# External Potentials and Ehrenfest Relations in
Lagrangian Field Theories

Rayn Samson<EMAIL_ADDRESS>Department of Physics, Grinnell College,
Grinnell, Iowa 50112, USA

###### Abstract

This paper develops a general method to construct Ehrenfest-like relations for
Lagrangian field theories when an external, coordinate-dependent scalar
potential is applied. To do so, we derive continuity equations in which the
spatial and temporal derivatives of the potential can be interpreted as a
source of field momentum and field energy, respectively. For a non-
relativistic Schrödinger field theory, these continuity equations yield
Ehrenfest’s theorem for energy, linear momentum, and angular momentum. We then
derive a relativistic counterpart for these relations using complex Klein-
Gordon fields coupled with an electric potential.

##  I Introduction

It is well understood in classical mechanics that the conservation of linear
momentum in a system only holds when no external force is acting on it. Put in
the language of Noether’s theorem, a system is not symmetric under translation
when a spatially dependent potential acts on it—in which case the conservation
of linear momentum is broken. However, much of the richness in classical
mechanics comes from understanding how mechanical momentum changes in time
when such a potential is applied. This is captured by Newton’s second law,

$\frac{d}{dt}\mathbf{p}=-\nabla V.$ (1)

Of course, this expression reduces to a statement of momentum conservation
when the potential $V$ is not spatially dependent. While this result can be
easily derived as the equation of motion in a classical Lagrangian of the form
$L(\mathbf{q},\dot{\mathbf{q}},t)$, there has so far been no systematic way to
produce analogous results for the field momentum of a field Lagrangian
$\mathcal{L}(\varphi_{1},\dots,\varphi_{n},\partial_{\mu}\varphi_{1},\dots,\partial_{\mu}\varphi_{n})$.
This paper develops such a method by first considering a non-relativistic
field theory that describes the Schrödinger equation with an external,
coordinate-dependent potential in Section III. Conservation of energy and
momentum does not hold in general for such a system, but a close examination
of the potential’s response to coordinate transformations yields Ehrenfest’s
theorem for linear and angular momentum as well as a similar result for
energy. We then use a generalized version of the same method in Section IV to
derive analogous results in a complex Klein-Gordon field theory with minimal
electric potential coupling. In this paper, such results are referred to as
“Ehrenfest” or “Ehrenfest-like” relations owing to their structural
similarities with the original results obtained from the Schrödinger field
theory. However, the content of relativistic Ehrenfest relations has an
entirely different interpretation from the non-relativistic case.

###  I.1 Notation and Conventions

We will be working in flat Minkowski spacetime with Cartesian coordinates. The
mostly positive signature $(-+++)$ is used. Coordinate derivatives are
abbreviated with a comma before the appropriate index, e.g.
$M_{,\mu}\equiv\partial_{\mu}M$. Greek indices run through all four spacetime
coordinates, while latin indices run through the three spatial coordinates. We
will switch freely between spatial index notation and vector notation. For
example, the four-divergence is given by

$\partial_{\mu}\equiv\begin{pmatrix}\partial_{0}\\\ \nabla\end{pmatrix},$ (2)

where $\nabla=(\partial_{1},\partial_{2},\partial_{3})$ may be represented
with the open spatial index operator $\partial_{i}$.

##  II Noether Currents

In the context of classical field theory, Noether’s theorem describes a
fascinating equivalence between continuous symmetries of a Lagrangian system
and its conserved quantities. Starting with an action of the form

$\mathcal{S}=\int\mathcal{L}(\varphi,\partial_{\mu}\varphi)\,d^{4}x,$ (3)

where $\mathcal{L}$ is a Lagrangian density (henceforth: “Lagrangian”) that
depends on a field $\varphi$ and its derivatives. Using the principle of
stationary action, the field $\varphi$ must satisfy the Euler-Lagrange field
equation of motion

$-\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\right)+\frac{\partial{\mathcal{L}}}{\partial{\varphi}}=0.$
(4)

A “symmetry” of the system is a transformation that yields the same physical
predictions. A transformation of the field,

$\varphi\to\varphi+\delta\varphi,$ (5)

is a symmetry if it leaves Eq. (4) invariant. This holds most generally when
the transformation deforms the Lagrangian by a divergence of some function
$K^{\mu}$:

$\mathcal{L}\Big{(}\varphi+\delta\varphi,\partial_{\mu}(\varphi+\delta\varphi)\Big{)}-\mathcal{L}(\varphi,\partial_{\mu}\varphi)=\partial_{\mu}K^{\mu}.$
(6)

(We will later argue that the reverse is not true: invariance of Eq. (4) does
not imply symmetry under a given transformation.) Calculating the left-hand
side of Eq. (6) with a series expansion and simplifying with the equation of
motion yields

$\mathcal{L}\Big{(}\varphi+\delta\varphi,\partial_{\mu}(\varphi+\delta\varphi)\Big{)}-\mathcal{L}(\varphi,\partial_{\mu}\varphi)=\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi\right).$
(7)

Equating this to the surface term $\partial_{\mu}K^{\mu}$ yields the vanishing
divergence

$\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi-K^{\mu}\right)=0,$
(8)

which defines a conservation law in the form of a continuity equation. The
term in parentheses defines a Noether current. Importantly, if the action
depends on more than one field, we would need to add the response of each
field to the transformation when defining the Noether current. In this paper,
we will examine Lagrangian systems that depend on two fields which are complex
conjugates of each other, so Noether currents will take the form

$J^{\mu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\delta\varphi+\frac{\partial{\mathcal{L}}}{\partial{\overline{\varphi}_{,\mu}}}\delta\overline{\varphi}-K^{\mu}.$
(9)

We now proceed to discuss how the conservation of these Noether currents is
affected by an external scalar potential.

##  III Schrödinger Field Theory

Suppose we are given a physical system that is influenced by an external,
coordinate-dependent potential. One such example is a system described by the
Schrödinger equation

$-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi=i\hbar\frac{\partial{\Psi}}{\partial{t}}.$
(10)

Crucially, Eq. (10) does not specify the source or dynamics of the potential
$V$, which is why we say that it is external to the theory. To formulate this
equation in terms of field theory, consider a Schrödinger
LagrangianFranklin2017 TakahashiRop1986

$\mathcal{L}=\frac{\hbar^{2}}{2m}\partial_{j}\overline{\Psi}\partial^{j}\Psi+V\overline{\Psi}\Psi-\frac{i\hbar}{2}\left(\overline{\Psi}\dot{\Psi}-\Psi\dot{\overline{\Psi}}\right),$
(11)

where the fields are $\Psi$ and $\overline{\Psi}$. The equation of motion
associated with the latter,

$-\frac{\partial{}}{\partial{t}}\left(\frac{\partial{\mathcal{L}}}{\partial{\dot{\overline{\Psi}}}}\right)-\frac{\partial{}}{\partial{x^{j}}}\left(\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,j}}}\right)+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}}}=0,$
(12)

reduces to Eq. (10). To account for the potential’s response to a coordinate
transformation, we treat the Lagrangian as an explicit function of $V$ in
addition to the fields and their derivatives:

$\mathcal{L}=\mathcal{L}(\Psi,\overline{\Psi},\partial_{\mu}\Psi,\partial_{\mu}\overline{\Psi},V).$
(13)

Despite the awkward dependence on the potential, this Lagrangian still admits
a $U(1)$ symmetry. Consider the complex rotation $\Psi\to e^{i\theta}\Psi$
(and cc). For small $\theta$ we have $e^{i\theta}\approx 1+i\theta$, and so
the fields respond with

$\Psi\to\Psi+i\theta\Psi\quad\text{and}\quad\overline{\Psi}\to\overline{\Psi}-i\theta\overline{\Psi}.$
(14)

The associated conservation law becomes a statement of probability
conservation,

$\frac{\partial{}}{\partial{t}}|\Psi|^{2}+\nabla\cdot\mathbf{j}=0,$ (15)

where $|\Psi|^{2}\equiv\overline{\Psi}\Psi$ is interpreted as a probability
density and $\mathbf{j}$ is the probability current defined byGriffiths2018-vs

$\mathbf{j}\equiv\frac{\hbar}{2mi}\left(\overline{\Psi}\nabla\Psi-\Psi\nabla\overline{\Psi}\right).$
(16)

This result is unaffected by the presence of $V$ in the Lagrangian, but the
same is not true for coordinate transformations. Consider a spacetime
translation

$x^{\mu}\to x^{\mu}+a^{\mu},$ (17)

for some (small) constant $a^{\mu}$. The fields are scalars, so they transform
as

$\Psi\to\Psi+a^{\mu}\partial_{\mu}\Psi\quad\text{and}\quad\overline{\Psi}\to\overline{\Psi}+a^{\mu}\partial_{\mu}\overline{\Psi},$
(18)

and crucially, so does the potential:

$V\to V+a^{\mu}\partial_{\mu}V.$ (19)

As a scalar, the Lagrangian also transforms as
$\mathcal{L}\to\mathcal{L}+a^{\mu}\partial_{\mu}\mathcal{L}$. Thus the change
in the Lagrangian (which we will denote as $\delta\mathcal{L}$) calculated by
treating $\mathcal{L}$ as a scalar is given by

$\delta\mathcal{L}=\partial_{\mu}(a^{\mu}\mathcal{L}),$ (20)

which gives the surface term. Care must be taken to understand what is meant
here: as a scalar, the Lagrangian does changes by an overall divergence, and
so the equations of motion are invariant even in the presence of $V$. However,
computing $\delta\mathcal{L}$ from a first-order series expansion in all
arguments of the transformed Lagrangian does not yield an overall divergence,
and dependence on $V$ is the culprit:

$\delta\mathcal{L}=a^{\nu}\left[\partial_{\mu}\left(\frac{\partial{\mathcal{L}}}{\partial{\Psi_{,\mu}}}\Psi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,\mu}}}\overline{\Psi}_{,\nu}\right)+\frac{\partial{\mathcal{L}}}{\partial{V}}\partial_{\nu}V\right].$
(21)

We can thus see that $\mathcal{L}$ is not translationally symmetric, and hence
there is no conservation law to be obtained from a vanishing divergence. That
is, invariance of a potential-dependent Lagrangian under a coordinate
transformation does not imply a symmetry under that transformation. However,
we can still set Eqs. (20) and (21) equal to each other and see what is
obtained instead of a conservation law. After discarding the overall constant
$a^{\nu}$, we define the energy-momentum tensor in a standard way:

$T^{\mu}_{\,\,\,\nu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\Psi_{,\mu}}}\Psi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\Psi}_{,\mu}}}\overline{\Psi}_{,\nu}-\delta^{\mu}_{\,\,\,\nu}\mathcal{L}.$
(22)

We still say that $T^{\mu}_{\,\,\,\nu}$ represents four Noether currents
containing the energy ($\nu=0$) and momentum ($\nu=1,2,3$) densities stored in
the fields. However, the divergence of $T^{\mu}_{\,\,\,\nu}$ is not in general
zero. Instead, the derivatives of the potential act as a “source” in the
continuity equations

$\partial_{\mu}T^{\mu}_{\,\,\,\nu}=-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial_{\nu}V.$
(23)

The conservation of these currents depends on the presence of an external
potential and its derivatives. In analogy with classical mechanics, the
momentum of a system is conserved only when there is no coordinate-dependent
external potential acting on it. When an external potential is applied,
momentum is no longer conserved and the change in momentum is summarized by
Newton’s second law: $d\mathbf{p}/dt=-\nabla V$. The equivalent result in non-
relativistic quantum mechanics would be Ehrenfest’s theorem for momentum,
$d\langle\hat{\mathbf{p}}\rangle/dt=\langle-\nabla V\rangle$. Section III.1
will show that this is one of two statements contained in Eq. (23).

###  III.1 Ehrenfest Relations

Having defined the energy-momentum tensor associated with the Schrödinger
Lagrangian, it is helpful to write out the components explicitly:

$\displaystyle T^{0}_{\,\,\,0}$
$\displaystyle=-\frac{\hbar^{2}}{2m}\partial^{i}\overline{\Psi}\partial_{i}\Psi-V\overline{\Psi}\Psi$
(24) $\displaystyle T^{i}_{\,\,\,0}$
$\displaystyle=\frac{\hbar^{2}}{2mc}\left(\dot{\overline{\Psi}}\partial^{i}\Psi+\dot{\Psi}\partial^{i}\overline{\Psi}\right)$
(25) $\displaystyle T^{0}_{\,\,\,i}$ $\displaystyle=-\frac{i\hbar
c}{2}\left(\overline{\Psi}\partial_{i}\Psi+\Psi\partial_{i}\overline{\Psi}\right)$
(26) $\displaystyle T^{i}_{\,\,\,j}$
$\displaystyle=\frac{\hbar^{2}}{2m}\left(\partial^{i}\overline{\Psi}\partial_{j}\Psi+\partial^{i}\Psi\partial_{j}\overline{\Psi}\right)-\delta^{i}_{\,j}\mathcal{L}.$
(27)

The momentum density is given by the $T^{0}_{\,\,\,j}$ components, where
$j=1,2,3$ represents components along each spatial coordinate. By noting that
$\partial\mathcal{L}/\partial V=\overline{\Psi}\Psi$ is the probability
density, the sourced continuity equation says

$\partial_{\mu}T^{\mu}_{\,\,\,j}=\overline{\Psi}(-\partial_{j}V)\Psi,$ (28)

where it can be seen that the right-hand side represents a force expectation
upon integration. For comparison, we put the left-hand side into integral
form, simplify using the divergence theorem, and assume boundary terms vanish.
This yields

$\displaystyle\int\\!\\!\partial_{\mu}T^{\mu}_{\,\,\,j}\,d^{3}x$
$\displaystyle=$
$\displaystyle\int\\!\\!\left(\frac{1}{c}\frac{\partial{T^{0}_{\,\,\,j}}}{\partial{t}}+\frac{\partial{T^{i}_{\,\,\,j}}}{\partial{x^{i}}}\right)d^{3}x$
(29) $\displaystyle=$
$\displaystyle\frac{d}{dt}\int\\!\frac{1}{c}T^{0}_{\,\,\,j}\,d^{3}x$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{2}\frac{d}{dt}\int\\!\\!\left(\overline{\Psi}\partial_{j}\Psi-\Psi\partial_{j}\overline{\Psi}\right)\,d^{3}x$
$\displaystyle=$
$\displaystyle-\frac{i\hbar}{2}\frac{d}{dt}\int\\!\\!\big{(}2\overline{\Psi}\partial_{j}\Psi-\partial_{j}(\overline{\Psi}\Psi)\big{)}\,d^{3}x$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\int\\!\overline{\Psi}\\!\left(\frac{\hbar}{i}\partial_{j}\right)\\!\Psi\,d^{3}x$
$\displaystyle=$ $\displaystyle\frac{d\langle\hat{p}_{j}\rangle}{dt},$

where the momentum operator $\hat{p}_{j}\equiv\frac{\hbar}{i}\partial_{j}$ has
been identified. We thus conclude that Eq. (28) in integral form is
Ehrenfest’s theorem for momentum,Takahashi1986

$\frac{d\langle\hat{\mathbf{p}}\rangle}{dt}=\langle-\nabla V\rangle.$ (30)

There are several points to make regarding this familiar result obtained by
unfamiliar means. Locally, the negative gradient of an external potential—the
external force—acts as a source of field momentum, which is captured by Eq.
(28). Globally, this results in Ehrenfest’s theorem for momentum. Moreover,
this result does not require a probabilistic interpretation of the
wavefunction, nor does it reference the field equations associated with Eq.
(11). This indicates that Eq. (23), which makes no reference to a specific
choice of Lagrangian, holds more generally for field theories with an external
potential.

There is now a clear motivation to ask about the remaining Noether current
representing energy density. The sourced continuity equation for this
component is

$\partial_{\mu}T^{\mu}_{\,\,\,0}=-\frac{1}{c}\overline{\Psi}\frac{\partial{V}}{\partial{t}}\Psi.$
(31)

Putting the left-hand side in integral form as before, we now have an
expression in terms of the Hamiltonian operator:

$\displaystyle\int\\!\\!\partial_{\mu}T^{\mu}_{\,\,\,0}\,d^{3}x$
$\displaystyle=$
$\displaystyle\int\\!\\!\left(\frac{1}{c}\frac{\partial{T^{0}_{\,\,\,0}}}{\partial{t}}+\frac{\partial{T^{i}_{\,\,\,0}}}{\partial{x^{i}}}\right)d^{3}x$
(32) $\displaystyle=$
$\displaystyle\frac{1}{c}\frac{d}{dt}\int\\!T^{0}_{\,\,\,0}\,d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{c}\frac{d}{dt}\int\\!\\!\left(-\frac{\hbar^{2}}{2m}\partial^{i}\overline{\Psi}\partial_{i}\Psi-V\overline{\Psi}\Psi\right)d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{c}\frac{d}{dt}\int\\!\left(\frac{\hbar^{2}}{2m}\overline{\Psi}\partial^{i}\partial_{i}\Psi-V\overline{\Psi}\Psi\right)d^{3}x$
$\displaystyle=$
$\displaystyle-\frac{1}{c}\frac{d}{dt}\int\\!\\!\overline{\Psi}\\!\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+V\right)\\!\Psi\,d^{3}x$
$\displaystyle=$ $\displaystyle-\frac{1}{c}\frac{d\langle\hat{H}\rangle}{dt}.$

Thus integrating both sides of Eq. (31) and dividing out the factor of $-1/c$,
we have an Ehrenfest-like relation for energy:

$\frac{d\langle\hat{H}\rangle}{dt}=\left\langle\frac{\partial{V}}{\partial{t}}\right\rangle.$
(33)

Just as the spatial derivatives of an external potential are sources of linear
momentum, the time derivative of an external potential acts as a source of
energy. Thus the expected energy is not a constant in time when a time-
dependent potential is applied, but we still have a formal expression to track
its rate of change. In Section III.2, we demonstrate one example of this
result holding true for an infinite square well with a moving floor.

We now briefly cover the case of angular momentum, which yields a similar
Ehrenfest relation. Consider a coordinate rotation

$x^{\mu}\to x^{\mu}+R^{\mu}_{\,\,\,\nu}x^{\nu},$ (34)

where $R^{\mu}_{\,\,\,\nu}$ represents an infinitesimal rotation of the
spatial coordinates. We can show that there are six associated Noether
currents

$\mathcal{M}^{\mu\nu\sigma}\equiv x^{\nu}T^{\mu\sigma}-x^{\sigma}T^{\mu\nu},$
(35)

where $T^{\mu\nu}$ is the energy-momentum tensor as defined before with raised
indices. A detailed derivation is provided in Appendix A.2. We can use the
Levi-Civita symbol to pick out the components representing the angular
momentum density along each spatial coordinate,

$\ell_{i}\equiv\frac{1}{c}\varepsilon_{ijk}x^{j}T^{0k}=\frac{1}{2c}\varepsilon_{ijk}\mathcal{M}^{0jk}.$
(36)

On one hand, we can now directly compute the divergence of Eq. (35) with the
product rule and pick out the terms of interest,

$\frac{1}{2}\varepsilon_{ijk}\partial_{\mu}\mathcal{M}^{0jk}=\overline{\Psi}\tau_{i}\Psi,$
(37)

where $\tau_{i}\equiv\varepsilon_{ijk}x^{i}(-\partial^{j}V)$ is the $i$th
component of the torque associated with $V$. It is thus clear that integration
along all components of Eq. (37) yields the expected torque,
$\langle\boldsymbol{\tau}\rangle$. On the other hand, computing the integral
in a similar fashion to Eqs. (29) and (32) yields

$\displaystyle\frac{1}{2}\varepsilon_{ijk}\\!\\!\int\\!\\!\partial_{\mu}\mathcal{M}^{0jk}\,d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{2}\varepsilon_{ijk}\\!\\!\int\\!\\!\left(\frac{1}{c}\frac{\partial{\mathcal{M}^{0jk}}}{\partial{t}}+\frac{\partial{\mathcal{M}^{\ell
jk}}}{\partial{x^{\ell}}}\right)d^{3}x$ (38) $\displaystyle=$
$\displaystyle\frac{1}{2c}\varepsilon_{ijk}\frac{d}{dt}\\!\int\\!\\!\mathcal{M}^{0jk}\,d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{2c}\varepsilon_{ijk}\frac{d}{dt}\\!\int\\!\\!\left(x^{j}T^{0k}-x^{k}T^{0j}\right)d^{3}x$
$\displaystyle=$
$\displaystyle\frac{d}{dt}\\!\int\\!\\!\overline{\Psi}\left(\varepsilon_{ijk}\hat{x}^{j}\hat{p}^{k}\right)\Psi\,d^{3}x$
$\displaystyle=$ $\displaystyle\frac{d\langle\hat{L}_{i}\rangle}{dt},$

where $\hat{L}_{i}\equiv\varepsilon_{ijk}\hat{x}^{j}\hat{p}^{k}$ is the
angular momentum operator. Comparing this with the integral of Eq. (37), we
have Ehrenfest’s theorem for angular momentum:Griffiths2018-vs

$\frac{d\langle\hat{\mathbf{L}}\rangle}{dt}=\langle\boldsymbol{\tau}\rangle.$
(39)

This derivation serves as an important template for deriving analogous results
in the relativistic Klein-Gordon setting, which is the subject of Section IV.

###  III.2 Moving Floor Infinite Square Well

We demonstrate one instance of the result in Eq. (33) using the time-varying
potential

$V(x,t)=\begin{cases}\tilde{V}(t)&\text{if }0<x<L\\\
\infty&\text{otherwise}\end{cases}$ (40)

for some differentiable function $\tilde{V}(t)$. This reduces to the more
familiar stationary floor infinite square well when $\tilde{V}(t)=0$. With
boundary condition $\Psi(0,t)=\Psi(L,t)=0$, the full solution to the time-
dependent Schrödinger equation can be expanded into a complete orthonormal
basis with time-varying coefficients

$\Psi(x,t)=\sum_{k=1}^{\infty}a_{k}(t)\sin\left(\frac{k\pi x}{L}\right).$ (41)

The coefficients $a_{k}(t)$ absorb the overall normalization and satisfy the
residual requirementFranklin2013

$0=\frac{\hbar^{2}k^{2}\pi^{2}}{2mL^{2}}a_{k}(t)-i\hbar\dot{a}_{k}(t)+a_{k}(t)\tilde{V}(t).$
(42)

The solution to this set of differential equations is

$a_{k}(t)=A_{k}e^{-E_{k}t/\hbar}e^{-iw(t)/\hbar},$ (43)

where $E_{k}=\frac{\hbar^{2}k^{2}\pi^{2}}{2mL^{2}}$ are the energy states of
the stationary square well and

$w(t)\equiv\int_{t^{\prime}=0}^{t}\tilde{V}(t^{\prime})\,dt^{\prime}$ (44)

is the additional phase picked up due to the time-dependent potential that
does not depend on $k$. We can thus write the solution

$\Psi(x,t)=e^{-iw(t)/\hbar}\sum_{k=1}^{\infty}A_{k}e^{-iE_{k}t/\hbar}\sin\left(\frac{k\pi
x}{L}\right),$ (45)

and the additional phase will disappear in any expression containing
$|\Psi|^{2}$. With this, we can calculate

$\displaystyle\frac{d\langle\hat{H}\rangle}{dt}$
$\displaystyle=\frac{d}{dt}\\!\int_{0}^{L}\\!\overline{\Psi}\left(\frac{-\hbar^{2}}{2m}\frac{\partial{{}^{2}}}{\partial{x^{2}}}+\tilde{V}(t)\right)\Psi\,dx$
$\displaystyle=\frac{-\hbar^{2}}{2m}\frac{d}{dt}\\!\\!\int_{0}^{L}\\!\\!\\!\overline{\Psi}\frac{\partial{{}^{2}\Psi}}{\partial{x^{2}}}\,dx+\frac{d}{dt}\\!\\!\int_{0}^{L}\\!\\!\\!\tilde{V}(t)|\Psi|^{2}dx.$
(46)

To handle the first integral, we use the orthonormal decomposition

$\displaystyle\int_{0}^{L}\\!\\!\overline{\Psi}\frac{\partial{{}^{2}\Psi}}{\partial{x^{2}}}\,dx$
$\displaystyle=$
$\displaystyle\\!\int_{0}^{L}\\!\\!\\!-\delta_{jk}\\!\left(\frac{k\pi}{L}\right)^{2}\overline{A}_{j}A_{k}e^{i(E_{j}-E_{k})t/\hbar}$
(47) $\displaystyle\quad\quad\times\sin\left(\frac{j\pi
x}{L}\right)\sin\left(\frac{k\pi x}{L}\right)dx$ $\displaystyle=$
$\displaystyle-\left(\frac{j\pi}{L}\right)^{2}|A_{j}|^{2}\\!\\!\int_{0}^{L}\\!\\!\sin^{2}\left(\frac{j\pi
x}{L}\right)dx$ $\displaystyle=$
$\displaystyle-\left(\frac{j\pi}{L}\right)^{2}.$

Since there is no time dependence here, the first term in Eq. (46) vanishes.
To handle the second term, we pass the time derivative into the integral and
use the product rule:

$\displaystyle\frac{d}{dt}\\!\int_{0}^{L}\\!\\!\tilde{V}(t)|\Psi|^{2}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{L}\\!\frac{\partial{\tilde{V}(t)}}{\partial{t}}|\Psi|^{2}dx+\tilde{V}(t)\\!\\!\int_{0}^{L}\\!\\!\frac{\partial{}}{\partial{t}}|\Psi|^{2}dx$
(48) $\displaystyle=$
$\displaystyle\int_{0}^{L}\\!\overline{\Psi}\frac{\partial{\tilde{V}(t)}}{\partial{t}}\Psi\,dx$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\\!\overline{\Psi}\frac{\partial{V(x,t)}}{\partial{t}}\Psi\,dx$
$\displaystyle=$
$\displaystyle\left\langle\frac{\partial{V(x,t)}}{\partial{t}}\right\rangle.$

We have thus verified that Eq. (33) holds for this system for any
differentiable motion of the floor $\tilde{V}(t)$.

##  IV Complex Klein-Gordon Field Theory with Minimal Electric Potential
Coupling

The general outline of the previous section appears to be a way to generate
Ehrenfest-like relations directly from a Lagrangian, without reference to the
probabilistic interpretation of the wavefunction. We use this as motivation to
develop analogous results in a relativistic setting using complex Klein-Gordon
fields coupled with an electric potential. We use a minimal coupling given by
the covariant derivative

$D_{\mu}\equiv\partial_{\mu}-i\frac{q}{\hbar}A_{\mu},$ (49)

where $q$ determines the coupling strength and

$A^{\mu}\equiv\begin{pmatrix}V/c\\\ \mathbf{A}\end{pmatrix}$ (50)

is the electromagnetic four-potential (note that $V$ now has a dimension of
energy per unit charge). Since we are only interested in the scalar potential
term, we set $\mathbf{A}=0$. In order to treat the potential as external to
the theory, we will further omit the
$\frac{1}{4\mu_{0}}F^{\alpha\beta}F_{\alpha\beta}$ term used in a scalar QED
action. With these modifications, the Lagrangian we use is

$\displaystyle\mathcal{L}$
$\displaystyle=\overline{\left(D_{\mu}\phi\right)}\left(D^{\mu}\phi\right)+\frac{m^{2}c^{2}}{\hbar^{2}}\overline{\phi}\phi$
$\displaystyle=\overline{\phi}_{,\mu}\phi_{,}^{\,\,\mu}+i\frac{qV}{\hbar
c^{2}}\left(\overline{\phi}\dot{\phi}-\phi\dot{\overline{\phi}}\right)+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\,\overline{\phi}\phi.$
(51)

Using the d’Alembertian operator $\square\equiv\partial_{\mu}\partial^{\mu}$,
the field equations are

$\displaystyle-\square\overline{\phi}+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\overline{\phi}+2i\frac{qV}{\hbar
c^{2}}\dot{\overline{\phi}}+\frac{iq\dot{V}}{\hbar c^{2}}\overline{\phi}=0$
(52)
$\displaystyle-\square\phi+\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\phi-2i\frac{qV}{\hbar
c^{2}}\dot{\phi}-\frac{iq\dot{V}}{\hbar c^{2}}\phi=0.$ (53)

As before, we will first find the conservation law associated with a $U(1)$
symmetry, $\phi\to e^{i\theta}\phi$. The conserved current is

$J^{\mu}=i\theta\left(\overline{\phi}\partial^{\mu}\phi-\phi\partial^{\mu}\overline{\phi}-2\frac{iq}{\hbar}A^{\mu}\overline{\phi}\phi\right).$
(54)

The 0-component can be simplified by separating the fields into their spatial
and time-dependent components

$\phi=\psi(\mathbf{r})e^{-iEt/\hbar},$ (55)

and the complex conjugate. This allows us to write the time derivatives of the
fields as

$\dot{\phi}=\frac{E}{i\hbar}\phi\quad\text{and}\quad\dot{\overline{\phi}}=-\frac{E}{i\hbar}\overline{\phi}.$
(56)

After setting units, the conservation law reads

$\frac{\partial{}}{\partial{t}}\rho+\nabla\cdot\mathbf{j}=0,$ (57)

where

$\rho\equiv\frac{2q}{\hbar^{2}c^{2}}(E-qV)\overline{\phi}\phi$ (58)

is interpreted as an electric charge density,Winter1959  and

$\mathbf{j}\equiv
i\frac{q}{\hbar}(\phi\nabla\overline{\phi}-\overline{\phi}\nabla\phi)$ (59)

is the corresponding electric current. Since electric charge conservation
appears precisely where probability conservation appeared in the Schrödinger
field theory, we anticipate the same replacement to appear in the continuity
equations associated with the energy-momentum tensor. Section IV.1 shows that
this is indeed the case.

###  IV.1 Relativistic Ehrenfest Relations

Given our Lagrangian that depends on $\phi$, its complex conjugate, their
derivatives, and the electric potential $V$

$\mathcal{L}=\mathcal{L}(\phi,\overline{\phi},\partial_{\mu}\phi,\partial_{\mu}\overline{\phi},V),$
(60)

we can follow the exact same procedure outlined from Eq. (17) to Eq. (23).
Again defining the energy-momentum tensor as

$T^{\mu}_{\,\,\,\nu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\phi_{,\mu}}}\phi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\phi}_{,\mu}}}\overline{\phi}_{,\nu}-\delta^{\mu}_{\,\,\,\nu}\mathcal{L},$
(61)

the continuity equation still reads as Eq. (23). However, the factor of
$\partial\mathcal{L}/\partial V$ in the source term is not as straightforward
to calculate. Computing it explicitly gives

$\displaystyle\frac{\partial{\mathcal{L}}}{\partial{V}}$ $\displaystyle=$
$\displaystyle\frac{iq}{\hbar
c^{2}}\left(\overline{\phi}\dot{\phi}-\phi\dot{\overline{\phi}}\right)-2\frac{q^{2}V}{\hbar^{2}c^{2}}\overline{\phi}\phi$
(62) $\displaystyle=$ $\displaystyle\frac{iq}{\hbar
c^{2}}\left(2\frac{E}{i\hbar}\overline{\phi}\phi\right)-2\frac{q^{2}V}{\hbar^{2}c^{2}}\overline{\phi}\phi$
$\displaystyle=$
$\displaystyle\frac{2q}{\hbar^{2}c^{2}}(E-qV)\overline{\phi}\phi$
$\displaystyle=$ $\displaystyle\rho,$

where we used Eq. (56) to simplify the time derivatives as before. Thus the
derivatives of the potential are weighted by the electric charge density in
the source term of Eq. (23), exactly taking the place of probability density
in the Schrödinger theory. The continuity equation is thus

$\partial_{\mu}T^{\mu}_{\,\,\,\nu}=-\rho\partial_{\nu}V,$ (63)

where the energy-momentum tensor components are

$\displaystyle T^{0}_{\,\,\,0}$
$\displaystyle=-\frac{1}{c^{2}}\dot{\overline{\phi}}\dot{\phi}-\nabla\overline{\phi}\cdot\nabla\phi-\left(\frac{m^{2}c^{2}}{\hbar^{2}}-\frac{q^{2}V^{2}}{\hbar^{2}c^{2}}\right)\overline{\phi}\phi$
(64) $\displaystyle T^{i}_{\,\,\,0}$
$\displaystyle=\frac{1}{c}\left(\dot{\overline{\phi}}\partial^{i}\phi+\dot{\phi}\partial^{i}\overline{\phi}\right)$
(65) $\displaystyle T^{0}_{\,\,\,i}$
$\displaystyle=-\frac{1}{c}\left(\dot{\overline{\phi}}\partial_{i}\phi+\dot{\phi}\partial_{i}\overline{\phi}\right)+i\frac{qV}{\hbar
c}\Big{(}\overline{\phi}\partial_{i}\phi-\phi\partial_{i}\overline{\phi}\Big{)}$
(66) $\displaystyle T^{i}_{\,\,\,j}$
$\displaystyle=\partial^{i}\overline{\phi}\partial_{j}\phi+\partial^{i}\phi\partial_{j}\overline{\phi}-\delta^{i}_{\,\,\,j}\mathcal{L}.$
(67)

The component $T^{0}_{\,\,\,0}$ is the Hamiltonian density or energy density
(up to a sign) stored in the fields. Upon integration, there is no longer a
familiar expression like the expected Hamiltonian in Eq. (32). Instead, we
write the field Hamiltonian directly in terms of $T^{0}_{\,\,\,0}$:

$H\equiv\int-T^{0}_{\,\,\,0}\,d^{3}x.$ (68)

With this definition and assuming boundary terms vanish as before, the $\nu=0$
continuity equation in integral form reads

$\frac{d}{dt}H=\int\\!\\!\rho\frac{\partial{V}}{\partial{t}}\,d^{3}x.$ (69)

Structurally, this is the same as Eq. (33) with a charge density replacing
probability density. In a similar fashion, we directly use the field momentum
components

$P_{i}\equiv\int\frac{1}{c}T^{0}_{\,\,\,i}\,d^{3}x,$ (70)

and $\mathbf{P}=(P_{1},P_{2},P_{3})$. Noting that $\mathbf{E}\equiv-\nabla V$
defines an electric field, the $\nu=1,2,3$ continuity equations yield

$\frac{d}{dt}\mathbf{P}=\int\\!\\!\rho\mathbf{E}\,d^{3}x.$ (71)

Stated in words, the rate of change of the field momentum is equal to the
external Coulomb force acting on a charge distribution. This is a fitting
counterpart to Ehrenfest’s theorem, which relates the rate of change of
expected momentum to the external force acting on a probability distribution.
Of course, Ehrenfest’s theorem for momentum is itself a quantum mechanical
counterpart of Newton’s second law, which relates the rate of change of
mechanical momentum to the external force acting on a mass distribution.

A similar result can be shown for angular momentum. We still have the six
Noether currents associated with a spatial coordinate rotation

$\mathcal{M}^{\mu\nu\sigma}\equiv x^{\nu}T^{\mu\sigma}-x^{\sigma}T^{\mu\nu},$
(72)

from which the angular momentum density components are

$\ell_{i}=\frac{1}{2c}\varepsilon_{ijk}\mathcal{M}^{0jk}.$ (73)

Now defining the field angular momentum

$L_{i}\equiv\int\ell_{i}\,d^{3}x,$ (74)

it is straightforward to show that

$\frac{1}{2}\varepsilon_{ijk}\int\partial_{\mu}\mathcal{M}^{\mu
jk}\,d^{3}x=\frac{d}{dt}L_{i},$ (75)

assuming boundary terms vanish so that only the $\mu=0$ term survives
integration. Compare this to an explicit calculation of the divergence of the
four-current using the product rule, which after simplifying with Eq. (63)
yields

$\frac{1}{2}\varepsilon_{ijk}\partial_{\mu}\mathcal{M}^{\mu
jk}=\varepsilon_{ijk}x^{j}(-\partial^{k}V)\rho,$ (76)

where the right-hand side is a torque per unit charge multiplied by the charge
density $\rho$. The right-hand side is a torque per unit charge multiplied by
the charge density $\rho$. Integrating and comparing with Eq. (76) we have a
relativistic Ehrenfest relation for angular momentum,

$\frac{d}{dt}\mathbf{L}=\int(\rho\mathbf{r}\times\mathbf{E})\,d^{3}x.$ (77)

Here $\rho\mathbf{r}=\rho(x^{1},x^{2},x^{3})$ represents a dipole moment
vector, and the electric field is defined as before. Since the integrand is
weighted by a charge density, we say that the change in field angular momentum
is equal to the external torque acting on a charge distribution.

##  V Conclusion

This paper examined the effect of external scalar potentials on Lagrangian
systems. When the potential is coordinate-dependent, such systems are not
symmetric under a coordinate transformation and hence conservation of energy
and momentum do not hold. By accounting for the response of the potential to
spacetime translation, however, we find a continuity equation wherein the
derivatives of the potential act as a source of field energy and field
momentum. In integral form, these continuity equations express familiar
Ehrenfest relations directly from a Schrödinger Lagrangian. This approach
makes no appeal to the probabilistic interpretation of the wavefunction, nor
does it require explicit reference to the Schrödinger equation. We thus
postulate that Eq. (23) is a general way to derive Ehrenfest-like relations
for a Lagrangian acted on by an external, coordinate-dependent scalar
potential.

We then demonstrated one example of Ehrenfest-like relations for a complex
Klein-Gordon Lagrangian coupled to an electric potential. These results for
relativistic spin 0 particles have an electric charge density $\rho$ taking
the place of the probability density $|\Psi|^{2}$ from the Schrödinger
Lagrangian. A natural progression of this work would be to replicate these
results for a Dirac Lagrangian describing spin 1/2 particles.

It should be noted that the derivation leading to Eq. (23) is not the only way
to derive the Ehrenfest relations shown this paper, although other methods are
far less elegant. One could instead explicitly calculate
$\partial_{\mu}T^{\mu}_{\,\,\,\nu}$ using Eqs. (24–27) for the Schrödinger
tensor and Eqs. (64–67) for the Klein-Gordon tensor. This process is much more
involved and requires extensive use of the field equations in order to arrive
at the desired continuity equations. However, the correspondence of both
methods lends support to the notion of treating a Lagrangian as an explicit
function of an external scalar potential when one is applied.

*

##  Appendix A Noether Currents From Coordinate Transformations

We discuss more generally the standard Noether currents spacetime translations
and spatial rotations for a Lagrangian of two fields $\varphi$ and
$\overline{\varphi}$. Generalizing to a Lagrangian with any number of fields
is straightforward.

###  A.1 Spacetime Translation

The infinitesimal spacetime translation $x^{\mu}\to x^{\mu}+a^{\mu}$ has the
linearized field response

$\varphi(x^{\mu}+a^{\mu})\approx\varphi(x^{\mu})+a^{\mu}\partial_{\mu}\varphi,$
(78)

and the complex conjugate. After calculating the response of the Lagrangian
(I) as a scalar function of the coordinates and (II) as a function of the
fields and their derivatives (and simplifying with field equations), we see
that the associated Noether current is

$J^{\mu}=a^{\nu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\varphi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\varphi}_{,\mu}}}\overline{\varphi}_{,\nu}-\delta^{\mu}_{\,\,\nu}\mathcal{L}\right).$
(79)

In taking the divergence of this current, $\partial_{\mu}$ only acts on the
expression inside the parentheses. This allows us to write the term inside as
four Noether currents expressed as a mixed second-rank stress tensor,

$T^{\mu}_{\,\,\,\nu}\equiv\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\varphi_{,\nu}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\varphi}_{,\mu}}}\overline{\varphi}_{,\nu}-\delta^{\mu}_{\,\,\nu}\mathcal{L}.$
(80)

We say that the four currents are associated with the energy and linear
momentum densities stored in the fields. To see this explicitly, first note
that

$T^{0}_{\,\,\,0}=\frac{\partial{\mathcal{L}}}{\partial{\dot{\varphi}}}\dot{\varphi}+\frac{\partial{\mathcal{L}}}{\partial{\dot{\overline{\varphi}}}}\dot{\overline{\varphi}}-\mathcal{L},$
(81)

and we define

$\pi(x^{\mu})\equiv\frac{\partial{\mathcal{L}}}{\partial{\dot{\varphi}}}\quad\overline{\pi}(x^{\mu})\equiv\frac{\partial{\mathcal{L}}}{\partial{\dot{\overline{\varphi}}}}$
(82)

to be the canonical momentum densities conjugate to the fields $\varphi$ and
$\overline{\varphi}$ respectively.Peskin:1995ev  We then see that this
component of the stress tensor defines the Hamiltonian density $\mathcal{H}$
of the system,

$-\mathcal{H}\equiv\pi(x^{\mu})\dot{\varphi}+\overline{\pi}(x^{\mu})\dot{\overline{\varphi}}-\mathcal{L}=T^{0}_{\,\,\,0}.$
(83)

Thus the Hamiltonian, or total energy of the system, is found by integrating
the above expression over all three-space,

$H\equiv\int\\!\\!\mathcal{H}\,d^{3}x=-\int\\!\\!T^{0}_{\,\,\,0}\,d^{3}x.$
(84)

Similarly, the spatial components give the physical linear momentum stored in
the field,

$\mathcal{P}_{i}\equiv[\pi(x^{\mu})\partial_{i}\varphi+\overline{\pi}(x^{\mu})\partial_{i}\overline{\varphi}]=\frac{1}{c}T^{0}_{\,\,\,i},$
(85)

and

$P_{i}\equiv\int\\!\\!\mathcal{P}_{i}\,d^{3}x=\int\\!\\!\frac{1}{c}T^{0}_{\,\,\,i}\,d^{3}x.$
(86)

###  A.2 Spatial Rotation

A rotation of spatial coordinates in classical mechanics is associated with
the conservation of angular momentum. A rotation of spatial coordinates can be
written in terms of a $4\times 4$ rotation matrix $A$ which transforms our
position vector with $x^{\mu}\to Ax^{\mu}$ but leaves the time component
unchanged. In the case of an infinitesimal transformation, the rotation matrix
can be decomposed into

$A\approx I+R,$ (87)

where $I$ is the identity and $R$ is an antisymmetric matrix with an
infinitesimal parameter (omitted here for simplicity). We can represent this
rotation as a mixed second-rank tensor $A^{\mu}_{\,\,\,\nu}$ applied to our
position vector, and using the decomposition (87) we have

$x^{\mu}\to x^{\mu}+R^{\mu}_{\,\,\,\nu}x^{\nu}.$ (88)

The field response to this transformation is approximated by Taylor expansion
about $x^{\mu}$:

$\varphi(x^{\mu}+R^{\mu}_{\,\,\,\nu}x^{\nu})\approx\varphi(x^{\mu})+(\partial_{\mu}\varphi)R^{\mu}_{\,\,\,\nu}x^{\nu},$
(89)

and complex conjugate. The Lagrangian, as a scalar function of coordinates,
responds with

$\mathcal{L}(x^{\mu}+R^{\mu}_{\,\,\,\nu}x^{\nu})\approx\mathcal{L}(x^{\mu})+(\partial_{\mu}\mathcal{L})R^{\mu}_{\,\,\,\nu}x^{\nu}.$
(90)

To write the last term as an overall divergence, note that

$\displaystyle\partial_{\mu}(\mathcal{L}R^{\mu}_{\,\,\,\nu}x^{\nu})$
$\displaystyle=(\partial_{\mu}\mathcal{L})R^{\mu}_{\,\,\,\nu}x^{\nu}+\mathcal{L}R^{\mu}_{\,\,\,\nu}(\partial_{\mu}x^{\nu})$
$\displaystyle=(\partial_{\mu}\mathcal{L})R^{\mu}_{\,\,\,\nu}x^{\nu}+\mathcal{L}R^{\mu}_{\,\,\,\nu}\delta^{\nu}_{\,\,\,\mu}$
$\displaystyle=(\partial_{\mu}\mathcal{L})R^{\mu}_{\,\,\,\nu}x^{\nu}+\mathcal{L}R^{\mu}_{\,\,\,\mu}$
$\displaystyle=(\partial_{\mu}\mathcal{L})R^{\mu}_{\,\,\,\nu}x^{\nu},$ (91)

where the last step holds since $R^{\mu}_{\,\,\,\nu}$ is antisymmetric and
hence the diagonal components vanish. Thus we can write the change in the
Lagrangian given in Eq. (90) as an overall divergence,

$\mathcal{L}(x^{\mu}+R^{\mu}_{\,\,\,\nu}x^{\nu})-\mathcal{L}(x^{\mu})=\partial_{\mu}(\mathcal{L}R^{\mu}_{\,\,\,\nu}x^{\nu}).$
(92)

A similar calculation can be made for Eq. (89). The Noether current for
rotations is then

$J^{\mu}=R^{\sigma}_{\,\,\,\nu}x^{\nu}\left(\frac{\partial{\mathcal{L}}}{\partial{\varphi_{,\mu}}}\varphi_{,\sigma}+\frac{\partial{\mathcal{L}}}{\partial{\overline{\varphi}_{,\mu}}}\overline{\varphi}_{,\sigma}-\delta^{\mu}_{\,\,\,\sigma}\mathcal{L}\right).$
(93)

The term in parentheses is just the energy-momentum tensor
$T^{\mu}_{\,\,\,\sigma}$ found in (80). After moving the $\sigma$ index, this
reduces to

$J^{\mu}=R_{\sigma\nu}x^{\nu}T^{\mu\sigma}.$ (94)

We want to exploit the fact that $R_{\sigma\nu}$ is an antisymmetric tensor
being multiplied into a third-rank contravariant tensor. First note that

$x^{\nu}T^{\mu\sigma}=\frac{1}{2}(x^{\nu}T^{\mu\sigma}+x^{\sigma}T^{\mu\nu})+\frac{1}{2}(x^{\nu}T^{\mu\sigma}-x^{\sigma}T^{\mu\nu}),$
(95)

where the first (second) term is symmetric (antisymmetric) under
$\nu\leftrightarrow\sigma$ interchange. When we multiply the expression above
by $R_{\sigma\nu}$ only the antisymmetric components survive the summation, so
we have

$J^{\mu}=\frac{1}{2}R_{\sigma\nu}(x^{\nu}T^{\mu\sigma}-x^{\sigma}T^{\mu\nu}).$
(96)

Discarding the constant term out front as before, we are left with six Noether
currents

$\mathcal{M}^{\mu\nu\sigma}\equiv x^{\nu}T^{\mu\sigma}-x^{\sigma}T^{\mu\nu}.$
(97)

To calculate the divergence, we use the product rule and find

$\displaystyle\mathcal{M}_{,\mu}^{\,\,\,\mu\nu\sigma}$
$\displaystyle=x^{\nu}\partial_{\mu}T^{\mu\sigma}-x^{\sigma}\partial_{\mu}T^{\mu\nu}+T^{\nu\sigma}-T^{\sigma\nu}$
$\displaystyle=x^{\nu}\\!\\!\left(-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial^{\sigma}V\right)-x^{\sigma}\\!\\!\left(-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial^{\nu}V\right)+T^{\nu\sigma}-T^{\sigma\nu},$
(98)

where we have in general assumed that the Lagrangian depends on an external
potential so that Eq. (23) holds. We have also not assumed a symmetric energy-
momentum tensor, since that was the case for both systems studied in this
paper. However, note that the linear momentum density is stored in the
$T^{0k}$ components, so the $\mathcal{M}^{\mu jk}$ components will be relevant
for studying angular momentum. The divergence of these terms is then

$\mathcal{M}_{,\mu}^{\,\,\,\mu
jk}=x^{j}\\!\\!\left(-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial^{k}V\right)-x^{k}\\!\\!\left(-\frac{\partial{\mathcal{L}}}{\partial{V}}\partial^{j}V\right),$
(99)

where now $T^{jk}-T^{kj}=0$ since the spatial components of the energy-
momentum tensor are in general symmetric. This result is used to write Eqs.
(37) and (76) using the Levi-Civita symbol. Thus conservation of angular
momentum only holds when a spherically symmetric (if any) potential is
applied. In general, however, the torque density in Eq. (99) is a source of
angular momentum from which Ehrenfest-like relations can be derived.

###### Acknowledgements.

I would first like to thank Joel Franklin, my primary thesis advisor for this
project at Reed College. Joel discovered a way to derive Ehrenfest’s theorem
for momentum by explicitly calculating the divergence of the energy-momentum
tensor associated with the Schrödinger Lagrangian, which inspired me to
formulate the general expression in Eq. (23). Joel also came up with the
choice of the relativistic Lagrangian Eq. (51) among many other key
suggestions in this paper. I would also like to thank Alexander Moll, my
secondary thesis advisor, for providing extensive comments that led to this
work.

## References

  * (1) Joel Franklin, _Classical field theory_ (Cambridge University Press, Cambridge, England, 2017).
  * (2) Y. Takahashi and C. Ropchan, “The first order Schrödinger equation, the symmetrization of the stress tensor and the spin-orbit coupling,” Progress of Theoretical Physics 76 (6), 1187–1197 (1986).
  * (3) David J Griffiths and Darrell F Schroeter, _Introduction to Quantum Mechanics_ , 3rd edition (Cambridge University Press, Cambridge, England, 2018).
  * (4) Yasushi Takahashi, “Energy-momentum tensors in relativistic and non-relativistic classical field theory,” Fortschritte der Physik 34 (5), 323–344 (1986).
  * (5) Joel Franklin, _Computational Methods for Physics_ (Cambridge University Press, 2013).
  * (6) Rolf G. Winter, “Klein paradox for the Klein-Gordon equation,” American Journal of Physics 27 (5), 355–358 (1959).
  * (7) Michael Edward Peskin and Daniel V. Schroeder, _An Introduction to Quantum Field Theory_ (Westview Press, 1995), reading, USA: Addison-Wesley (1995) 842 p.


# Rectification at Graphene-Semiconductor Interfaces: Zero-Gap Semiconductor
Based Diodes

S. Tongay1,2,3, M. Lemaitre1, X. Miao2, B. Gila2,3, B. R. Appleton2,3, and A.
F. Hebard2 1 Materials Science and Engineering, University of Florida,
Gainesville, Florida 32611  2 Department of Physics, University of Florida,
Gainesville, FL 32611  3 Nanoscience Institute for Medical and Engineering
Technology, University of Florida, Gainesville, FL 32611

###### Abstract

Using current-voltage ($I$-$V$), capacitance-voltage ($C$-$V$) and electric
field modulated Raman measurements, we report on the unique physics and
promising technical applications associated with the formation of Schottky
barriers at the interface of a one-atom-thick zero-gap semiconductor
(graphene) and conventional semiconductors. When chemical vapor deposited
graphene is transferred onto n-type Si, GaAs, 4H-SiC and GaN semiconductor
substrates, there is a strong van der Waals attraction that is accompanied by
charge transfer across the interface and the formation of a rectifying
(Schottky) barrier. Thermionic emission theory in conjunction with the
Schottky-Mott model within the context of bond-polarization theory provides a
surprisingly good description of the electrical properties. Applications, such
as to sensors where in forward bias there is exponential sensitivity to
changes in the Schottky barrier height due to the presence of absorbates on
the graphene or to analogue devices for which Schottky barriers are integral
components are promising because of graphene’s mechanical stability, its
resistance to diffusion, its robustness at high temperatures and its
demonstrated capability to embrace multiple functionalities.

###### pacs:

72.80.Vp, 81.05.Ue, 73.30.+y, 73.40.Ei

##  I 1\. INTRODUCTION

Single atom layers of carbon (graphene) have been studied intensively after
becoming experimentally accessible with techniques such as mechanical
exfoliationNovoselov et al. (2004), thermal decomposition on SiC
substratesBerger et al. (2004) and chemical vapor deposition (CVD) Li et al.
(2009); Kim et al. (2009). Graphene is a zero-gap semiconductor (ZGS) with an
exotic linearly dispersing electronic structure, high optical transparency,
exceptional mechanical stability, resilience to high temperatures and an in-
plane conductivity with unusually high mobilityCastroNeto et al. (2009).
Accordingly, graphene has been proposed as a novel material for incorporation
into devices ranging from Schottky light emitting diodes (LEDs) Li et al.
(2010a); Fan et al. (2011); Chen et al. (2011) to field effect transistors
(FETs) Chung et al. (2010); Lin et al. (2009). Although integration of
graphene into semiconductor devices is appealing, there is still very little
known about the interface physics at graphene-semiconductor junctions. To this
end, graphene/Si junctions showing successful solar cell operation have been
produced by transferring either CVD-prepared Li et al. (2010a) or exfoliated
Chen et al. (2011) graphene sheets onto Si substrates. The resulting diodes
have shown ideality factors (measure of deviation from thermionic emission)
varying from $\sim$1.5 Li et al. (2010a) which is close to the ideal value of
unity, to values in the range $\sim$5-30 on exfoliated graphene Chen et al.
(2011) implying that additional non-thermionic current carrying processes
exist at the graphene/Si interface. Nevertheless these promising results point
to the need for additional research on integrating graphene with
technologically important semiconductors.

Figure 1: (a) Graphene/semiconductor diode sample geometry where the $J-V$
characteristics were measured between ohmic contact (ground) and graphene
(high) (b) Hall bar geometry for measurements of the carrier density of
graphene. In this configuration the graphene does not make contact with the
semiconductor. (c) Optical image of the graphene/Au/SiO2 \- graphene/Si
transition edge after the graphene transfer. (d) Scanning electron microscope
image of Cu foils after the CVD graphene growth showing formation of grain
sizes large with respect to the 10 $\mu$m scale bar.

Here we report rectification (diode) effects at ZGS-semiconductor (i.e
graphene-semiconductor) interfaces on a surprisingly wide variety of
semiconductors. In addition to current-voltage measurements we utilize, Hall,
capacitance-voltage and electric field modulated Raman techniques to gain
heretofore unrecognized insights into the unique physics occurring at the
graphene/semiconductor interface. We find that when CVD-prepared graphene
sheets are transferred onto n-type Si, GaAs, 4H-SiC and GaN semiconductor
substrates, equilibration of the Fermi level throughout the system gives rise
to a charge transfer between the graphene and semiconductor, thereby creating
strong rectification (Schottky effect) at the interface. We find that
graphene’s Fermi level ($E_{F}^{gr}$) is subject to variation during charge
transfer across the graphene-semiconductor interface as measured by in-situ
Raman spectroscopy measurements, unlike conventional metal-semiconductor
diodes where the Fermi level (EF) of the metal stays constant due to a high
density of states at the Fermi level. These effects become particularly
pronounced at high reverse bias voltages when the induced negative charge in
the graphene is sufficient to increase $E_{F}^{gr}$ and give rise to increased
current leakage. Our observations and interpretation based on a modification
of thermionic emission theory not only provide a new understanding for the
development of high frequency, high power, and high temperature Schottky based
devices, such as metal-semiconductor field effect transistors (MESFETs) and
high electron mobility transistors (HEMTs), but also allow us to integrate
graphene into semiconductor devices while simultaneously preserving the
superior properties of the graphene and avoiding chemical-structural
modifications to the semiconductor.

##  II 2\. EXPERIMENTAL METHODS

Our diodes are fabricated by transferring large scale graphene sheets grown by
chemical vapor deposition (CVD) directly onto the semiconductor under
investigation and allowing Van der Waals attraction to pull the graphene into
intimate contact with the semiconductor. Large-area single layer graphene
sheets were synthesized on Cu foils via a multi-step, low-vacuum CVD process
similar to that used in Ref. Li et al. (2010b). A quartz tube furnace
operating in CVD-mode was loaded with 25-50 $\mu$m-thick Cu foils (Puratronic,
99.9999% Cu), evacuated to 4 mTorr, and subsequently heated to 500∘C under a
25 sccm flow of H2 at 325 mTorr. After 30 minutes soak, the temperature was
raised to 1025∘C for 60 minutes to promote Cu grain growth (mean grain size
exceeds 5 mm2 determined by optical microscopy). An initial low-density
nucleation and slow growth phase was performed at 1015∘C for 100 minutes with
a mixture of CH4 and H2 at a total pressure of 90 mTorr and flows of $\leq$
0.5 and 2 sccm, respectively. Full coverage was achieved by dropping the
temperature to 1000∘C for 10 minutes while increasing the total pressure and
methane flow to 900 mTorr and 30 sccm, respectively. A 1.5 $\mu$m-thick film
of PMMA (MicroChem, 11$\%$ in Anisole) was then spin-cast onto the Cu foils at
2000 rpm for 60 seconds. The exposed Cu was etched in an O2 plasma to remove
unwanted graphene from the backside of the samples. The PMMA supported films
were then etched overnight in a 0.05 mg/L solution of Fe(III)NO3 (Alfa Aesar)
to remove the copper. The graphene-PMMA films were then washed in deionized
water, isopropyl alcohol (IPA), and buffered oxide etch for 10 minutes, each.
After growth and transfer, the graphene films were characterized and
identified using a Horiba-Yvon micro-Raman spectrometer with green, red and UV
lasers.

Commercially available semiconducting wafers were purchased from different
vendors. $n$-Si and $n$-GaAs samples were doped with P (2-6$\times$$10^{15}$
cm-3) and Si (3-6$\times$$10^{16}$ cm-3) respectively. Epilayers of $n$-GaN
and $n$-4H-SiC, 3-6 $\mu$m-thick, were grown on semi-insulating sapphire
substrates with Si (1-3$\times$$10^{16}$ cm-3) and N (1-3$\times$$10^{17}$
cm-3) dopants. During the sample preparation and before the graphene transfer,
the wafers were cleaned using typical surface cleaning techniques. Ohmic
contacts to the semiconductors were formed using conventional ohmic contact
recipesHan et al. (2002); Sze (1981); Hao et al. (1996); Ruvimov et al.
(1996). Multilayer ohmic contacts were thermally grown at the back/front side
of the semiconductor and were annealed at high temperatures using rapid
thermal annealing. After the ohmic contact formation, a 0.5-1.0 $\mu$m thick
SiOx window was grown on various semiconductors using a plasma enhanced
chemical vapor deposition (PECVD) system, and $\sim$ 500 nm thick gold
electrodes were thermally evaporated onto SiOx windows at 5$\times 10^{-7}$
Torr. The graphene contacting areas were squares with sides in the range 500
$\mu$m to 2000 $\mu$m. Application of IPA improves the success rate of the
graphene transfer and does not affect the measurements presented here. After
depositing the graphene/PMMA films, the samples were placed in an acetone
vapor rich container for periods ranging from 10 minutes to $\sim$10 hours.
The acetone bath allows slow removal of the PMMA films without noticeable
deformation of the graphene sheets.

Prior to the graphene transfer there is an open circuit resistance between the
Au contacts and the semiconductor. After the transfer of the PMMA-graphene
bilayer, the graphene makes simultaneous connection to the Au contacts and the
semiconductor as evidenced by the measured rectifying I-V characteristics.
Since the diodes made with the PMMA-graphene bilayer show essentially the same
rectifying characteristics as the samples in which the PMMA has been dissolved
away, we conclude that the carbon layer on the PMMA (shown by Raman
measurements to be graphene) is making intimate contact with the
semiconductor.

Figure 2: (a) Raman spectra of CVD-grown graphene on Cu foils and (b) graphene
after transfer onto various semiconductor substrates. Graphene sheets show
large $I_{2D}/I_{G}$ ratio, and after the transfer the graphene becomes
slightly disordered. (c-d) Raman G and 2D peaks measured respectively on
graphene/Cu and on graphene/semiconductor combinations indicated in the legend
of panel (b).

A schematic for our graphene based diodes is shown in Fig. 1(a); the backside
of the semiconductor substrate is covered with an ohmic contact and the
graphene sheet transfered onto Cr/Au contacts grown on SiOx windows. After the
transfer, the graphene and semiconductor adhere to each other in an intimate
Van der Waals contact in the middle of the open window, and the Cr/Au contact
pad provides good electrical contact with the graphene. Our ohmic contact
arrangements allow current density versus voltage ($J$-$V$) and capacitance
versus voltage ($C$-$V$) measurements to be taken separately. $J$-$V$
measurements were taken in dark room conditions using a Keithley 6430 sub-
femptoamp source-meter, and $C$-$V$ measurements were taken using an HP 4284A
capacitance bridge. The electric field modulated Raman measurements were made
on the same configuration. Four-terminal transport and Hall measurements
however were performed with an intervening layer of SiOx (Fig. 1(b)) using a
physical property measurement system (PPMS), at room temperature in magnetic
fields up to 7 Tesla.

##  III 3\. RESULTS AND DISCUSSION

###  III.1 A. Raman measurements

Figure 3: In-situ Raman spectra taken on Graphene/GaN junctions as a function
of applied bias: 0V (black line), +1V (red line) and -10V (blue line).

In Fig. 2(a-d), we show typical Raman spectroscopy data taken on graphene
sheets grown onto Cu foils by CVD deposition before and after transferring
onto semiconductors. The presented scans have been reproduced at more than 20
random spots and are good representations of the quality of the graphene on
the Cu foils before transfer and on the semiconductor surface after transfer.
In the literature the quality of graphene sheets is measured by a large 2D to
G intensity ratio ($I_{2D}$/$I_{G}$) and a low D peak intensity ($I_{D}$).
Single layer graphene is expected to show $I_{2D}$/$I_{G}>2$, and the amount
of disorder in the sheets is often correlated with $I_{D}$. In our samples, we
observe $I_{2D}$/$I_{G}\geq 2$ and a negligible D peak amplitude. However
after graphene transfer to the semiconductor substrate, we observe that
$I_{D}$ becomes apparent while $I_{2D}$/$I_{G}$ remains the same (Fig. 2(b)).
The increase in $I_{D}$ reflects the lower sheet mobility of CVD-grown
graphene and gives rise to weak localization effects at low temperatures Cao
et al. (2010). Moreover, because of the low solubility of carbon in Cu,
graphene growth onto Cu foils is known to be self-limiting Li et al. (2009)
therefore allowing large-area single layers of graphene to be grown onto Cu
foil surfaces. After the graphene growth, the backside of the Cu foils have
been exposed to O2 plasma to remove unwanted graphene and checked with Raman
spectroscopy. This step assures that bi-layer (or multi-layer) graphene is not
formed on PMMA/graphene after etching the Cu foils (see Experimental methods).

The Raman spectrum of exfoliated graphene transferred onto Si/SiO2 substrates
has been previously studied as a function of applied bias Das et al. (2008).
It has been found that the G and 2D peaks of graphene are sensitive to the
Fermi energy (carrier density) of graphene and allow one to estimate the bias-
induced changes in E${}_{F}^{gr}$. Considering the typical operating voltages
of Schottky junctions, the low carrier density in graphene, and the associated
bias dependence of E${}_{F}^{gr}$, we have also measured the Raman spectrum of
graphene transferred onto GaN as a function of applied bias. Our Raman
measurements differ from those reported in Ref. Das et al. (2008) in the
following three ways: (1) we are using CVD-prepared rather than exfoliated
graphene, (2) the graphene is in direct contact with GaN rather than oxidized
Si, and (3) the graphene is measured in situ as part of a Schottky rather than
a gated FET. In Fig. 3, we show the evolution of the Raman spectrum as a
function of applied bias. While G and 2D are almost identical with the same
peak positions at 0V and 1V, in reverse bias at 10V, the G band shifts to
higher (by $\sim$6 $\pm$3 cm-1) and the 2D band shifts to lower (by $\sim$7
$\pm$3 cm-1) wavenumbers. The relative shifts in the Raman peaks along with a
reduction of the 2D/G peak ratio from 2.6 (at 0 V) to 1.2 (at 10 V) imply that
graphene sheets transferred onto GaN become electron doped. Considering the
previous results reported on graphene/SiO2 Das et al. (2008), the shift in EF
can be estimated to be in the range $\sim$0.2-0.5 eV.

###  III.2 B. Hall measurements

Figure 4: Rxy versus magnetic field data taken at 300 K. Typically sample
mobilities are in the range 1400-2100 cm2/Vs and carrier densities (holes) in
the range 2-8$\times 10^{12}$ cm2.

Hall measurements show that the Hall mobility of the graphene sheets used in
our diodes is in the range 1400-2100 cm2/Vs, and that the sheets are hole
doped with carrier densities in the range 2-8$\times 10^{12}$ cm-2 (Fig. 4).
The presence of extrinsic residual doping in exfoliated graphene has been
previously reportedNovoselov et al. (2004) and attributed to residual water
vapor (p type) or NH3 (n-type). In both cases annealing reduces the
concentration of the dopants and forces $E_{F}^{gr}$ closer to the neutrality
point. For our CVD prepared graphene, the presence of residual impurity doping
can be attributed to a lowering of $E_{F}^{gr}$ due to hole doping of the
graphene during the Fe(III)NO3 etching-transfer processSu et al. (2011).

###  III.3 C. Current-voltage measurements

Schottky diodes are expected to pass current in the forward bias
(semiconductor is negatively biased) while becoming highly resistive in
reverse bias (semiconductor is positively biased). As seen in Fig. 5(a-d),
$J$-$V$ (main panels) and log$J$-$V$ (insets) data taken on various
graphene/n-type semiconductor junctions display strong rectification. This
rectification is a consequence of Schottky barrier formation at the interface
when electrons flow from the semiconductor to the graphene as the Fermi
energies equilibrate (Fig. 7(b)).

Figure 5: Room temperature current density-voltage characteristics show
Schottky rectification at the (a) graphene/$n$-Si, (b) $n$-GaAs, (c)
$n$-4H-SiC and (d) $n$-GaN interfaces. Insets: Semilogarithmic leaf plots,
log$J$-$V$, reveal a thermionic emission dominated current density in forward
bias that spans at least two decades of linearity (dotted lines) allowing us
to extract out the Schottky barrier height recorded in Table. I.

In principle, any semiconductor with electron affinity ($\chi_{e}$) smaller
than the work function of the metal $(\Phi_{metal})$ can create rectification
at a metal-semiconductor (M-S) interface with Schottky barrier height,
$\phi_{SBH}=\Phi_{metal}-\chi_{e}$, given by the Schottky-Mott model. Electron
transport over the Schottky barrier at the M-S interface is well described by
thermionic emission theory (TE) with the expression;

$J(T,V)=J_{s}(T)[\exp({eV}/{\eta k_{B}T})-1],$ (1)

where $J(T,V)$ is the current density across the graphene/semiconductor
interface, V the applied voltage, T the temperature and $\eta$ the ideality
factor Sze (1981). The prefactor, $J_{s}(T)$ is the saturation current density
and is expressed as $J_{s}=A^{*}T^{2}\exp({-e\phi_{SBH}}/{k_{B}T})$, where
$e\phi_{SBH}$ is the zero bias Schottky barrier height (SBH) and $A^{*}$ is
the Richardson constant.

Figure 6: (a) The temperature dependence of the current ($I$) versus voltage
($V$) curves measured across a graphene/GaAs junction from 250 K up to 320 K
with 10 K intervals separating each isotherm. The arrows indicate the
direction of increasing temperature. (b) The temperature dependence of $I$-$V$
curves taken on graphene/GaAs junctions at different temperatures. (c)
Extracted $I_{s}$ values from Fig.6(b) are plotted in terms of
ln$I_{s}$/$T^{2}$ versus 1000/T.

When electronic transport across the barrier is dominated by thermionic
emission described by Eq. 1, semilogarithmic plots of the $J$-$V$ curves
should display a linear region in forward bias. As seen in the insets of Fig.
5(a-d) the overlying straight line segments of our measurements typically
reveal 2-4 decades of linearity, thus allowing us to extract $J_{s}$ and
$\eta$ for each diode. The deviations from linearity at higher bias are due to
series resistance contributions from the respective semiconductors. The
temperature-dependent data for the graphene/GaAs diode (Fig. 6(a-b)) show that
for both bias directions, a larger (smaller) current flows as the temperature
is increased (decreased) and the probability of conduction electrons
overcoming the barrier increases (decreases). In forward bias, the TE process
manifests itself as linear “log$J$-V curves”(Fig. 6(b)) and linear
“$\ln(I_{s}(T)/T^{2})$ versus $T^{-1}$ curves”(Fig. 6(c)) where
$I_{s}(T)=J_{s}(T)A$. The SBH is calculated directly from the slope of this
linear dependence. By repeating these temperature-dependent measurements for
the four different diodes, we find that the SBH ($\phi^{JV}_{SBH}$) values at
the graphene/semiconductor interfaces are 0.86 eV, 0.79 eV, 0.91 eV and 0.73
eV for Si, GaAs, SiC and GaN respectively (Table I). While the overall reverse
current density increases as $T$ is increased, we notice that at high reverse
bias the magnitude of the breakdown voltage $V_{b}$ decreases linearly with
temperature (see boxed region in upper left hand corner of Fig. 6(b)) implying
that $V_{b}$ has a positive breakdown coefficient and that the junction
breakdown mechanism is mainly avalanche multiplicationSze (1981).

Table 1: Extracted SBHs, doping densities, and corresponding graphene work function values on various graphene/semiconductor junctions |  $\phi_{SBH}^{JV}$ |  $\phi_{SBH}^{CV}$ |  $N_{D}^{CV}$ |  $N_{D}^{Hall}$ |  $\Phi_{gr}$
---|---|---|---|---|---
Junction type |  [eV] |  [eV] |  [cm-3] |  [cm-3] |  [eV]
Graphene/nSi |  0.86 |  0.92 |  $4.0\times 10^{15}$ |  $3.0\times 10^{15}$ |  4.91
Graphene/nGaAs |  0.79 |  0.91 |  $3.5\times 10^{16}$ |  $3.0\times 10^{16}$ |  4.89
Graphene/n4H-SiC |  0.91 |  N A |  N A |  $1.0\times 10^{16}$ |  4.31
Graphene/nGaN |  0.73 |  N A |  N A |  $1.0\times 10^{17}$ |  4.83

Schottky barrier values are well described using either the Bardeen or
Schottky limits. In the Bardeen limit, the interface physics is mostly
governed by interface states which, by accumulating free charge, change the
charge distribution at the interface and cause $E_{F}$ of the semiconductor to
be fixed (Fermi level pinning). Accordingly, the SBH shows weak dependence on
the work function of the metals used for contacts, as is found for example in
GaAs Sze (1981). On the other hand, the wide band gap semiconductors SiC and
GaN are well described by the Schottky-Mott (S-M) limit,

$\phi_{SBH}=\Phi_{gr}-\chi_{e},$ (2)

where $\Phi_{gr}$ is the work function of the graphene and $\chi$ is the
electron affinity of the semiconductor. Using the extracted values of
$\phi_{SBH}$, and electron affinity values ($\chi_{Si}\sim 4.05$ eV,
$\chi_{GaAs}\sim 4.1$ eV, $\chi_{4H-SiC}\sim 3.4$ eV and $\chi_{GaN}\sim 4.1$
eV), we calculate $\Phi_{gr}$ (Table I). The calculated values of the work
function are typically higher than the accepted values ($\sim$4.6 eV) of
graphene when $E_{F}$ is at the Dirac point (K point). The deviation from this
ideal graphene work function can be attributed to the lowering of $E_{F}$ due
to hole doping of the graphene during the Fe(III)NO3 etching-transfer
processSu et al. (2011) (Fig. 2(c-d)) together with the fact that the graphene
is in physical contact with the gold electrodesYu et al. (2009) (Fig. 4).

Although the SBHs on Si, GaAs and GaN can be roughly explained within the S-M
model, in reality GaAs surfaces have a high density of surface states and thus
exhibit characteristic Fermi level pinning. In the Bardeen limit, GaAs based
diodes generally have SBHs in the range of 0.75-0.85 eV as observed in our
measurements, and proper interpretation of the SBH on GaAs/graphene junctions
requires the Bardeen model. Subsequent to the placement of the graphene on the
semiconductor surface, there is charge separation and concomitant formation of
induced dipoles at the interface. According to bond polarization theoryTung
(2001a, b), the SBH is determined by charge separation at the boundary between
the outermost layers of the metal (here, a single layer carbon sheet) and the
semiconductor. Our results are in good agreement with the findings of our
earlier work on graphite and MLG junctions where the layer in closest
proximity to the semiconductor surface is a single sheet of carbon atomsTongay
et al. (2009, 2011). On the other hand, barriers formed on the 4H-SiC
substrates give an unphysically low value for $\Phi_{gr}$ (see Table I) and
therefore cannot be explained by either model.

Next we turn our attention to reverse bias characteristics when the
semiconductor (graphene) is positively (negatively) charged. In conventional
metal-semiconductor Schottky diodes, the work function of the metal is pinned
independent of bias voltage due to the high density of states at $E_{F}$ while
in the reverse (forward) bias the Fermi energy of the semiconductor shifts
down (up) allowing observed rectification via an increase (decrease) in the
built-in potential ($V_{bi}$). Unlike conventional metals, graphene’s work
function ($\Phi_{gr}$) is a function of bias Das et al. (2008), and for large
voltage values the SBH does not stay constant. When Schottky diodes are
forward biased, they pass large currents at voltages well below $\sim$1 V and
and small decreases in the Fermi level of graphene cannot be distinguished
from voltage drops associated with a series resistance. Said in another way,
the deviation from linearity in the semilogarithmic plots of Fig. 5(a-d) for
forward bias could be due to a combination of a series resistance becoming
important at high currents together with a small increase in $\Phi_{gr}$ and a
downward shift in $E_{F}$ for the positively charged graphene. However, in
reverse bias where the applied voltage can be larger than 10 V, $E_{F}$ starts
changing dramaticallyYu et al. (2009) and the fixed SBH assumption clearly no
longer holds. In reverse bias when the graphene electrodes are negatively
charged, $E_{F}$ increases and $\Phi_{gr}$ decreases causing the SBH height to
decrease as the reverse bias is increased. As observed in the insets of Fig.
5(a-d) this effect causes the total reverse current to increase as the
magnitude of the bias is increased, thus preventing the Schottky diode from
reaching reverse current saturation. This non-saturating reverse current has
not been observed in graphite based Schottky junctions due to the fixed Fermi
level of graphite Tongay et al. (2009).

Figure 7: Capacitive response of the graphene based Schottky diodes,
determination of the built-in potential, $V_{bi}$, and the proposed Schottky
band diagram. (a) Plots of the inverse square capacitance ($1/C^{2}$) versus
applied bias ($V$) graphene/$n$-Si (Red squares) and $n$-GaAs (green circles)
at 300 K and 100 Hz show a linear dependence implying that the Schottky-Mott
model provides a good description. The intercept on the abscissa gives the
built-in potential ($V_{bi}$) which can be correlated to the Schottky barrier
height while the slope of the linear fit gives
$2/eN_{D}\epsilon_{s}\epsilon_{0}$. Extracted $\phi_{SBH}$ and $N_{D}$ values
are listed in Table. I

###  III.4 D. Capacitance-voltage measurements

Capacitance-voltage $C$-$V$ measurements made in the reverse bias mode are
complementary to $J$-$V$ measurements and provide useful information about the
distribution and density $N_{D}$ of ionized donors in the semiconductor and
the magnitude of the built-in potential $V_{bi}$. For a uniform distribution
of ionized donors within the depletion width of the semiconductor, the
Schottky-Mott relationship between $1/C^{2}$ and the reverse bias voltage
$V_{R}$, satisfies the linear relationship,
$1/C^{2}=2(V_{R}+V_{bi})/eN_{D}\epsilon_{s}\epsilon_{0}$, which as shown in
Fig. 7(a) is observed to hold for graphene/GaAs and graphene/Si junctions.
Linear extrapolation to the intercept with the abscissa gives the built-in
potential, $V_{bi}$, which is related to $\phi_{SBH}$ via the expression,
$\phi_{SBH}=V_{bi}+e^{-1}k_{b}T$ln$(N_{c}/N_{D})$ Sze (1981). Here $N_{c}$ is
the effective density of states in the conduction band, $N_{D}$ is the doping
level of the semiconductor, and the slope of the linear fitting to $1/C^{2}$
versus $V_{R}$ gives the doping density of the semiconductor. We list
$\phi_{SBH}^{CV}$ and $N_{D}$ values for the graphene/GaAs and graphene/Si
junctions in Table. I.

We note from Table 1 that the extracted $\phi^{CV}_{SBH}$ values on the Si and
GaAs junctions are generally higher than $\phi^{JV}_{SBH}$. The discrepancy
between the SBHs determined by the two methods can be attributed to: (a) the
existence of a thin oxide or residue at the graphene/semiconductor interface,
and/or (b) Schottky barrier inhomogeneity. Graphene sheets transferred onto
SiO2 are known to have charge puddles mostly due to the inhomonegous doping
either originating from natural graphite (mechanical exfoliation transfer) or
from chemicals used during the graphene production or transfer (CVD graphene
transfer) process. Since the SBH is sensitive to the EF of graphene, patches
with different charge densities (doping) are expected to have an impact on the
SBH and hence the $J$-$V$ characteristics of the graphene diodes.

An important difference between the $C$-$V$ and $J$-$V$ techniques is that the
$C$-$V$ measurements probe the average junction capacitance at the interface
thereby yielding an average value for the SBH, while the $J$-$V$ measurements
give a minimum value for the SBH, since electrons with thermionic emission
probabilities exponentially sensitive to barrier heights choose low barrier
patches (less p-doped graphene patches) over higher patches (more p-doped
graphene patches)Tung (2001b). While $C$-$V$ measurements give reasonable
values of the SBH for graphene/GaAs and graphene/Si, we have not been able to
obtain reliable $C$-$V$ measurements for graphene deposited on GaN and SiC
because of high series resistance in these wide band gap semiconductors.

The linearity of the $C$-$V$ measurements shown in Fig. 7 is consistent with
the Schottky-Mott model and the abrupt junction approximation, which assumes
that the density of ionized donors $N_{D}$ is constant throughout the
depletion width of the semiconductor. This good agreement invites a more
quantitative analysis of the Fermi energy shifts in the graphene that are the
source of the non-saturating reverse bias currents discussed in the previous
subsection. We begin by writing the electron charge density per unit area $Q$
on the graphene as

$Q=en_{induced}=C_{dep}(V_{bi}+V_{R}),$ (3)

where

$C_{dep}=\sqrt{\frac{e\epsilon_{s}\epsilon_{0}N_{D}}{2\left(V_{bi}+V_{R}\right)}},$
(4)

is the Schottky-Mott depletion capacitance, $n_{induced}$ is the number of
electrons per unit area and $V_{R}$ is the magnitude of the reverse bias
voltage. Combining these two equations gives the result,

$n_{induced}=\sqrt{\epsilon_{s}\epsilon_{0}N_{D}(V_{bi}+V_{R})/{2e}}.$ (5)

The above expression provides an estimate of the number of carriers per unit
area associated with the electric field within the depletion width but does
not account for extrinsic residual doping described by the carrier density
$n_{0}$ on the graphene before making contact with the semiconductor. The
processing steps used to transfer the CVD grown graphene from Cu substrates to
semiconductor surfaces typically results in p-doped material with $n_{0}\sim
5\times 10^{12}$cm-2 as inferred from Hall data (Fig. 4) taken at 300 K.
Accordingly, the final carrier density including contributions from the as-
made graphene and the charge transfers associated with the Schottky barrier
($V_{bi}$ and the applied voltage $V_{R}$) reads as,

$n_{final}=n_{0}-n_{induced},$ (6)

Using the well-known expression for graphene’s Fermi energyCastroNeto et al.
(2009) we can write

$E_{F}=-\hbar\left|v_{F}\right|k_{F}=-\hbar\left|v_{F}\right|\sqrt{\pi(n_{0}-n_{induced})},$
(7)

which in combination with Eq. 5 becomes

$E_{F}=-\hbar\left|v_{F}\right|\sqrt{\pi(n_{0}-\sqrt{\epsilon_{s}\epsilon_{0}N_{D}(V_{bi}+V_{R})/{2e}})},$
(8)

To calculate typical shifts in $E_{F}$, we use parameter values
$\epsilon_{0}=8.84\times 10^{-14}$ F/cm2, $\hbar=6.5\times 10^{-16}$ eV s,
$e=1.6\times 10^{-19}$ C, $v_{F}=1.1\times 10^{8}$ cm/s, $V_{bi}\sim 0.6$ V
and $\epsilon_{s}\sim 10$ for a typical semiconductor. Thus the Fermi energy
of the as-made graphene with $n_{0}\sim 5\times 10^{12}$ cm-2 is calculated
from $E_{F}=-\hbar\left|v_{F}\right|\sqrt{\pi n_{0}}$ to be $-0.287$ eV below
the charge neutrality point, a shift associated with the aforementioned
p-doping during processing. When the graphene is transferred to the
semiconductor, equilibration of the chemical potentials and concomitant
formation of a Schottky barrier (Fig. 7) results in a transfer of negative
charge to the graphene and an increase in $E_{F}$ (calculated from Eq. 8 for
$V_{R}=0$) to be in the range 3 to 11 meV for $N_{D}$ in the range $1\times
10^{16}$ to $1\times 10^{17}$ cm-3. The application of a typical 10 V reverse
bias (see Figs. 5 and 6) creates significantly larger Fermi energy shifts
which from Eq. 8 give $E_{F}$ in the range $-0.271$ to $-0.233$ eV for the
same factor of ten variation in $N_{D}$. The corresponding shifts from the
pristine value of $-0.287$ eV are in the range 15 - 53 meV and thus bring EF
closer to the neutrality point. These numerical calculations show that for our
n-doped semiconductors, it is relatively easy to induce Fermi energy shifts on
the order of 50 meV with the application of a sufficiently high reverse bias
voltage. An upward shift in $E_{F}$ of 50 meV causes a reduction in
$\Phi_{gr}$ by the same amount. Since the electron affinity of the
semiconductor remains unchanged, the Schottky-Mott constraint of Eq. 2
enforces the same reduction in $\phi_{SBH}$ thus leading to a greater than 5%
reduction in the measured SBH’s shown in Table I. We note that the induced
shift in graphene’s EF as determined by the in-situ Raman spectroscopy
measurements (Fig.3) is larger ($\Delta$EF$\sim$200-500 meV) than our
theoretical estimation ($\Delta$EF$\sim$50 meV).

The discrepancy between the theoretical estimate of $\Delta$EF and the
experimental values might be attributed to: (1) the existence of an interface
capacitance induced by dipoles at the graphene/semiconductor interface (within
bond polarization theory) causing deviation from the ideal Schottky-Mott
capacitance relation given by Eq. 4 and (2) the estimate of $\Delta$EF using
relative peak shifts in the G and 2D peak positions for graphene deposited on
Si/SiO2 Das et al. (2008) might be different than the change in Fermi level
for graphene transferred onto semiconductors.

###  III.5 E. Modification of thermionic emission theory

As discussed in the previous sections, since the EF of graphene electrode is
sensitive to the applied bias across the graphene/semiconductor interface, the
SBH at the interface becomes bias dependent especially for large reverse
voltages. However, extracting the SBH from $J$-$V$ characteristics using Eq. 1
which involves extrapolating current density to zero bias saturation current
(Js) yields the putative zero bias barrier height (Table.1). In this section,
we present a simple modification to the Richardson equation (Eq. 1)
considering the shift in EF of graphene induced by applied bias. The modified
Richardson equation preserves the original functional form of Eq. 1 but allows
one to estimate the SBH at fixed voltages.

The voltage-dependent SBH ($\Phi_{SBH}(V)$) can be written as,

$e\Phi_{SBH}=e\Phi^{0}_{SBH}+e\Delta\Phi_{SBH}\left(V\right)\\\
=e\Phi^{0}_{SBH}-\Delta E_{F}\left(V\right)\\\ $ (9)

where e$\Phi$${}^{0}_{SBH}$ is the zero bias SBH and e$\Delta\Phi_{SBH}(V)$ is
the correction to the SBH at fixed voltage V. The change in the Fermi energy
$\Delta E_{F}\left(V\right)$ is opposite to e$\Delta\Phi_{SBH}(V)$, i.e.,
$\Delta E_{F}\left(V\right)=-e\Delta\Phi_{SBH}(V)$, as seen in Fig. 7b. Thus
for reverse bias (addition of electrons to the graphene) we use Eq. 5 in Eq.
7, together with the inequality $n_{induced}<<n_{0}$ to calculate

$\begin{split}&e\Delta\Phi_{SBH}(V_{R})=-\Delta E_{F}(V_{R})=\\\ &\hbar
v_{F}\left[\sqrt{\pi(n_{0}-n_{induced})}-\sqrt{\pi n_{0}}\right]\\\
&\approx-\frac{1}{2}\hbar v_{F}\sqrt{\pi n_{0}}\frac{n_{induced}}{n_{0}}\\\
&=-\frac{1}{2}\hbar
v_{F}\sqrt{\frac{\pi\epsilon_{s}\epsilon_{0}N_{D}(V_{bi}+V_{R})}{2en_{0}}}\end{split}$
(10)

Adding the reverse and forward current densities as is done in standard
treatments of the diode equationSze (1981) yields the total current density
across the graphene/semiconductor interface,

$\begin{split}&J(V)=\\\
&A^{*}T^{2}\texttt{exp}\left(-\frac{e\Phi^{0}_{SBH}+e\Delta\Phi_{SBH}\left(V\right)}{k_{B}T}\right)\biggl{[}\texttt{exp}\left(\frac{eV}{k_{B}T}\right)-1\biggr{]}\end{split}$
(11)

Here, we note that the original form of the Richardson equation is preserved
with slight modifications to the saturation current term which is given as;

$J_{s}=A^{*}T^{2}\texttt{exp}\left(-\frac{e\Phi^{0}_{SBH}+e\Delta\Phi_{SBH}(V)}{k_{B}T}\right)$
(12)

with $\Delta\Phi_{SBH}(V)$ for reverse bias given by Eq. 10.

In our conventional $J$-$V$ analysis using Eq. 1, the zero-bias saturation
current Js is extracted by extrapolating the current density to zero bias
limit. In this limit, the correction to the SBH is expected to be zero, since
the graphene is not subject to applied bias and hence the Fermi level does not
shift from the original value. However, using the extrapolated zero-bias
saturation current density, one can extract out the SBH and the correction to
the SBH at fixed bias (V) can be taken into account by the additional term
($\Delta\Phi_{SBH}(V)$ in Eq. 12).

##  IV Conclusion

In summary, we have used current-voltage and capacitance-voltage measurements
to characterize the Schottky barriers formed when graphene, a zero-gap
semiconductor, is placed in intimate contact with the n-type semiconductors:
Si, GaAs, GaN and SiC. The good agreement with Schottky-Mott (S-M) physics
within the context of bond-polarization theory is somewhat surprising since
the S-M picture has been developed for metal-semiconductor interfaces, not for
single atomic layer ZGS-semiconductor interfaces discussed here. Moreover, due
to a low density of states, graphene’s Fermi level shifts during the charge
transfer across the graphene-semiconductor interface. This shift does not
occur at metal-semiconductor or graphite-semiconductor interfaces where
$E_{F}$ remains fixed during Schottky barrier formation and the concomitant
creation of a built-in potential, $V_{bi}$ with associated band bending (see
Fig. 7). Another major difference becomes apparent when under strong reverse
bias. According to our in-situ Raman spectroscopy measurements, large voltages
across the graphene/semiconductor interface change the charge density and
hence the Fermi level of graphene as determined by relative changes in the G
and 2D peak positions. The bias-induced shift in the Fermi energy (and hence
the the work function) of the graphene causes significant changes in the diode
current. Considering changes in the barrier height associated with bias
induced Fermi level shift, we modify the thermionic emission theory allowing
us to estimate the change in the barrier height at fixed applied bias. The
rectification effects observed on a wide variety of semiconductors suggest a
number of applications, such as to sensors where in forward bias there is
exponential sensitivity to changes in the SBH due to the presence of
absorbates on the graphene or to MESFET and HEMT devices for which Schottky
barriers are integral components. Graphene is particularly advantageous in
such applications because of its mechanical stability, its resistance to
diffusion, its robustness at high temperatures and its demonstrated capability
to embrace multiple functionalities.

We thank N. Savage for her help during preparation of the diagrams used in
this manuscript and Prof. A. Rinzler and Dr. M. McCarthy for technical
assistance. This work is supported by the Office of Naval Research (ONR) under
Contract Number 00075094 (BA) and by the National Science Foundation (NSF)
under Contract Number 1005301 (AFH).

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