# Free Electron Theory for Thin Metal Films

Philip B. Allen<EMAIL_ADDRESS>Department of Physics and
Astronomy, Stony Brook University, Stony Brook, New York 11794-3800, USA

###### Abstract

Quantum free electrons, i.e. plane waves, with wavevector $\vec{k}$, and
occupancy constrained by the Pauli exclusion principle, are explained in all
introductory texts about solids. A free-electron description works
surprisingly well for many properties of “simple” metals. It is assumed that
the interior of the metal is essentially infinite, and surfaces are presumed
irrelevant. Over the past 30 years, experiments that visualize surfaces have
revolutionized solid state physics, stimulating new theory and applications.
Therefore, a basic question is, how can the free electron picture be applied
to properties of solids where surfaces play a prominent role? Various versions
of an extended free-electron theory are used, but not always explained
pedagogically. This paper focusses on idealized metallic films. Three versions
(an oversimplified one and two stages of improvement) of a free-electron
description of metal films are given. These versions are illustrated in detail
for the specific example of a slab of aluminum with six layers of atoms.

##  I Introduction

Not long after the electron was discovered and recognized as a particle, Drude
Drude (1900) showed a way to understand electrical conduction in metals using
a classical description of electrons wandering freely (except for occasional
scattering) in a homogeneous medium. Atoms, and electrical charge, are
ignored. In the early days of quantum mechanics, Sommerfeld Sommerfeld (1928)
gave the quantum version of Drude’s model. The free-electron wave functions
used in this “Drude-Sommerfeld model” are

$\psi_{\vec{k}}=\frac{1}{\sqrt{V}}e^{i\vec{k}\cdot\vec{r}}=\frac{1}{\sqrt{V}}e^{i(k_{x}x+k_{y}y+k_{z}z)}$
(1)

where $V$ is the volume of the metal. The state $\vec{k}$ has energy

$\epsilon_{\vec{k}}=\frac{\hbar^{2}|\vec{k}|^{2}}{2m}$ (2)

as follows from the Schroedinger equation for a non-relativistic electron of
mass $m$ and no interactions, $\mathcal{H}=p^{2}/2m$. Since electrons obey
Fermi statistics, the Pauli exclusion principle applies, and quantum states
$\psi_{\vec{k}}$ can be occupied by at most one electron of spin up and one of
spin down. The set of vectors $\vec{k}$ is known as “reciprocal space,” or
$\vec{k}$-space. This picture gives a surprisingly useful model of bulk
behavior of “simple metals” (like Na, Mg, Al, and Pb), in situations where
surfaces can be ignored.

Thin crystalline films have become increasingly important in the last 30
years, stimulated by the invention of the scanning tunneling microscope Binnig
and Rohrer (1982, 1987). This led to remarkable improvements in film growth,
new visualizations and spectroscopies, and new applications. The free-electron
description can be modified to describe how electrons behave in films of
simple metals where the surfaces parallel to the films can no longer be
ignored. In metallic films, electrons propagate freely in two dimensions,
parallel to the film, but have discrete “quantum well states” (QWS’s) in the
perpendicular direction. The experimental spectra shown in Fig. 1 contain
evidence of QWS’s evolving with film thickness. Such effects have been studied
by experiment and theory Mudrik  _et al._ (1993); Chiang (2000); Miller  _et
al._ (2009); Miller and Chiang (2011), and treated pedagogically Milun  _et
al._ (2002); Atkinson and Slavin (2008); Tringides  _et al._ (2007). The aim
of the present article is to go deeper into the pedagogy and explore deeper
extensions of the free-electron model.

Figure 1:  On the left is a schematic of a film on a substrate. The figure on
the right is taken from a paper by Wu et al. Wu  _et al._ (2015). They made
films of aluminum (Al), carefully grown on a crystalline silicon substrate.
The thickness $H$ of those films ranges from 1.2 to 2.1 nm. The horizontal
($x$ and $y$) dimensions $L$ are many times larger. The measurement shows
features that change as the number of Al monolayers (ML) increases. An
extension of the free electron picture gives a qualitative and very useful
explanation of this behavior, as will be discussed in later sections.

##  II Periodic Boundary Conditions

In Eq. 1 the plane wave state $\psi_{\vec{k}}$ is normalized so that
$\int_{V}d\vec{r}|\psi_{k}(\vec{r})|^{2}=1$. A finite volume $V$ is used, but
apart from normalization, $\psi_{\vec{k}}$ appears, at first sight, unaltered
by the boundary. There is, however, an important alteration: the wave-vector
$\vec{k}$ is discretized.

Actual boundaries do alter the wave function, and destroy the propagating wave
nature close to the boundary. This is correctly ignored when the aim is to
describe bulk properties. But discretization of $\vec{k}$ is necessary to
quantify the number of states $\psi_{\vec{k}}$ available for the electrons
contained in $V$. This is done by “periodic boundary conditions” (periodic
BC’s), introduced by Born and von Karman Born and von Karman (1912) in 1912.

Consider a finite but otherwise perfect crystal. For simplicity, let all atoms
be identical, with one atom per unit cell, at locations
$\vec{\ell}=n_{a}\vec{a}+n_{b}\vec{b}+n_{c}\vec{c}$, where $n_{\alpha}$ are
integers in the range $1\leq n_{\alpha}\leq N$. Thus there are $N^{3}$ atoms.
The vectors $\vec{a},\vec{b},\vec{c}$ define the edges of the crystal’s “unit
cell”. Each atom has $Z$ “free” electrons (e.g. 1 for alkali metals like Na,
K, $\ldots$; 3 for Al, Ga, etc.); so there are $Z$ electrons per unit cell.
The unit cell has volume $V_{\rm cell}=\vec{a}\cdot(\vec{b}\times\vec{c})$.
Then there are $n_{\rm el}=Z/V_{\rm cell}$ (potentially free) electrons per
unit volume.

It is physically impossible, but mathematically sensible, to have $N^{3}$
regularly spaced atoms in a finite volume $V$ with no boundaries. The
mathematical trick is to re-imagine the volume as a 3-dimensional torus. For
clarity, and later use in metal films, consider first a corresponding one or
two dimensional perfect finite crystal. In one dimension, the chain of $N$
atoms with spacing $|\vec{a}|=a$ is mapped onto a circle of circumference
$Na=L$. The circle is a 1-dimensional torus. The point $r+Na$ on the circle is
the same as the point $r$, so $\psi_{k}(r+Na)$ must equal $\psi_{k}(r)$. This
constrains $\exp(ikNa)$ to be 1, so the one-dimensional $k$-vector is
quantized in the form $k=2\pi m/Na$ where $m$ can be any integer. The number
of $k$-states with $-\pi/a\leq k<\pi/a$ is $N$, the same as the number of
atoms in the 1d crystal. If the number of atoms $N$ is increased, the density
of $k$ states increases, so that the maximum occupied energy is unchanged. The
states $\psi_{k}$ are otherwise unaffected by the boundary.

The smallest $N$ $k$-vectors ($-\pi/a\leq k<\pi/a$) define the one-dimensional
version of the “Brillouin zone” (BZ). When $k$ lies outside the BZ, it is a
close relative of a state $k-G$ inside the BZ, where $G=2\pi m/a$ for integer
$m$, and $G$ is referred to as a “reciprocal lattice vector”. “Close relative”
means that when $\psi_{\vec{k}}$ is evaluated only at atomic positions
$\vec{r}=\vec{\ell}$, any value of $\vec{k}$ outside the BZ can be replaced by
a vector $\vec{k}+\vec{G}$ inside the BZ without altering $\psi_{\vec{k}}$ at
atomic positions. The definition of $\vec{G}$ for dimension $>1$ will be given
soon.

In two dimensions, the torus is a doughnut, tiled by $N^{2}$ parallelograms of
sides $\vec{a}$ and $\vec{b}$. The coordinate $\vec{r}+N\vec{a}$ is the same
as $\vec{r}$, so $\psi_{\vec{k}}(\vec{r}+N\vec{a})$ is the same as
$\psi_{\vec{k}}(\vec{r})$, and similarly for $\vec{b}$. This places two
constraints on $\vec{k}$. Define a vector $\vec{A}$ to have
$\vec{A}\cdot\vec{a}=2\pi$ and direction perpendicular to $\vec{b}$
($\vec{A}\cdot\vec{b}=0$). Similarly, define $\vec{B}$ to have
$\vec{B}\cdot\vec{b}=2\pi$ and direction perpendicular to $\vec{a}$. The
periodic constraints
$\exp(iN\vec{k}\cdot\vec{a})=\exp(iN\vec{k}\cdot\vec{b})=1$ are satisfied if
the $k$-vector is quantized as $\vec{k}=(m_{A}\vec{A}+m_{B}\vec{B})/N$. The
number of $\vec{k}$-states in the parallelgram of sides $\vec{A}$ and
$\vec{B}$ is $N^{2}$, the same as the number of atoms in the idealized 2-d
crystal. This parallelogram is the Brillouin zone (BZ) in two dimensions.

Similar arguments for 3-d crystals are given in all introductory texts. The
unit cell has volume $V_{\rm cell}=(\vec{a}\times\vec{b})\cdot\vec{c}$, and
the crystal has $N^{3}$ atoms in volume $V=N^{3}V_{\rm cell}$. Periodic BC’s
are the constraint $\psi_{\vec{k}}(\vec{r}+N\vec{a})=\psi_{\vec{k}}(\vec{r})$,
and similarly for translations by $N\vec{b}$ and $N\vec{c}$. This puts three
constraints on the $\vec{k}$-vectors. Choose three primitive
$\vec{k}$-vectors, $\vec{A},\vec{B},\vec{C}$, where

$\vec{A}=2\pi\frac{\vec{b}\times\vec{c}}{V_{\rm cell}};\
\vec{B}=2\pi\frac{\vec{c}\times\vec{a}}{V_{\rm cell}};\
\vec{C}=2\pi\frac{\vec{a}\times\vec{b}}{V_{\rm cell}}.$ (3)

Then $\vec{A}\cdot\vec{a}=2\pi$ and
$\vec{A}\cdot\vec{b}=\vec{A}\cdot\vec{c}=0$, and similarly for $\vec{B}$ and
$\vec{C}$. These are the “primitive” reciprocal lattice vectors. The vectors

$\vec{G}=n_{A}\vec{A}+n_{B}\vec{B}+n_{C}\vec{C}$ (4)

for any integers $n_{A},n_{B},n_{C}$, are the reciprocal lattice vectors. The
three special vectors $\vec{A},\vec{B},\vec{C}$ define a primitive cell in
$\vec{k}$-space, the “Brillouin zone” (BZ). Any $\vec{k}$-vector outside the
BZ can be uniquely related to a vector $\vec{k}+\vec{G}$ inside the BZ, by the
fact that both of these vectors share the same phase factor
$e^{i\vec{k}\cdot\vec{\ell}}=e^{i(\vec{k}+\vec{G})\cdot\vec{\ell}}$ under
translation by a lattice vector $\vec{\ell}$. The states
$\psi_{\vec{k}}(\vec{r})$ and $\psi_{\vec{k}+\vec{G}}(\vec{r})$ behave the
same way under translations by $\vec{\ell}$ in spite of being different plane-
waves.

The BZ has volume $V_{\rm
BZ}=\vec{A}\cdot(\vec{B}\times\vec{C})=(2\pi)^{3}/V_{\rm cell}$. This uses the
identity

$\vec{a}\cdot(\vec{b}\times\vec{c})=\frac{(2\pi)^{3}}{\vec{A}\cdot(\vec{B}\times\vec{C})}.$
(5)

With these definitions, there is a simple definition of the plane waves
$\psi_{\vec{k}}$ that obey the periodic BC’s. Define $\vec{k}$ to have the
form

$\vec{k}\equiv\frac{1}{N}(n_{A}\vec{A}+n_{B}\vec{B}+n_{C}\vec{C}).$ (6)

Then

$\psi_{\vec{k}}(\vec{r}+N\vec{a})=e^{2\pi
in_{A}}\psi_{\vec{k}}(\vec{r})=\psi_{\vec{k}}(\vec{r}),$ (7)

and similarly for translations by $N\vec{b}$ and $N\vec{c}$. The integers
$n_{A},n_{B},n_{C}$ are unrestricted. For any allowed $\vec{k}$ (i.e.
satisfying Eq. 6), the plane-wave state $\psi_{\vec{k}}(\vec{r})$ obeys
periodic BC’s. The set of $N^{3}$ $\vec{k}$-vectors inside the parallelopiped
defined by $\vec{A},\vec{B},\vec{C}$ defines the 3-d BZ. Each $\vec{k}$-point
in the BZ has its own set of phase factors under translations by $\vec{\ell}$.
The $\vec{k}$-vectors in the BZ exhaust all the allowed phase factors.
Normally the Brillouin zone is chosen to be centered around the wavevector
$\vec{k}=0$, but any other center gives a shifted BZ with wavevectors having
the same complete set of phase factors.

In all dimensions $d$, the size of the unit cell in $\vec{r}$-space multiplied
by the size of the primitive cell in $\vec{k}$-space (the Brillouin zone) is
$(2\pi)^{d}$,

$V_{r}^{(d)}=\frac{(2\pi)^{d}}{V_{k}^{(d)}}.$ (8)

“Size” here means length in 1d, area in 2d, and volume in 3d; it is denoted as
$V_{r}^{(d)}$, or usually just $V_{d}$ or $V$.

##  III Free Electrons

For free electrons, the Fermi wavevector $k_{F}$ is the radius of the object
(line, circle, sphere in $d=1,2,3$) that contains all the $\vec{k}$-points
occupied at $T=0$. For monatomic crystals,

$V_{k,{\rm
FS}}^{(d)}=\frac{Z}{2}V_{k}^{(d)}=\frac{Z}{2}\frac{(2\pi)^{d}}{V_{r}^{(d)}}=\frac{n_{d}}{2}(2\pi)^{d},$
(9)

where $n_{d}=Z/V_{r}^{(d)}$ is the density of electrons. The factor $1/2$ in
Eq. 9 accounts for equal occupancy of both spin states. The Fermi wavevector
$k_{F}$ in $d=1,2,3$ is then

$\displaystyle 2k_{F}=\pi n_{1}\ {\rm or}\ k_{F}$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}n_{1}\ {\rm in}\ d=1$ $\displaystyle\pi
k_{F}^{2}=2\pi^{2}n_{2}\ {\rm or}\ k_{F}$ $\displaystyle=$ $\displaystyle(2\pi
n_{2})^{1/2}\ {\rm in}\ d=2$
$\displaystyle\frac{4\pi}{3}k_{F}^{3}=4\pi^{3}n_{3}\ {\rm or}\ k_{F}$
$\displaystyle=$ $\displaystyle(3\pi^{2}n_{3})^{1/3}\ {\rm in}\ d=3$ (10)

where the left hand side of each is $V_{k,FS}^{(d)}$, the $k$-space size of
the Fermi surface in dimension $d$.

As an example, consider an idealized 1-d chain of $N$ monovalent atoms, like
Na or K. The length of the chain is $L=Na$, where the nearest neighbor
distance $a$ is roughly twice the radius of the outer $s$-state of the neutral
atom. In vacuum, the outermost $s$-electron is weakly bound, and in the chain,
it is essentially released. In first approximation, charge is ignored, because
the positive charge of the ions left behind and the negative charge of the
electrons are assumed to cancel each other. The $N$ free electrons each occupy
a spin up or down state $\psi_{k}=e^{ikx}/\sqrt{L}$. The lowest energy
configuration has $N/2$ doubly occupied $k$-states. Because $Z=1$, the BZ is
half-filled. The $N/2$ doubly occupied states have $-\pi/2a\leq k\leq+\pi/2a$,
or $-k_{F}\leq k\leq+k_{F}$ where $k_{F}=\pi/2a$ is the Fermi wavevector. The
highest energy of the occupied states is $\epsilon_{F}=\hbar^{2}k_{F}^{2}/2m$,
the Fermi energy.

If atoms are divalent, they release two electrons, and there are $N$ doubly
occupied states. The Fermi wavevector of the 1-d chain is now $k_{F}=\pi/a$,
filling the BZ. Trivalent atoms releasing three free electrons have
$k_{F}=3\pi/2a$, filling the “first BZ” and half-filling the next one.

##  IV Nearly Free Electrons

In reality, outer electrons of atoms in a crystal are not truly free. They
interact with the ions and with each other. The exact quantum many-body states
$\Psi(\vec{r}_{1},\vec{r}_{2},\ldots)$ are too complex to either compute or
use. A trick for dealing with this, called the “empirical pseudopotential
method”, was developed Cohen and Bergstresser (1966) in the 1960’s. The many-
body wavefunction is approximated by an antisymmetrized sum of products of
one-electron states. Then a carefully constructed approximate one-electron
Schroedinger equation is written:

$\left[\frac{p^{2}}{2m}+V_{\rm
ps}(\vec{r})\right]\psi_{\vec{k}}(\vec{r})=\epsilon_{\vec{k}}\psi_{\vec{k}}(\vec{r})$
(11)

The Fourier representation of the “pseudopotential” $V_{\rm ps}$ is

$V_{\rm ps}(\vec{r})=\sum_{\vec{G}}V_{\rm
ps}(\vec{G})e^{i\vec{G}\cdot\vec{r}},$ (12)

where the Fourier coefficients $V_{\rm ps}(\vec{G})$ are adjusted using
experimental data. The potential has the periodicity of the lattice because
the vectors $\vec{G}$ are reciprocal lattice vectors. This empirical method is
now less used, because sophisticated semi-empirical procedures (e.g. “density-
functional theory” or DFT) have been developed which enable an effective one-
electron potential $V(\vec{r})$ to be computed without adjustable parameters.
Application to Al surfaces is given in Ref. Huang, 2008. However, the old
empirical method gives equally good or better results if fitted well to good
experiments, especially for the so-called “simple metals” (Na, Mg, Al, Pb,
etc., with only $s$ or $p$ valence electrons). For these, the pseudopotential
is small, i.e. $|V_{\rm ps}(\vec{G})|/\epsilon_{F}\sim 0.2$ or less.

###  IV.1 One dimensional crystals

To simplify even further, consider the truncated Schroedinger equation

$\left[\frac{p^{2}}{2m}+V_{\rm ps}(e^{2\pi ix/a}+e^{-2\pi
ix/a})\right]\psi_{k}(x)=E_{k}\psi_{k}(x).$ (13)

This is a one-dimensional version of Eqs. 11 and 12. The periodic potential is
simplified to have only $G=\pm 2\pi/a$, the smallest non-zero reciprical
lattice vectors. The potential oscillates in space as $\cos(2\pi x/a)$ with a
small coefficient $2V_{\rm ps}$. Start from the non-interacting (plane wave)
states $\psi_{k}$, the 1-d version of Eq. 1. Then perturbation theory says the
energy becomes $E_{k}=\epsilon_{k}+\Delta\epsilon_{k}$. When the shift is
small, second-order perturbation theory gives the answer

$\Delta\epsilon_{k}=\sum_{G=\pm 2\pi/a}\frac{|V_{\rm
ps}(G)|^{2}}{\frac{\hbar^{2}}{2m}\left[k^{2}-(k+G)^{2}\right]}.$ (14)

This is small as long as the denominator isn’t too small. When $k=\pm
G/2=\pm\pi/a$, one of the two denominators vanishes, and a more accurate
solution is needed. The exact solution for $k=\pi/a$ (and
$\epsilon_{\pi/a}=(\hbar\pi/a)^{2}/2m$) is

$E_{\pi/a}=\epsilon_{\pi/a}\pm V_{\rm ps},$ (15)

and the same for $k=-\pi/a$.

Figure 2:  Free and nearly free electrons in 1d. Free electrons have
$\epsilon_{k}\propto k^{2}$. The dashed vertical line is the Brillouin zone
boundary at k=G/2, where nearly free electrons have a gap $\Delta E=2|V_{\rm
ps}|$. The vertical axis is electron energy in units of
$\hbar^{2}(G/2)^{2}/2m$. The pseudopotential $V_{\rm ps}=V(k=G)$ was chosen to
have magnitude 0.1 in the same units.

The full answer is illustrated in Fig. 2. In the case $Z=1$, the occupied
states occupy the region from $k=0$ to $k=\pi/2a$, half-way to the Brillouin
zone (BZ) boundary at $G/2=\pi/a$. The free electron approximation describes
the occupied states very well. For $Z=2$, the occupied states fill the BZ. It
is no longer a metal, but an insulator with a gap $2V_{\rm ps}$. The free
electron approximation no longer works for low energy excitations. For $Z=3$,
the occupied states have a Fermi surface at $k=3\pi/a$, half-way beyond the
first BZ boundary; the free electron approximation is fine for electrons near
the Fermi energy, but fails for deeper lying states. Most properties will be
correctly described by free electrons for 1d chains of atoms with Z loosely
bound electrons, except when Z is even, in which case the weak pseudopotential
is sufficient to make an insulator.

###  IV.2 Two dimensional crystals

Unlike $d=1$, actual samples of monolayers and thin films with $d=2$ are
becoming available thanks to recent advances in sythesis. An actual $2d$
monolayer (a metal, unlike graphene) is likely to have a triangular lattice.
This is the structure of a single layer of an hexagonal close packed (hcp)
crystal, or a single layer perpendicular to the (111) direction of a face-
centered cubic (fcc) crystal. Translation vectors can be chosen as

$\vec{a}=a\left(\frac{\sqrt{3}}{2},-\frac{1}{2}\right)\ {\rm and}\
\vec{b}=a\left(\frac{\sqrt{3}}{2},+\frac{1}{2}\right),$ (16)

where $a=|\vec{a}|=|\vec{b}|$ is the nearest-neighbor distance. The vectors
$\vec{a}$ and $\vec{b}$ are 60o apart in the $\hat{x}$, $\hat{y}$ plane. There
is one atom per $2d$ primitive cell, at positions
$\vec{\ell}=n_{a}\vec{a}+n_{b}\vec{b}$. The corresponding reciprocal lattice
vectors are

$\vec{A}=\frac{4\pi}{\sqrt{3}a}\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\
{\rm and}\
\vec{B}=\frac{4\pi}{\sqrt{3}a}\left(\frac{1}{2},+\frac{\sqrt{3}}{2}\right)$
(17)

The BZ is a hexagon with edges perpendicular to the midpoint of reciprocal
lattice vectors like $\vec{A}$ and $\vec{B}$, as shown in Fig. 3 The shortest
distance to the BZ boundary is $|\vec{A}|/2=2\pi/\sqrt{3}a$, or 1.1547 in
units of $\pi/a$. The longest is 1.3333 in units of $\pi/a$. The area of the
unit cell is $\sqrt{3}a^{2}/2$. From Eq. 10, the Fermi wavevector (in units of
$\pi/a$) is 0.857 ($Z$=1), 1.2125 ($Z$=2), and 1.4850 ($Z$=3). Just as in
$d=1$, a 2$d$ monovalent metal should have a Fermi surface (in this case, a
circle) unaffected by a weak pseudopotential, and a trivalent metal should
have states near the Fermi surface unaffected. The Fermi circle of a $2d$
metal with $Z=2$ intersects the BZ boundary at 12 points, deforming the Fermi
surface. However, most of the points on this deformed Fermi surface may still
be nearly free electron states with a small deformation $\Delta\vec{k}$ away
from a Fermi circle.

Figure 3:  Free electrons in 2d have circular Fermi surfaces. The hexagon is
the surface of the Brillouin zone of a triangular close-packed 2-d lattice,
with reciprocal lattice vectors $\vec{A}$ and $\vec{B}$. The blue, red, and
green circles are Fermi circles for $Z=$1, 2, and 3 electrons per atom.

###  IV.3 Three dimensional crystals

Typical metals that are reasonably well described by a free electron model are
Na, Mg, Al and Pb with $Z$=1,2,3, and 4. Al and Pb have fcc structure, with 4
atoms in a cube of volume $(\sqrt{2}a)^{3}$, where $\sqrt{2}a$ is the cubic
lattice constant. The Fermi surfaces are shown in Fig. 4.

Figure 4:  Free electrons in 3d have spherical Fermi surfaces. Sodium with
$Z=1$ has a body-centered cubic crystal structure with $Z=1$ electron per unit
cell. Its Fermi sphere fits inside its BZ. The other three metals shown have
$Z=2,3,4$ and Fermi spheres that intersect the surface of their BZ’s.
Magnesium has an hexagonal close-packed crystal structure with two atoms (and
$2Z=4$ electrons) in the unit cell. Aluminum and lead have face-centered cubic
(fcc) structures with $Z=3$ or 4 electrons in the unit cell. Figures are from
ref. Choy  _et al._ , 2000.

##  V Simplified pictures of free electrons in slabs

Figure 5:  Three (111) layers of an fcc crystal. The nearest neighbor distance
is $a=|\vec{a^{\prime}}|=|\vec{b}^{\prime}|$. Layers 2 and 3 (blue and yellow
atoms) lie at distances $h=\sqrt{2/3}a$ and $2h$ above layer 1. Image adopted
from ref. Acharya and Rahman, 2018.

Free-electron descriptions of metal slabs have often been published, for
example, in Refs. Schulte, 1976; Trivedi and Ashcroft, 1988; Pitarke and
Eguiluz, 2001; Wu and Zhang, 2008; Han and Liu, 2009. The example used in this
paper is a slab of six layers ($n_{L}=6)$ of aluminum. It is an idealized
piece of a single (face-centered cubic, fcc) crystal, sliced parallel to the
(111) plane. The plane is labeled (111) because it is perpendicular to the
$(x,y,z)=(1,1,1)$ direction of the cubic crystal. Each layer of the slab is a
plane of triangular close packed atoms, arranged in rhombic unit cells
described by Eq. 16. The atoms are at positions
$\vec{\ell}=n_{a}\vec{a}+n_{b}\vec{b}$. The second and third layers are
identical to the first, except shifted by 1/3 and 2/3 of the diagonal
($\vec{a}+\vec{b}$) of the rhombic cell. The fourth layer is exactly on top of
the first layer. The layer spacing in the $\hat{z}$ direction, for the fcc
structure, is $\sqrt{2/3}a$, where $a$ denotes the nearest neighbor distance
(not the cubic lattice constant which is $\sqrt{2}a$). The slab layer spacing
will relax slightly near the surface, but this is hard to measure or compute
and is ignored here for simplicity. A 2d projection is shown in Fig. 5. A
3-dimensional repeat unit, with one atom per cell, has translation vectors

$\displaystyle\vec{a}^{\prime}$ $\displaystyle=$ $\displaystyle
a\left(\frac{\sqrt{3}}{2}\hat{x}-\frac{1}{2}\hat{y}\right)\ \
\vec{b}^{\prime}=a\left(\frac{\sqrt{3}}{2}\hat{x}+\frac{1}{2}\hat{y}\right)$
(18) $\displaystyle{\rm and}\ \ \vec{c}^{\
\prime}=\frac{1}{3}\left(\vec{a}+\vec{b}+\sqrt{6}a\hat{z}\right).$

There are several ways to choose electron wavefunctions for a slab that are
related to the free electron wavefunctions of bulk crystals. Three will now be
illustrated:
A. The method closest to the bulk description uses periodic BC’s. It differs
from bulk results because, while there are many cells ($N^{2}$) in the
$\vec{a},\ \vec{b}$ plane, there is a small number $n_{L}$ in the
perpendicular direction. The quantization of $(k_{x},k_{y})$ is very dense,
but $k_{z}$-quantization is sparce.
B. An improvement replaces periodic BC’s with hard-wall BC’s. The potential is
a square-well in the $\hat{z}$-direction with infinite sides.
C. A further improvement uses a square well with finite sides so that
electrons interior to the well can be emitted to the vacuum if given enough
energy.

###  V.1 Slab with Periodic boundary conditions

When treating bulk free electron systems, the crystal is described as if it
had dimensions $N\vec{a}$, $N\vec{b}$, and $N\vec{c}$. The number $N$ is
large. A compromise is needed when doing computations; often $N^{3}$ is taken
to be as small as $10^{6}$. For (111) films, the dimensions are
$N\vec{a}^{\prime}$, $N\vec{b}^{\prime}$, and $n_{L}\vec{c}^{\ \prime}$, where
$n_{L}$, the actual number of layers in the film, is small. It is convenient
to use a choice of reciprocal lattice vectors slightly different from Eq. 3,
and more like Eq. 17,

$\displaystyle\vec{A}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{\sqrt{3}a}\left(\frac{1}{2}\hat{x}-\frac{\sqrt{3}}{2}\hat{y}\right),\
\vec{B}^{\prime}=\frac{4\pi}{\sqrt{3}a}\left(\frac{1}{2}\hat{x}+\frac{\sqrt{3}}{2}\hat{y}\right)$
(19) $\displaystyle{\rm and}\
\vec{C}^{\prime}=\sqrt{\frac{3}{2}}\frac{2\pi}{a}\hat{z}.$

These vectors obey $\vec{A}^{\prime}\cdot\vec{a}^{\prime}=2\pi$,
$\vec{A}^{\prime}\cdot\vec{b}^{\prime}=0$, etc. Quantization of
$\vec{k}$-vectors is described by

$\displaystyle\ \ \ \vec{k}=\vec{k}_{\parallel}+\vec{k}_{\perp},\ {\rm where}$
$\displaystyle\vec{k}_{\parallel}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\left(n_{A}\vec{A}^{\prime}+n_{B}\vec{B}^{\prime}\right)\
{\rm and}\ \vec{k}_{\perp}=\frac{1}{n_{L}}\left(n_{C}\vec{C}^{\prime}\right).$

The vectors $\vec{k}_{\parallel}$ are very dense in the $xy$ plane, while the
vectors $\vec{k}_{\perp}$ are discrete and well spaced in the $z$ direction.
The free electron plane-wave states $\psi_{\vec{k}}$ factorize as

$\psi_{\vec{k},{\rm
PBC}}(\vec{r})=\frac{1}{\sqrt{V}}e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}\times
e^{ik_{\perp}z}$ (21)

These are a complete basis of states $\psi_{\vec{k}}$. They have energy

$\epsilon_{\vec{k},{\rm
PBC}}=\epsilon_{\vec{k}_{\parallel}}+\frac{\hbar^{2}}{2m}\left(\frac{2\pi}{H}\right)^{2}n_{c}^{2}$
(22)

where $H=\sqrt{2/3}n_{L}a=14.03\AA$ is the width of the 6-layer slab. The
discrete energies proportional to $n_{c}^{2}$ (where
$n_{c}=\ldots,-2,-1,0,1,2,\ldots$) split the free electron band ($\propto
k^{2}$) into separate two-dimensional bands, each $\propto k_{\parallel}^{2}$,
and separated in energy. The separated bands of a slab are often called
“quantum well states” (QWS).

For a 6-layer film of (111) aluminum, the occupied states are contained in a
region of $\vec{k}$-space shown in Fig. 6. Each discrete value of
$\vec{k}_{\perp}$ has an infinite two-dimensional ($k_{x},k_{y}$) plane of
corresponding $\vec{k}_{\parallel}$’s. The occupied states are in circular
pieces whose radii $k_{F,n}$ reduce as $|n|$ increases. In first
approximation, the radii are fixed by
$\hbar^{2}k_{F,n}^{2}/2m+\epsilon_{\perp}(n_{c})=\epsilon_{F}$, where
$\epsilon_{\perp}(n_{c})$ is the second term of Eq. 22, and $\epsilon_{F}$ is
the bulk Fermi energy. However, the slab Fermi energy differs a bit from the
bulk Fermi energy. To compute $\epsilon_{F,{\rm slab}}$, guess the maximum
value of $n$ for which $\epsilon_{n}<\epsilon_{F}$, and then sum over $n_{c}$
to compute what value of $\epsilon_{F,{\rm slab}}$ gives the correct total
number of occupied states. The computation has to be repeated for different
guesses of the maximum $n$ until everything is self-consistent. For the six
layer slab of Al, using periodic BC’s, there are 7 partially occupied slabs
($n_{c}$ goes from -3 to +3). The Fermi wavevector for the $n=0$ central slab
is increased by 0.3% compared to bulk and the Fermi energy (the same for all
$k_{z}$-layers) is increased by 0.6%.

Figure 6:  The “Fermi surface” of a 6 layer slab of (111) sheets of Al, with
periodic BC’s. The occupied states lie on circular sheets stacked at values of
$k_{z}=n(2\pi/H)$ where $n$ goes from -3 to 3, and $H$ is the slab thickness,
set to 6 times the layer spacing. The “Fermi surface” is the set of 7 circles.
The (purely symbolic) sphere is close to the free electron Fermi surface of
bulk Al. When the number of layers increases, the number of circles in the
sphere increases. In the bulk limit, the circles are so dense that the
occupied states are correctly viewed as filling the free electron Fermi
sphere.

###  V.2 Slab with Hard Walls

Periodic BC’s give discrete bands of $k$-states of the slab. The discreteness
is correct, but details are wrong. The main error is that it gives discrete
waves $e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}e^{ik_{\perp}z}$ which
propagate (incorrectly) in the $\hat{z}$ direction as well as correctly in the
$\hat{x},\hat{y}$ directions. A more realistic alternative is “hard-wall”
BC’s. An implementation is given by Trivedi and Ashcroft Trivedi and Ashcroft
(1988).

Instead of using wavefunctions with periodicity $H$ in the $z$-direction,
imagine instead that there is a confining potential,

$V_{\rm HW}(z)=\begin{cases}0,&\text{if $0<z<H$},\\\
\infty,&\text{otherwise}.\end{cases}$ (23)

The Schroedinger equation is

$\mathcal{H}\psi=\left[\frac{p^{2}}{2m}+V_{\rm HW}(z)\right]\psi=E\psi.$ (24)

The solutions are

$\psi_{\vec{k}_{\parallel},n}^{\rm
HW}(\vec{r})=\sqrt{\frac{2}{V}}e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}\times\sin\left(\frac{n\pi
z}{H}\right),$ (25)

where $n=1,2,\ldots$. The factor of $\sqrt{2}$ appears because the integral of
$|\sin(n\pi/z)|^{2}$ is half as big as the integral of
$|e^{ik_{\perp}z}|^{2}$. The factor $n\pi/H$ is $k_{\perp}$. However, it does
not propagate in the $z$ direction; it is a standing wave. The energies are

$\epsilon_{\vec{k}_{\parallel},n,{\rm
HW}}=\epsilon_{\vec{k}_{\parallel}}+\frac{\hbar^{2}}{2m}\left(\frac{\pi}{H}\right)^{2}n^{2}.$
(26)

For even values of $n$, the energies are the same as in Eq. 22; odd values of
$n$ have energies between; and, unlike for periodic BC’s where $n_{C}$ can be
positive, negative, or 0, for hard-wall BC’s, only positive (and non-zero)
values of $n$ are permitted.

Figure 7:  The “Fermi surface” of a 6 layer slab of (111) sheets of Al, with
hard-wall BC’s. The occupied states lie on the yellow circular sheets stacked
at values of $k_{z}$ equal to $n(\pi/H)$ where $n$ goes from 1 to 7, and $H$
is 6 times the layer spacing. The Fermi surface is the set of 7 circles. When
the number of layers increases, the number of circles increases. In the bulk
limit, the circles are so dense that the occupied states essentially fill the
upper half of the free electron Fermi sphere. The spacing of the $k_{z}$
layers is smaller by 2 than in the previous figure. A more realistic figure of
the same type is in Fig. 1 of Ref. Czoschke  _et al._ , 2005.

Results are shown in Fig. 7. The method of hard-wall BC’s brings the free-
electron picture a step closer to reality than the fully periodic BC method. A
further improvement is discussed in the next section.

###  V.3 Soft Walls Surrounded by Vacuum

The work function of the (111) surface of Al is $\Phi(111)$=4.26 eV Eastment
and Mee (1973). Electron states with energy less than $\Phi(111)$ above the
Fermi level $\epsilon_{F}^{\rm slab}$ are confined in the slab. Higher energy
electrons will escape, if initially inside the slab, unless they relax down to
lower energy confined states. The work function is measured as the minimum
photon energy $\hbar\omega$ which can eject an electron from the material. In
an Al film, an electron at the Fermi energy $\epsilon_{F}^{\rm slab}$ can be
ejected if $\hbar\omega>\Phi(111)$. An electron initially outside and directed
toward the slab, will penetrate a bit, and then either be reflected back out
or temporarily absorbed. To extend the free-electron model to cover these
cases, replace the hard walls of the slab by soft walls. Previously, the
position $z=0$ was at the lower surface of the slab. Now it is more convenient
to choose $z=0$ to be at the center of the slab.

$V_{\rm SW}(z)=\begin{cases}0&\text{if $|z|<H/2$},\\\
U_{0}&\text{otherwise},\end{cases}$ (27)

where

$U_{0}=\epsilon_{F,{\rm slab}}+\Phi(111).$ (28)

The solutions
$\psi_{\vec{k}}(\vec{r},z)=\exp(i\vec{k}_{\parallel}\cdot\vec{r})u(z)$ have
wave-vectors $\vec{k}_{\parallel}$ parallel to the plane. The boundary
condition, that $\psi$ is continuous at the slab edges $z=\pm H/2$, requires
the wavevector $\vec{k}_{\parallel}$ to be the same both inside and outside
the slab.

####  V.3.1 Confined states

Electrons with energy less than $U_{0}$ are confined, but have finite,
exponentially decaying, amplitudes outside the slab. The wavefunctions are

$\displaystyle\psi_{\vec{k}_{\parallel}}^{\rm conf}$ $\displaystyle\propto$
$\displaystyle\begin{cases}&e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}\sin\left(k_{\perp}^{\rm
SW}z\right)\ ({\rm odd})\\\
&e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}\cos\left(k_{\perp}^{\rm
SW}z\right)\ ({\rm even})\\\ \end{cases}$ $\displaystyle{\rm if}\ |z|\leq
H/2,\ {\rm and}$ $\displaystyle\psi_{\vec{k}_{\parallel}}^{\rm conf}$
$\displaystyle\propto$ $\displaystyle
e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}e^{-\lambda(|z|-H/2)}$ (29)
$\displaystyle{\rm if}\ |z|>H/2$

This generalizes the wavefunctions in Eq. 25. However, the wavenumbers were
easily found to be $k_{\perp}^{\rm HW}=n\pi/H$ in the hard-wall case, but the
soft-wall wavenumbers $k_{\perp}^{\rm SW}$ need more careful application of
the boundary conditions. The answers are similar in both cases, as is shown in
Table 1.

n | $k_{\perp}^{\rm HW}a$ | $\epsilon_{\perp}^{\rm HW}$ | $k_{\perp}^{\rm SW}a$ | $\epsilon_{\perp}^{\rm SW}$
---|---|---|---|---
1 | 0.64 | 0.19 | 0.60 | 0.17
2 | 1.28 | 0.76 | 1.20 | 0.67
3 | 1.92 | 1.72 | 1.80 | 1.50
4 | 2.57 | 3.06 | 2.39 | 2.66
5 | 3.21 | 4.78 | 2.99 | 4.15
6 | 3.85 | 6.88 | 3.58 | 5.96
7 | 4.49 | 9.37 | 4.17 | 8.07
8 | 5.13 | 12.24 | 4.74 | 10.46
9 | 5.77 | 15.49 | 5.31 | 13.09
10 | 6.41 | 19.12 | 5.81 | 15.70
Table 1: For the 6-layer Al(111) slab, these are the perpendicular wavenumbers
and energies for the first 10 hard-wall (HW), and all the confined soft-wall
(SW) cases.

The wavefunctions in Eq. 29 have energies determined by the Schroedinger
equation $(p^{2}/2m+V_{\rm SW})\psi_{\rm SW}=E\psi_{\rm SW}$, where

$\displaystyle E-\frac{(\hbar k_{\parallel})^{2}}{2m}$ $\displaystyle=$
$\displaystyle\begin{cases}\frac{(\hbar k_{\perp}^{\rm SW})^{2}}{2m}\ {\rm
if}\ |z|<H/2\\\ U_{0}-\frac{(\hbar\lambda)^{2}}{2m}\ {\rm
otherwise}\end{cases}$ $\displaystyle{\rm where}$ $\displaystyle\
U_{0}=\epsilon_{F,{\rm slab}}+\Phi(111)$ (30)

The equality of $E$ inside and outside gives one relation, $(k_{\perp}^{\rm
SW})^{2}+\lambda^{2}=2mU_{0}/\hbar^{2}$, between the wavenumber
$k_{\perp}^{\rm SW}$ and the damping $\lambda$. A second relation comes from
continuity of $\psi_{\rm SW}$ and $d\psi_{\rm SW}/dz$ at the walls. This
permits both $k_{\perp}^{\rm SW}$ and $\lambda$ to be fixed. There are
multiple solutions, labeled by $n$, similar to the hard-wall case. For hard
walls $k_{\perp}=n\pi/W$, and any integer $n>0$ is allowed. The soft-wall case
has a further restriction, that the energy E of the states of Eqs. 29,30
cannot exceed the energy of the vacuum, $U_{0}=\epsilon_{F,{\rm
slab}}+\Phi(111)$. There are a finite number of such states $n$. There are
infinitely many additional states with $E>U_{0}$, but they are not damped by
$e^{-\lambda|z|}$, because they are unconfined, i.e. truly free outside the
slab. Equation 29 does not apply; they are discussed in the next section.

For the 6-layer Al(111) case, using the bulk Fermi energy (11.62 eV) and the
measured work function ($\Phi(111)$=4.26eV), there are 10 solutions. The
lowest energy solutions have energies and wavefunctions quite close to the
hard-wall solutions. This is shown in Table 1. By the fifth highest solution,
the differences are quite noticable. This is illustrated in Fig. 8. Figure 9
shows the energies of all 10 soft-wall states, and the corresponding hard-wall
states. In the hard-wall case, solutions with higher integers $n$ continue for
$n\rightarrow\infty$, but are not shown because their energies are beyond the
top of the graph.

The soft-wall method extends free-electron theory as far as it can go in thin
slabs, unless more accurate (and more complicated) potentials $V(z)$ are
known. Sophisticated one-electron theory, at great computational expense, can
produce 3-dimensional potentials $V(x,y,z)$ for a slab, or a film grown on a
surface. Then in principle, the $x$\- and $y$-dependences could be averaged
out, to get an accurate $V(z)$. But if the 3-dimensional potential is
available, the full wavefunctions $\psi(x,y,z)$, and corresponding energies,
would have been found during the computation of the potential. There would be
little motivation or need for a free-electron theory.

Figure 8:  Wavefunctions of band $n=5$ for hard-wall boundary conditions
(dashed) and soft-wall boundary conditions (solid line). Figure 9:  Hard-wall
(dashed curves) and soft-wall (solid curves) electron energies as a function
of the 2-dimensional wavevector $k_{\parallel}=|\vec{k}_{\parallel}|$. The
soft-wall energies terminate at the energy of the upper dashed line at
$U_{0}=\epsilon_{F}+\Phi$=15.87 eV. Beyond that energy, there are vacuum
states which may penetrate the slab but are not described by the same
equations. The energies colored black are $n=1$, maroon are $n=2$, blue are
$n=3$, etc. For $n$=1 or 2, hard- and soft-wall states are almost
indistinguishable. The 10th soft-wall state is just below the vacuum level,
barely visible at the upper left. The 10th and higher hard-wall states are
above the top of the graph.

####  V.3.2 Unconfined states

From the Schroedinger eqn. 24, the unconfined states with $E>U_{0}$ have
wavefunctions

$\psi_{\vec{k}_{\parallel}}^{\rm unconf}=\begin{cases}C_{\rm
in}e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}e^{ik_{\perp}^{\rm in}z}\
{\rm if}\ |z|<H/2\\\ C_{\rm
out}e^{i\vec{k}_{\parallel}\cdot\vec{r}_{\parallel}}e^{ik_{\perp}^{\rm out}z}\
{\rm otherwise}.\end{cases}$ (31)

The component $\vec{k}_{\parallel}$ of the wavevector parallel to the slab is
the same both inside and outside the slab. The component $k_{\perp}$ has
different values, $k_{\perp}^{\rm in}$ and $k_{\perp}^{\rm out}$. The energy
$E(\vec{k})$ has different forms inside and outside of the slab:

$E(\vec{k})=\frac{(\hbar k_{\parallel})^{2}}{2m}+\begin{cases}\frac{(\hbar
k_{\perp}^{\rm in})^{2}}{2m}\\\ U_{0}+\frac{(\hbar k_{\perp}^{\rm
out})^{2}}{2m}\end{cases}$ (32)

The equality of $E(\vec{k})$ inside and outside gives the relation,

$(k_{\perp}^{\rm in})^{2}-(k_{\perp}^{\rm out})^{2}=2mU_{0}/\hbar^{2},$ (33)

between the perpendicular wavenumbers $k_{\perp}^{\rm in}$ inside, and
$k_{\perp}^{\rm out}$ outside the slab.

Unlike the case of confined states, the wavevector inside, $k_{\perp}^{\rm
in}$, is not constrained by the width $H$ of the slab; it takes essentially
continuous values. More precisely, the full system is considered, for
normalization purposes, to lie in a finite volume $V=L^{3}$. The slab is
pictured as occupying the center ($-H/2<z<H/2$) of the space in the $z$
direction $(-L/2<z<L/2)$. Periodic boundary conditions give
$\vec{k}_{\parallel}$ the values $2\pi n_{x}/L$ and $2\pi n_{y}/L$ in the $x$
and $y$ directions, and $k_{\perp}^{\rm out}$ the values $2\pi n_{z}/L$. The
wavevector $k_{\perp}^{\rm in}$ is then fixed by Eq. 33. The coefficient
$C_{\rm out}$, to lowest order in $H/L$, is $\sqrt{(}1/V)$, and the
coefficient $C_{\rm in}$ is similar in size, but depends on $k_{\perp}^{\rm
in}$ and $k_{\perp}^{\rm out}$ in a complicated way that depends on matching
$\psi$ at the boundaries.

####  V.3.3 Unconfined standing waves

The states $\psi^{\rm unconf}$, both inside and outside, have $E$ independent
of the signs of $\vec{k}_{\parallel}$ and $k_{\perp}$, so the complete
wavefunction can consist of a variety of linear combinations of simple waves.
The next section deals with waves incident on the slab from outside. This
section deals with a symmetric standing wave, which illuminates the states
nicely, but is not easy to realize experimentally. The wavefunction has the
form

$\displaystyle\psi_{\rm in}$ $\displaystyle=$
$\displaystyle\frac{A}{\sqrt{V}}e^{i\vec{k}_{\parallel}\cdot\vec{r}}\cos(k_{\perp}^{\rm
in}z)$ $\displaystyle\psi_{\rm out}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{V}}e^{i\vec{k}_{\parallel}\cdot\vec{r}}\cos(k_{\perp}^{\rm
out}z\pm\phi),$ (34)

where $\psi_{\rm in}$ has $|z|<H/2$ and $\psi_{\rm out}$ has $|z|>H/2$. The
coefficients $A$ and $\phi$ are real numbers to be fixed by boundary
conditions. The plus sign on $\phi$ is used for $z>H/2$, and the minus sign
for $z<-H/2$. Continuity of $\psi$ and $d\psi/dz$ across the boundaries gives
the two equations necessary to determine $A$ and $\phi$,

$\displaystyle A\cos(k_{\perp}^{\rm in}H/2)$ $\displaystyle=$
$\displaystyle\cos(k_{\perp}^{\rm out}H/2+\phi),$ $\displaystyle
Ak_{\perp}^{\rm in}\sin(k_{\perp}^{\rm in}H/2)$ $\displaystyle=$
$\displaystyle k_{\perp}^{\rm out}\sin(k_{\perp}^{\rm out}H/2+\phi).$

The ratio of these equations eliminates $A$ and gives the equation for the
phase $\phi$,

$k_{\perp}^{\rm in}\tan(k_{\perp}^{\rm in}H/2)=k_{\perp}^{\rm
out}\tan(k_{\perp}^{\rm out}H/2+\phi)$ (36)

To illustrate this, consider a state with energy 3eV above the vacuum level
$U_{0}$=15.9 eV. The energy is $E$=18.9eV=$(\hbar^{2}k_{\perp}^{\rm
in})^{2}/2m$, which gives $k_{\perp}^{\rm in}H/2=15.6$. Then from Eq. 33,
$k_{\perp}^{\rm out}H/2=6.22$. Equation 36 then gives $\phi=-6.44$, and Eq.
LABEL:eq:bcsw gives $A=-0.980$. The wavefunction is shown in Fig. 10.

Figure 10:  The symmetric standing-wave wavefunction (Eq. 34) for the state
with energy 3eV higher than the vacuum energy $U_{0}$. The slab goes from
$z=-H/2$ to $z=H/2$. The sample volume $V$ is replaced by 1.

####  V.3.4 Unconfined external incident waves

Fabrication of free-standing thin slabs (like the present example of Al) is
making rapid progress, largely thanks to to advances in imaging of surfaces.
One way to examine such a slab is to probe with a monochromatic electron beam.
An electron incident from the left has an amplitude T for being transmitted,
and R for being reflected from the surface at $z=-H/2$. The transmitted
electron wave then travels inside, and has an amplitude $T^{\prime}$ for being
transmitted out through the surface at $z=+H/2$ and $R^{\prime}$ for being
reflected back in the slab. It will then either be transmitted into the vacuum
at $z<-H/2$ or reflected again back into the slab. The final result is
described by a wavefunction $\psi(z)$ which is a right-moving traveling wave
in the right-hand vacuum ($z>H/2$) and a mix of left- and right-moving
traveling waves in the other regions. For an incident wave traveling in the
positive $z$ direction ($k_{\perp}^{\rm out}>0$ and $\vec{k}_{\parallel}=0$),
the wavefunction is

$\psi(z)\propto
e^{i\vec{k}_{\parallel}\cdot\vec{r}}\begin{cases}e^{ik_{\perp}^{\rm
out}z}+\mathcal{R}e^{-ik_{\perp}^{\rm out}z}\ {\rm if}\ z<-H/2\\\
\mathcal{A}e^{ik_{\perp}^{\rm in}z}+\mathcal{B}e^{-ik_{\perp}^{\rm in}z}\ {\rm
if}\ |z|<H/2\\\ \mathcal{T}e^{ik_{\perp}^{\rm out}z}\ {\rm if}\
z>H/2\end{cases}$ (37)

There are four boundary conditions that determine the four unknowns
($\mathcal{A},\mathcal{B},\mathcal{R},\mathcal{T}$), namely continuity of
$\psi$ and $d\psi/dz$ at $z=-H/2$ and at $z=+H/2$. These give the four
equations

$\displaystyle e^{-ik^{\prime}H/2}+\mathcal{R}e^{ik^{\prime}H/2}$
$\displaystyle=$ $\displaystyle\mathcal{A}e^{-ikH/2}+Be^{ikH/2}$
$\displaystyle e^{-ik^{\prime}H/2}-\mathcal{R}e^{ik^{\prime}H/2}$
$\displaystyle=$
$\displaystyle\frac{k}{k^{\prime}}\left[\mathcal{A}e^{-ikH/2}-Be^{ikH/2}\right]$
$\displaystyle\mathcal{T}e^{ik^{\prime}H/2}$ $\displaystyle=$
$\displaystyle\mathcal{A}e^{ikH/2}+Be^{-ikH/2}$
$\displaystyle\mathcal{T}e^{ik^{\prime}H/2}$ $\displaystyle=$
$\displaystyle\frac{k}{k^{\prime}}\left[\mathcal{A}e^{ikH/2}-Be^{-ikH/2}\right]$

Here a simplified notation uses $k=k_{\perp}^{\rm in}$ and
$k^{\prime}=k_{\perp}^{\rm out}$. Adding the first and the second equations
gives an equation involving only the two unknowns $\mathcal{A},\mathcal{B}$.
Subtracting the third from the fourth gives another such equation. Solving the
two linear equations gives

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle
2e^{-i(k+k^{\prime})H/2}/(1-k/k^{\prime})\mathcal{D}$
$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle-2e^{i(k-k^{\prime})H/2}/(1+k/k^{\prime})\mathcal{D}$
$\displaystyle\mathcal{D}$ $\displaystyle=$
$\displaystyle+\frac{k-k^{\prime}}{k+k^{\prime}}e^{ikH}-\frac{k+k^{\prime}}{k-k^{\prime}}e^{-ikH}.$
(39)

The remaining two unknowns $\mathcal{R},\mathcal{T}$ can then be evaluated,

$\displaystyle\mathcal{R}$ $\displaystyle=$
$\displaystyle-e^{-ik^{\prime}H}+\frac{2}{\mathcal{D}}\left[\frac{e^{-i(k+k^{\prime})H}}{1-k/k^{\prime}}-\frac{e^{i(k-k^{\prime})H}}{1+k/k^{\prime}}\right]$
$\displaystyle\mathcal{T}$ $\displaystyle=$
$\displaystyle\frac{2}{\mathcal{D}}e^{-ik^{\prime}H}\left[\frac{1}{1-(k/k^{\prime})}-\frac{1}{1+(k/k^{\prime})}\right]$
(40)

Figure 11 shows the result for $\psi$ from Eqs. 39 and 40, using the same
energy as in the standing wave of Fig. 10, namely 3eV above the vacuum energy.
The real part of $\psi$ (the solid black curve of Fig. 11) is very similar to
the standing wave of Fig. 10.

Figure 11:  The wavefunction (Eq. 37, ignoring the $x,y$ dependence) when a
plane wave with energy 3eV higher than the vacuum energy $U_{0}$ is incident
from the left, perpendicular to the slab. The black solid and dashed curves
are the real and imaginary parts of the total wave. The red solid and dashed
curves are the real and imaginary parts of the reflected wave.

##  VI Summary

Bulk crystals have surfaces which can be ignored when thinking about bulk
properties like electrical and thermal conductivity. The low energy bulk
properties of “simple” metals (those with only $s$ and $p$ electrons near the
Fermi level) conform reasonably well with the standard free electron model.
However, their surfaces are important for many other considerations, and have
properties (e.g. surface states) outside the scope of a free electron picture.

Crystalline thin films differ in some ways from their bulk versions. Low
energy properties of films of “simple” metals are modified by not having
electron propagation in the perpendicular ($z$) direction. Instead, there are
“quantum well states” (QWS) whose nature is not badly described by modifying
the free electron theory appropriately. The simplest modification, using
periodic boundary conditions in the $z$ direction, while unrealistic, gives a
sensible first approach to the QWS modification of electron states. An almost
equally simple modification, confining the $z$ motion by a potential well
$V(z)$ with infinite barriers at the film boundaries, improves the
correspondence with reality. However, it is still unrealistic except when the
Fermi level is not too far above the bottom of the $sp$ bands. A nice further
improvement is found if the potential well $V(z)$ has a finite barrier set by
the measured work function of the bulk metal.

The resulting free electron picture of metallic films does not describe many
of the important complications of real films. However, it gives a good
pedagogical introduction to metal films. Comparison of actual properties of
films with the free electron model is a good way to start understanding the
important further complications.

## References

  * Drude (1900) Paul Drude, “Zür Elektronentheorie der Metalle,” Annalen der Physik 306, 566 (1900).
  * Sommerfeld (1928) A. Sommerfeld, “Zur Elektronentheorie der Metalle auf Grund der Fermischen Statistik,” Zeitschrift für Physik 47, 1 (1928).
  * Binnig and Rohrer (1982) G. K. Binnig and H. Rohrer, “Scanning tunneling microscopy,” Helv. Phys. Acta 55, 726–735 (1982).
  * Binnig and Rohrer (1987) G. K. Binnig and H. Rohrer, “Scanning tunneling microscopy—from birth to adolescence,” Rev. Mod. Phys. 59, 615–625 (1987).
  * Mudrik  _et al._ (1993) M. Mudrik, S. S. Cohen, and N. Croitoru, “Electron conductivity of very thin metal films,” Thin Solid Films 226, 140–143 (1993).
  * Chiang (2000) T.-C. Chiang, “Photoemission studies of quantum well states in thin films,” Surface Science Reports 39, 181–235 (2000).
  * Miller  _et al._ (2009) T. Miller, M. Y. Chou, and T.-C. Chiang, “Phase relations associated with one-dimensional shell effects in thin metal films,” Phys. Rev. Lett. 102, 236803 (2009).
  * Miller and Chiang (2011) T. Miller and T.-C. Chiang, “Quantum electronic stability of atomically uniform films,” in  _Thin Film Growth_ , edited by Zexian Cao (Woodhead Publishing, Cambridge, U.K., 2011) Chap. 2, pp. 22–51.
  * Milun  _et al._ (2002) M. Milun, P. Pervan, and D. P. Woodruff, “Quantum well structures in thin metal films: simple model physics in reality?” Reports on Progress in Physics 65, 99 (2002).
  * Atkinson and Slavin (2008) W. A. Atkinson and A. J. Slavin, “A free-electron calculation for quantum size effects in the properties of metallic islands on surfaces,” Am. J. Phys. 76, 1099–1101 (2008).
  * Tringides  _et al._ (2007) M. C. Tringides, M. Jalochowski, and E. Bauer, “Quantum size effects in metallic nanostructures,” Physics Today 60, 50–54 (2007).
  * Wu  _et al._ (2015) Xuefeng Wu, Chaoqiang Xu, Kedong Wang, and Xudong Xiao, “Systematic investigation of pseudogaps in In, Al, and Pb islands,” Phys. Rev. B 92, 035434 (2015).
  * Born and von Karman (1912) M. Born and T. von Karman, “Über Schwingungen im Raumgittern,” Physikalishe Zeitschrift 13, 297–309 (1912).
  * Cohen and Bergstresser (1966) M. L. Cohen and T. K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev. 141, 789–796 (1966).
  * Huang (2008) Gui Qin Huang, “Electronic structures, surface phonons, and electron-phonon interactions of Al(100) and Al(111) thin films from density functional perturbation theory,” Phys. Rev. B 78, 214514 (2008).
  * Choy  _et al._ (2000) T.-S. Choy, J. Naset, S. Hershfield, and C. J. Stanton, (2000), the Fermi surface database, http://www.phys.ufl.edu/fermisurface.
  * Acharya and Rahman (2018) Shree Ram Acharya and Talat S Rahman, “Toward multiscale modeling of thin-film growth processes using SLKMC,” Journal of Materials Research 33, 709–719 (2018).
  * Schulte (1976) F. K. Schulte, “A theory of thin metal films: electron density, potentials and work function,” Surface Science 55, 427–444 (1976).
  * Trivedi and Ashcroft (1988) N. Trivedi and N. W. Ashcroft, “Quantum size effects in transport properties of metallic films,” Phys. Rev. B 38, 12298 (1988).
  * Pitarke and Eguiluz (2001) J. M. Pitarke and A. G. Eguiluz, “Jellium surface energy beyond the local-density approximation: Self-consistent-field calculations,” Phys. Rev. B 63, 045116 (2001).
  * Wu and Zhang (2008) Biao Wu and Zhenyu Zhang, “Stability of metallic thin films studied with a free electron model,” Phys. Rev. B 77, 035410 (2008).
  * Han and Liu (2009) Yong Han and Da-Jiang Liu, “Quantum size effects in metal nanofilms: Comparison of an electron-gas model and density functional theory calculations,” Phys. Rev. B 80, 155404 (2009).
  * Czoschke  _et al._ (2005) P. Czoschke, Hawoong Hong, L. Basile, and T.-C. Chiang, “Quantum size effects in the surface energy of Pb/ Si (111) film nanostructures studied by surface x-ray diffraction and model calculations,” Phys. Rev. B 72, 075402 (2005).
  * Eastment and Mee (1973) R. M. Eastment and C. H. B. Mee, “Work function measurements on (100), (110) and (111) surfaces of aluminium,” J. Phys. F: Metal Phys. 3, 1738 (1973).