Surface Modes and Semiconductor Transport: Theory of the Dember Effect
Maja Kr\v cmar and Wayne M. Saslow
Exact solution of the linearized equations for steady-state transport in semiconductors yields two modes that vary exponentially in space, one involving screening (without entropy production) and one involving diffusion and recombination (with entropy production). Neither mode is quasineutral. For constant surface photoexcitation with generation of electrons and holes, the steady-state response is a linear combination of these modes, subject to global electroneutrality. The resultant charge separation produces a voltage difference across the sample (the Dember effect). %The conditions for large Dember voltages are obtained. Application to other transport situations are briefly discussed.
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\title
%{Surface Modes and Semiconductor Transport: Theory of the Dember Effect}
%{Screening Mode, Diffusion Mode, and the Dember Effect: Violation of Quasineutrality}
%{The Dember Effect: Violation of Quasineutrality in Semiconductor Transport}
{Violation of Quasineutrality in Semiconductor Transport: The Dember Effect}

\author{Maja Kr\v cmar and Wayne M. Saslow}
\address{Department of Physics, Texas A\&M University, College Station,
Texas 77840-4242}

%\vspace{1cm}

\maketitle


\begin{abstract}Exact solution of the linearized equations for steady-state
transport in semiconductors yields two modes that vary exponentially in space,
one involving screening (without entropy production) and one involving diffusion
and recombination (with entropy production).  Neither mode is quasineutral.  For
constant surface photoexcitation with generation of electrons and holes, the
steady-state response is a linear combination of these modes, subject to global
electroneutrality. The resultant charge separation produces a voltage difference
across the sample (the Dember effect).

%The conditions for large Dember voltages are obtained. Application to other transport situations are briefly discussed.

\end{abstract}
\pacs{PACS numbers: 72.20.-i, 72.20 Jv, 72.40.+w, 73.50.Pz}
]

%\section{Introduction}

The Dember effect is the voltage difference that develops across the bulk of a
material that steadily absorbs light at one surface.\cite{dember}  It has been
observed in insulators,\cite{ber1,ber2} high resistivity
semiconductors,\cite{bad,berg} and $C_{60}$ thin films.\cite{dipan} An extensive
discussion of photovoltages, both at surfaces and in the bulk, is given in
Ref.\onlinecite{kronikshap}.

An intuitive physical picture of the Dember effect is as follows.\cite{chaz} 
Incident light produces equal numbers of electrons and holes at the surface. The
higher mobility electrons travel further than the lower mobility holes.  The
resultant charge separation amounts to a dipole layer.  As a consequence the
illuminated surface is at the higher voltage, in agreement with experiment.
However, conventional theories of the Dember effect,\cite{kronikshap,chaz}
which assume quasineutrality (local electroneutrality), yield no dipole layer;
moreover, they yield a non-zero net charge.

The present work obtains exact solutions for the surface modes of the linearized
macroscopic transport equations.  It differs from previous analytic work in
that: (1) It explicitly satisfies the principles of irreversible thermodynamics,
which constrain the form for the charge-carrier fluxes and recombination rates
by the condition that the rate of entropy production be non-negative; (2) In
solving the transport equations, it employs both electron and hole densities $n$
and $p$ as variables, rather than assuming deviations from equilibrium satisfy
quasineutrality ($\delta n\approx \delta p$); (3) Using the resultant two
steady-state surface modes, it explicitly enforces overall electroneutrality.

One surface mode corresponds to ordinary screening by a multi-charge-carrier
system (i.e. Debye-H\"uckel screening), without fluxes or entropy production:
the {\it screening mode.} The other surface mode has both electron and hole flux
as well as entropy production: the {\it diffusion-recombination mode.} The
characteristic length $l$ of the screening mode typically is shorter than the
characteristic length $L$ for the diffusion-recombination mode.  Neither mode is
locally electroneutral.  Applied to the Dember effect, the light generates a
diffusion-recombination mode with a relatively extended negative charge and a
screening mode with a relatively compact positive charge, in agreement with the
physical picture given above.

{\bf Irreversible thermodynamics.} Consider a uniform semiconductor with an
ideal surface having no extrinsic surface states and no charged intrinsic
states. The recombination rate $r$ is the same for both electrons and holes. 
With $u$ the energy density, $T$ the temperature, $s$ the entropy density, and
$\tilde\mu_e$ and $\tilde\mu_h$ the electron and hole electrochemical
potentials, the fundamental thermodynamic differential for this system is
\begin{equation}
du=Tds+\tilde\mu_e dn+\tilde\mu_h dp.
\label{1}
\end{equation}
With $\mu_e$ and $\mu_h$ the chemical potentials (to be distinguished from the
diffusivities $\mu_n$ and $\mu_p$),
\begin{equation}
\tilde\mu_e\equiv\mu_e-e\phi, \qquad \tilde\mu_h\equiv\mu_h+e\phi.  
\label{2}
\end{equation}
Here the electrical potential $\phi$ satisfies Poisson's equation
\begin{equation}
\nabla^2\phi=-{\rho\over\epsilon_0\epsilon}
=-{e\over\epsilon_0\epsilon}(p-n+N_d^+ - N_a^-),
\label{3}
\end{equation}
where $\epsilon$ is the semiconductor dielectric constant, $\epsilon_0$ is the
permittivity of free space, and the charge density $\rho=e(p-n+N_d^+ - N_a^-)$.
$N_d^+$ and $N_a^-$ are the respective densities of ionized donors and
acceptors, and $\tilde\mu_e$ and $\tilde\mu_h$ are often called quasi-Fermi
energies.

The conservation laws for this system are
\begin{equation}
\partial_t u+\partial_i j_i^u=0, \quad
\partial_t s+\partial_i j_i^s={{\cal P}\over T}\ge 0, 
\label{4}
\end{equation}
\begin{equation}
\partial_t n+\partial_i j_i^n=r, \quad \partial_t p+\partial_i j_i^p=r. 
\label{5}
\end{equation}
Here $j_i^u$ is the energy flux density, $j_i^s$ is the entropy flux
density, ${\cal P}$ is the local rate of heat production (${\cal P}/T$ is local
rate of entropy production), $j_i^n$ is the electron number flux density, and
$j_i^p$ is the hole number flux density.  The fluxes, $r$, and $\cal P$ are to
be determined.

The time-derivatives of (\ref{1}) and (\ref{4})--(\ref{5}) lead to 
\begin{eqnarray}
0\le {\cal P}&=&-\partial_i j_i^u +T\partial_i j_i^s
-\tilde\mu_e(r-\partial_i j_i^n)-\tilde\mu_h(r-\partial_i j_i^p)
\nonumber\\
&=&-\partial_i(j_i^u - Tj_i^s - \tilde\mu_e j_i^n - \tilde\mu_h
j_i^p)\nonumber\\
& &-j_i^s\partial_i T - j_i^n\partial_i\tilde\mu_e - j_i^p\partial_i\tilde\mu_h
-r(\tilde\mu_e+\tilde\mu_h). 
\label{6}
\end{eqnarray}

Expressing ${\cal P}$ as a non-negative quadratic form requires that 
$j_i^u=Tj_i^s+\tilde\mu_e j_i^n+\tilde\mu_h j_i^p$; that 
\begin{equation}
j_i^s =-\alpha_T\partial_iT=-{\kappa\over T}\partial_iT, 
\quad 
\label{7}
\end{equation}
where $\kappa\ge 0$ is the thermal conductivity; that  
\begin{eqnarray}
j_i^n=-\alpha_{nn}\partial_i\tilde\mu_e-\alpha_{np}\partial_i\tilde\mu_h,\nonumber\\
j_i^p=-\alpha_{pn}\partial_i\tilde\mu_e-\alpha_{pp}\partial_i\tilde\mu_h, 
\label{8}
\end{eqnarray}
where $\alpha_{nn}\ge 0, \alpha_{pp}\ge 0$, and 
$\alpha_{nn}\alpha_{pp}\ge \alpha_{np}\alpha_{pn}$ (the $\alpha$'s 
are related to the diffusivities, and $\alpha_{np}=\alpha_{pn}$ by the Onsager 
symmetry principle\cite{Onsager-LL}); and that 
\begin{equation}
r=-\lambda(\tilde\mu_e+\tilde\mu_h)=-\lambda(\mu_e+\mu_h). 
\label{9}
\end{equation}
where $\lambda\ge 0$ is related to the electron and hole lifetimes $\tau_n$ and
$\tau_p$. In equilibrium $r=0$, so (\ref{9}) implies the well-known results
that $\tilde\mu_e=-\tilde\mu_h$ and $\mu_e=-\mu_h$ in equilibrium.  We
now express $r$, $j_i^n$, and $j_i^p$ in more conventional form.  

Linearizing (\ref{9}) about equilibrium with $\delta
\mu_e=(\partial\mu_e/\partial n)\delta n$ and $\delta
\mu_h=(\partial\mu_h/\partial p)\delta p$ yields
\begin{equation}
r=-{\delta n\over\tau_n} -{\delta p\over\tau_p}, 
\quad \tau_n^{-1}\equiv \lambda{\partial\mu_e\over\partial n}, 
\quad \tau_p^{-1}\equiv \lambda{\partial\mu_h\over\partial p}.
\label{10}
\end{equation}
For a dilute semiconductor, as considered here, 
$(\partial\mu_e/\partial n)\approx k_B T/n_0$ and 
$(\partial\mu_h/\partial p)\approx k_B T/p_0$, so
$\tau_n=(n_0/k_B T\lambda)$ and $\tau_p=(p_0/k_B T\lambda)$.  

Now set $\alpha_{np}=\alpha_{pn}\approx 0$,  
$\alpha_{nn}=D_n(\partial n/\partial \mu_e)$, and 
$\alpha_{pp}=D_p(\partial p/\partial \mu_h)$, where $D_n$
and $D_p$ are the electron and hole diffusivities.  Next, define 
the electron and hole mobilities $\mu_n$ and $\mu_p$, which satisfy 
the Einstein relations: 
\begin{equation}
\mu_n=
{eD_n\over n}{\partial n\over\partial \mu_e}\approx {eD_n\over k_B T}, 
\quad \mu_p=
{eD_p\over p}{\partial p\over\partial \mu_h}\approx {eD_p\over k_B T}, 
\label{11}
\end{equation}
Then in one-dimension $j_i^n$ and $j_i^p$ become\cite{ash,sze}  
\begin{eqnarray}
j^n_x=-D_n\partial_x n+\mu_n n\partial_x\phi, \nonumber\\
j^p_x=-D_p\partial_x p-\mu_p p\partial_x\phi. 
\label{12}
\end{eqnarray}

Note that Poisson's equation linearizes to 
\begin{eqnarray}
\nabla^2\delta\phi&=&-{e\over\epsilon_0\epsilon}(\delta p-\delta n)
={e^2\over\epsilon_0\epsilon}({\partial n\over \partial\mu_e}+
{\partial p\over \partial\mu_h})\delta\phi\nonumber\\
& &-{e\over\epsilon_0\epsilon}({\partial p\over \partial\mu_h}\delta\tilde\mu_h
-{\partial n\over \partial\mu_e}\delta\tilde\mu_e).
\label{13}
\end{eqnarray}

%\subsection {Surface Screening Mode}

{\bf Static surface solutions in one-dimension.} One steady-state solution that
automatically satisfies (\ref{5}), (\ref{8}), and (\ref{9}) has
$\delta\tilde\mu_e=\delta\tilde\mu_h=0$ and $j_i^n=j_i^p=0$.  Thus the system
is in local equilibrium, with no entropy production.  This corresponds to
ordinary screening, and thus we call it the {\it screening mode,} with subscript
$S$.  From (\ref{13}), its potential $\delta\phi_S$ satisfies
\begin{equation}
\nabla^2\delta\phi_S
={e^2\over\epsilon_0\epsilon}({\partial p\over \partial\mu_h}+
{\partial n\over \partial\mu_e})\delta\phi_S=q_S^2\delta\phi_S, 
\label{14}
\end{equation}
where 
\begin{equation}
q_S^2=q_{Sn}^2+q_{Sh}^2, 
\quad q_{Sn}^2\equiv{e^2\over\epsilon_0\epsilon}{\partial n\over \partial\mu_e}, 
\quad q_{Sp}^2\equiv{e^2\over\epsilon_0\epsilon}{\partial p\over \partial\mu_h}.
\label{15}
\end{equation}
For the present system, $q_{Sn}^2=(e^2n_0/\epsilon_0\epsilon k_B T)$,
$q_{Sp}^2=(e^2p_0/\epsilon_0\epsilon k_B T)$, so $q_S^2=(e^2/\epsilon_0\epsilon
k_B T)(n_0+p_0)$.  The solution to (\ref{14}) that goes to zero as
$x\rightarrow\infty$ is
\begin{equation}
\delta\phi_S=A_S\exp{(-q_S x)}. 
\label{16}
\end{equation}
$l\equiv q_S^{-1}$ is the {\it screening length.} From (\ref{16}) and
(\ref{13}), the screening mode has charge density
\begin{equation}
\delta\rho_S=-\epsilon_0\epsilon\nabla^2\phi_S
=-(\epsilon_0\epsilon) q_S^2 A_S\exp{(-q_S x)}.
\label{17}
\end{equation}
Here $\delta n_S=(\partial n/\partial\mu_e)\delta\mu_e
=(en_0/k_B T)\delta\phi_S$ and 
$\delta p_S=(\partial n/\partial\mu_h)\delta\mu_h
=-(en_0/k_B T)\delta\phi_S$, so $\delta n_S/\delta p_S=-n_0/p_0$. 

%\subsection {Surface Diffusion-recombination Mode}

The second steady-state solution to (\ref{5}), (\ref{8}), (\ref{9}) and
(\ref{13}) has $\delta\tilde\mu_e$ and $\delta\tilde\mu_h$ non-zero, so the
fluxes $j_i^n$ and $j_i^p$ are non-zero.  With $\alpha_{np}=\alpha_{pn}=0$,
combining (\ref{5}), (\ref{8}) and (\ref{9}) yields
\begin{eqnarray}
-\alpha_{nn}\partial_x^2\delta\tilde\mu_e
=-\lambda(\delta\tilde\mu_e+\delta\tilde\mu_h), \nonumber\\
-\alpha_{pp}\partial_x^2\delta\tilde\mu_h
=-\lambda(\delta\tilde\mu_e+\delta\tilde\mu_h).
\label{18}
\end{eqnarray}
Let us denote properties of this {\it diffusion-recombination mode} by the
subscript $D$.  Now set
\begin{equation}
q_{Dn}^2\equiv {\lambda\over\alpha_{nn}}={1\over D_n\tau_n}, \quad
q_{Dp}^2\equiv {\lambda\over\alpha_{pp}}={1\over D_p\tau_p}.   
\label{19}
\end{equation}
Then, with
\begin{equation}
\delta\tilde\mu_e=A_{Dn}\exp{(-q_D x)}, \quad \delta\tilde\mu_h=A_{Dp}\exp{(-q_D x)},
\label{20}
\end{equation}
(\ref{18}) yields  
\begin{equation}
q_D^2=q_{Dn}^2+q_{Dp}^2  
\label{21}
\end{equation}
and $A_{Dp}/q_{Dp}^2=A_{Dn}/q_{Dn}^2$. 

The corresponding electrostatic potential $\delta\phi_D$ takes the form 
\begin{equation}
\delta\phi_D=A_D\exp{(-q_D x)},
\label{22}
\end{equation}
where 
\begin{equation}
{A_{Dp}\over q_{Dp}^2}={A_{Dn}\over q_{Dn}^2}
={q_S^2-q_D^2\over q_{Sp}^2q_{Dp}^2-q_{Sn}^2q_{Dn}^2}eA_D.  
\label{23}
\end{equation}
$L\equiv q_D^{-1}$ is the {\it diffusion-recombination length,} normally
called the diffusion length.  By (22) and (3), the diffusion-recombination mode
possesses charge density
\begin{equation}
\delta\rho_D=-(\epsilon_0\epsilon) q_D^2 A_D\exp{(-q_D x)}.
\label{24}
\end{equation}
Moreover,  
$\delta n_D/\delta p_D=(n_0/p_0)(q_{Sp}^2-q_{Dn}^2)/(q_{Sn}^2-q_{Dp}^2)$.  This 
is independent of $n_0$ for fixed $n_0p_0=n_i^2$. 

%\subsection{Solution for Dember Effect}

{\bf Surface mode amplitudes.}  Let light incident at the surface $x=0$ 
produce equal electron and hole fluxes $G$, so $j_i^n=j_i^p=G$ at $x=0$.  On
setting $\alpha_{np}=\alpha_{pn}=0$, (\ref{8}) and (\ref{20}) yield the
diffusion-recombination mode amplitudes
\begin{equation}
A_{Dn}={G\over q_D\alpha_{nn}}, \quad A_{Dp}={G\over q_D\alpha_{pp}}. 
\label{25}
\end{equation}

Overall electroneutrality, or
$0=\int_0^{\infty}dx(\delta\rho_S(x)+\delta\rho_D(x))$, implies that 
\begin{equation}
A_S=-{q_D\over q_S}A_D.
\label{26}
\end{equation}

%\subsection{Discussion of the Solutions}

From (\ref{17}) and (\ref{26}), and from (\ref{23}-\ref{25}), the total charge
density is
\begin{equation}
\delta\rho(x)={Ge^2\over k_BT}\ {(\mu_n-\mu_p)\over \mu_n\mu_p}\ 
{q_S e^{-q_S x}-q_D e^{-q_D x}\over q_S^2-q_D^2}, 
\label{27}
\end{equation}
where the mobilities have been employed.  Typically $\mu_n>\mu_p$, so
$\rho(x=0)$ is positive, as expected if the higher mobility charge-carrier
preferentially leaves the vicinity of the surface.  Note that $\delta\rho(x)$
changes in sign as $x$ increases, as needed to produce a dipole layer.

From (\ref{17}) and (\ref{24}), the total electrostatic potential is 
\begin{equation}
\delta\phi(x)={Ge^2\over \epsilon_0\epsilon k_BT}\ 
{(\mu_n-\mu_p)\over \mu_n\mu_p}\ 
{q_D^{-1}e^{-q_D x}-q_S^{-1}e^{-q_S x}\over q_S^2-q_D^2}.
\label{28}
\end{equation}

To compare with experiment, (\ref{28}) gives the Dember voltage 
$\phi_{Dem}\equiv\delta\phi(x=0)$ as 
\begin{equation}
\phi_{Dem}={Ge^2\over \epsilon_0\epsilon k_BT}
\ {(\mu_n-\mu_p)\over \mu_n\mu_p}\ 
{1\over q_S q_D}{1\over q_S+q_D}.
\label{29}
\end{equation}

{\bf Discussion.} Eq.(29) makes numerous predictions. First, since typically
$\mu_n>\mu_p$, $\phi_{Dem}$ is positive, as expected for a dipole layer with
positive charge closer to the surface. Further, since typically $q_D\ll q_S$,
the term $q_S q_D(q_S+q_D)\approx q_S^2 q_D$.  The dependences of $q_S$ on $n$
and $\epsilon$, and of $q_D$ on $\mu$ and $\tau$, show that $\phi_{Dem}$ varies
inversely with carrier density $n$, as the square root of the characteristic
recombination time $\tau$, as the inverse square root of the characteristic
mobility $\mu$, and is independent of $\epsilon$.

The dependence on $n$ explains why $\phi_{Dem}$ is observed in materials with
relatively low carrier densities.  The dependence on $\tau$ indicates that
$\phi_{Dem}$ is larger for longer recombination times; the dependence on $\mu$
indicates that $\phi_{Dem}$ is larger for lower mobilities.  

\begin{figure}
\centerline{\psfig{file=d_potrho.eps,width=1\linewidth}}
\caption{Photo-induced electrical potential $\delta\phi$ from (28) and charge 
density $\delta\rho$ from (27).} 
\label{fig:1}
\end{figure}

\begin{figure}
\centerline{\psfig{file=d_carr.eps,width=1\linewidth}}
\caption{Electron and hole number densities $n$ and $p$ corresponding to Figure 1.} 
\label{fig:2}
\end{figure}

Figures 1-4 are computed for an intrinsic semiconductor with equilibrium carrier
number density $n_0=p_0=n_i=10^{16}$~m$^{-3}$, recombination time
$\tau=10^{-5}$~s, electron mobility $\mu_n=1000$~cm$^{2}$/(V-s), hole mobility
$\mu_p=200$~cm$^{2}$/(V-s), room temperature $k_B T=0.0253$~eV, and dielectric
constant $\epsilon=10$.  These values give $l=26.4~\mu$m and $L=64.8~\mu$m. From
(28), the ratio of the contributions of the diffusion mode and the screening
mode to the Dember voltages is $q_D^{-1}/q_S^{-1}=L/l$. Taking
$G=10^{17}$~m$^{-2}$s$^{-1}$ gives $\phi_{Dem}=9.22$~mV, of which the diffusion
mode is responsible for 15.56~mV and the screening mode $-6.34$~mV.  The theory,
which is valid for $\delta\tilde\mu_e$ and $\delta\tilde\mu_h$ small relative to
$k_B T$, becomes invalid for significantly larger $G$. 

Fig.1 presents $\delta\phi$ and $\delta\rho$.  Note that $\delta\phi$ is
monotonic, whereas $\delta\rho$ changes sign, so that the system can be overall
neutral.  Fig.2 presents the number densities $n$ and $p$ associated with
$\delta\rho$ of Fig.1.  The curves sketched (but not computed) in 
Ref.\onlinecite{chaz} (Fig.V.3) agree qualitatively with Figs.1 and 2.   

The individual charge densities $\rho$ due to the diffusion-recombination and
screening modes are presented in Fig.3.  Note that $\delta n_S/\delta p_S=-1$
and $\delta n_D/\delta p_D=1.306$ for the present parameters. (Increasing both
mobilities by a factor of ten would give $\delta n_D/\delta p_D=1.022$.) Hence
quasineutrality is not a good approximation for either mode. The charge density
for the screening mode is positive and concentrated near the surface, whereas
the charge density for the diffusion-recombination mode is negative and more
extended. Note that numerical solutions of the macroscopic transport equations
for semiconductors normally do not assume quasineutrality, so they retain the
full physics.\cite{kormay}  However, they do not permit extensive physical
interpretation.

Fig.4 presents the bulk charge density $\delta\rho$ for $\mu_n$ and $\mu_p$ as
above ($\mu_p/\mu_n=0.2$), and for three additional values of $\mu_p/\mu_n$. 
$\delta\rho$ and $\phi_{Dem}$ increase with the difference in the electron and
hole mobilities.

\begin{figure}
\centerline{\psfig{file=d_modes.eps,width=1\linewidth}}
\caption{Charge densities for the screening and diffusion modes, from (17) and 
(24).} 
\label{Fig.3}
\end{figure}

\begin{figure}
\centerline{\psfig{file=d_mobil.eps,width=1\linewidth}}
\caption{Charge density $\delta\rho$ from (27) for different hole mobilities.} 
\label{Fig.4}
\end{figure}

%\section{Summary}

Surface solutions also are relevant to current flow across interfaces.  Already
in the nineteenth century, Maxwell noted that for current flow between two
single-carrier materials charge builds up at the interface; however, without a
more microscopic theory it is not possible to determine how much charge is
associated with each material.  A similar effect must occur at a
semiconductor-metal interface, even without the complexities associated with
modification of the Schottky barrier. Surface solutions likely are present in
recent numerical studies showing that spin injection in a semiconductor (e.g.,
by preferential absorption of polarized light) persists across a $p-n$
junction.\cite{zutic}  Such a system, with up and down electrons and with
unpolarized holes, should have three surface modes: one screening mode and two
diffusion-recombination modes.  The rapid falloff of spin polarization at one
end of the system is evidence for surface solutions induced by what the authors
of Ref.~\onlinecite{zutic} consider to be an artificial boundary condition.

The present multi-carrier system, with a finite diffusion-recombination length
$L$, is non-neutral only over a finite distance from the disturbance.  On the
other hand, a multi-carrier system with no recombination (and thus
$L\rightarrow\infty$) can be non-neutral in the bulk, far from a
disturbance.\cite{sas}

We would like to thank A. Zangwill for a critical reading of the manuscript. WMS
would like to acknowledge the Department of Energy for its support from DOE
Grant No. DE-FG03-96ER45598.  

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\begin{figure}
\epsffile{d_potrho.eps}
\caption{Photo-induced electrical potential and charge density.} 
\label{Fig.1}
\end{figure}

\begin{figure}
\epsffile{d_carr.eps}
\caption{Overall number densities.} 
\label{Fig.2}
\end{figure}

\begin{figure}
\epsffile{d_modes.eps}
\caption{Mode densities.} 
\label{Fig.3}
\end{figure}

\begin{figure}
\epsffile{de_mobil.eps}
\caption{Overall charge density for various relative mobilities.} 
\label{Fig.4}
\end{figure}

\bibitem{shock} W.Shockley, and W.T. Read, Jr., Phys. Rev. {\bf 87}, 835(1952).

\bibitem{sas} W.M. Saslow, Surface Modes for Semiconductor with no
Depletion or Accumulation Layer, (unpublished).

If the free surface of a semiconductor is illuminated by orthogonally incident 
light that produces electron and hole photocarriers, the potential difference 
will appear between the illuminated and unilluminated semiconductor surface.
This potential difference is first discovered in experiments by 
Dember,\cite{dember1,dember2} and it is referred to as the Dember voltage.

The photoproduction of electrons and holes at the surface of an
illuminated semiconductor was essential to the classic
Haynes-Shockley experiment establishing that the charge carriers in
semiconductors are of two varieties, one positive (holes) and the other
negative (electrons). 

After it s discovery in 1931, the Dember effect is later studied by many 
authors in experiments on
insulating crystals\cite{ber1,ber2} and high resistivity semiconductors,\cite{bad,berg}
through the photovoltage measurements. More recently,
this effect is also studied in $C_{60}$ thin films.\cite{dipan}
Many characteristics of Dember effect can be found in Ref.[\onlinecite{jap}].


The intuitive origin of the Dember effect can be easily understood as follows.\cite{chaz}
The photocarriers, electrons and holes, have different mobilities within
the semiconductor, and usually the electron mobility $\mu_n$ exceeds the hole
mobility $\mu_p$. Thus, the electrons will evacuate from the region near the 
illuminated surface faster then holes. This will lead to the excess of holes
near the surface, and excess of electrons further from it. The established
charge distribution will cause the electric field to appear within the sample,
in direction that will tend to stop further separation of carriers, i.e. it will act 
to slow the electrons down while it speeds up the holes. The competitive effects,
the carrier diffusion and induced internal electric field, will result in the 
relaxation of the carrier density to their equilibrium values further from 
the illuminated surface.

From the above discussion, it can be seen that the photocarrier distribution is
determined by the two characteristic length: first, that describes the carrier fluxes
due to the their different mobilities, and second that describes the electric field
effect which tends to screens out the flux charge distribution.

The previous standard Dember effect theories have made the drastic assumption of 
local electroneutrality, arguing that the electric field effect would keep the
carriers close together within the sample, despite their 
different mobilities.\cite{chaz,gold}
The carrier flow within the sample would then be described via a single characteristic
length, yielding a mode that decays exponentially in space. This result is reviewed 
below in Overview.  
Moreover, the standard theories use electron and hole number density as natural variables
for semiconductor description. In description of the Dember effect, these are not 
the natural variables for the following reason. If the local electro neutrality is 
not assumed, the standard approach would yield the two characteristic modes which
would have mixed flux and screening properties.\cite{sas} 
Thus, since the illuminated system 
is in non-equilibrium, both of these modes will yield positive entropy production.
However, the positive entropy production should come from the fluxes only. There
must be no entropy production associated with screening, since semiconductors
screen external or internal charge distributions without dissipation of energy.

This article has two goals.
The first goal is to obtain the Dember voltage and describe the carrier charge densities
going beyond the local neutrality assumption. In other words, we wont to obtain both 
the characteristic length due to carrier fluxes, and characteristic length due to 
their screening. 
The second goal is to find the macroscopic equations describing a semiconductor 
using the electrochemical potentials as the natural variables. This enables us to
construct the solutions that satisfy the correct entropy production requirements 
for both flux and screening mode.

We consider the case of steady illumination of semi-infinite sample
with no depletion or accumulation layers in equilibrium,
 in a one-dimensional geometry. We assume that
the illumination is uniform over the surface and absorbed with very large absorption
coefficient (surface absorption only), as well as that the low-injection conditions apply.
Under the low-injection condition, the number density of the injected carriers is 
smaller than their corresponding equilibrium number densities.\cite{sze}

Using the electron and hole electrochemical potentials as
the primary variables, we find two exponentially decaying modes.  The {\it flux
mode} carries equal electron and hole fluxes (and thus no net electric
current),
dissipates energy, and has charge associated with it.  The {\it screening mode}
carries no fluxes and dissipates no energy, but does have charge associated
with
it.  On illumination, the diffusion-recombination mode is generated in proportion to the
illumination; to maintain electroneutrality near the surface (but not local
electroneutrality), accompanying the diffusion-recombination mode is the screening mode.

\subsection{Overview}

We now briefly review the Dember effect result obtained under the local electro
neutrality assumption.\cite{chaz} The  sample with ideal surface is considered.
The incident light is absorbed by the surface only, and the steady-state 
conditions are assumed. 

For comparison purposes with our results that follow
in the next sections, we use the following notation. The hole number density
is denoted as $n_1$, whereas the electron number density is denoted as $n_2$.
The equilibrium carrier number density is $n_i$, and the recombination time is $\tau$.

The equations for the carrier number densities in steady-state are
$$
0=D_p{d^2 n_1\over d x^2}+
\mu_p{d\over d x}(n_1{d\phi\over d x})-{n_1-n_i\over\tau},
$$ 
\begin{equation}
\ \ \ \ \ 0=D_n{d^2 n_2\over d x^2}-
\mu_n{d\over d x}(n_2{d\phi\over d x})-{n_2-n_i\over\tau},
\label{ch5,1}
\end{equation}
where $D_n$ and $D_p$ are electron and hole diffusion constants, and $\phi$ is electrostatic
potential within the sample. The local
electro neutrality requires that $n_1=n_2$. Also, Poisson's equation requires that
$d^2\phi/d x^2=-e(n_1-n_2)/(\epsilon_0\epsilon)$. In principle, the coupled
equations in (\ref{ch5,1}) should be solved together with Poisson's equation,
since in general, the terms depending on $\phi$ can be large. At this point, 
in standard theories, the terms depending upon $\phi$ in (\ref{ch5,1}) are 
eliminated by replacing the two coupled equations of (\ref{ch5,1}) with
one equation independent on $\phi$. In the remained equation, the local
electro neutrality is imposed, yielding a single equation that describes both 
electrons and holes
\begin{equation}
0=(D_n\mu_p+D_p\mu_n){d^2 n_1\over d x^2}-(\mu_n+\mu_p){n_1-n_i\over\tau}.
\label{ch5,2}
\end{equation}
This procedure is called {\it ambipolar approximation}.\cite{sze,smith}

The solution to (\ref{ch5,2}) is given by
\begin{equation}
n_1=n_i+C\exp{(-x/L)},\ \ \ \ \ \ 
\ \ \ L={({D_n\mu_p+D_p\mu_n\over \mu_n+\mu_p}\tau)}^{1/2}
\label{ch5,3}
\end{equation}
where $C$ is constant, and $L$ is {\it ambipolar diffusion length} 
Requiring that the carrier number current densities at the surface $x=0$) are equal
to the surface generation rate $G$, we have 
$$
j_1=-D_p{dn_1(0)\over x}-\mu_p n_1(0){d\phi(0)\over dx}=G,
$$ 
\begin{equation}
\ \ \ j_2=-D_n{dn_1(0)\over x}+\mu_n n_1(0){d\phi(0)\over dx}=G.
\label{ch5,4}
\end{equation}
Eliminating the term with $d\phi(0)/dx$ from (\ref{ch5,4}), one finds
$C=G\tau/L$. Once the solution for carrier number densities is obtained,
the Dember voltage can be determined. Eliminating the recombination terms from
(\ref{ch5,1}), one finds
\begin{equation}
(D_n-D_p){d^2n_1\over dx^2}=(\mu_n+\mu_p){d\over dx}(n_1{d\phi\over dx}),\ \ 
\hbox{thus}\ \ \ \phi_{D}={D_n-D_p\over \mu_n+\mu_p}\ln{[1+{G\tau\over L n_i}]}.
\label{ch5,5}
\end{equation}
In (\ref{ch5,5}), $\phi_D$ denotes the potential difference between the
illuminated and unilluminated end of a semiconductor, i.e. the Dember voltage.

The generation rate $G$ in general depends on the optical power, energy
and quantum efficiency (number of carrier pairs created by a single photon) 
of incident light, as well as on the surface area of the illuminated sample.\cite{sze}
If the low-injection conditions apply, (\ref{ch5,5}) can be linearized
to obtain\cite{smith}
\begin{equation}
\phi_{D}={D_n-D_p\over \mu_n+\mu_p}\ {G\tau\over L n_i}.
\label{ch5,6}
\end{equation}


\section{Dember Effect in Semiconductor}

The constant $\lambda$ can be determined from the requirement that 
the recombination rate (\ref{ch5,15}) can be written in standard form\cite{sze}
\begin{equation}
r=-{1\over \tau}(\delta n_1+\delta n_2).
(\hbox{intrinsic semiconductor})
\label{ch5,171}
\end{equation}
Employing (\ref{ch5,08}) and (\ref{ch5,081}), it can be easily seen that in the
intrinsic semiconductor case
\begin{equation}
\lambda={1\over \tau}{n_i\over k_BT}.
\label{ch5,172}
\end{equation}

To obtain the surface flux and screening mode we solve the equations
(\ref{ch5,05}) and (\ref{ch5,06}) coupled with the Poisson's equation (\ref{ch5,07})
employing (\ref{ch5,08}), (\ref{ch5,12}), (\ref{ch5,13}) and (\ref{ch5,15}).
Employing (\ref{ch5,02}) and (\ref{ch5,08}), 

The screening part of $\delta\phi$ is the homogeneous solution of
(\ref{ch5,181}), since for it $\delta\tilde\mu_e,\delta\tilde\mu_h=0$ (${\cal
P}=0$). The flux part of $\delta\phi$ is the particular solution of
(\ref{ch5,181}) (${\cal P}>0$) that can be obtained after
$\delta\tilde\mu_1,\ \delta\tilde\mu_2$ are determined.

We first consider the macroscopic transport equations for an intrinsic
semiconductor, emphasizing the role of entropy production to obtain the
electron-hole recombination rate $r$ in a convenient form.  We then solve these
equations in the linear regime to obtain the properties of the screening and
diffusion-recombination modes, which we apply to the Dember effect.  These modes
also also are generated when current flows across an interface between a
semiconductor and a metal, and provide a way to view current flow across a p-n
junction.

From (\ref{ch5,01}) and in agreement with (\ref{ch5,10}) it follows that
\begin{equation}
j_i^u=Tj_i^s+\tilde\mu_e j_{ie}+\tilde\mu_h j_{ih}. 
\label{ch5,16}
\end{equation}

Employing (\ref{ch5,11})--(\ref{ch5,16}) in (\ref{ch5,10}) gives
\begin{equation}
0\le {\cal P}={\kappa\over T}(\partial_i T)^2 +D_{e}(\partial_i\tilde\mu_e)^2
2D_{eh}(\partial_i\tilde\mu_e)(\partial_i\tilde\mu_h)
+D_{h}(\partial_i\tilde\mu_h)^2 + \lambda(\tilde\mu_e+\tilde\mu_h)^2.
\label{ch5,17}
\end{equation}

 The constants $\alpha_{ee}$ and $\alpha_{hh}$ can be determined from the requirement
that the hole and electron fluxes can be written in standard forms\cite{ash}
$$
j_e=-D_n\partial_x n_e-\mu_n n_{e0}(-\partial_x\phi)=
-\alpha_{ee}{\partial\mu_e\over\partial n_e}\partial_x n_e-\alpha_{ee}e(-\partial_x\phi),
$$
\begin{equation}
j_h=-D_p\partial_x n_1+\mu_p n_{h0}(-\partial_x\phi)=
-\alpha_{hh}{\partial\mu_h\over\partial n_h}\partial_x n_h+\alpha_{hh}e(-\partial_x\phi)
\label{ch5,173}
\end{equation}
From (\ref{ch5,173}) we find that
\begin{equation}
\alpha_{hh}={\mu_p n_{i}\over e}\ \ \  (D_p=\alpha_{hh}{\partial\mu_h\over\partial n_h}), 
\ \ \hbox{and}\ \ \
\alpha_{ee}={\mu_n n_{i}\over e}\ \ \ (D_n=\alpha_{ee}{\partial\mu_e\over\partial n_e}).
\label{ch5,174}
\end{equation}
The partial derivatives in (\ref{ch5,172}) are evaluated at equilibrium, since
we consider small variations from equilibrium, and take the values 
\begin{equation}
{\partial n_e\over \partial\mu_e}={n_{e0}\over k_BT}, \quad 
{\partial n_h\over \partial\mu_h}={n_{h0}\over k_BT}. \quad (n_{e0}=n_{h0}=n_i)
\label{ch5,081}
\end{equation}
We linearize to obtain $\delta n_h={\partial n_h/\partial\mu_h}\delta\mu_h$, 
$\delta n_e={\partial n_e/\partial\mu_e}\delta\mu_e$.  

\begin{equation}
T\partial_t s=\partial_t \varepsilon-
\tilde\mu_e \partial_t n_e -\tilde\mu_h \partial_t n_h.
\label{ch5,09}
\end{equation}
Rewriting (\ref{ch5,09}) with (\ref{ch5,03})--(\ref{ch5,05}) gives

The wavevector $q_s$ is given by
\begin{equation}
q_s^2={e^2\over\epsilon_0\epsilon}({\partial n_e\over \partial\mu_e}+
{\partial n_e\over \partial\mu_e})={e^2\over\epsilon_0\epsilon}
{2n_i\over k_BT}.
\label{ch5,23}
\end{equation}

due to the diffusion-recombination mode is particular solution of
(\ref{ch5,181}), and thus satisfies
\begin{equation}
\nabla^2\phi_D=-{e\over\epsilon_0\epsilon}(\delta n_1-\delta n_2)_D=
{e^2\over\epsilon_0\epsilon}({\partial n_1\over \partial\mu_1}+
{\partial n_2\over \partial\mu_2})\delta\phi_D
-{e\over\epsilon_0\epsilon}({\partial n_1\over \partial\mu_1}\delta\tilde\mu_1
-{\partial n_2\over \partial\mu_2}\delta\tilde\mu_2).
\label{ch5,29}
\end{equation}
Thus, employing (\ref{ch5,26}), $\delta\phi_D$ can be written as
\begin{equation}
\delta\phi_D=A_{D3}\exp{(-q_D x)},
\label{ch5,30}
\end{equation}

$A_{D3}$ is constant.
Employing (\ref{ch5,23}), (\ref{ch5,26}), (\ref{ch5,28})
 and (\ref{ch5,30}) in (\ref{ch5,29}), $A_{D3}$ is given by

 

The carrier fluxes in non-equilibrium are proportional to the intensity of the
applied illumination. These can only be provided by the diffusion-recombination
mode.  However, in addition, the diffusion-recombination mode possesses charge.
For overall electro neutrality, the screening mode is excited  in an amount that
compensates for the charge of the diffusion-recombination mode.

Let the injected generation flux at the illuminated surface $x=0$ is,
as before, the carrier generation rate $G$.
Since the electron and hole fluxes are equal for all $x$,
the boundary condition at $x=0$ satisfied by the fluxes is
\begin{equation}
\ \ \ \ \ \ \ \ j_1=G,\ \ \ \hbox{or}\ \ \j_2=G.\ \ \ \ \ \ (x=0)
 \label{ch5,330}
\end{equation}
Since from (\ref{ch5,12}) $j_1=-\alpha_{11}\partial_x\delta\tilde\mu_1-\alpha_{12}
\partial_x\delta\tilde\mu_2$, employing (\ref{ch5,330})
we find
\begin{equation}
G=(\alpha_{11}+\beta\alpha_{12})q_DA_{D1},\ \ \ \ \ 
\hbox{or}\ \ \ \ \ \ A_{D1}={G\over q_D\alpha_{11}}
\label{ch5,33}
\end{equation}
after $\alpha_{12}=\alpha_{21}$ is neglected.

From (\ref{ch5,36}), the charge density at the surface $x=0$ is given by
\begin{equation}
\rho(x=0)={e^2\over k_BT}G\ {(\mu_n-\mu_p)\over \mu_n\mu_p}\ 
{1\over q_s+q_D}.
\label{ch5,38}
\end{equation}
Therefore, if $\mu_n>\mu_p$, the charge density near the illuminated surface is 
positive, since electrons, as more mobile, faster evacuated from the surface region. 
Oppositely, if $\mu_p>\mu_n$, the charge density near the illuminated surface is
 negative, since then holes, as more mobile, faster evacuate from the surface region.

and the electrostatic potential is 
decreasing function of $x$. Thus, the induced internal electric field acts in 
positive $x$--direction, tending to slow down the electrons while speeding up the holes.
If $\mu_n<\mu_p$ the Dember voltage is negative, and electrostatic potential
is increasing function of $x$ such that $\delta\phi(x\rightarrow \infty)\rightarrow 0$.
Thus, the induced internal electric field acts in negative $x$--direction, now 
slowing down the holes while speeding up the electrons.

The electrostatic potential is positive, decreasing function of the distance,
as we discussed in the previous section. The charge density is decreasing function
that changes the sign, due to the competitive effects between the flux and
flux--induced electric field. The electron number density changes more rapidly than the 
hole number density, since $\mu_n>\mu_p$.

\section{Speculation about Non-Intrinsic Semiconductors}

If a semiconductor also has donors, then there is an additional positive charge
density $n_D$ due to ionized donors.  There will then be two
diffusion-recombination modes.  I suspect that one of these will be like the
previous one, but one will be of very short range. In addition, there will be a
more complex statement about flux production at the surface.  This will fix the
amplitudes of the two diffusion-recombination modes. Once that is done, as
before overall electroneutrality will determine the amplitude of the screening
mode.

It is not clear how the ``old'' diffusion-recombination mode will be modified by
$n_D$.  It may have to include some $n_D$ and its effect on the relaxation time
$\tau_e$.

a new approach for description of the optically generated carriers on free
surfaces of intrinsic semiconductors. We have considered steady-state and
low-injection conditions, in one-dimensional geometry. Abandoning the
quasi-neutrality assumption, we have obtained the two characteristic lengths
that describe the system: the diffusion-recombination mode inverse wavevector,
and the screening mode inverse wavevector. The diffusion-recombination mode
coming from the electron and hole fluxes induced by the illumination carries
non-zero net charge. For overall electro neutrality of the system, the screening
mode is induced to screen the diffusion-recombination mode charge.

We have described the system in terms of electron and hole electrochemical
potentials, which enabled us to satisfy certain entropy production requirements.
These are that the diffusion-recombination mode only increases the system's
entropy, whereas the screening mode induced by the diffusion-recombination mode
charge distribution has no entropy production associated with it.

We have assumed ideal, uncharged semiconductor surfaces. If the surface has
surface states, upon the illumination the charge in these surface states will be
in proportion to the intensity of the light. Overall neutrality must still hold,
now involving the surface states, the diffusion-recombination mode, and the
screening mode. (Note that, if there is surface charge due to surface states in
absence of incident illumination, its field is screened out by the surface mode
alone.)

A_{D}=A_{Dh}({e\over\epsilon_0\epsilon})
{\biggl((\partial n_h/\partial\mu_h)-(\alpha_{ee}/\alpha_{hh}(\partial n_e/\partial\mu_e)\biggr)
\over q_s^2-q_D^2}
\equiv\gamma A_{D1}.

\begin{figure}
\centerline{\psfig{file=polarization.ps,width=1\linewidth}}
\caption{Calculated spin polarization ($\alpha$)
profiles for different voltages.
The upper figure is for the solar cell. The lower figure is for an injection
of spin by a ferromagnetic electrode at $x=2\mu {\rm m}$. In both cases
$\alpha$ extends beyond the transition region, while the strong
dependence of $\alpha$ on $V$ is seen only in the solar cell. The dashed
line in both figures is the doping profile.
}
\label{fig:2}
\end{figure}

one in  The {\it
screening mode} has a charge density but no electron or hole flux, with
wavevector $q_s^2=q_{se}^2+q_{sh}^2$, where $q_{se}$ and $q_{sh}$ are the
inverse screening lengths for electrons and holes alone.  The {\it
diffusion-recombination mode} has a charge density and equal electron and hole
fluxes, with wavevector $q_D^2=q_{De}^2+q_{Dh}^2$, where $q_{De}$ and $q_{Dh}$
are the inverse diffusion-recombination lengths for electrons and holes alone. 
Light absorbed at one surface generates a diffusion-recombination mode (in
proportion to the light intensity) and a screening mode (to assure overall
neutrality). The resultant charge separation produces a voltage difference
across the sample (the Dember effect).  Applications to other situations are 
discussed. 

To summarize, we have solved the linearized equations for macroscopic transport
in semiconductors, without assuming quasineutrality.  

To ensure that the recombination terms satisfy the principles of irreversible
thermodynamics, it was essential to employ the electrochemical potentials. There
are two surface modes, each involving both carriers: one is the usual screening
mode associated with Debye-H\"uckel screening; the other is a
diffusion-recombination mode. Application to the Dember effect yields agreement
with the conventional physical descriptions of the phenomenon -- specifically,
the change in sign of the charge density -- in contrast to previous approaches,
and specific quantitative predictions.  The theory predicts that the Dember
effect (voltage due to photoexcitation) is most easily observed in systems with
a low free carrier density, a slow recombination rate, and low (and unequal)
mobilities.

Thus slower 
recombination favors $\phi_D$, but slower scattering disfavors $\phi_D$. 

To close we note that, for systems driven out of equilibrium, space-independent
violations of quasineutrality can occur if the charge-carriers do not
recombine.\cite{sas}

72.20.-i    - conducting phenomena in semiconductors and insulators

72.20 Jv  -  charge carriers: generation, recombination, lifetime and
trapping

72.40.+w Photoconduction and photovoltaic effects (semiconductors)

73.50.Pz Photoconduction and photovoltaic effects (thin films)

cond-mat0107152.extracted_bib
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{\bf 33}, 207 (1932).


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(1994).


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{\bf 37}, 1 (1999). 


\bibitem{chaz} Jean-No\"{e}l Chazalviel, {\it Coulomb Screening by Mobile Charges,}
%Application to Materials Science, Chemistry, and Biology
Birkh\"{a}user, Boston (1999).


\bibitem{Onsager-LL} L.D. Landau and E.M. Lifshitz, {\it Statistical Physics,} 
2nd edition, Addison-Wesley, Reading, MA (1969).  See pp.374-378. 


\bibitem{ash} Neil W. Ashcroft and N. David Mermin, {\it Solid State
Physics,} Holt, Rinehart and Winston, New York (1975).


\bibitem{sze} S.M. Sze, {\it Physics of Semiconductor Devices,} John Wiley \&
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\bibitem{zutic} I. Zutic, J. Fabian, and S. Das Sarma, preprint 
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\bibitem{gold} S. Robert Goldman, Kalman Kalikstein and Bernard Kramer,
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London (1978).


\bibitem{shock1} W.Shockley, {\it Electrons and Holes in Semiconductors,} 
D.Van Nostrand, Princeton (1950). 

\begin{figure}
\epsffile{d_potrho.eps}
\caption{Photo-induced electrical potential and charge density.} 
\label{Fig.1}

\bibitem{shock} W.Shockley, and W.T. Read, Jr., Phys. Rev. {\bf 87}, 835(1952).


\bibitem{sas} W.M. Saslow, Surface Modes for Semiconductor with no
Depletion or Accumulation Layer, (unpublished).

If the free surface of a semiconductor is illuminated by orthogonally incident 
light that produces electron and hole photocarriers, the potential difference 
will appear between the illuminated and unilluminated semiconductor surface.
This potential difference is first discovered in experiments by 
Dember,\cite{dember1,dember2} and it is referred to as the Dember voltage.

The photoproduction of electrons and holes at the surface of an
illuminated semiconductor was essential to the classic
Haynes-Shockley experiment establishing that the charge carriers in
semiconductors are of two varieties, one positive (holes) and the other
negative (electrons). 

After it s discovery in 1931, the Dember effect is later studied by many 
authors in experiments on
insulating crystals\cite{ber1,ber2} and high resistivity semiconductors,\cite{bad,berg}
through the photovoltage measurements. More recently,
this effect is also studied in $C_{60}$ thin films.\cite{dipan}
Many characteristics of Dember effect can be found in Ref.[\onlinecite{jap}].


The intuitive origin of the Dember effect can be easily understood as follows.\cite{chaz}
The photocarriers, electrons and holes, have different mobilities within
the semiconductor, and usually the electron mobility $\mu_n$ exceeds the hole
mobility $\mu_p$. Thus, the electrons will evacuate from the region near the 
illuminated surface faster then holes. This will lead to the excess of holes
near the surface, and excess of electrons further from it. The established
charge distribution will cause the electric field to appear within the sample,
in direction that will tend to stop further separation of carriers, i.e. it will act 
to slow the electrons down while it speeds up the holes. The competitive effects,
the carrier diffusion and induced internal electric field, will result in the 
relaxation of the carrier density to their equilibrium values further from 
the illuminated surface.

From the above discussion, it can be seen that the photocarrier distribution is
determined by the two characteristic length: first, that describes the carrier fluxes
due to the their different mobilities, and second that describes the electric field
effect which tends to screens out the flux charge distribution.

The previous standard Dember effect theories have made the drastic assumption of 
local electroneutrality, arguing that the electric field effect would keep the
carriers close together within the sample, despite their 
different mobilities.\cite{chaz,gold}
The carrier flow within the sample would then be described via a single characteristic
length, yielding a mode that decays exponentially in space. This result is reviewed 
below in Overview.  
Moreover, the standard theories use electron and hole number density as natural variables
for semiconductor description. In description of the Dember effect, these are not 
the natural variables for the following reason. If the local electro neutrality is 
not assumed, the standard approach would yield the two characteristic modes which
would have mixed flux and screening properties.\cite{sas} 
Thus, since the illuminated system 
is in non-equilibrium, both of these modes will yield positive entropy production.
However, the positive entropy production should come from the fluxes only. There
must be no entropy production associated with screening, since semiconductors
screen external or internal charge distributions without dissipation of energy.

This article has two goals.
The first goal is to obtain the Dember voltage and describe the carrier charge densities
going beyond the local neutrality assumption. In other words, we wont to obtain both 
the characteristic length due to carrier fluxes, and characteristic length due to 
their screening. 
The second goal is to find the macroscopic equations describing a semiconductor 
using the electrochemical potentials as the natural variables. This enables us to
construct the solutions that satisfy the correct entropy production requirements 
for both flux and screening mode.

We consider the case of steady illumination of semi-infinite sample
with no depletion or accumulation layers in equilibrium,
 in a one-dimensional geometry. We assume that
the illumination is uniform over the surface and absorbed with very large absorption
coefficient (surface absorption only), as well as that the low-injection conditions apply.
Under the low-injection condition, the number density of the injected carriers is 
smaller than their corresponding equilibrium number densities.\cite{sze}

Using the electron and hole electrochemical potentials as
the primary variables, we find two exponentially decaying modes.  The {\it flux
mode} carries equal electron and hole fluxes (and thus no net electric
current),
dissipates energy, and has charge associated with it.  The {\it screening mode}
carries no fluxes and dissipates no energy, but does have charge associated
with
it.  On illumination, the diffusion-recombination mode is generated in proportion to the
illumination; to maintain electroneutrality near the surface (but not local
electroneutrality), accompanying the diffusion-recombination mode is the screening mode.

\subsection{Overview}

We now briefly review the Dember effect result obtained under the local electro
neutrality assumption.\cite{chaz} The  sample with ideal surface is considered.
The incident light is absorbed by the surface only, and the steady-state 
conditions are assumed. 

For comparison purposes with our results that follow
in the next sections, we use the following notation. The hole number density
is denoted as $n_1$, whereas the electron number density is denoted as $n_2$.
The equilibrium carrier number density is $n_i$, and the recombination time is $\tau$.

The equations for the carrier number densities in steady-state are
$$
0=D_p{d^2 n_1\over d x^2}+
\mu_p{d\over d x}(n_1{d\phi\over d x})-{n_1-n_i\over\tau},
$$ 
\begin{equation}
\ \ \ \ \ 0=D_n{d^2 n_2\over d x^2}-
\mu_n{d\over d x}(n_2{d\phi\over d x})-{n_2-n_i\over\tau},
\label{ch5,1}
