideally. However,
one can consider to use another ebit as a resource (dashed, in red in Fig. 5).
In that case, depending on the output of their local BSMs, Alice and Bob
apply, to their respective part of the second ebit, the local operators
$\sigma_{a}$ and $\sigma_{b}$, respectively. In this way, they end up with a
shared post-measurement state that is indeed the measured eigenstate,
rendering the measurement ideal as well as localizable. This means that if
they repeat the measurement procedure immediately afterwards (by using another
extra ebit), although the single local outcomes $a$ and $b$ may differ, they
combine to yield again the same result.

In fact, in Ref. [19], it was proven that BSM is the only measurement of two
qubits that is at the same time localizable and ideal (besides the trivial
case of the product observables). For the degenerate case, the same result
about localizabily holds, although the measurement cannot be ideal. The
intuitive reason is that BSM is special because it completely erases the
information about the pre-measurement states, which is a necessary condition
to implement an ideal localizable measurement [19] (see also section IV.4).

Let us emphasize that the BSM is an exceptional case, and should thus not be
taken as a general example of joint measurement.

###  IV.3 Elegant Joint Measurements

The Elegant Joint Measurement (EJM) is an observable whose eigenvectors form a
basis of partially entangled states (all with the same degree of entanglement)
with particular symmetry properties [28] (see also [29] for a classification
of joint entangled measurements of two qubits). Indeed, their reduced states
point at the vertices of the regular tetrahedron inscribed in the Bloch
sphere. The states assume the functional form:
$\Phi_{j}=\frac{\sqrt{3}+1}{2\sqrt{2}}|{\vec{m}_{j},-\vec{m}_{j}}\rangle+\frac{\sqrt{3}-1}{2\sqrt{2}}|{-\vec{m}_{j},\vec{m}_{j}}\rangle$,
where $\vec{m}_{j}$ are the vectors pointing at the vertices of the
tetrahedron.

Figure 6: Scheme to implement a localizable EJM using three additional ebits
as a resource. On Alice’s side, first the unitary $U^{\dagger}$ is applied to
the first two qubits, then BSMs are performed between the qubits 1-3 and 2-4.

We now show that nonlocal EJMs are localizable using three ebits as a
resource. With reference to Fig. 6, assume that $\psi_{i}$ is one of the
eigenstates of the EJM observable. All the three additional ebits are prepared
in a $\Phi^{+}$ Bell state. As in Fig. 6, Alice first applies the operator
$U^{\dagger}$ on the first pair of qubits, where $U$ is the unitary operator
whose columns are the four EJM states. Then she performs two BSMs on the two
pairs of qubits 1-3 and 2-4, obtaining the outcomes $a_{1}$ and $a_{2}$,
respectively. Bob, instead, performs a BSM on two of his qubits and, depending
on the the outcome $b$ he obtains, he measures the other two qubits applying
$O_{b}$ defined as following. If $b=0$, this corresponds to a measurement in
the Z-Z product basis
$\\{|{+z,+z}\rangle,|{+z,-z}\rangle,|{-z,+z}\rangle|{-z,-z}\rangle\\}$; if
$b=1$, in the Y-Z Bell basis
$\\{|{+y,+z}\rangle\pm|{-y,-z}\rangle,|{+y,-z}\rangle\pm|{-y,+z}\rangle\\}$;
if $b=2$, in the Z-Y Bell basis
$\\{|{+z,+y}\rangle\pm|{-z,-y}\rangle,|{+z,-y}\rangle\pm|{-z,+y}\rangle\\}$;
finally, if $b=3$, he measures in the Y-Y product basis
$\\{|{+y,+y}\rangle,|{+y,-y}\rangle,|{-y,+y}\rangle|{-y,-y}\rangle\\}$. Here
$|{\pm y}\rangle$ and $|{\pm z}\rangle$ are the two possible eigenstates of a
qubit measured in the $Y$ and in the $Z$ direction, respectively. The result
of the nonlocal measurement can be retrieved by an observer (after the
classical communication) via the expression

$\displaystyle\psi_{B_{2}}=\sigma_{a_{1}}\otimes\sigma_{a_{2}}U^{\dagger}\mathds{1}\otimes\sigma_{b}\psi_{i},$
(3)

where $\psi_{B_{2}}$ is the state of the two qubits of Bob to which he applies
$O_{b}$ (i.e., the two bottom right in Fig. 6). EJMs are therefore localizable
using three additional ebits (but cannot be ideal).

Based on numerical evidence, we put forward the following:

######  Conjecture 2.

EJM are essentially the only localizable measurement of two qubits with
entangled eigenstates that can be measured with three ebits (and no less) as
resources.888In fact, besides the EJM, we find three further bases that can be
related to EJMs. We are investigating this in further detail and a manuscript
is in preparation.

###  IV.4 Review of general results

In this subsection we summarize some general results presented above and from
the literature.

######  Theorem 1.

All localizable and ideal measurements must completely erase all local
information about the pre-measurement state in order to be ideal.

The proof is provided in Ref. [19].

######  Lemma 1.

The only localizable and ideal nonlocal measurement of two qubits is the BSM
(and product measurements).

The proof is provided in Ref. [19]. In section IV.2, we explicitly showed that
to implement a localizable BSM one requires an extra ebit as a resource, but
to implement it ideally one needs two ebits. Moreover, for N-partite systems
on can prove:

######  Lemma 2.

GHZ measurements on $N$ qubits are localizable and ideal.

A GHZ measurement is defined as the observable whose eigenstates form the
(n-partite) GHZ basis (see, e.g, [30]). This lemma follows as a trivial
multipartite extension of the construction for BSMs in IV.2.

The main result concerning localizability is the following:

######  Theorem 2.

All nonlocal measurements (of any dimension and any number of parties) are
localizable.

The constructive proof is provided in Ref. [22]. As already noted, in general
these measurements cannot be ideal and their implementation requires an
infinite amount of resources. However, one can show:

######  Lemma 3.

There are localizable measurements that require only finite entangled
resources.

Concrete examples are provided in sections IV.1, IV.2, and IV.3.

In conclusion, given enough resources, “impossible” measurements are always
localizable (Theorem 2), hence physically possible, but cannot in general be
ideal (Theorem 1).

##  V Discussion and outlook

We have shown (i) that Sorkin’s impossible measurements, initially raised as a
problem of QFT, have a non-relativistic limit as quantum nonlocal measurements
that lead to signaling between separated parties. They should therefore be
addressed as a general foundational problem of quantum physics, independently
of QFT. We then showed (ii) that the problem of impossible measurements is
solved (in non-relativistic quantum mechanics) by considering localizable
measurements of nonlocal variables. That is, those measurements that can be
carried out at a distance without violating no-signaling, and which we deem,
therefore, physically possible. Moreover, (iii) we have recalled that almost
all localizable measurements are non-ideal, i.e., are not immediately
reproducible. We emphasize that from this point of view the well-known BSM is
exceptional and, thus, should not be used to build our intuition about joint
measurements. Finally, (iv) we reviewed and developed further the theory of
localizable measurements, bridging the gap between results put forward by
different communities that had, seemingly, not been connected so far.

Sorkin’s impossible measurements in QFT have recently received a great deal of
attention, mostly thanks to a proposed solution that, at last, puts forward an
explicit measurement scheme for QFT [13, 14, 15, 16]. That approach, however,
has so far not addressed explicitly the possibility of measuring nonlocal
variables, and only considered the fields to be measured by coupling them to a
probe (a quantum field itself) that then subluminally mediates the
interactions within the extended region M (see also [31]).999This is
equivalent to a semilocalizable measurement in the parlance of [27, 32], i.e.,
one that can be carried out by sending one-way communication between the
parties. Similar schemes were also developed in [20]. We believe that by
introducing entangled probes all the results here presented could be
eventually reproduced in the language of algebraic QFT, but this remains to be
done. These considerations can thus perhaps bring novel insights in helping
guide the development of a complete theory of measurements in QTF, i.e. one
that yields outcomes and prescribes how to update the state afterwards.

Moreover, even in standard quantum mechanics, joint measurements remain poorly
understood, an issue that has recently been listed among the most relevant
open problems in the foundations of quantum physics [33]. Only lately, some
attempts to systematize joint measurements have been put forward [34, 29], but
much has still to be done. The results surveyed in this paper show that with
enough resources, every nonlocal measurement is localizable, although it is
almost never ideal. We further showed several cases in which the amount of
resources is finite and relatively small, making this proposal physically
compelling. This suggests that localizability could help characterize joint
measurements by the necessary amount of resources (i.e. entanglement) that
they require. Such an approach can single out some measurements (or classes
thereof) with special proprieties.101010Some work aimed to characterize the
amount of resources need for localizability was already provided in Refs. [24,
35], but only in the asymptotic limit For instance, BSM is the only possible
measurement (up to local unitaries) to be localizable and ideal. Perhaps other
measurements, like the EJM, can be singled out in a similar way (cf.
Conjecture 2). We are addressing this problem more systematically and a
manuscript addressing the classification of joint measurements by their
property of localizability is currently in preparation.

A further interesting question would be to analyze Sorkin’s impossible
measurements in different theoretical frameworks. This could help understand
the minimal requirements for a theory to exhibit this kind of problems,
namely, what is fundamentally the source of impossible measurements? This
paper shows that this is not a problem of relativistic quantum physics per se.
One might then think that the source may be nonlocality, but the most nonlocal
of the no-signaling (fictional) theories, so-called “boxworld”, exclusively
admits separable joint measurements [36] and therefore does not have a
Sorkin’s problem. On the other hand, a recent work claims that impossible
measurements can be found in a classical limit (i.e., non-quantum) of QFT
[37], also seemingly ruling out that this is a genuine quantum effect.

Finally, in the axiomatic approach to quantum theory based on property-
lattices [38], Piron’s theorem states that, assuming certain natural axioms,
like the covering law (the projection of a pure state is necessarily a pure
state), the lattice of quantum properties is isomorphic to the closed
subspaces of a linear space, possibly with superselection rules [39]. One of
us, NG, proved that the lattice of non-local properties is atomic (it has pure
states) [40]. Here we proved that it is also ortho-modular (each property has
an orthogonal one). But since we showed that (almost all) nonlocal
measurements cannot be ideal, the covering law is a doubtful axiom. As it is
the latter that ensures linearity in quantum physics, could this be a hint
that one should relax linearity in QFT?

###### Acknowledgements.

We thank Sandu Popescu, Daniel Collins, Christopher Fewster, Tein van der
Lugt, Kuntal Sengupta, Mirjam Weilenmann, and Pavel Sekatski for useful
discussions and for pointing out relevant references. This research was
supported by the FWF (Austrian Science Fund) through an Erwin Schrödinger
Fellowship (Project J 4699), and the Swiss National Science Foundation via the
NCCR-SwissMap.

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# Screening effect in Spin-Hall Devices

M. Creff  LSI, École Polytechnique, CEA/DRF/IRAMIS, CNRS, Institut
Polytechnique de Paris, 91120 Palaiseau, France  E. Olive  GREMAN UMR-CNRS
7347, Universit de Tours, INSA Centre Val de Loire, Parc de Grandmont, 37200
Tours, France  J.-E. Wegrowe<EMAIL_ADDRESS>LSI, École
Polytechnique, CEA/DRF/IRAMIS, CNRS, Institut Polytechnique de Paris, 91120
Palaiseau, France

###### Abstract

The stationary state of the spin-Hall bar is studied in the framework of a
variational approach that includes non-equilibrium screening effects. The
minimization of the power dissipated in the system is performed with taking
into account the spin-flip relaxation and the global constrains due to the
electric generator and global charge conservation. The calculation is
performed in both approximations of negligible spin-flip scattering and strong
spin-flip scattering. In both cases, the expressions of the spin-accumulation
and the longitudinal and transverse pure spin-currents are derived
analytically. Due to the small value of the Debye-Fermi screening length, the
spin-accumulation is shown to be linear in $y$ (across the device), linear in
the electric field imposed by the generator, and inversely proportional to the
temperature for non-degenerate conductors.

##  I Introduction

Spin-accumulation can be produced at the edges of a conducting bar at zero
external magnetic field, by injecting an electric current into a non-magnetic
material with high spin-orbit coupling Awschalom ; Jungwirth ; Valenzuela ;
Otani ; Gambardella ; Bottegoni . This effect is called Spin Hall Effect
(SHE). It has been predicted some decades ago Dyakonov , and described in the
framework of various theoretical models Dyakonov2 ; Hirsch ; Zhang ; Tse ;
Maekawa ; Hoffmann ; Saslow ; SHE ; Sinova . However, a description that takes
into account the non-equilibrium nature of the electric screening occurring in
the SHE is still an open problem. One of the main difficulty is the same as
for the classical Hall-effect boundary ; Trefenthen ; Heremans ; Geometry ;
Connection ; Perrott ; Nanowire ; Calcul ; Solin ; SolinJAP : it is due to the
fact that the values of the charge accumulation at the edges of the Hall bar
are not directly imposed by the external constraints. Instead, for the
stationary state, the accumulation of electric charges and spins at the edges
is generated by the system itself, in reaction to the action of the internal
magnetic field (the spin-orbit effective magnetic field), together with to
stationary current injected from the electric generator, and for a given
geometry. As a consequence, the determination of a unique solution to the
drift-diffusion equations at stationary state based on the local boundary
conditions is problematic Tse .

As shown in a series of publications that led to the present workEPL1 ; EPL2 ;
Benda ; JAP1 ; JAP2 ; JAP3  \- it is possible to take into account the non-
equilibrium nature of the electric screening in the case of a ideal Hall bar
on the basis of the least dissipation principle Onsager_Diss ; Bruers ;
MinDiss . The stationary state can then be deduced from the minimization of
the power under the global constraints, instead of solving independently the
drift-diffusion equations. This approach has been applied successfully in the
case of the classical Hall effect without spin JAP1 ; JAP2 ; JAP3 . It has
been shown that the charges accumulated at the edges are not static but
generate a non-uniform longitudinal current $\delta J_{x}(y)$, which has not
been derived before. Physically, this surface current is responsible for the
well-known robustness of the Hall voltage, which can be measured while using a
good or a bad voltmeter (i.e. with huge variations of the electric leakage at
the edges Hall ), because the electric charges accumulated are renewed
permanently despite the zero transverse current. Furthermore, if a secondary
passive circuit is contacted the two edges of the Hall bar, the variational
method shows that the electric current injected is mainly carried by the
longitudinal surface currents $\delta J_{x}(y)$, instead of the transverse
current $J_{y}$ JAP3 .

The model developed in the present work is the application of the variational
method to the SHE. The model is based on two spin-channel description of the
electric currents, that is well-known in the studies of spin transport in the
context of giant magnetoresistance Johnson ; Wyder ; Fert ; PRB2000 ; Schmidt
; Jedema ; JPhys17 . The spin-Hall bar is then equivalent to a superimposition
of two subsystems composed of two classical Hall bars, activated by an
effective spin-orbit magnetic field. The sub-systems are the two spin-
dependent electronic carriers, and the effective magnetic field is acting with
identical amplitude and opposite sign on each spin-channel. The consequence of
these symmetries on the electric transport is that the electric charges
accumulated at the edges have an opposite sign for each channel, so that the
sum over the two channel is zero (if the symmetry is strictly respected), and
the difference gives the spin-accumulation observed. The application of the
variational method presented below shows that the same symmetry also applies
for the out-of-equilibrium spin currents, as sketched in Figure 1 and Figure
2. The transverse and the non-homogeneous longitudinal currents are both pure
spin-currents.

The paper is organized in three sections: after the presentation of the model
and formalism (Section II below), the derivation of the spin-accumulation and
currents is performed in the case of negligible spin-flip scattering in
section III. The Joule functional is minimized under the global constrains,
and the stationarity condition for the electric current $\vec{J}$ is obtained.
The accumulation of electric charges and spin are then derived by integration
of the Maxwell-Gauss equation. In section IV, the same derivation is performed
in the case of strong spin-flip scattering, by adding the spin-flip
dissipation into the Joule functional. In order to deal with analytical
expressions, the spin-flip scattering length $l_{sf}$ is assumed to be small
with respect to the width of the Hall-bar $\ell$. It is shown that there is a
solution of continuity between the two limits ($\ell\ll l_{sf}$ and $\ell\gg
l_{sf}$). From a more quantitative viewpoint, the small value of the Debye
length scale $\lambda_{D}$ leads to the following surprising result: the spin-
flip scattering does not modify significantly the profile of the spin-
accumulation $n^{Sp}(y)$, nor that of the longitudinal spin-current
$J^{Sp}_{x}(y)$. Three main properties of the spin-accumulation should thus be
expected: (a) the linearity with $y$, (b) the linearity with the applied
voltage $\Delta V$, and (c) the $1/T$ dependence to the temperature for non-
degenerate conductors (as an expression of the paramagnetic behavior of the
spin-accumulation). The results are summarized in the conclusion (section V).

Figure 1:  Schematic representation of the two channel model (corresponding to
spin $\uparrow$ and spin $\downarrow$) for a spin-Hall-bar. The device is
contacted to an electric generator that imposes a longitudinal stationary
current $J_{x}^{0}$ (not shown). The charge accumulations $\pm\delta
n_{\updownarrow}$ and the inhomogeneous part of the longitudinal currents
$\delta J_{x\updownarrow}(y)=J_{x\updownarrow}(y)-J_{x}^{0}$ are represented
at both edges for each channel, together with the transverse current
$J_{y\updownarrow}$ flowing from one edge to the other. Figure 2:
Superimposition of the two channels represented for symmetric device and
environment ($C_{E}=0$ in the text). The electric generator and the injected
current $J_{x}^{0}$ are not shown for clarity. The system is characterized by
the absence of total charge accumulation, the spin-accumulation
$n^{Sp}(y)=\delta n_{\uparrow}(y)-\delta n_{\downarrow}(y)$ with opposite
directions at the two edges, the inhomogeneous part of the longitudinal spin-
currents $J^{Sp}_{x}(y)=J_{x\uparrow}(y)-J_{x\downarrow}(y)$, and the
transverse spin-current $J^{Sp}_{y}=J_{y\uparrow}(y)-J_{y\downarrow}(y)$.

##  II The model

The system is a long bar of width $\ell$ and thickness $d$, composed of a thin
layer of a non-magnetic material with strong spin-orbit coupling contacted to
an electric generator. We assume that the bar is invariant by translation
along its longitudinal axis $x$ (i.e. we do not take into account the region
close to the generator, nor the perturbation caused by possible lateral
contacts), that the lateral edges are symmetric, that the device is planar (no
charge transport in the $z$ direction), and that the polarization axis of the
spins is oriented along the $z$ direction. In the framework of the two channel
model, the charge carriers are separated into two populations, that are the
charge carriers of spin up $\uparrow$ and the charge carriers of spin down
$\downarrow$, with the respective charge densities $n_{\uparrow}$ and
$n_{\downarrow}$.

The conductor is characterized by a density of charge carriers $n^{Q}\equiv
n_{\uparrow}+n_{\downarrow}=2n_{0}+\delta n^{Q}$, where $n_{0}$ is the density
defined by electroneutrality at equilibrium (i.e. without current injected).
Accordingly $n_{0}$ is uniform and does not depend on the spin-channel. On the
other hand $\delta n^{Q}(y)$ is the charge accumulation, which is not a
function of $x$ due to the invariance by translation, and is the sum of the
charge accumulation for each spin-channel: $\delta n^{Q}(y)=\delta
n_{\uparrow}(y)+\delta n_{\downarrow}(y)$. The Coulomb interaction is
described by the electric potential $V$, that follows the Poisson law
$\nabla^{2}V=-q\delta n^{Q}/\epsilon$, where $q$ is the charge of the
carriers, $\epsilon$ is the permittivity of the material, and
$\vec{\nabla}=\\{\partial_{x},\partial_{y}\\}$ is the gradient operator in 2D.

The energy of the system is then defined by the two chemical potentials
$\mu_{\uparrow}$ and $\mu_{\downarrow}$ such that:

$\mu_{\updownarrow}=\frac{kT}{q}\,\ln{\left(\frac{n_{\updownarrow}}{n_{0}}\right)}+V+\mu_{\updownarrow}^{ch}$
(1)

where $k$ is the Boltzmann constant, $T$ is equal to the temperature of the
material in the case of non-degenerate semiconductors, or $T$ is equal to the
Fermi temperature $T\equiv T_{F}$ in the case of degenerate semiconductors and
metals RqueT . The first term in the right hand side accounts for the
diffusion of the carriers (this term is justified in the framework of the
local equilibrium approximation Rubi ; JPhys17 ). The second term describes
the electric potential $V$. The third term $\mu_{\updownarrow}^{ch}$ is the
pure chemical potential that accounts for the spin-flip scattering, and is the
main parameter for the description of giant magnetoresistance effects Johnson
; Wyder ; Fert ; Jedema ; Schmidt ; PRB2000 . In the following, we assume that
$\mu_{\updownarrow}^{ch}$ is uniform in order to treat uniquely the spin-
accumulation due to the SHE. The electric field reads
$\vec{\mathcal{E}}=-\vec{\nabla}V=\mathcal{E}_{x}^{0}\,\vec{e}_{x}+\mathcal{E}_{y}\,\vec{e}_{y}$,
where the $x$-component $\mathcal{E}_{x}^{0}$ is constant (due to the
invariance along $x$), so that the Poisson law is reduced to
$\partial_{y}\mathcal{E}_{y}=q\delta n^{Q}/\epsilon$. Note that the electric
field does not depend on the spin-channel, so that the electric potential $V$
couples the two channels in Eq.(1). We can re-write the Poisson law with the
help of the chemical potential Eq.(1) as follow:

$\nabla^{2}\mu_{\updownarrow}-\lambda_{D}^{2}\,\frac{qn_{0}}{\epsilon}\,\nabla^{2}\ln{\left(\frac{n_{\updownarrow}}{n_{0}}\right)}+\frac{q\delta
n^{Q}}{\epsilon}=0$ (2)

where $\lambda_{D}=\sqrt{\frac{kT\epsilon}{q^{2}n_{0}}}$ is the well-known
Debye-Fermi screening length.

On the other hand, the transport equations for the charge carriers are given
by the Ohm’s law for each channels:

$\vec{J}_{\updownarrow}=-\hat{\sigma}_{\updownarrow}\,\vec{\nabla}\mu_{\updownarrow}=-qn_{\updownarrow}\hat{\eta}_{\updownarrow}\,\vec{\nabla}\mu_{\updownarrow},$
(3)

where $\hat{\sigma}_{\updownarrow}\equiv
qn_{\updownarrow}\hat{\eta}_{\updownarrow}$ is the spin-dependent conductivity
tensor, and $\eta_{\updownarrow}$ is by definition the electric mobility
tensor.

As a consequence of both the Onsager reciprocity relations Onsager  and the
property of the spin-orbit effective fields, the mobility tensor takes the
following form in the orthonormal basis
$\\{\vec{e}_{x_{\uparrow}},\vec{e}_{y_{\uparrow}},\vec{e}_{x_{\downarrow}},\vec{e}_{y_{\downarrow}}\\}$
(see Fig.1):

$\hat{\eta}_{\updownarrow}=\eta\begin{pmatrix}1&\theta_{so}&0&0\\\
-\theta_{so}&1&0&0\\\ 0&0&1&-\theta_{so}\\\ 0&0&\theta_{so}&1\\\
\end{pmatrix}$ (4)

where $\eta$ is the mobility of the material and $\theta_{so}$ is the spin-
Hall angle. the transport equation reads:

$\vec{J}_{\updownarrow}=-qn_{\updownarrow}\eta\left(\vec{\nabla}\mu_{\updownarrow}\mp\theta_{so}\,\vec{e_{z}}\times\vec{\nabla}\mu_{\updownarrow}\right),$
(5)

This equation is equivalent to the Dyakonov-Perel equation of the spin-Hall
effect JPhys17 . It is convenient to rewrite Eq.(5) in the following forms:

$\displaystyle-
qn_{\updownarrow}\eta(1+\theta_{so}^{2})\partial_{x}\mu_{\updownarrow}$
$\displaystyle=J_{x\updownarrow}\mp\theta_{so}J_{y\updownarrow}$ (6)
$\displaystyle-
qn_{\updownarrow}\eta(1+\theta_{so}^{2})\partial_{y}\mu_{\updownarrow}$
$\displaystyle=J_{y\updownarrow}\pm\theta_{so}J_{x\updownarrow}$ (7)
$\displaystyle\|\vec{J}_{\updownarrow}\|^{2}\equiv
J_{x\updownarrow}^{2}+J_{y\updownarrow}^{2}$
$\displaystyle=(qn_{\updownarrow}\eta)^{2}\,(1+\theta_{so}^{2})\,\|\vec{\nabla}\mu_{\updownarrow}\|^{2}.$
(8)

where the symbols $\mp$ and $\pm$ refer to the spin-channel $\uparrow$ for the
upper sign and spin-channel $\downarrow$ for the lower sign. The power
dissipated by the charge carriers in each spin-channel is the Joule power :

$P_{J\updownarrow}=S_{\text{lat}}\int_{-\ell}^{\ell}qn_{\updownarrow}\eta\|\vec{\nabla}\mu_{\updownarrow}\|^{2}dy=\frac{S_{lat}}{qn_{0}\eta(1+\theta_{so}^{2})}\int_{-\ell}^{\ell}\frac{n_{0}}{n_{\updownarrow}}\|\vec{J}_{\updownarrow}\|^{2}dy.$
(9)

where we have introduced the lateral surface $S_{lat}=L_{x}\,d$, where $L_{x}$
is the length of the Hall bar along the $x$ direction.

On the other hand, the power dissipated by the spin-flip scattering (i.e. the
transition of an electric charge from one spin-channel to the other) per unit
of volume reads $\mathcal{L}\,\Delta\mu^{2}$, where
$\Delta\mu=\mu_{\uparrow}-\mu_{\downarrow}$ and $\mathcal{L}$ is the Onsager
transport coefficient describing the spin-flip relaxation rate in the
framework of the two channel modelWyder ; Fert ; PRB2000 ; JPhys17 ; EPL2 .
Note that the relaxation rate $\dot{\psi}=\mathcal{L}\Delta\mu$ is by
definition the source term in the continuity equation for the spin-dependent
electric charges (in the absence of charge accumulation): $q\partial
n_{\updownarrow}/\partial
t=-\vec{\nabla}.\vec{J}_{\updownarrow}\mp\dot{\psi}$. Accordingly, the
coefficient $\mathcal{L}$ is related to the measured spin-flip scattering
length $l_{sf}$ by the relation JPhys17

$\mathcal{L}=\frac{\sigma_{0}}{2l_{sf}^{2}},$ (10)

where $\sigma_{0}$ is the conductivity of the material. In the framework of
the variational approach, and ignoring temperature gradient, the stationary
state is defined by the principle of least power dissipation Bruers ; MinDiss
. In other terms, the distribution of charge densities and currents is
determined by the state of minimum power production, taking into account the
global constraints applied to the system. These constraints are due to the
presence of the electric generator, the electrostatic boundary conditions, and
the symmetries of the device EPL1 ; EPL2 ; JAP1 ; JAP2 ; JAP3 . Due to the
electroneutrality and the fact that the device is symmetric, the integral of
the total charge accumulated in the two channels cancels out
$\int_{-\ell}^{\ell}(\delta n_{\uparrow}+\delta n_{\downarrow})dy=0$. Or in
other terms:

$\frac{1}{2\ell}\,\int_{-\ell}^{\ell}\left(n_{\uparrow}(y)+n_{\downarrow}(y)\right)dy=2n_{0}$
(11)

On the other hand, the generator injects a constant current through the device
(of constant lateral surface $S_{lat}$), si that the integrated current is
constant:

$\frac{1}{2\ell}\,\int_{-\ell}^{\ell}\left(J_{x\uparrow}(y)+J_{x\downarrow}(y)\right)dy=2J_{x}^{0}$
(12)

##  III Spin Hall Effect with negligible spin-flip scattering

The approximation of negligible spin-flip scattering corresponds to the case
$l_{sf}\gg\ell$ where $\ell$ is the width of the spin-Hall bar. Let use define
for convenience the reduced power:
$\tilde{P}_{J\updownarrow}=\frac{q\eta(1+\theta_{so}^{2})}{S_{\text{lat}}}\,P_{J\updownarrow}=\int_{-\ell}^{\ell}\frac{J_{x\updownarrow}^{2}+J_{y\updownarrow}^{2}}{n_{\updownarrow}}dy$.
We shall use the Lagrange multipliers $\lambda_{J}$ and $\lambda_{n}$ in order
to take into account the constraints respectively Eq.(11) and Eq.(12) in the
minimization of the functional $\mathcal{F}$:

${\mathcal{F}}_{J\updownarrow}[J_{x\updownarrow},J_{y\updownarrow},n_{\updownarrow}]=\int_{-\ell}^{\ell}\left(\frac{J_{x\updownarrow}^{2}+J_{y\updownarrow}^{2}}{n}-\lambda_{J}\,\left(J_{x\uparrow}+J_{x\downarrow}\right)-\lambda_{n}\,\left(n_{\uparrow}+n_{\downarrow}\right)\right)dy$
(13)

The minimum of the reduced Joule power
${\mathcal{F}}_{J}=\tilde{\mathcal{P}}_{J\uparrow}+\tilde{\mathcal{P}}_{J\downarrow}$
corresponds to :

$\frac{\delta\mathcal{F}_{J\updownarrow}}{\delta J_{x\updownarrow}}=0\
\Longleftrightarrow\ 2J_{x\updownarrow}=n_{\updownarrow}\lambda_{J},$ (14)
$\frac{\delta\mathcal{F}_{J\updownarrow}}{\delta J_{y\updownarrow}}=0\
\Longleftrightarrow\ J_{y\updownarrow}=0,$ (15)
$\frac{\delta\mathcal{F}_{J\updownarrow}}{\delta(n_{\updownarrow})}=0\
\Longleftrightarrow\
J_{x\updownarrow}^{2}+J_{y\updownarrow}^{2}=-\lambda_{n}n_{\updownarrow}^{2},$
(16)

Using Eqs.(11), Eqs.(12), and Eq.(14) leads to
$\lambda_{J}=\frac{2J_{x}^{0}}{n_{0}}$ so that
$J_{x\updownarrow}=\frac{n_{\updownarrow}}{n_{0}}J_{x}^{0}$ (and from Eq.(16)
we have furthermore $\lambda_{n}=-(J_{x}^{0}/n_{0})^{2}$). The stationarity
conditions for the currents are then defined by the two relations:

$J_{x\updownarrow}(y)=J_{x}^{0}\frac{n_{\updownarrow}(y)}{n_{0}}\quad\text{and}\quad
J_{y\updownarrow}=0.$ (17)

Like for the usual Hall effect, there is no transverse current in the device
for $\l_{sf}\gg\ell$. Let us define the inhomogeneous current $\delta
J_{x\updownarrow}=J_{x\updownarrow}-J_{x}^{0}$ produced by the spin-orbit
field. This longitudinal spin-current is a pure spin current, in the sense
that the unhomogeneous current of electric charge $\delta J_{x}^{Q}$ is zero
and the current of spins $J_{x}^{Sp}$ is proportional to the spin accumulation
$n^{Sp}(y)$:

$\delta J_{x}^{Q}=\delta J_{x\uparrow}+\delta
J_{x\downarrow}=0\qquad\text{and}\qquad J_{x}^{Sp}=\delta J_{x\uparrow}-\delta
J_{x\downarrow}=J_{x}^{0}\frac{n^{Sp}}{n_{0}}.$ (18)

The explicit expression of $J_{x\updownarrow}(y)$ is obtained if we know the
expression of the densities $n_{\updownarrow}(y)$. This is the aim of the
following paragraph.

Inserting the stationarity conditions Eq.(17) into Eq.(6) and Eq.(7) we deduce
$\partial_{x}\mu_{\updownarrow}=\partial_{x}\mu=\frac{-J_{x}^{0}}{qn_{0}\eta(1+\theta_{so}^{2})}$
and
$\partial_{y}\mu_{\updownarrow}=\mp\frac{\theta_{so}J_{x}^{0}}{qn_{0}\eta(1+\theta_{so}^{2})}$.
These two terms are constant so that $\nabla^{2}\mu_{\updownarrow}=0$ and the
chemical potentials $\mu_{\updownarrow}$ are harmonic functions. Equation (2)
reduces to:

$\lambda_{D}^{2}\partial_{y}^{2}\ln\left(1+\frac{\delta
n_{\updownarrow}}{n_{0}}\right)=\frac{\delta n^{Q}}{n_{0}}.$ (19)

In order to evaluate the global boundary conditions for the electric field we
have to integrate the Maxwell equation
$\vec{\nabla}\cdot\vec{\mathcal{E}}=\partial_{y}\mathcal{E}_{y}=\frac{q}{\varepsilon}\delta
n^{Q}$ at a given point $y_{0}$ inside the material :

$\mathcal{E}_{y}(y_{0})=-\partial_{y}V(y_{0})=\frac{E^{\infty}}{2}-\frac{q}{2\varepsilon}\int_{-\ell}^{\ell}\delta
n^{Q}(y)sgn(y-y_{0})dy,$ (20)

where $E^{\infty}$ accounts for possible electric charges in the environment
of the spin-Hall bar (this is the case if a conducting or insulating layer is
deposited on one side of the spin-Hall bar). Inserting the chemical potential
Eq.(1), the transport equation Eq.(7), and the stationary conditions Eq.(17),
yields:

$\pm\lambda_{SH}^{0}+\lambda_{D}^{2}\partial_{y}\ln\left(\frac{n_{\updownarrow}}{n_{0}}\right)(y_{0})-\frac{C_{E}}{2}+\frac{1}{2n_{0}}\,\int_{-\ell}^{\ell}\delta
n^{Q}(y)sgn(y-y_{0})dy=0,$ (21)

where the asymmetry of the electric environment is described by the
characteristic length $C_{E}=\frac{\varepsilon E^{\infty}}{qn_{0}}$. We have
also introduced the spin-Hall characteristic length:

$\lambda_{SH}^{0}=\frac{\varepsilon}{q^{2}n_{0}^{2}\eta}\,\frac{\theta_{so}}{1+\theta_{so}^{2}}\,J_{x}^{0}=\frac{\lambda_{D}^{2}}{kT}\left(\frac{\theta_{so}}{1+\theta_{so}^{2}}\right)\frac{J_{x}^{0}}{n_{0}\eta}.$
(22)

At the first order in $\delta n_{\updownarrow}/n_{0}$, the difference of both
spin-channels in Eqs.(21) gives the spin-accumulation $n^{Sp}=\delta
n_{\uparrow}-\delta n_{\downarrow}$, linear in $y$:

$\frac{n^{Sp}(y)}{n_{0}}=-2\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\,y=-2\left(\frac{\theta_{so}}{1+\theta_{so}^{2}}\right)\,\left(\frac{q\Delta
V}{kT}\right)\frac{y}{L},$ (23)

where we have introduced on the right hand side the longitudinal voltage
$\Delta V$ in order to compare the energy imposed by the generator $q\Delta V$
with the thermal energy $kT$, as illustrated in Figure 3b. The voltage $\Delta
V$ is measured along the axis $x$ over the distance $L$, and is given by the
relation $\Delta V=\mathcal{E}_{x}^{0}L=L\,J_{x}^{0}/\sigma_{0}$.

Note that, according to Eq.(23), the spin-current $J^{Sp}_{x}$ (Eq.(18)) is
proportional to the square of the injected current density
$\left(J_{x}^{0}\right)^{2}$. The Joule heating is thus expected to be a power
four $\left(J_{x}^{0}\right)^{4}$, which is not without consequence for the
temperature distribution inside the device Beach .
In order to summarize, we see that three main properties of the spin-
accumulation are derived in the limit $l_{sf}\gg\ell$: (a) the linearity with
$y$, (b) linearity with the applied voltage $\Delta V$, and (c)
proportionality with $1/T$ for non-degenerate conductors. From a more
quantitative viewpoint, assuming a small sample in which $L$ and $\ell$ are of
the same order, the spin-accumulation at the border is of the order of
$n^{Sp}(y=\ell)/n_{0}\approx\theta_{so}\,q\Delta V/kT$, where $\theta_{so}$ is
of the order of $10^{-3}-10^{-4}$ and the maximum of $q\Delta V$ is a fraction
of $eV$. Accordingly, the ratio $n^{Sp}(y=\ell)/n_{0}$ can reach $0.1$, e.g.
at low temperature (as shown in Figure 3b).

Furthermore, if both the device and its environment are symmetric we have
$C_{E}=0$. The sum of the two Eqs.(21), at the first order in $\delta
n_{\updownarrow}/n_{0}$, gives

$\frac{\delta n^{Q}(y)}{n_{0}}=0,$ (24)

and the relation $\delta n_{\uparrow}=-\delta n_{\downarrow}$ is verified.
This corresponds to symmetric SHE devices, as sketched in Fig.1 and Fig.2.

However, if the device or its environment are not symmetric (typically if a
dielectric is placed on one side of the device), $C_{E}\neq 0$ and there is an
asymmetry of the charge accumulation and spin-polarization between the two
edges. The following charge accumulation appears :

$\frac{\delta
n^{Q}(y)}{n_{0}}=\frac{C_{E}}{\lambda_{D}\sqrt{2}}\frac{sh\left(\frac{\sqrt{2}}{\lambda_{D}}y\right)}{ch\left(\frac{\sqrt{2}}{\lambda_{D}}\ell\right)},$
(25)

and residual spin-Hall voltage is generated. This result seems to be in
agreement with that derived in the reference DiVentra .

It is worth noting that if a ferromagnetic layer is present on one edge of the
spin-Hall bar (e.g. a ferromagnetic 3d metal, or a magnetic oxide like YIG) or
a spin-injection mechanism is present also on one edge of the spin-Hall bar,
the charge accumulation $\delta n^{Q}$ will depend on the magnetization state
$\vec{M}$ of the ferromagnet, or on the state of the spin-injector (also
denoted by $\vec{M}$ for convenience). The asymmetry of $\delta
n_{\updownarrow}$ will then lead to a spin-dependent voltage where the
parameter $C_{E}$ depends on $\vec{M}$. This voltage can mimic the so-called
“inverse spin-Hall” effect.

##  IV Approximation of localized spin-flip scattering at the edges
($l_{sf}\ll\ell$)

The previous section describes the case of negligible spin-flip scattering,
i.e. large spin-diffusion length with respect to the width of the Spin-Hall
bar. In this section we will explore the opposite limit, for short spin-
diffusion length with respect to the spin-Hall bar : $l_{sf}\ll\ell$. This
situation can be described by a spin-flip scattering that occurs locally at
the edges. The approximation of local spin-flip scattering leads to a constant
transverse current $J_{y\updownarrow}$, so that simple analytical results can
be derived. This approximation is also usual in practice, since the samples
are often larger than few tens of microns.

Let us first define $\Delta\mu_{0}$ such that:

$\Delta\mu_{0}\ =\ \int_{-\ell}^{+\ell}dy\,\partial_{y}\mu_{\uparrow}\
-\int_{-\ell}^{+\ell}dy\,\partial_{y}\mu_{\downarrow}$ (26)

The subscript $0$ points-out that the potential difference is evaluated
between $y=+\ell$ and $y=-\ell$. Note that due to the translation invariance
along $x$ we have: $\partial_{x}\Delta\mu=0$ and
$\partial_{x}\mu_{\uparrow}=\partial_{x}\mu_{\downarrow}$. In the framework of
our approximation, the power dissipated by the spin-flip scattering is a
constant, given by $P_{\text{sf}}\ =\ v\,\mathcal{L}\,\Delta\mu_{0}^{2}$,
where $v=2S_{lat}\ell$ is the volume of the device. Inserting the transport
equations (6) and (7) into Eq.(26) gives:

$\Delta\mu_{0}\ =\ \frac{-\Delta A}{q\eta(1+\theta_{so}^{2})}$ (27)

where $\Delta A$ is given by the integrals:

$\Delta
A=\int_{-\ell}^{+\ell}dy\,\frac{J_{y\uparrow}+\theta_{so}J_{x\uparrow}}{n_{\uparrow}}-\
\int_{-\ell}^{+\ell}dy\,\frac{J_{y\downarrow}-\theta_{so}J_{x\downarrow}}{n_{\downarrow}}.$
(28)

The reduced dissipated power
$\tilde{P}=\frac{q\eta(1+\theta_{so}^{2})}{S_{\text{lat}}}\,P$ then reads:

$\tilde{P}\ =\ \tilde{P}_{J}+\tilde{P}_{\text{sf}}\ =\
\int_{-\ell}^{+\ell}\left(\frac{J_{x\uparrow}^{2}+J_{y\uparrow}^{2}}{n_{\uparrow}}+\frac{J_{x\downarrow}^{2}+J_{y\downarrow}^{2}}{n_{\downarrow}}\right)dy\
+\alpha\ \left(\Delta A\right)^{2}$ (29)

expressed as a function of the control parameter $\alpha$, defined by:

$\alpha=\frac{\mathcal{L}\,2\ell
n_{0}}{\sigma_{0}(1+\theta_{so}^{2})}=\frac{n_{0}\ell}{l_{sf}^{2}(1+\theta_{so}^{2})},$
(30)

where we used Eq.(10) for the expression of $\alpha$ as a function of the
spin-flip scattering length $l_{sf}$. The characteristic length $l_{sf}$ is
well-known in the context of the giant magnetoresistance effects. It is
usually greater than the Debye-Fermi length $\lambda_{D}$ and it is range
between few nanometers to few micrometers. The functional of the dissipated
power now reads:

${\mathcal{F}}[J_{x\updownarrow},J_{y\updownarrow},n_{\updownarrow}]=\int_{-\ell}^{\ell}\left[\left(\frac{J_{x\uparrow}^{2}+J_{y\uparrow}^{2}}{n_{\uparrow}}+\frac{J_{x\downarrow}^{2}+J_{y\downarrow}^{2}}{n_{\downarrow}}\right)-\lambda_{J}\,\left(J_{x\uparrow}+J_{x\downarrow}\right)-\lambda_{n}\,\left(n_{\uparrow}+n_{\downarrow}\right)\right]dy+\alpha\left(\Delta
A\right)^{2},$ (31)

and its minimization leads to the stationarity conditions. The functional
derivation as a function of $J_{x\updownarrow}$ gives:

$\frac{\delta{\mathcal{F}}}{\delta J_{x\updownarrow}}=0\ \Longleftrightarrow\
\pm 2\,\alpha\,\theta_{so}\Delta
A+2J_{x\updownarrow}=n_{\updownarrow}\lambda_{J}$ (32)

The Lagrange coefficient $\lambda_{J}$ is determined using the global
conditions Eq.(11) and Eq.(12) on currents and charges for the sum of the two
channels:

$\lambda_{J}=\frac{2J_{x}^{0}}{n_{0}}$ (33)

Re-inserting Eq.(33) into Eq.(32) gives the longitudinal currents as a
function of the spin-dependent density of electric charges:

$J_{x\updownarrow}(y)=\frac{n_{\updownarrow}(y)}{n_{0}}J_{x}^{0}\mp\alpha\theta_{so}\,\Delta
A.$ (34)

On the other hand, the minimization as a function of $J_{y\updownarrow}$ gives
the current across the spin-Hall bar:

$\frac{\delta{\mathcal{F}}}{\delta J_{y\uparrow}}=0\ \Longleftrightarrow\
J_{y\updownarrow}=\mp\alpha\Delta A$ (35)

This result shows that the transverse current is a pure spin-current, in
agreement with the results known from the direct resolution of the drift-
diffusion equationsDyakonov ; Zhang ; Tse ; Maekawa ; SHE . Indeed, the charge
current vanishes $J_{y}^{Q}\equiv J^{0}_{y\uparrow}+J^{0}_{y\downarrow}=0$,
and the spin-current reads $J^{Sp}_{y}\equiv
J^{0}_{y\uparrow}-J^{0}_{y\downarrow}=2J^{0}_{y\uparrow}$.

Inserting the expression of the transverse current Eq.(35) into the
expressions of the longitudinal current Eq.(34) divided by the density
$n_{\updownarrow}$, the sum over the two channels then reads:

$\displaystyle\frac{J_{x\uparrow}}{n_{\uparrow}}+\frac{J_{x\downarrow}}{n_{\downarrow}}=\frac{2J_{x}^{0}}{n_{0}}+\theta_{so}J_{y}^{0}\left(\frac{1}{n_{\uparrow}}-\frac{1}{n_{\downarrow}}\right).$
(36)

In order to simplify as much as possible the physical interpretation, we
continue the derivation at the first order in the accumulation $\delta
n_{\updownarrow}/n_{0}$ (which is a realistic case). This approximation leads
also to the first order in $\delta J_{x\updownarrow}/J_{x}^{0}$. Equation (28)
then becomes $\Delta
A\approx\frac{4\ell}{n_{0}}\left(J_{y}^{0}+\theta_{so}J_{x}^{0}\right)$ so
that the stationarity condition Eq.(35) reads:

$J_{y\updownarrow}=\mp\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\theta_{so}J_{x}^{0}$
(37)

where we have introduced the dimensionless parameter :

$\tilde{\alpha}=\frac{4\ell\alpha}{n_{0}}=\frac{4\ell^{2}}{l_{sf}^{2}(1+\theta_{so}^{2})}.$
(38)

The curve $J^{Sp}_{y}$ (Eq.(37) with Eq.(38)) is plotted in Figure 3a as a
function of the ratio $\ell/l_{sf}$. We see that the current $J_{y}^{0}$ is
composed of the same current as that expected in a Corbino disk of the same
material Benda ; Madon , i.e.
$J_{y\updownarrow}^{Corbino}=\mp\theta_{so}J_{x}^{0}$, but weighted by the
coefficient $\tilde{\alpha}/(\tilde{\alpha}+1)$, where the control parameter
$\tilde{\alpha}$ is given by Eq.(38).

  * •

Note that in the case of small spin-flip scattering, i.e. $l_{sf}/\ell>1$, the
control parameter $\tilde{\alpha}$ goes to zero and
$J_{y\updownarrow}\rightarrow 0$, which is indeed the result obtained in the
approximation treated in the previous section.

  * •

In the other limit of strong spin-flip scattering - i.e. small spin-diffusion
length $l_{sf}/\ell\ll 1$ \- the ratio $\tilde{\alpha}/(1+\tilde{\alpha})$ is
close to $1$, and the spin-currents $J_{y\updownarrow}$ are maximum, like in
the Corbino disk. The presence of this transverse pure-spin current with
$\tilde{\alpha}=1$ is the major feature of the SHE as described in previous
theories (i.e. without taking into account the out-of-equilibrium electric
screening). The present work describes the transition between the two regimes
from large to small values of the parameter $l_{sf}/\ell$.

The transverse charge and spin currents then read:

$J_{y}^{Q}(y)=0\qquad{\text{a}nd}\qquad
J_{y}^{Sp}(y)=2J^{0}_{y}=-2\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\theta_{so}\,J_{x}^{0}$
(39)

where $J_{y}^{0}=J_{y\uparrow}^{0}=-J_{y\downarrow}^{0}$.

Figure 3:  (a) Transverse pure spin current $J_{y}^{Sp}$ (Eq.(39) in the text)
as a function the ratio $\ell/l_{sf}$ of the width of the Spin-Hall bar over
the spin-flip scattering length (Eq.(39) and Eq.(38)). (b) Longitudinal pure
spin current $J_{x}^{Sp}$ as a function of the temperature (Eq.(41)) and
(Eq.(23)) for different values of the injected voltage $q\Delta V$, expressed
in $meV$. The amplitude is calculated for $y=L=\ell$. The currents are
normalized by the product of the injected current $2J_{x}^{0}$ by the spin-
Hall angle $\theta_{so}$, at the first order in $\theta_{so}$.

The longitudinal spin-currents Eq.(34) with Eq.(35) now read:

$J_{x\updownarrow}(y)=\left(\frac{n_{\updownarrow}(y)}{n_{0}}\mp\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\theta_{so}^{2}\right)J_{x}^{0}.$
(40)

The solution Eq.(17) found for $J_{x\updownarrow}$ in the case without spin-
flip is now corrected by a small term proportional to $\theta_{so}^{2}$. The
longitudinal charge current and the longitudinal spin current are given by the
expressions:

$J_{x}^{Q}(y)=J_{x}^{0}\,\left(2+\frac{\delta
n^{Q}(y)}{n_{0}}\right)\qquad{\text{a}nd}\qquad
J_{x}^{Sp}(y)=J_{x}^{0}\,\left(\frac{n^{Sp}(y)}{n_{0}}-2\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\theta_{so}^{2}\right)$
(41)

In conclusion, the stationary state for the spin-Hall bar for $l_{sf}\ll\ell$
and at the first order in $\delta n_{\updownarrow}/n$ is simply defined by
equations Eq(37) and Eq.(40), where the expression of $n_{\updownarrow}$ is
given by the solution of the Poisson law Eq.(2). At the first order in
$\theta_{so}$, the stationary states obtained in the previous section is
recovered: there is a solution of continuity between the two opposite
approximations used in this study. The following paragraphs are devoted to the
calculation of the densities $n_{\updownarrow}(y)$.

Injecting Eq.(37) and Eq.(40) into Eq.(7) in order to obtain
$\partial_{y}\mu$, the Poisson law Eq.(2) reads now, at the first order in
$\delta n_{\updownarrow}/n_{0}$:

$\,\partial_{y}^{2}(\delta
n_{\updownarrow})\mp\frac{\lambda_{SH}}{\lambda_{D}^{2}}\left(1\pm\theta_{so}^{2}\right)\,\partial_{y}(\delta
n_{\updownarrow})-\frac{\delta n^{Q}}{\lambda_{D}^{2}}=0,$ (42)

where we have define the spin-Hall characteristic length under spin-flip
scattering as:

$\lambda_{SH}\equiv\frac{\epsilon}{q^{2}\eta
n_{0}^{2}}\frac{J_{y}^{0}}{1+\theta_{so}^{2}}=-\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\,\lambda_{SH}^{0}.$
(43)

where $\lambda_{SH}^{0}$ is the spin-Hall length with negligible spin-flip
scattering defined in Eq.(22). Since the measured Spin-Hall angle verifies
$\theta_{so}\ll 1$, Eq.(42) can be approximated at the first order in
$\theta_{so}$. Summing and subtracting the equations Eq.(42) for the two
channels, we have the coupled equations:

$\frac{\partial_{y}^{2}n^{Sp}}{n_{0}}-\frac{\lambda_{SH}}{\lambda_{D}^{2}}\,\frac{\partial_{y}\delta
n^{Q}}{n_{0}}=0,$ (44) $\frac{\partial_{y}^{2}\delta
n^{Q}}{n_{0}}-\frac{\lambda_{SH}}{\lambda_{D}^{2}}\,\frac{\partial_{y}n^{Sp}}{n_{0}}-\frac{2}{\lambda_{D}^{2}}\,\frac{\delta
n^{Q}}{n_{0}}=0$ (45)

In the case of symmetrical device and electrical environment ($C_{E}=0$), the
spin-accumulation is a odd function of $y$, and the solutions of equations
Eq.(44) and Eq.(45) are:

$\frac{n^{Sp}(y)}{n_{0}}=a\,y+b\,sh\left(\frac{y}{\lambda_{m}}\right)$ (46)

and

$\frac{\delta n^{Q}(y)}{n_{0}}=c+d\,ch\left(\frac{y}{\lambda_{m}}\right)$ (47)

where $a$, $b$, $c$ and $d$ are the integration constants defined by the
symmetry of the device and the electromagnetic properties of its environment.
The fact that the charge accumulation $\delta n^{Q}(y)$ is an even function of
$y$ means that the voltage difference $\Delta V=0$ is zero between the two
edges of the device, like in the case without spin-flip (for $C_{E}=0$). The
typical spin-Hall diffusion length $\lambda_{m}$ is given by the expression:

$\lambda_{m}=\frac{1}{\sqrt{\left(\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\right)^{2}+\frac{2}{\lambda_{D}^{2}}}}$
(48)

This expression can be simplified due to the small value of the Debye-Fermi
length $\lambda_{D}$. Indeed, the maximal value is of the order of some few
microns, so that typically $\lambda_{D}^{-2}>10^{10}$ $m^{-2}$. On the other
hand, the quantitative evaluation of the ratio
$\lambda_{SH}^{0}/\lambda_{D}^{2}\approx\theta_{so}\,\left(\frac{q\Delta
V}{kT}\right)\frac{1}{L}$ has been performed in the previous section below
Eq.(23). This quantity depends on the intensity of the voltage $\Delta V$
imposed by the generator, but it is limited for experimental reasons
(electromigration), and also due to the limitation of our model (classical or
semi-classical diffusive process for which $n^{Sp}/n_{0}<1$). But even if we
where able to push the value of $\lambda_{SH}^{0}/\lambda_{D}^{2}$ to few
$10^{4}$ $m^{-1}$, its square is still negligible with respect to
$\lambda_{D}^{-2}$. Since we have $\tilde{\alpha}/(1+\tilde{\alpha})\leq 1$
whatever the value of the ratio $l_{sf}/\ell$, we expect
$\lambda_{m}\approx\lambda_{D}/\sqrt{2}$ in usual experimental situation.

Figure 4:  Spin accumulation $n^{Sp}$ (Eq.(23) in the text) as a function the
ratio $y/\ell$ for different values of the ratio $q\Delta V/kT$, from $q\Delta
V/kT=2$ to $q\Delta V/kT=12$. The spin-accumulation is normalized by the
product of the density of carriers at equilibrium $2n_{0}$ by the spin-Hall
angle $\theta_{so}$. The curve for $q\Delta V/kT=-12$ shows that reversing the
electric polarity reverses the sign of the spin-accumulation.

Taking into account the symmetry of the device and its environment
($C_{E}=0$), the global condition Eq.(11), and the result of section III for
the limit $\ell/l_{sf}\ll 1$, we have:

$a=-4\frac{\lambda_{m}^{2}}{\lambda_{D}^{2}}\,\left(\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\right)\approx-2\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}},$
(49)

so that the first term appearing in the right-hand side of Eq.(46) corresponds
the spin-accumulation without spin-flip obtained for Eq.(23) in the previous
section. The second term is given by:

$b=-2\frac{\ell\lambda_{m}^{2}}{sh\left(\frac{\ell}{\lambda_{m}}\right)}\,\left(\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\right)^{3}\,\left(\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\right)^{2}.$
(50)

The non-linear correction $b\,sh(y/\lambda_{m})$ appearing in Eq.(46) has a
maximum value of the order of $\lambda_{D}^{2}\,a^{2}$, much smaller than the
linear part (in usual experimental situations), which is of the order of
$\ell\,a$. Equation (46) thus reduced to $\frac{n^{Sp}(y)}{n_{0}}\approx
a\,y$, which is the result obtained in the previous section for the case
without spin-flip, as shown in Figure 3b and Figure 4. On the other hand, the
constant for the charge accumulation Eq.(47) is:

$c=-2\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\,\lambda_{m}^{2}\,\left(\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\right)^{2},$
(51)

which maximum value of the order of $\lambda_{D}^{2}a^{2}$, i.e. small, and

$d=2\frac{\lambda_{m}\ell}{sh\left(\frac{\ell}{\lambda_{m}}\right)}\,\left(\frac{\lambda_{SH}^{0}}{\lambda_{D}^{2}}\right)^{2}\,\left(\frac{\tilde{\alpha}}{1+\tilde{\alpha}}\right),$
(52)

so that the maximum value of the second term $d\,ch(y/\lambda_{m})$ in the
right-hand side of Eq.(47) is also of the order $\lambda_{D}^{2}a^{2}$.

Accordingly, it appears that the role of the spin-flip scattering is
negligible for both spin accumulation and charge accumulation. The unique
sizable spin-flip scattering effect in the SHE is thus the generation of the
transverse spin-current $J_{y}^{Sp}$ (already well-known in the usual drift-
diffusion description of the SHE). The result about the linear dependence on
$y$ confirms the observations performed by Bottegoni et al. Bottegoni  on
devices for which the spin-diffusion length $l_{sf}$ is large in absolute
value, but still smaller or of the same order than the width $\ell$ of the
spin-Hall bar. The measurements show that the spin-accumulation is linear in
$y$, linear in $\Delta V$, and inversely proportional to the temperature.

##  V Conclusion

The stationary state of the spin-Hall bar has been studied in the framework of
a variational approach that includes non-equilibrium screening effects. The
minimization of the Joule power dissipated in the system is performed with
taking into account the spin-flip relaxation and the two global constrains
(global galvanostatic conditions and global electroneutrality). The
calculation is performed within the two limiting cases that are the negligible
spin-flip scattering limit $l_{sf}/\ell\gg 1$ and the strong spin-flip
scattering limit $l_{sf}/\ell\ll 1$ (where $l_{sf}$ is the spin-flip
scattering length and $\ell$ is the width of the Spin-Hall bar). In both
cases, the profile of the spin-accumulation $n^{Sp}(y)$ and the spin-currents
$\vec{J}_{\updownarrow}(y)$ can be described analytically. The two limits
coincide, and simple expressions are given at the first order in the spin-
accumulation.

In the approximation of negligible spin-flip scattering, the main result is
the absence of transverse currents $J_{y\updownarrow}=0$ with a longitudinal
spin current $J_{x}^{Sp}(y)$ proportional to the spin accumulation
$n^{Sp}(y)$, while the charge current $J_{x}^{Q}$ is zero (which is the usual
definition of a “pure spin current”). The spin-accumulation $n^{Sp}(y)$ is
shown to be linear in $y$ (across the device), linear in the electric field
imposed by the generator along the $x$ axis (i.e. linear to the injected
current $J_{x}^{0}$), and inversely proportional to the temperature for non-
degenerate conductors. Note that the temperature dependence that is given by
the ratio $\theta_{so}/kT$ is the out-of-equilibrium reminiscence of the Curie
law that describes a paramagnetic system, for which the external magnetic
field has been replaced by a weak internal spin-orbit field.

In the case of strong spin-flip scattering, the main difference with the case
of negligible spin-flip is the presence of a transverse pure spin-currents
$J_{y}^{Sp}$, which is flowing across the sample (as predicted by the drift-
diffusion theory). This current is weighted by the factor
$\tilde{\alpha}/(1+\tilde{\alpha})$ (Eq.(37)), where
$\tilde{\alpha}\approx(2\ell/(l_{sf})^{2}$ (Eq.(30)). This multiplying factor
describes the transition from a strong spin-flip scattering regime to a weak
spin-flip scattering regime, for which the transverse spin-current vanishes.

The most surprising result of this study is that the spin-flip scattering does
not change significantly the stationary state of the Spin-Hall effect (for
standard experimental situations), except by the presence of the a transverse
pure-spin-current $J_{y}^{Sp}$. Indeed, the longitudinal spin current
$J_{x}^{Sp}(y)$ is approximatively the same as for the case without spin-flip
scattering for two reasons. The first reason is that the supplementary term
due to spin-flip scattering is at the second order in the spin-Hall angle
$\theta_{so}^{2}$, and this term is negligible in general. The second reason
is that the spin-accumulation $n^{Sp}(y)$ is also approximatively unchanged
whatever the value of the ratio $\ell/l_{sf}$, due to the small value of the
Debye-Fermi length scale $\lambda_{D}$.

Furthermore, it is shown that if the Spin-Hall bar is asymmetric, a spin-
dependent voltage is generated between the two edges. This voltage could then
be measured if a layer (either oxide, semiconductor, or metal) is deposited on
one side of the device.

In conclusion, in all cases, the stationary state is defined by a spin-
accumulation $n^{Sp}(y)$ (and the inhomogeneous part of the pure-spin-current
$J_{x}^{Sp}\propto n^{Sp}(y)$) which is linearly distributed across the spin-
Hall-bar (i.e. along the $y$ direction), linear in the applied electric
potential $\Delta V$, and proportional to the inverse of the temperature for
non-degenerate conductors. These surprising characteristics are observed in
the measurements performed by Bottegoni et al Bottegoni  in a series of
measurements, for which $\ell/l_{sf}$ is slightly greater than one

##  VI Acknowledgement

We thank Felix Faisant for his important contribution to the initial
development of this work, Pierre-Michel déjardin for the stochastic
description, and Serge Boiziau for his help and for valuable discussions.

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