# On Sub-optimality of Random Binning for Distributed Hypothesis Testing

Shun Watanabe1  1Department of Computer and Information Sciences, Tokyo
University of Agriculture and Technology, Japan,
<EMAIL_ADDRESS>
###### Abstract

We investigate the quantize and binning scheme, known as the Shimokawa-Han-
Amari (SHA) scheme, for the distributed hypothesis testing. We develop tools
to evaluate the critical rate attainable by the SHA scheme. For a product of
binary symmetric double sources, we present a sequential scheme that improves
upon the SHA scheme.

##  I Introduction

In [1], Berger introduced a framework of statistical decision problems under
communication constraint. Inspired by his work, many researchers studied
various problems of this kind [2, 3, 4, 5, 6, 7, 8]; see [9] for a thorough
review until 90s. More recently, the problem of communication constrained
statistics has regained interest of researchers; see [10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24] and references therein.

In this paper, we are interested in the most basic setting of hypothesis
testing in which the sender and the receiver observe correlated sources
$X^{n}$ and $Y^{n}$, respectively, and the observation $(X^{n},Y^{n})$ is
distributed according to the product of the null hypothesis $P_{XY}$ or the
alternative hypothesis $Q_{XY}$; the sender transmit a message at rate $R$,
and the receiver decide the hypotheses based on the message and its
observation.

In [2], Ahlswede and Csiszár introduced the so-called quantization scheme, and
showed that the quantization scheme is optimal for the testing against
independence, i.e., the alternative hypothesis is $Q_{XY}=P_{X}\times P_{Y}$.
In [3], Han introduced an improved version of the quantization scheme that
fully exploit the knowledge of the marginal types. In [25], Shimokawa, Han,
and Amari introduced the quantize and binning scheme, which we shall call the
SHA scheme in the following. For a long time, it has not been known if the SHA
is optimal or not. In [13], Rahman and Wagner showed that the SHA scheme is
optimal for the testing against conditional independence.111In fact, for the
testing against conditional independence, it suffices to consider a simpler
version of quantize and binning scheme than the SHA scheme.  More recently,
Weinberger and Kochman [17] (see also [26]) derived the optimal exponential
trade-off between the type I and type II error probabilities of the quantize
and binning scheme. Since the decision is conducted based on the optimal
likelihood decision rule in [17], their scheme may potentially provide a
better performance than the SHA scheme; however, since the expression involves
complicated optimization over multiple parameters, and a strict improvement
has not been clarified so far.

A main objective of this paper is to investigate if the SHA scheme is optimal
or not. In fact, it turns out that the answer is negative, and there exists a
scheme that improves upon the SHA scheme. Since the general SHA scheme is
involved, we focus on the critical rate, i.e., the minimum rate that is
required to attain the same type II exponent as the Stein exponent of the
centralized testing. Then, under a certain non-degeneracy condition, we will
prove that the auxiliary random variable in the SHA scheme must coincide with
the source $X$ itself, i.e., we must use the binning without quantization. By
leveraging this simplification, for the binary symmetric double sources
(BSDS), we derive a closed form expression of the critical rate attainable by
the SHA scheme. Next, we will consider a product of the BSDS such that each
component of the null hypothesis and the alternative hypothesis is “reversely
aligned.” Perhaps surprisingly, it turns out that the SHA is sub-optimal for
such hypotheses. Our improved scheme is simple, and it uses the SHA scheme for
each component in a sequential manner; however, it should be emphasized that
we need to conduct binning of each component separately, and our scheme is not
a naive random binning.

##  II Problem Formulation

We consider the statistical problem of testing the null hypothesis
$\mathtt{H}_{0}:P_{XY}$ on finite alphabets ${\cal X}\times{\cal Y}$ versus
the alternative hypothesis $\mathtt{H}_{1}:Q_{XY}$ on the same alphabet. The
i.i.d. random variables $(X^{n},Y^{n})$ distributed according to either
$P_{XY}^{n}$ or $Q_{XY}^{n}$ are observed by the sender and the receiver; the
sender encodes the observation $X^{n}$ to a message by an encoding function

$\displaystyle\varphi_{n}:{\cal X}^{n}\to{\cal M}_{n},$

and the receiver decides whether to accept the null hypothesis or not by a
decoding function

$\displaystyle\psi_{n}:{\cal M}_{n}\times{\cal
Y}^{n}\to\\{\mathtt{H}_{0},\mathtt{H}_{1}\\}.$

When the block length $n$ is obvious from the context, we omit the subscript
$n$. For a given testing scheme $T_{n}=(\varphi,\psi)$, the type I error
probability is defined by

$\displaystyle\alpha[T_{n}]:=P\bigg{(}\psi(\varphi(X^{n}),Y^{n})=\mathtt{H}_{1}\bigg{)}$

and the type II error probability is defined by

$\displaystyle\beta[T_{n}]:=Q\bigg{(}\psi(\varphi(X^{n}),Y^{n})=\mathtt{H}_{0}\bigg{)}.$

In the following, $P(\cdot)$ (or $Q(\cdot)$) means that $(X^{n},Y^{n})$ is
distributed according to $P_{XY}^{n}$ (or $Q_{XY}^{n}$).

In the problem of distributed hypothesis testing, we are interested in the
trade-off among the communication rate and the type I and type II error
probabilities. Particularly, we focus on the optimal trade-off between the
communication rate and the exponent of the type II error probability when the
type I error probability is vanishing, which is sometimes called “Stein
regime”.

###### Definition 1

A pair $(R,E)$ of the communication rate and the exponent is defined to be
achievable if there exists a sequence $\\{T_{n}\\}_{n=1}^{\infty}$ of testing
schemes such that

$\displaystyle\lim_{n\to\infty}\alpha[T_{n}]=0$

and

$\displaystyle\limsup_{n\to\infty}\frac{1}{n}\log|{\cal M}_{n}|$
$\displaystyle\leq R,$
$\displaystyle\liminf_{n\to\infty}-\frac{1}{n}\log\beta_{n}[T_{n}]$
$\displaystyle\geq E.$

The achievable region ${\cal R}$ is the set of all achievable pair $(R,E)$.

For a given communication rate $R$, the maximum exponent is denoted by

$\displaystyle E(R):=\max\\{E:(R,E)\in{\cal R}\\}.$

When there is no communication constraint, i.e., $R\geq\log|{\cal X}|$, the
sender can send the full observation without any encoding, and the receiver
can use a scheme for the standard (centralized) hypothesis testing. In such a
case, the Stein exponent $D(P_{XY}\|Q_{XY})$ is attainable. We define the
critical rate as the minimum communication rate required to attain the Stein
exponent:

$\displaystyle R_{\mathtt{cr}}:=\inf\\{R:E(R)=D(P_{XY}\|Q_{XY})\\}.$

##  III On Evaluation of SHA Scheme

When the communication rate $R$ is not sufficient to send $X$, the standard
approach is to quantize $X$ using a test channel $P_{U|X}$ for some finite
auxiliary alphabet ${\cal U}$. For $P_{UXY}=P_{U|X}P_{XY}$ and
$Q_{UXY}=Q_{U|X}Q_{XY}$ with $Q_{U|X}=P_{U|X}$, let

$\displaystyle E(P_{UXY}\|Q_{UXY})$
$\displaystyle:=\min\big{\\{}D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|Q_{UXY}):P_{\tilde{U}\tilde{X}}=P_{UX},P_{\tilde{U}\tilde{Y}}=P_{UY}\big{\\}}.$

In [3], a testing scheme based on quantization was proposed, and the following
achievability bound was derived: for any test channel $P_{U|X}$,

$\displaystyle E(R)\geq E(P_{UXY}\|Q_{UXY}).$

In [25], a testing scheme based on quantization and binning was proposed, and
the following achievability bound was derived: for any test channel $P_{U|X}$
such that $R\geq I(U\wedge X|Y)$,

$\displaystyle E(R)$
$\displaystyle\geq\min\bigg{[}\min_{P_{\tilde{U}\tilde{X}\tilde{Y}}\in{\cal
P}_{\mathtt{b}}(P_{U|X})}D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|Q_{UXY})$
$\displaystyle~{}~{}~{}~{}~{}~{}+|R-I(U\wedge
X|Y)|^{+},E(P_{UXY}\|Q_{UXY})\bigg{]},$

where $|t|^{+}=\max[t,0]$, and

$\displaystyle{\cal P}_{\mathtt{b}}(P_{U|X})$
$\displaystyle:=\big{\\{}P_{\tilde{U}\tilde{X}\tilde{Y}}:$
$\displaystyle~{}~{}~{}P_{\tilde{U}\tilde{X}}=P_{UX},P_{\tilde{Y}}=P_{Y},H(\tilde{U}|\tilde{Y})\geq
H(U|Y)\big{\\}}.$

Particularly, by taking $P_{U|X}$ to be noiseless test channel, i.e., $U=X$,
we have

$\displaystyle E(R)$ $\displaystyle\geq E_{\mathtt{b}}(R)$ (1)
$\displaystyle:=\min\bigg{[}\min_{P_{\tilde{X}\tilde{Y}}\in{\cal
P}_{\mathtt{b}}}D(P_{\tilde{X}\tilde{Y}}\|Q_{XY})+|R-H(X|Y)|^{+},$
$\displaystyle~{}~{}~{}~{}~{}~{}D(P_{XY}\|Q_{XY})\bigg{]},$ (2)

where

$\displaystyle{\cal
P}_{\mathtt{b}}:=\big{\\{}P_{\tilde{X}\tilde{Y}}:P_{\tilde{X}}=P_{X},P_{\tilde{Y}}=P_{Y},H(\tilde{X}|\tilde{Y})\geq
H(X|Y)\big{\\}}.$

In the following, we focus on the critical rate. Under some mild conditions,
in order to attain the Stein exponent, we must take $U=X$ in the quantization-
bining bound, i.e., no quantization. In the rest of this section, we prove
this claim.

Suppose that $P_{XY}$ and $Q_{XY}$ have full support. Let $\hat{\Lambda}$ be a
function on ${\cal X}\times{\cal Y}$ defined by222We suppose that alphabets
are ${\cal X}=\\{0,\ldots,m_{\mathsf{x}}\\}$ and ${\cal
Y}=\\{0,\ldots,m_{\mathsf{y}}\\}$; taking $y=0$ as a special symbol in (3) is
just notational convenience.

$\displaystyle\hat{\Lambda}(x,y):=\log\frac{P_{XY}(x,y)}{Q_{XY}(x,y)}-\log\frac{P_{XY}(x,0)}{Q_{XY}(x,0)}.$
(3)

###### Proposition 2

Suppose that, for every $x\neq x^{\prime}$, rows $\hat{\Lambda}(x,\cdot)$ and
$\hat{\Lambda}(x^{\prime},\cdot)$ are distinct. Then,
$E(P_{UXY}\|Q_{UXY})=D(P_{XY}\|Q_{XY})$ only if there does not exist
$u\in{\cal U}$ and $x\neq x^{\prime}$ such that
$P_{U|X}(u|x)P_{U|X}(u|x^{\prime})>0$.

###### Proof:

Since we assumed that $P_{XY}$ and $Q_{XY}$ have full support and since
$P_{U|X}=Q_{U|X}$, $P_{UXY}$ and $Q_{UXY}$ have the same support, which we
denote by ${\cal A}$. Let ${\cal
A}^{\prime}=\mathtt{supp}(P_{UX})=\mathtt{supp}(Q_{UX})$. Note that $P_{UY}$
and $Q_{UY}$ have full support. Let ${\cal L}\subseteq{\cal P}({\cal A})$ be
the linear family of distributions $P_{\tilde{U}\tilde{X}\tilde{Y}}$
satisfying

$\displaystyle\sum_{(u,x,y)\in{\cal
A}}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)\delta_{\bar{u}\bar{x}}(u,x)=P_{UX}(\bar{u},\bar{x})$

for every $(\bar{u},\bar{x})\in{\cal A}^{\prime}$ and

$\displaystyle\sum_{(u,x,y)\in{\cal
A}}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)\delta_{\bar{u}\bar{y}}(u,y)=P_{UY}(\bar{u},\bar{y}),$

for every $(\bar{u},\bar{y})\in{\cal U}\times{\cal Y}$. Then, we can write
$E(P_{UXY}\|Q_{UXY})$ as

$\displaystyle
E(P_{UXY}\|Q_{UXY})=\min_{P_{\tilde{U}\tilde{X}\tilde{Y}}\in{\cal
L}}D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|Q_{UXY}).$ (4)

If

$\displaystyle E(P_{UXY}\|Q_{UXY})=D(P_{XY}\|Q_{XY})=D(P_{UXY}\|Q_{UXY}),$

then $P_{UXY}$ must be the (unique) minimizer in (4). Then, the I-projection
theorem (cf. [27, Theorem 3.2]) implies that the minimizer $P_{UXY}$ lies in
the exponential family generated by $Q_{UXY}$ and the constraints of ${\cal
L}$, i.e., it has of the form

$\displaystyle P_{UXY}(u,x,y)$ $\displaystyle\propto
Q_{UXY}(u,x,y)\exp\bigg{[}\sum_{\bar{u},\bar{x}}\theta_{\bar{u}\bar{x}}\delta_{\bar{u}\bar{x}}(u,x)+\sum_{\bar{u},\bar{y}}\theta_{\bar{u}\bar{y}}\delta_{\bar{u}\bar{y}}(u,y)\bigg{]}.$

for some $\theta_{\bar{u}\bar{x}},\theta_{\bar{u}\bar{y}}\in\mathbb{R}$.
Equivalently, this condition can be written as

$\displaystyle\log\frac{P_{UXY}(u,x,y)}{Q_{UXY}(u,x,y)}=g_{1}(u,x)+g_{2}(u,y)$

for some functions $g_{1}$ on ${\cal A}^{\prime}$ and $g_{2}$ on ${\cal
U}\times{\cal Y}$. If there exist $u\in{\cal U}$ and $x\neq x^{\prime}$ such
that $P_{U|X}(u|x)P_{U|X}(u|x^{\prime})>0$, then, for every $y\in{\cal Y}$, we
have

$\displaystyle\hat{\Lambda}(x,y)-\hat{\Lambda}(x^{\prime},y)$
$\displaystyle=\bigg{[}\log\frac{P_{XY}(x,y)}{Q_{XY}(x,y)}-\log\frac{P_{XY}(x,0)}{Q_{XY}(x,0)}\bigg{]}$
$\displaystyle~{}~{}~{}-\bigg{[}\log\frac{P_{XY}(x^{\prime},y)}{Q_{XY}(x^{\prime},y)}-\log\frac{P_{XY}(x^{\prime},0)}{Q_{XY}(x^{\prime},0)}\bigg{]}$
$\displaystyle=\bigg{[}\log\frac{P_{UXY}(u,x,y)}{Q_{UXY}(u,x,y)}-\log\frac{P_{UXY}(u,x,0)}{Q_{UXY}(u,x,0)}\bigg{]}$
$\displaystyle~{}~{}~{}-\bigg{[}\log\frac{P_{UXY}(u,x^{\prime},y)}{Q_{UXY}(u,x^{\prime},y)}-\log\frac{P_{UXY}(u,x^{\prime},0)}{Q_{UXY}(u,x^{\prime},0)}\bigg{]}$
$\displaystyle=\bigg{[}g_{1}(u,x)+g_{2}(u,y)-g_{1}(u,x)-g_{2}(u,0)\bigg{]}$
$\displaystyle~{}~{}~{}-\bigg{[}g_{1}(u,x^{\prime})+g_{2}(u,y)-g_{1}(u,x^{\prime})-g_{2}(u,0)\bigg{]}$
$\displaystyle=0,$

which contradict the assumption that $\hat{\Lambda}(x,\cdot)$ and
$\hat{\Lambda}(x^{\prime},\cdot)$ are distinct. ∎

Since the quantization-binning bound is at most $E(P_{UXY}\|Q_{UXY})$, when
the assumption of Proposition 2 is satisfied, $U=X$ is the only choice of test
channel to attain the Stein exponent.333Even though we can take a redundant
test channel $P_{U|X}$ such that
$\mathtt{supp}(P_{U|X}(\cdot|x))\cap\mathtt{supp}(P_{U|X}(\cdot|x^{\prime}))=\emptyset$
for any $x\neq x^{\prime}$, such a test channel does not improve the
attainable exponent.

In Proposition 2, the assumption that every raws of $\hat{\Lambda}(x,y)$ are
distinct cannot be replaced by the assumption that every raws of the log-
likelihood function

$\displaystyle\Lambda(x,y)=\log\frac{P_{XY}(x,y)}{Q_{XY}(x,y)}$

are distinct; for instance, when $P_{XY}$ and $Q_{XY}$ are given by

$\displaystyle P_{XY}$
$\displaystyle=\frac{1}{24a}\left[\begin{array}[]{ccc}a&2a&3a\\\ 2a&3a&3a\\\
a&3a&6a\end{array}\right],$ $\displaystyle Q_{XY}$
$\displaystyle=\frac{1}{14a}\left[\begin{array}[]{ccc}2a&a&a\\\ a&a&a\\\
a&2a&4a\end{array}\right]$

for some $a>0$, then every raws $\Lambda(x,\cdot)$ are distinct but
$\hat{\Lambda}(1,\cdot)=\hat{\Lambda}(2,\cdot)$. In fact, as is shown in the
following Proposition 3, the symbols $x=1$ and $x=2$ in this example can be
merged while maintaining $E(P_{UXY}\|Q_{UXY})=D(P_{XY}\|Q_{XY})$. Even though
Proposition 3 is not used in the rest of the paper, it may be of independent
interest.

###### Proposition 3

Let $\kappa:{\cal X}\to{\cal U}$ be a function such that
$\kappa(x)=\kappa(x^{\prime})$ if and only if
$\hat{\Lambda}(x,\cdot)=\hat{\Lambda}(x^{\prime},\cdot)$, and let
$U=\kappa(X)$. Then, for any $P_{\tilde{U}\tilde{X}\tilde{Y}}$ satisfying
$P_{\tilde{U}\tilde{X}}=P_{UX}$ and $P_{\tilde{U}\tilde{Y}}=P_{UY}$, we have

$\displaystyle D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|Q_{UXY})$
$\displaystyle=D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|P_{UXY})+D(P_{UXY}\|Q_{UXY}).$
(5)

Particularly, we have

$\displaystyle E(P_{UXY}\|Q_{UXY})=D(P_{XY}\|Q_{XY}).$ (6)

###### Proof:

Since (6) follows from (5), it suffices to prove (5). For each $u$, fix
$x_{u}\in{\cal X}$ such that $\kappa(x_{u})=u$. Then, we have

$\displaystyle
D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|Q_{UXY})-D(P_{\tilde{U}\tilde{X}\tilde{Y}}\|P_{UXY})-D(P_{UXY}\|Q_{UXY})$
$\displaystyle=\sum_{u,x,y}\big{(}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)-P_{UXY}(u,x,y)\big{)}\log\frac{P_{UXY}(u,x,y)}{Q_{UXY}(u,x,y)}$
$\displaystyle=\sum_{u,x,y}\big{(}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)-P_{UXY}(u,x,y)\big{)}\log\frac{P_{XY}(x,y)}{Q_{XY}(x,y)}$
$\displaystyle=\sum_{u,x,y}\big{(}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)-P_{UXY}(u,x,y)\big{)}\hat{\Lambda}(x,y)$
$\displaystyle=\sum_{u,x,y}\big{(}P_{\tilde{U}\tilde{X}\tilde{Y}}(u,x,y)-P_{UXY}(u,x,y)\big{)}\hat{\Lambda}(x_{u},y)$
$\displaystyle=\sum_{u,y}\big{(}P_{\tilde{U}\tilde{Y}}(u,y)-P_{UY}(u,y)\big{)}\hat{\Lambda}(x_{u},y)$
$\displaystyle=0,$

where the third equality follows from $P_{\tilde{X}}=P_{X}$ and the last
equality follows from $P_{\tilde{U}\tilde{Y}}=P_{UY}$. ∎

###### Remark 4

Introducing the function $\Lambda(x,y)$ as in (3) is motivated by the problem
of distributed function computing. Note that, in order to conduct the
likelihood ratio test, the receiver need not compute the log-likelihood ratios
$(\Lambda(x_{i},y_{i}):1\leq i\leq n)$ for a given observation
$(\bm{x},\bm{y})\in{\cal X}^{n}\times{\cal Y}^{n}$; instead, it suffices to
compute the summation

$\displaystyle\Lambda_{n}(\bm{x},\bm{y})=\sum_{i=1}^{n}\Lambda(x_{i},y_{i}).$

In the terminology of distributed computing [28, Definition 4], we can verify
that the function $\Lambda_{n}$ is $\overline{{\cal
X}}_{\hat{\Lambda}}$-informative.

##  IV Binary Symmetric Double Source (BSDS)

Let ${\cal X}={\cal Y}=\\{0,1\\}$, and let $P_{XY}$ and $Q_{XY}$ be given by

$\displaystyle P_{XY}=\left[\begin{array}[]{cc}\frac{1-p}{2}&\frac{p}{2}\\\
\frac{p}{2}&\frac{1-p}{2}\end{array}\right],~{}~{}~{}Q_{XY}=\left[\begin{array}[]{cc}\frac{1-q}{2}&\frac{q}{2}\\\
\frac{q}{2}&\frac{1-q}{2}\end{array}\right]$

for $0\leq p,q\leq 1$ and $p\neq q$. Since
$\hat{\Lambda}(0,0)=\hat{\Lambda}(1,0)=0$ and

$\displaystyle\hat{\Lambda}(0,1)=\log\frac{(1-q)p}{(1-p)q}\neq\log\frac{(1-p)q}{(1-q)p}=\hat{\Lambda}(1,1),$

the assumption of Proposition 2 is satisfied. Thus, we need to set $U=X$ in
the quantization-binning bound to attain the Stein exponent. In the
minimization of (1), since $P_{\tilde{X}}=P_{X}$ and $P_{\tilde{Y}}=P_{Y}$ are
the uniform distribution, the only free parameter is the crossover probability
$P_{\tilde{Y}|\tilde{X}}(1|0)=P_{\tilde{Y}|\tilde{X}}(0|1)=\tilde{p}$. Then,
we can easily find that the binning bound for BSDS is

$\displaystyle E_{\mathtt{b}}(R)$
$\displaystyle=\left\\{\begin{array}[]{ll}\min[|R-h(p)|^{+},D(p\|q)]&\mbox{ if
}h(p)\leq h(q)\\\ D(p\|q)&\mbox{ if }h(p)>h(q)\end{array}\right.$ (9)

for $R\geq h(p)$. Furthermore, this bound implies that the critical rate
attainable by the SHA scheme is

$\displaystyle
R_{\mathtt{cr}}\leq\left\\{\begin{array}[]{ll}h(p)+D(p\|q)&\mbox{ if }h(p)\leq
h(q)\\\ h(p)&\mbox{ if }h(p)>h(q)\end{array}\right..$ (12)

##  V Product of BSDS

Next, let us consider the case where $P_{XY}=P_{X_{1}Y_{1}}\times
P_{X_{2}Y_{2}}$ and $Q_{XY}=Q_{X_{1}Y_{1}}\times Q_{X_{2}Y_{2}}$, and
$P_{X_{i}Y_{i}}$ and $Q_{X_{i}Y_{i}}$ for $i=1,2$ are DSBSs with parameters
$0\leq p_{1},p_{2},q_{1},q_{2}\leq 1$, respectively. Particularly, we consider
the case where $p_{1}=q_{2}$, $p_{2}=q_{1}$, where $p_{1}\neq q_{1}$. Note
that, for the product source, the function defined by (3) can be decomposed as

$\displaystyle\hat{\Lambda}(x_{1}x_{2},y_{1}y_{2})=\hat{\Lambda}_{1}(x_{1},y_{1})+\hat{\Lambda}_{2}(x_{2},y_{2}),$

where

$\displaystyle\hat{\Lambda}_{i}(x_{i},y_{i})=\log\frac{P_{X_{i}Y_{i}}(x_{i},y_{i})}{Q_{X_{i}Y_{i}}(x_{i},y_{i})}-\log\frac{P_{X_{i}Y_{i}}(x_{i},0)}{Q_{X_{i}Y_{i}}(x_{i},0)}.$

Furthermore, we have $\hat{\Lambda}_{1}(1,1)=-\hat{\Lambda}_{1}(0,1)$,
$\hat{\Lambda}_{2}(0,1)=-\hat{\Lambda}_{1}(0,1)$, and
$\hat{\Lambda}_{2}(1,1)=\hat{\Lambda}_{1}(0,1)$. Thus, by denoting
$a=\hat{\Lambda}_{1}(0,1)$, we have

$\displaystyle\hat{\Lambda}(x_{1}x_{2},y_{1}y_{2})=\left[\begin{array}[]{cccc}0&-a&a&0\\\
0&a&a&2a\\\ 0&-a&-a&-2a\\\ 0&a&-a&0\end{array}\right],$

and the assumption of Proposition 2 is satisfied. Thus, we need to set $U=X$
in the quantization-binning bound to attain the Stein exponent.

Since $p_{1}=q_{2}$ and $p_{2}=q_{1}$, the conditional entropies $H_{P}(X|Y)$
and $H_{Q}(X|Y)$ with respect to $P_{XY}$ and $Q_{XY}$ satisfy
$H_{P}(X|Y)=H_{Q}(X|Y)$. Thus, we find that the inner minimization of the
binning bound is attained by $P_{\tilde{X}\tilde{Y}}=Q_{XY}$, and

$\displaystyle E_{\mathtt{b}}(R)$
$\displaystyle=\min\big{[}|R-h(p_{1})-h(p_{2})|^{+},D(p_{1}\|q_{1})+D(p_{2}\|q_{2})\big{]}.$

Furthermore, this bound implies that the critical rate attainable by the SHA
scheme is

$\displaystyle R_{\mathtt{cr}}\leq
h(p_{1})+h(p_{2})+D(p_{1}\|q_{1})+D(p_{2}\|q_{2}).$ (13)

##  VI Sequential Scheme for Product of BSDS

Now, we describe a modified version of SHA scheme for the product of BSDS. For
$i=1,2$, let $T_{\mathtt{b}}^{[i]}=(\varphi^{[i]},\psi^{[i]})$ be the SHA
scheme (without quantization) for $P_{X_{i}Y_{i}}$ versus $Q_{X_{i}Y_{i}}$.
For a given rate $R$, we split the rate as $R=R_{1}+R_{2}$, and consider
sequential scheme $\tilde{T}_{\mathtt{b}}=(\varphi,\psi)$ constructed from
$(T_{\mathtt{b}}^{[1]},T_{\mathtt{b}}^{[2]})$ as follows. Upon observing
$\bm{x}=(\bm{x}_{1},\bm{x}_{2})$, the sender transmit
$m_{1}=\varphi^{[1]}(\bm{x}_{1})$ and
$m_{2}=\varphi^{[2]}(\bm{x}_{2})$.444Note that $m_{1}$ and $m_{2}$ include
indices of random binning as well as marginal types of $\bm{x}_{1}$ and
$\bm{x}_{2}$. Then, upon receiving $(m_{1},m_{2})$ and observing
$\bm{y}=(\bm{y}_{1},\bm{y}_{2})$, the receiver decides
$\psi(m_{1},m_{2},\bm{y})=\mathtt{H}_{0}$ if
$\psi^{[1]}(m_{1},\bm{y}_{1})=\mathtt{H}_{0}$ and
$\psi^{[2]}(m_{2},\bm{y}_{2})=\mathtt{H}_{0}$. The type I and type II error
probabilities of this sequential scheme $\tilde{T}_{\mathtt{b}}$ can be
evaluated as

$\displaystyle\alpha[\tilde{T}_{\mathtt{b}}]$
$\displaystyle=P\bigg{(}\psi^{[1]}(\varphi^{[1]}(X_{1}^{n}),Y_{1}^{n})=\mathtt{H}_{1}\vee\psi^{[2]}(\varphi^{[2]}(X_{2}^{n}),Y_{2}^{n})=\mathtt{H}_{1}\bigg{)}$
$\displaystyle\leq\sum_{i=1}^{2}P\bigg{(}\psi^{[i]}(\varphi^{[i]}(X_{i}^{n}),Y_{i}^{n})=\mathtt{H}_{1}\bigg{)}$

by the union bound, and

$\displaystyle\beta[\tilde{T}_{\mathtt{b}}]$
$\displaystyle=Q\bigg{(}\psi^{[1]}(\varphi^{[1]}(X_{1}^{n}),Y_{1}^{n})=\mathtt{H}_{0},~{}\psi^{[2]}(\varphi^{[2]}(X_{2}^{n}),Y_{2}^{n})=\mathtt{H}_{0}\bigg{)}$
$\displaystyle=\prod_{i=1}^{2}Q\bigg{(}\psi^{[i]}(\varphi^{[i]}(X_{i}^{n}),Y_{i}^{n})=\mathtt{H}_{0}\bigg{)}.$

Thus, if the type I error probabilities of each scheme is vanishing, then the
type I error probability $\alpha[\tilde{T}_{\mathtt{b}}]$ is also vanishing.
On the other hand, the type II exponent of the sequential scheme is the
summation of the type II exponent of each scheme.

As in Section V, suppose that $p_{1}=q_{2}$ and $p_{2}=q_{1}$. Furthermore,
let $h(q_{1})<h(p_{1})$. Then, from the above argument, and (9), we find that
the following exponent is attainable by the sequential scheme:

$\displaystyle\tilde{E}_{\mathtt{b}}(R)=D(p_{1}\|q_{1})+\min[|R_{2}-h(p_{2})|^{+},D(p_{2}\|q_{2})]$
(14)

for $R_{1}\geq h(p_{1})$ and $R_{2}\geq h(p_{2})$. By setting $R_{1}=h(p_{1})$
and $R_{2}=h(p_{2})+D(p_{2}\|q_{2})$, we can derive the following bound on the
critical rate:

$\displaystyle R_{\mathtt{cr}}\leq h(p_{1})+h(p_{2})+D(p_{2}\|q_{2}).$ (15)

Interestingly, the bound on the critical rate in (15) improves upon the bound
in (13).

From an operational point of view, this improvement can be explained as
follows. In the SHA scheme (without quantization), we use the minimum entropy
decoder to compute an estimate $\hat{\bm{x}}$ of $X^{n}=\bm{x}$; then, if the
joint type $\mathsf{t}_{\hat{\bm{x}}\bm{y}}$ is close to $P_{XY}$, we accept
the null hypothesis. When the empirical conditional entropy $H(\bm{x}|\bm{y})$
is such that $H_{P}(X|Y)\gnsim H(\bm{x}|\bm{y})$, even if there exists
$\hat{\bm{x}}$ satisfying $H(\bm{x}|\bm{y})\geq H(\hat{\bm{x}}|\bm{y})$, the
type II testing error does not occur though the receiver may erroneously
compute $\hat{\bm{x}}\neq\bm{x}$.555Note that $H_{P}(X|Y)\gnsim
H(\bm{x}|\bm{y})\geq H(\hat{\bm{x}}|\bm{y})$ implies that
$\mathsf{t}_{\hat{\bm{x}}\bm{y}}$ is not close to $P_{XY}$. Consequently, the
type II testing error may occur only if $(X^{n},Y^{n})=(\bm{x},\bm{y})$
satisfies $H(\bm{x}|\bm{y})\gtrsim H_{P}(X|Y)$.

For the product of BSDSs, when the types of modulo sums
$\bm{x}_{1}+\bm{y}_{1}$ and $\bm{x}_{2}+\bm{y}_{2}$ are
$H(\mathsf{t}_{\bm{x}_{1}+\bm{y}_{1}})\simeq h(\tau_{1})$ and
$H(\mathsf{t}_{\bm{x}_{2}+\bm{y}_{2}})\simeq h(\tau_{2})$, the type II testing
error may occur only if $h(\tau_{1})+h(\tau_{2})\gtrsim h(p_{1})+h(p_{2})$;
see Fig. 1a(a). In the modified SHA scheme, $\bm{x}_{1}$ and $\bm{x}_{2}$ are
binned and estimated separately. Because of this, the constraint of type II
testing error event become more stringent, i.e., $h(\tau_{1})\gtrsim h(p_{1})$
and $h(\tau_{2})\gtrsim h(p_{2})$; see Fig. Fig. 1a(b).

(a)

(b)

Figure 1: The meshed area describes the range of $h(\tau_{1})$ and
$h(\tau_{2})$ such that the type II testing error may occur when
$H(\mathsf{t}_{\bm{x}_{1}+\bm{y}_{1}})\simeq h(\tau_{1})$ and
$H(\mathsf{t}_{\bm{x}_{2}+\bm{y}_{2}})\simeq h(\tau_{2})$ for (a) SHA scheme
and (b) modified SHA scheme.

##  VII Discussion

In this paper, we have developed tools to evaluate the critical rate
attainable by the SHA scheme, and exemplified that the SHA scheme is sub-
optimal for a product of BSDSs. A future problem is to generalize the idea of
sequential scheme in Section VI and develop a new achievability scheme for the
distributed hypothesis testing. Another future problem is a potential utility
of the scheme in [17]; as we mentioned in Section I, it has a potential to
improve upon the SHA scheme, but strict improvement is not clear since the
expression involves complicated optimization over multiple parameters.

Lastly, it should be pointed out that certain kinds of “structured” random
coding is known to improve upon the naive random coding in the context of the
mismatched decoding problem; eg. see [29, 30] and references therein. Even
though there is no direct connection between the two problems, it might be
interesting to pursue a mathematical connection between these problems.

## Acknowledgment

The author appreciates Marat Burnashev for valuable discussions at an early
stage of the work. The author also thanks Te Sun Han and Yasutada Oohama for
fruitful discussions.

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# Three-dimensional charge transport mapping by two-photon absorption edge
transient-current technique in synthetic single-crystalline diamond

C. Dorfer<EMAIL_ADDRESS>Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  D. Hits  Institute for
Particle Physics and Astrophysics, ETH Zurich, 8093 Zurich, Switzerland  L.
Kasmi  Institute of Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
G. Kramberger  Department of Experimental Particle Physics, Jožef Stefan
Institute and Department of Physics, University of Ljubljana, 1000 Ljubljana,
Slovenia  M. Lucchini  Dipartimento di Fisica, Politecnico Milano, 20133
Milano, Italy  M. Mikuž  Department of Experimental Particle Physics, Jožef
Stefan Institute and Department of Physics, University of Ljubljana, 1000
Ljubljana, Slovenia  R. Wallny  Institute for Particle Physics and
Astrophysics, ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

We demonstrate the application of two-photon absorption transient current
technique to wide bandgap semiconductors. We utilize it to probe charge
transport properties of single-crystal Chemical Vapor Deposition (scCVD)
diamond. The charge carriers, inside the scCVD diamond sample, are excited by
a femtosecond laser through simultaneous absorption of two photons. Due to the
nature of two-photon absorption, the generation of charge carriers is confined
in space (3-D) around the focal point of the laser. Such localized charge
injection allows to probe the charge transport properties of the semiconductor
bulk with a fine-grained 3-D resolution. Exploiting spatial confinement of the
generated charge, the electrical field of the diamond bulk was mapped at
different depths and compared to an X-ray diffraction topograph of the sample.
Measurements utilizing this method provide a unique way of exploring spatial
variations of charge transport properties in transparent wide-bandgap
semiconductors.

The development of electronic devices based on wide-bandgap semiconductors, as
for example, for power electronicsgreen_diamond  or charged particle detectors
Lukas_PSD , requires a detailed knowledge of the charge carrier transport
properties in those materials. In this letter we present an extension of the
known transient current technique (TCT) to obtain such knowledge in wide
bandgap materials. It is based on generating charge carriers in the bulk of a
wide-bandgap semiconductor through two-photon absorption (TPA) of a
femtosecond laser pulse. The transient current pulse formed by the movement of
the carriers in the electric field of the bulk is then observed and analyzed.

The laser beam enters through the edge of the sample, thus giving the name to
the method: edge-TCT. The ability to generate, within femtoseconds, spatially
confined charge carriers in any point within the specimen accessible by a
laser beam, allows to disentangle the influence of local variations in the
electric field from trapping of free charge carriers. This gives TPA edge-TCT
a significant advantage over the traditional TCT methods, where the carrier
are either confined in space at the surface (top-TCT) or distributed along the
beam line (edge-TCT). Among the traditional methods are above-bandgap laser
excitationgabrysch  or the use of alpha particles. Both types of signal
generation do not allow for a 3-dimensional scanning of the bulk. In addition,
the absence of independent knowledge of the carrier generation time in the
case of alpha particles further complicates the analysis.

The edge-TCT method was first developed for silicon devices for charged
particle trackingopa_kramberger2010 . It uses a laser beam with sub-bandgap
wavelength that excites the carriers through the device edge along a narrow
($\sim$10 $\rm\mu m$) path. The current pulse evolution is subsequently
observed on one of the narrow strip electrodes perpendicular to the laser
path. Recently the TPA edge-TCT has also been demonstrated in silicon devices
tpa2 ; tpa3 ; tpa4 . While in silicon devices the TPA edge-TCT requires a more
complex laser setup in comparison to the traditional sub-bandgap edge-TCT, in
wide-bandgap semiconductors the TPA edge-TCT is the only feasible method. This
is because the generation and fine tuning of a deep ultra-violet (DUV) laser
beam with a wavelength slightly below the bandgap energy, for example in
diamond samples, presents significant difficulties. It requires relatively
complex laser sources such as DUV dye lasers or multistage frequency doubling
with an optical parametric amplifier, which further increases the cost and
complicates the setup. In addition, charge in sub-bandgap wavelength
experiments is generated all along the light path and not spatially confined
around the focal point as in TPA edge-TCT.

Figure 1 shows a simplified overview of our setup. The Ti:Sapphire laser
system (Spectra-Physics Femtopower Pro V CEP) of the ETH Ultrafast Laser
Physics groupuflgroup ; lucchini_laser  provides 25-fs laser pulses with
800-nm wavelength and pulse energies of up to 2.2 mJ at a repetition rate of 1
kHz. Only a sub-percent reflection of the initial beam is sent to the setup in
which the second harmonic (400 nm) was generated by a 100-$\rm\mu m$ thick
barium borate (BBO) crystal. The beam is then sent through a filter to remove
residual IR components from the 400-nm wavelength light. Focusing the beam
with a $f$=30 mm achromatic doublet along the z-axis into the diamond results
in a focal point with full width at half maximum (FWHM) of 1.6 $\rm\mu m$ and
Rayleigh length of 28 $\rm\mu m$ as measured with the "knife edge" technique
in air. Typical pulse energies for the experiment ranged from 0.1-0.2 nJ with
pulse intensity stable to below 1%. Due to the high refractive index of
diamond, refraction and dispersive pulse broadening should be taken into
account. Refraction causes the focal point to elongate and shifts its position
forward in the direction of the beam when entering the diamond. While the
diameter remains approximately the same, the Rayleigh length increases by
roughly the refractive index of diamond. The z-position of the focal point
inside diamond is affected in a similar way. Therefore, in order to determine
the actual z-position inside the diamond, the z-displacement of the xyz-stage
was corrected by the index of refraction and referenced to the sample edge
through which the laser pulse enters. Dispersive pulse broadening causes an
increase of pulse length from 25 fs to roughly 100 fs after passing through
the optical elements and $\sim$2 mm of diamond. Based on observed laser power
dependence of the signal, we conclude that two-photon absorption (TPA) is the
prevalent mechanism of charge generation in diamond, which agrees with
published absorption constants Kozak12 .

The diamond under test was purchased from Element6111Element Six Technologies
Limited, Kings Ride Park, Ascot, Berkshire, SL5 8BP, United Kingdom. It was a
4660$\times$4790$\times$540 $\mu m^{3}$ scCVD diamond with the large surfaces
oriented perpendicular to [001] crystallographic orientation. A
4100$\times$4100 $\mu m^{2}$ Ti:W square pad was sputtered to both larger
surfaces through shadow mask sputtering after which the sample was annealed at
400 ∘C for 5 minutes in N${}_{\text{2}}$ atmosphere. In order to reduce
dispersion at the edges, two of the diamond smaller sides were polished. Since
the injected charge is spatially confined there was no need for a segmented
metallization as required in sub-bandgap excitation setups; therefore a simple
pad layout was chosen. The diamond lower pad was glued onto a small PCB and
its upper pad was wire-bonded to the input of the amplifier. A CIVIDECcividec
C2-HV (2 GHz, 40 dB) broadband amplifier/bias-T combination amplified the
signals coming from the diamond while providing the bias voltage to the
diamond. No active cooling or heating was used. All measurements were
performed at room temperature.

Figure 1: Schematic view of experimental setup

The measurements were fully automated. A computer controlled the 3-axis
movement of the sample while constantly logging the leakage current and
acquiring waveforms from the laser-triggered oscilloscope. The xyz-stage
consisted of 3 linear Newport stages with sub-micron resolution driven by a
Newport ESP301 motion controller. The 2 GHz bandwidth, 10 GS/s sampling rate
Tektronix MSO5204B oscilloscope allowed to digitize and record, synchronous to
the laser repetition rate, a series of 20 to 100 waveforms with 8-bit pulse
height resolution and a length of 200 ns.

The first step in the analysis was the averaging and baseline correction of
the waveforms. The latter was done by taking the average of a large number of
waveforms acquired in air outside the diamond and subtracting the result from
the average waveform at every scan point. This method allowed us not only to
find the baseline of the waveforms but also to remove noise from the laser
Pockels cells as well as recurring pickup of the amplifying circuit. Figure 2
shows a collection of waveforms taken at -1400 V bias voltage on the top
electrode at different distances from the bottom electrode (y-positions). The
rising edges of the signals indicates the initial drift of charge carriers
just after the laser pulse. Thereafter, the initial peak of the signals at
$y$-positions (350-500 $\rm\mu m$) is a result of both electron and hole
drift. The tail reaching up to $\sim$5.5 ns is caused by the electron cloud
drifting to the ground electrode. For $y$=300 $\rm\mu m$ the drift times of
electrons and holes to the respective electrodes are roughly the same and no
tail is seen.

Figure 2: Averaged and baseline corrected waveforms taken at 5 different
positions with varying $y$-coordinates and constant $x$=2500 $\mu m$ and
$z$=1000 $\mu m$ coordinates (see figure 3 for reference). The signals shown
are the average of 100 acquisitions with a baseline correction calculated from
1200 waveforms recorded outside the diamond region.

The collected charge for a scan point at position ($x$,$y$,$z$) can be
calculated as

$Q=\int_{t_{a}}^{t_{b}}I(x,y,z,t)dt$ (1)

where $t_{a}$ and $t_{b}$ denote the integration times. The integration window
was always chosen to integrate the full range of the waveforms (0-200 ns) in
order not to neglect the charges with a smaller drift speed than expected.
Performing this procedure for every point with step sizes of $\Delta x$=50
$\rm\mu m$ and $\Delta y$=25 $\rm\mu m$ and roughly 4400 measurement points
yields a 3-D (x,y at a given z) charge collection map shown in figure 3.

Figure 3: xy-scan of the diamond at z=1000 $\mu m$ showing collected charge at
a bias voltage of -500 V. The thin white lines show the silhouette of the
diamond and the two thick lines represent the contact electrodes.

The “prompt” current methodopa_kramberger2010  was used to measure the
electric field configuration inside the diamond. Immediately after a laser
pulse, electrons and holes that were generated in the voxel of the focal point
start drifting in the electric field of the diamond. This initial drift is
proportional to the average electric field in the immediate surroundings of
the injection voxel. Drifting charges in the diamond cause a current that is
measured at the amplifier output. While the drift of the carriers from the
injection voxel to the electrodes is characterized by pulse shapes equivalent
to the ones shown in figure 2, the information about the drift in the first
few hundred picoseconds is contained in the rising edge of the current pulses.
We measure this initial drift by computing a 200-ps integral around the center
of the rising edge. The integral helps to minimize the uncertainty (e.g. from
trigger jitter) in comparison to a single value measurement at the center of
the rising edge. The outcome of the integral is called prompt current. The
prompt current ($I_{prompt}$) is a measure that only depends on the weighting
field $\vec{E}_{w}$ and the mobilities of electrons and holes near the
injection point. This can be shown by using Shockley-Ramo’s formula for the
instantaneous currentramo39  multiplied by the amplifier gain A,

$\begin{split}I_{promt}(y,t\approx
0)=Aq(\vec{v}_{e}+\vec{v}_{h})\vec{E}_{w}\end{split}$ (2)

then substituting the definition for mobility
$\vec{v}_{e}+\vec{v}_{h}=[\mu_{e}(E)+\mu_{h}(E)]\cdot\vec{E}(y)$ and combining
all terms not depending on the electric field E in a constant c we obtain:

$\begin{split}I_{promt}(y,t\approx 0)=c[\mu_{e}(E)+\mu_{h}(E)]\cdot
E(y)\end{split}$ (3)

The strength of this approach is the ability to disentangle the measurement of
the electric field from trapping effects that occur during the drift of the
charges. Current pulses in figure 2 have an average 20-80% rise time of 350
ps, which at the given field corresponds to an average drift distance of 32
$\rm\mu m$. The carrier mean free path in electronic grade scCVD diamond is
much larger at such electric fields, usually it exceeds the sample thickness.
Therefore the trapping during the initial drift can be neglected in this
sample. In materials where trapping effects are not negligible compared to the
amount of generated charge, equation 3 has to be corrected for immediate
charge trapping during the initial drift. Figure 4 shows two xy-prompt current
maps at two different z-positions in the diamond.

Figure 4: Prompt current xy-maps for $z=1500\,\mu m$ and $z=1000\,\mu m$ at
bias voltage of -500 V. The red box in the $z=1500\,\mu m$ plot is the
fiducial area for the comparison between prompt current and X-ray topograph in
figure 7.

The plots show strong variations of the prompt current and thus the electric
field, which in turn indicates a strong, non-homogeneous variation of space
charge in the diamond bulk. In figure 5 the prompt current distributions in
figure 4 are plotted as a function of $y$ for several selected $x$, $z$
positions. For the profile at $z=1.5\,mm$ and $x=2.5\,mm$ the density of space
charge is high enough to leave the region at small $y$ without a sizable
field.

Figure 5: The prompt current profiles for a few selected $x$, $z$ positions as
indicated in the legend. For comparison the profiles were normalized to the
same area.

To see whether the variations in the prompt current are caused by structural
defects in the diamond, we compared regions of the prompt current xy-scans to
slices of an X-ray diffraction topograph taken at the BM05 beamline of the
ESRF in Grenobleesrf . The (400) Laue spot of the diamond visualizes defects
present in the bulk. Dark regions in the topograph in figure 6 correspond to a
high defect density.

Figure 6: White beam X-ray topograph of the diamond sample. Areas of higher
structural defect density appear darker. The outline of the pad metallization
is shown in blue. The z-positions of the xy-scans shown in figure 4 are
indicated in red. The red bars indicate the FWHM-extension of the focal
point’s squared intensity profile.

In order to make a comparison between the X-ray topograph and the xy-scan of
the prompt current that is not biased by the high field region around the
electrode, a fiducial area, shown as red box in figure 4, was selected. The
prompt current values were summed for each x-column in the fiducial area,
resulting in 1-dimensional distribution in figure 7. The image histogram of
the X-ray topograph was obtained by summing the contrast values of the pixels
enclosed by the red bar at z=1500 $\mu m$ in figure 6. The summation was done
along the z-axis for every pixel column along the x-axis. To account for the
non-homogenous intensity distribution of the focal point along the z-direction
the pixels were weighted with the squared intensity profile of the focal point
prior to summation. Both distributions (shown in figure 7) were normalized to
their maximum values for comparison. The electric field drop below 700 $\mu m$
and above 4000 $\mu m$ is due to the edge effect. In addition to this, we
observe a large drop in the electric field between x $\sim$1500 and x $\sim$
2800 $\mu m$. This drop coincides with a large dark region at the same
position in the x-ray topograph. The other dark regions in the x-ray topograph
do not cause significant disturbance in the electric field, indicating that
those defects do not affect the electric field.

Figure 7: Comparison of the x-projection of the red box in the prompt current
map of figure 4 (black) to the focal point’s squared intensity profile
weighted slice of the X-ray topograph shown as a red bar at z=1500 $\mu m$ in
figure 6. High values in the red curve signifies a high transparency and thus
a low defect concentration in the X-ray topograph. The blue lines represent
the end of the electrode pad metallization and the dashed black line the
fiducial area cut. Data outside the fiducial area cut is shown for
completeness.

Our results complement the X-ray diffraction topograpy. The X-ray diffraction
topography indicates the structural defects of the sample. Our method is
sensitive to variations of the electric field due to the space charge formed
by trapping on defects. It therefore indicates the location and the polarity
of the trapped charge. In addition, our method explores the defects in 3
dimensions while the X-ray topograph in our case is 2-dimensional. Figure 7
exhibits a correlation indicating a link between structural defects and charge
trapping centers. In our future work we will perform a more systematic study
of the correlation between the structural defects and their effect on charge
trapping and space charge distribution. In addition to correlation with X-ray
topography, it will be enlightening to explore correlations with other defect
imaging methods, such as the recently reported photoelectrical readout of
single defectssiyushev .

In summary, we demonstrated the application of the TPA edge-TCT technique in a
wide-bandgap semiconductor. We presented 3-D resolved measurements of the
charge collection and the electric field in the diamond sample. We have shown
that a correlation between the electric field in the diamond and the
dislocations observed in the X-ray topography image exists. In addition we
have demonstrated that the measurements of the charge collection and the
electric field can be achieved with an excellent position resolution in all 3
dimensions. This was previously difficult to achieve in wide-bandgap
materials, but is now accomplished through the use of two-photon absorption.

###### Acknowledgements.

We would like to thank Robert Stone from Rutgers University for providing us
with the diamond sample, Marcos Fernández García of the CERN Silicon Lab for
many useful discussions that led to these measurements. Special thanks to
Professor Ursula Keller at ETH for letting us use her group’s laser system and
Lukas Gallmann for his consulting on ultrafast laser physics. We would also
like to thank Professor Steven Johnson at ETH for providing us with a laser
and a space in his lab that allowed us to continue this experiment, and also
Martin Kubli and Larissa Boie for their advice and support there. This work
was supported by ETH grant ETH-51 15-1.

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