Brief theory of photoemission spectroscopy

Before starting to read this, stare at your surroundings for a few seconds and reflect on the most fundamental trait that every object around you shares. The floor you stand on, the chair you are sitting on, even the air you breathe is composed of small basic units called atoms. It is the arrangement of these atoms that forms an object and dictates the properties exhibited by it. As a rule of thumb, if the dimensions of the object are such that it can be seen with the naked eye, the laws of classical mechanics will rule the fate of the object and its behavior under the action of a force (or forces) is well understood. For objects with small dimensions, the laws of classical mechanics tend to lead to erroneous results (such as the 'ultraviolet catastrophe' at the beginning of the 1900s). A series of theoretical and experimental efforts performed by the brilliant minds of the time led to the development of quantum mechanics, a branch of physics that describes a body (or a system as is used for the quantum mechanical description) by means of the Schrödinger equation. The Schrödinger equation is a second order differential equation that implicitly alludes to the conservation of energy to obtain information about the system in question. In its full glory the equation has the mathematical form

$$\begin{eqnarray}&&{\hat{H}}\;{\rm \Psi}\left( {\boldsymbol{r}} ,\theta ,\phi ,t \right)={\rm i}\hbar \frac{\partial }{\partial t}{\rm \Psi}\left( {\boldsymbol{r}} ,\theta ,\phi ,t \right)\end{eqnarray} \tag{ 1.1 }$$

where $\hbar$ is the reduced Planck constant, i is the imaginary unit used in complex number analysis, $\hat{H}$ is the Hamiltonian ¹ that describe the quantum mechanical system and Ψ( r , θ, , t) is the wavefunction of the system purposely chosen by the author to be described by spherical coordinates ² . The wavefunction is a mathematical function that depends on both the spatial coordinates and time, and is used to describe the wave behavior of the particle. For some quantum mechanical systems, the states in which the particle can be found are independent of time, such states are called stationary states. Under the time-independent assumption, the simplified form of (1.1) is

$$\begin{eqnarray}&&{\hat{H}}\;{\rm \Psi}\left( {\boldsymbol{r}} \right)=E\;{\rm \Psi}\left( {\boldsymbol{r}} \right)\end{eqnarray} \tag{ 1.2 }$$

where E represents the total energy of the system. This equation is known as the non-relativistic time-independent Schrödinger equation. In essence, (1.2) is an eigenvalue problem and reads as 'when the Hamiltonian operator acts on the wavefunction Ψ( r ), it will yield a 'value' times the wavefunction'. That 'value' corresponds to the total energy of the system. While the statement on how the equation is read provides no physical insight, there are several features of (1.2) that merit serious consideration and must be understood prior to its implementation. First, as obtained from the Schrödinger equation, Ψ( r ) is meaningless as a laboratory observable; it is the square modulus of the wavefunction |Ψ( r )|² that has a physical interpretation. In fact, |Ψ( r )|² gives the probability of finding a particle in a specific electronic state ³ at a certain point in space via the expression

$$\begin{eqnarray}&&\mid {\rm \Psi}\left( {\boldsymbol{r}} \right){{\mid }^{2}}={{{\rm \Psi}}^{*}}\left( {\boldsymbol{r}} \right){\rm \Psi}\left( {\boldsymbol{r}} \right)\;\end{eqnarray} \tag{ 1.3 }$$

where the function ${{{\rm \Psi}}^{*}}$( r ) is the complex conjugate of Ψ( r ) as denoted by the symbol *.

Second, the Schrödinger equation is a linear equation, meaning that the total wavefunction describing the system can be written as a linear expansion of its eigenfunctions as

$$\begin{eqnarray}&&{\rm \Psi}\left( {\boldsymbol{r}} \right)=\sum \limits_{n=1}^{\infty } {{c}_{n}}{{\varphi }_{n}}\left( {\boldsymbol{r}} \right)\end{eqnarray} \tag{ 1.4 }$$

where c _n is a coefficient related to the probability distribution and _n ( r ) is a particular state of the system with a specific energy E _n . ⁴ What is the meaning of this expansion? Imagine that the system under study is that of a free electron (no net force acts on it) constrained to travel in a one-dimensional path. The total 'non-normalized' wavefunction Ψ(x) describing that system can be written as

$$\begin{eqnarray}&&{\rm \Psi}\left( x \right)={{c}_{1}}\;{{{\rm e}}^{{\rm i}kx}}+{{c}_{2}}\;{{{\rm e}}^{-{\rm i}kx}}\end{eqnarray} \tag{ 1.5 }$$

where the first term represents a traveling wave in the +x direction and the second term presents a traveling wave in the −x direction. It is this expression that leads to the treatment of the 'free' photoelectron as a fully symmetric free electron state. Hence, Ψ(x) can be thought as the 'total' wavefunction of the system, meaning that it contains all the possible states the electron can occupy.

Third and last, the solution of the Schrödinger equation has a strong dependence on the Hamiltonian describing the problem. The Hamiltonian is an operator that represents the total energy of the system. For a system of N particles it is written as

$$\begin{eqnarray}&&{\hat{H}}\left( {{{\boldsymbol{r}} }_{1}},{{{\boldsymbol{r}} }_{2}},{\mkern 1mu} \ldots {{{\boldsymbol{r}} }_{n}},t \right)=\sum \limits_{n=1}^{N} \frac{{{\hbar }^{2}}}{2{{m}_{n}}}\nabla _{n}^{2}+{\rm U}\left( {{{\boldsymbol{r}} }_{1}},{{{\boldsymbol{r}} }_{2}},{\mkern 1mu} \ldots {{{\boldsymbol{r}} }_{n}},t \right)\end{eqnarray} \tag{ 1.6 }$$

where m _n is the mass of the n particle and $\nabla _{n}^{2}$ is the Laplacian operator that will 'act' on the wavefunction describing the system. The first term of (1.6) represents the total kinetic energy of the system, which entirely depends on the particles' motion. The second term represents the potential energy of the system, which depends on the interactions of the particles with their surroundings and/or among themselves. In most quantum mechanical systems, the complexity of the Hamiltonian arises from the form of the potential energy function. One common goal each scientist strives for is that of simplifying complex problems. However, this must be achieved by carefully balancing the simplifying assumptions with the amount of information lost by their implementation. This is what has led to various reformulations of (1.6) when solving a specific quantum system. For instance, one of the simplest (which is not necessarily simple) cases one can tackle is that of a single particle, for which the Hamiltonian can be written as

$$\begin{eqnarray}&&{\hat{H}}\left( {\boldsymbol{r}} \right)=\frac{{{\hbar }^{2}}}{2m}\nabla _{r}^{2}+{\rm U}\left( {\boldsymbol{r}} \right).\end{eqnarray} \tag{ 1.7 }$$

Among the quantum mechanical systems described by (1.7) one can find the potential square well, the potential step, the harmonic oscillator, etc, where each scenario differs solely by the potential function U( r ) which the particle is moving through. However, a realistic quantum mechanical system is composed of more than one particle and, while it is true that the idealization inherent in (1.7) provides good insight on the quantum mechanical nature of the particle and its behavior under certain conditions, it is highly limited for the description of complex systems.

One approach that is related to the upcoming discussion in section 1.1.3. is the simplification of the quantum mechanical system described by (1.2) using the center of mass formulation for a two-particle system when the potential energy function depends on the relative distance between the particles (i.e. U( r ₂ −  r ₁)). Consider a quantum mechanical system that is composed of two particles with masses m ₁ and m ₂ at distances r ₁ and r ₂ from an origin O as shown in figure 1.1(a).

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The Hamiltonian for such a quantum mechanical system has the form

$$\begin{eqnarray}&&{\hat{H}}\left( {{{\boldsymbol{r}} }_{1}},{{{\boldsymbol{r}} }_{2}} \right)=-\frac{{{\hbar }^{2}}}{2{{m}_{1}}}\nabla _{{{{\boldsymbol{r}} }_{1}}}^{2}-\frac{{{\hbar }^{2}}}{2{{m}_{2}}}\nabla _{{{{\boldsymbol{r}} }_{2}}}^{2}+{\rm U}\left( {{{\boldsymbol{r}} }_{1}},{{{\boldsymbol{r}} }_{2}} \right).\end{eqnarray} \tag{ 1.8 }$$

The issue with (1.8) is that the description of the quantum mechanical system depends simultaneously on the coordinates r ₁ and r ₂. However, for a potential energy function that depends only on the relative distance between the two particles (such as that due to electrostatics) one could use the coordinate transformations

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {\boldsymbol{R}} & = & \frac{{{m}_{1}}{{{\boldsymbol{r}} }_{1}}+{{m}_{2}}{{{\boldsymbol{r}} }_{2}}}{{{m}_{1}}+{{m}_{2}}} \\ \;{\boldsymbol{r}} & = & {{{\boldsymbol{r}} }_{2}}-{{{\boldsymbol{r}} }_{1}} \\\end{array}\end{eqnarray} \tag{ 1.9 }$$

to describe the system in the center of mass frame as shown in figure 1.1(b). The execution of the coordinate transformations in (1.8) leads to a Hamiltonian of the form

$$\begin{eqnarray}&&{\hat{H}}\left( {\boldsymbol{R}} ,{\boldsymbol{r}} \right)=-\frac{{{\hbar }^{2}}}{2M}\nabla _{{\boldsymbol{R}} }^{2}-\frac{{{\hbar }^{2}}}{2\mu }\nabla _{{\boldsymbol{r}} }^{2}+{\rm U}\left( {\boldsymbol{r}} \right)\end{eqnarray} \tag{ 1.10 }$$

where

$$\begin{eqnarray}&&M={{m}_{1}}+{{m}_{2}}\end{eqnarray} \tag{ 1.11 }$$

is the total mass of the system and

$$\begin{eqnarray}&&\mu \equiv \frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}\end{eqnarray} \tag{ 1.12 }$$

is defined as the reduced mass (the mass of a fictitious particle that is equivalent to the average of the two-body problem). To clarify the meaning of this problem, let us raise the question: what does this change of coordinates mean? By straight comparison of (1.8) and (1.10), the center of mass formulation allows us to replace the system containing particles with masses m ₁ and m ₂ by another system in which the Hamiltonian is described by a 'free particle' with mass M and a 'fictitious' reduced particle with mass μ (figure 1.1(c)). This raises the following question: what is the advantage of this transformation? The advantage is that now the potential energy function is independent of the coordinate R ; it depends entirely on r which is the distance between the two particles. Moreover, since it is assumed that the particles do not interact with each other (i.e. they are moving in the same potential) one can separate the wavefunction as

$$\begin{eqnarray}&&{\rm \Psi}\left( {\boldsymbol{r}} ,{\boldsymbol{R}} \right)=\psi \left( {\boldsymbol{r}} ^{\prime} \right)\psi \left( {\boldsymbol{R}} \right).\end{eqnarray} \tag{ 1.13 }$$

These will decouple the coordinates and will allow for the independent description of each particle through the equations

$$\begin{eqnarray}&&-\frac{{{\hbar }^{2}}}{2M}\;\nabla _{R}^{2}\psi \left( {\boldsymbol{R}} \right)={{E}_{{\rm COM}}}\psi \left( {\boldsymbol{R}} \right)\end{eqnarray} \tag{ 1.14 }$$

$$\begin{eqnarray}&&-\frac{{{\hbar }^{2}}}{2\mu }\;\nabla _{r}^{2}\psi \left( {\boldsymbol{r}} \right)+{\rm U}\left( {\boldsymbol{r}} \right)=E{\mkern 1mu} \psi \left( {\boldsymbol{r}} \right).\end{eqnarray} \tag{ 1.15 }$$

The solution of (1.14) is that of a free particle, which is described by

$$\begin{eqnarray}&&\psi \left( {\boldsymbol{R}} \right)=A{{{\rm e}}^{{\rm i}kR}}.\end{eqnarray} \tag{ 1.16 }$$

This will allow for the calculation of the total momentum of the center of mass using

$$\begin{eqnarray}&&p=-{\rm i}\hbar {{\nabla }_{R}}\psi \left( {\boldsymbol{R}} \right)=\hbar k\end{eqnarray} \tag{ 1.17 }$$

which is constant. This means one can choose to work in the center of mass frame of reference (setting R  = 0) and the two-particle problem reduces to

$$\begin{eqnarray}&&{\rm \hat{H}}\left( {\boldsymbol{r}} \right)=-\frac{{{\hbar }^{2}}}{2\mu }\nabla _{r}^{2}+{\rm U}\left( {\boldsymbol{r}} \right)\end{eqnarray} \tag{ 1.18 }$$

which is that of a single particle with mass μ moving in a potential U( r ) (see section 1.1.3).

As mentioned previously, no connection to the photoemission process has been attempted here. These concepts are purely theoretical and are intended to help the reader grasp some basic ideas that originate from quantum mechanics (i.e. the use and interpretation of the Schrödinger equation, wavefunctions, etc). One must keep in mind that the culmination of quantum mechanics in the Schrödinger equation is still incomplete when it comes to photoemission. While one can apply the Schrödinger equation to the photoemission process, this is largely a single-particle approximation and photoemission from solids is, intrinsically, a many-body problem: the electrons, phonons and plasmons of the solid interact in the photoemission process.

The remaining subsections share a similar approach and will be devoted to reinforcing the concepts discussed here by applying them to different quantum mechanical systems. Let us begin with the simplest case and then develop more complex scenarios from basic principles, this will help (although it is not strictly required) in assembling the building blocks that will serve as the basis for PES.

Let us consider the one-dimensional 'particle in a box' problem to exemplify the discussion in section 1.1.1. Keep in mind that this problem, while fictional in nature, provides valuable insight into the nature of quantum mechanics. In fact, this model has been used to find approximate solutions for complex physical systems. Particular to the discussion of photoemission is that of quantum well states [1]. Our quantum system now consists of an electron with mass m _e moving along a line with length L as shown in figure 1.2(a).

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At the extreme points of the line there is an idealized force acting only at those two points and pointing in a direction such that the particle will remain trapped along the one-dimensional path. Mathematically, the problem can be described in terms of a potential function as depicted in figure 1.2(b). The potential at points x = 0 and x = L tends to infinity. This is equivalent to a strong force acting on the electron only at those points in such way that the electron will remain trapped in the line. Between those two coordinates the potential energy is zero. For convenience, this will be interpreted as if there is no net force acting on the particle ⁵ . For this particular problem (1.5) reduces to

$$\begin{eqnarray}&&-\frac{{{\hbar }^{2}}}{2{{m}_{e}}}\frac{{{\partial }^{2}}\psi \left( x \right)}{\partial {{x}^{2}}}+U\left( x \right)\psi \left( x \right)=E\psi \left( x \right).\end{eqnarray} \tag{ 1.19 }$$

But, since the particle is subject to no potential along its path, (1.19) takes the form

$$\begin{eqnarray}&&\frac{{{\partial }^{2}}\psi \left( x \right)}{\partial {{x}^{2}}}=-{{k}^{2}}\psi \left( x \right)\end{eqnarray} \tag{ 1.20 }$$

where

$$\begin{eqnarray}&&k=\sqrt{\frac{2{{m}_{e}}E}{\;{{\hbar }^{2}}}}\end{eqnarray} \tag{ 1.21 }$$

and is referred to as the wavenumber of the electron. By definition, the wavenumber is the magnitude of the wavevector, which physically describes the traveling direction of the electron wave. Nonetheless, (1.20) has a general solution of the form

$$\begin{eqnarray}&&\psi \left( x \right)=A{{{\rm e}}^{{\rm i}kx}}+B{{{\rm e}}^{-{\rm i}kx}}\end{eqnarray} \tag{ 1.22 }$$

where A and B are coefficients that need to be determined. Since the potential energy function at the point x = 0 is infinity, the particle can never be found at that location. This condition leads to the requirement that $\psi \left( x=0 \right)=0$ which will be true if and only if A = −B. These lead to the solution

$$\begin{eqnarray}&&\psi \left( x \right)=A\left( {{{\rm e}}^{{\rm i}kx}}-{{{\rm e}}^{-{\rm i}kx}} \right).\end{eqnarray} \tag{ 1.23 }$$

Using Euler's equation, the expression above can be rewritten as

$$\begin{eqnarray}&&\psi \left( x \right)=2{\rm i}A{\rm sin} \left( kx \right).\end{eqnarray} \tag{ 1.24 }$$

Similarly, the electrostatic potential at the point x = L is also infinity, which leads to the requirement that $\psi \left( x=L \right)=0$ which will be true if and only if sin(kL) = 0 or

$$\begin{eqnarray}&&k=\frac{n\pi }{L}\quad n=1,2,3,\ldots \end{eqnarray} \tag{ 1.25 }$$

where n denotes the quantization imposed by the boundary condition at x = L. Hence, the wavefunction for an electron subject to a potential of this form is given by

$$\begin{eqnarray}&&{{\psi }_{n}}\left( x \right)=2{\rm i}A{\rm sin} \left( \frac{n\pi x}{L} \right).\end{eqnarray} \tag{ 1.26 }$$

Since the result we seek is probabilistic in nature, it is customary to enforce that the area under the curve of |ψ _n (x)|² is equal to one (which is not the case in (1.26) as written). This will ensure that the sum of all probabilities for each point in space is equal to 100%. The technical term for this process is 'normalization' and the normalization condition is mathematically expressed as

$$\begin{eqnarray}&&\int \nolimits_{0}^{L} {{\left| {{\psi }_{n}}\left( x \right) \right|}^{2}}{\rm d}x=1.\end{eqnarray} \tag{ 1.27 }$$

The successful execution of (1.27) will lead to the result

$$\begin{eqnarray}&&2{\rm i}A=\sqrt{\frac{2}{L}}.\end{eqnarray} \tag{ 1.28 }$$

Therefore, the solution of the Schrödinger equation for a one-dimensional particle in a 'box' (in this case the particle being an electron) is

$$\begin{eqnarray}&&{{\psi }_{n}}\left( x \right)=\sqrt{\frac{2}{L}}{\rm sin} \left( \frac{n\pi x}{L} \right)\end{eqnarray} \tag{ 1.29 }$$

which leads to

$$\begin{eqnarray}&&{{\left| {{\psi }_{n}}\left( x \right) \right|}^{2}}=\frac{2}{L}{{{\rm sin} }^{2}}\left( \frac{n\pi x}{L} \right).\end{eqnarray} \tag{ 1.30 }$$

Let us carefully examine this solution. Figure 1.3 shows the probability of finding the electron within the line L for a state with n = 1, 2 and 3. For an electron in the ψ ₁ state, the probability distribution peaks at the middle of the line. What this means is that if one were to locate the electron, it is most likely that the electron will be found at $x=\frac{L}{2}$. However, this does not necessarily mean that the electron will be found at that location, it can still be anywhere along the line subject to a certain probability. For an electron in the state described by ψ ₂, the probability distribution is different and peaks at two distinct locations.

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If the electron is in the ψ ₂ state, one is more likely to find it with equal probability at x = $\frac{L}{4}$ and x = $\frac{3L}{4}$. For this state the electron can be anywhere along the line except at x = $\frac{L}{2}$. If one considers the electron in a state described by ψ ₃, an emerging pattern is visible. Note that at all times the electron is enclosed within a specific volume (a line in this case) despite the state that it is occupying. What changes in the enclosed volume is the probabilistic distribution that the electron will adopt. It is customary to refer to these distributions as electronic orbitals, which in essence are the collection of points in space in which an electron can be found even if the probability is small. In fact, an orbital can be thought as a density plot of |ψ _n (x)|² which is shown in figure 1.4. The stronger the intensity in the density plot, the higher the probability of finding the electron at that specific location.

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In essence, an orbital can be interpreted as those points in spatial space in which the electron in a specific state will 'hang around' with a certain probability. Since it is not adequate to refer to 'those points in spatial space in which the electron in a specific state will hang around' when having a discussion, it is universally accepted to denote the density plot of |ψ _n (x)|² with an orbital notation. Since a different quantum number will correspond to a different electronic state, we can use n to assign an orbital notation. One could refer to the state with quantum number n = 1 as an s orbital, n = 2 as a p orbital and n = 3 as d orbital ⁶ as shown in figure 1.4. Another feature that must be pointed out is that the orbital shape is also related to the quantum number n.

The information about the location of the electron within an enclosed volume can be useful and can provide important insights about the problem in question, even if it is probabilistic in nature. However, it is possible to obtain more information about the quantum mechanical system by placing (1.30) into (1.19). If such a process is executed, the result

$$\begin{eqnarray}&&\frac{{{\hbar }^{2}}}{2m}{{\left( \frac{n\pi }{L} \right)}^{2}}{{\psi }_{n}}\left( x \right)={{E}_{n}}{{\psi }_{n}}\left( x \right)\end{eqnarray} \tag{ 1.31 }$$

or

$$\begin{eqnarray}&&{{E}_{n}}=\frac{{{\hbar }^{2}}{{n}^{2}}{{\pi }^{2}}}{2m{{L}^{2}}}\end{eqnarray} \tag{ 1.32 }$$

will be obtained. I must point that the ψ _n (x) cancels in (1.31) only after the second derivative of the wavefunction is taken (i.e. the operator acts on the wavefunction). This is because before extracting a value such as energy, certain operations must first be executed ⁷ . Equation (1.32) tells us that for each electronic state described by ψ _n (x), the electron will have an energy E _n associated with it. What does this mean? It means that for each electronic state that the electron can have, that state will be characterized with an energy E _n . What significance does this have? It has many, but for the purpose of the author's intentions it means that even though one does not know the exact location of the electron, it is possible to characterize its state via a quantity that is known with high accuracy, its energy!

Since the use of the electron energies is advantageous in describing electronic states, is common to use an energy level diagram as shown in figure 1.5(a). There are several uses for an energy level diagram. It contains information about the energies that the electron will possess when occupying a specific electronic state. In nature any system will configure itself in such a way as to minimize the total energy of the system. This means that an electron trapped in this 'box' will automatically be in the state that requires the least energy (unless energy is supplied) which is the energy corresponding to n = 1. This means that the electron in the lowest energy state is described by ψ ₁(x), since that state will allow the system (electron) to be in the state with the minimum possible energy. This is the reason that this state is referred to as the 'ground state' of the system.

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It is very common to use the energy level diagrams to describe electronic excitations. If energy is applied to the system (via photons, heat, etc), the electron in the ground state can be excited from one energy level to another, as long as the energy given to the electron is equal to the difference in energy levels involved in the excitation. Similarly, a decay or de-excitation between energy levels can happen and the energy released (usually as a photon) must be equal to the difference in energy between the two levels involved. This is shown in figure 1.5(b). As we will see later, a particular electronic transition from one energy level to another is not guaranteed with the available energy. In fact, transitions are governed by 'selection rules'. These rules determine which electronic transitions are allowed and which are not, given the energy available to the system.

On a final note, the particle in a box problem allows to one summarize in a concise and clear manner the following points and concepts.

• 1.  
  A quantum mechanical system can be solved using the Schrödinger equation. Its solution yields the wavefunction of the electron in a particular state. In such case, ${{\psi }_{n}}\left( x \right)\psi _{n}^{*}\left( x \right)$ gives the probability distribution of finding the electron within an enclosed volume.
• 2.  
  The form of the potential energy U( r ) that the electron is subject to influences the form of the solution to the Schrödinger equation. In other words, the electrostatic potential essentially dictates the behavior of the electron in the quantum mechanical system.
• 3.  
  The density plot of |ψ _n (x)|² will be referred to as an orbital. An orbital is the collection of points in space in which the electron is most likely to be found when in a specific state as described by |ψ _n (x)|².
• 4.  
  Knowledge of the wavefunction leads to the calculation of the energy that the electron will have if found in a particular state. The energy values are known with high accuracy and are commonly described using an energy diagram.
• 5.  
  The quantization (or quantum number) appears due to the boundary conditions of the specific problem. In this example, they appear from the requirements that the electron remains within the line of length L at all times.
• 6.  
  The electron is said to be delocalized as it can be anywhere within the enclosed volume. Its particular connection to a specific location is merely probabilistic in nature.
• 7.  
  The electron will be found in the configuration that minimizes the energy of the system. In doing so, transitions from a high energy level to a lower energy level are possible with the emission of a photon with an energy equal to the difference in energy of the energy levels involved in the transition.

While most of these points are important for the description of PES, it is highly recommended that the reader devote time to the concept of energy levels and their relation with their respective wavefunctions, as these will be vital in understanding the photoemission process. Let us proceed now to a more realistic system, that of the hydrogen atom.

In section 1.1.2 it was discussed how an electron behaves when constricted to move freely in a linear path (i.e. no external forces acting on it) of length L. Now let us consider a more realistic (and the most simple) real system, the hydrogen atom. In this scenario, the electron (which carries an inherent negative charge) is subject to an electrostatic force (unlike the previous case in which it was free) due to the influence of the positively charged proton located at the nucleus of the atom. A schematic of the physical problem is shown in figure 1.6.

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The Hamiltonian of the system is given by

$$\begin{eqnarray}&&{\hat{H}}\left( {\boldsymbol{r}} \right)=-\frac{{{\hbar }^{2}}}{2\mu }\nabla _{r}^{2}+{\rm U}\left( {\boldsymbol{r}} \right)\end{eqnarray} \tag{ 1.33 }$$

where μ is the reduced mass of the system as discussed in section 1.1.1. The first term represents the kinetic energy of the system (as represented by a particle with mass μ). The second term U( r ) represents the system potential energy as described by the Columbic interaction

$$\begin{eqnarray}&&{\rm U}\left( {\boldsymbol{r}} \right)=-\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{{{e}^{2}}}{{\boldsymbol{r}} }\end{eqnarray} \tag{ 1.34 }$$

where e is the magnitude of the electron charge, ε ₀ is the permittivity constant in a vacuum and r is the distance between the electron and the proton. There are two points that must be understood before proceeding with this problem. First, the Hamiltonian was simplified by choosing the center of mass reference frame (see section 1.1.1). Second, the potential function representing the energy due to electrostatic interaction exhibits spherical symmetry (the distance | r | is symmetric with respect to nucleus position). Due to the symmetry inherent in the potential energy function, it is convenient to adopt the time-independent Schrödinger equation in a spherical coordinate. This choice of coordinate will have clear advantages in solving the problem; it will reduce its dimensionality (the number of variables to deal with) and it can be solved using the separation of variable methods.

It is strongly recommended to revise (and understand) the summary points at the end of section 1.1.2, as those will be applied to gain information about the hydrogen atom by analogy, not extensive calculation. If one were to pursue a rigorous solution to this system, let us just point out that just the expansion of (1.32) alone will lead to

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} -\frac{{{\hbar }^{2}}}{2\mu }\frac{1}{{{r}^{2}}}\left[ \frac{\partial }{\partial r}\left( {{r}^{2}}\frac{\partial }{\partial r}\psi \left( r,\theta ,\phi \right) \right)+\frac{1}{{\rm sin} \theta }\frac{\partial }{\partial \theta }\left( {\rm sin} \theta \frac{\partial }{\partial \theta }\psi \left( r,\theta ,\phi \right) \right)+\frac{1}{{{{\rm sin} }^{2}}\;\theta }\frac{{{\partial }^{2}}}{\partial {{\phi }^{2}}}\psi \left( r,\theta ,\phi \right) \right] \\ \quad +V\left( r \right)\psi \left( r,\theta ,\phi \right)=E\psi \left( r,\theta ,\phi \right) \\\end{array}\end{eqnarray} \tag{ 1.35 }$$

and such a problem (as fun as it looks) is not essential for the purpose of this work. Instead, a practical view will be attempted here. Luckily, from the author's point of view, (1.35) should serve three purposes: to contemplate the complexity that can arise from such a fundamental problem; to provide yet another example in which the Schrödinger equation can be solve exactly (and at what price!); and to inculcate humble thoughts in the minds of those in need of it. Perhaps embracing the points specified at the end of section 1.1.2 to attempt an analogy is not so catastrophic after all. If the reader has the time, energy and desire to review a formal derivation, the work in [6] is suggested. Otherwise, (1.35) can be separated into three different differential equations as

$$\begin{eqnarray}&&\begin{array}{ccccccccccccccc} \frac{{{\partial }^{2}}R\left( r \right)}{\partial {{r}^{2}}}+\frac{2}{r}\frac{\partial R\left( r \right)}{\partial r}-\frac{\lambda }{{{r}^{2}}}R\left( r \right)=\frac{2\mu }{{{\hbar }^{2}}}\left[ U\left( r \right)-E \right] \\ \frac{{{\partial }^{2}}{\rm \ \Theta\ }\left( \theta \right)}{\partial {{\theta }^{2}}}+{\rm cot} \theta \frac{\partial {\rm \ \Theta\ }\left( \theta \right)}{\partial \theta }+\left( \lambda -\frac{{{m}^{2}}}{{{{\rm sin} }^{2}}\;\theta \lambda } \right){\rm \ \Theta\ }\left( \theta \right)=0 \\ \frac{{{\partial }^{2}}{\rm \ \Phi\ }\left( \phi \right)}{\partial {{\phi }^{2}}}+{\rm \ \Phi\ }\left( \phi \right){{m}^{2}}=0 \\\end{array}\end{eqnarray} \tag{ 1.36 }$$

by assuming a wavefunction of the form

$$\begin{eqnarray}&&\psi \left( r,\theta ,\phi \right)=R\left( r \right){\mkern 1mu} {\rm \ \Theta\ }\left( \theta \right){\mkern 1mu} {\rm \ \Phi\ }\left( \phi \right).\end{eqnarray} \tag{ 1.37 }$$

Note that m ² and λ are just constants chosen to separate the coordinates r, θ and . The solutions for (1.35) will yield the functional form of R(r), Θ(θ) and Φ(). Each differential equation is subject to a boundary condition that must be enforced for each coordinate in order to have a valid solution. In the particle in a box problem one requires the electron to be confined on a line of length L, which leads to the quantization of the wavenumber k _n . In the hydrogen atom, it is required that the electron must obey quantized angular momentum conservation. This requirement will lead to the quantum number

$$\begin{eqnarray}&&n=1,2,3,\ldots \end{eqnarray} \tag{ 1.38 }$$

The other requirement is that the wavefunction describing the electron around the atom must be continuous. In the polar coordinate this is achieved by ensuring that Θ(θ) converges in the limits θ = 0 and θ = π, which will lead to the azimuthal quantum number

$$\begin{eqnarray}&&l=0,1,2,\ldots ,n-1.\end{eqnarray} \tag{ 1.39 }$$

In the axial coordinate this is achieved via the requirement that Φ() = Φ( + 2π) which leads to the magnetic quantum number

$$\begin{eqnarray}&&{{m}_{l}}=0,\pm 1,\pm 2,\ldots ,l.\end{eqnarray} \tag{ 1.40 }$$

The imposition of these conditions will lead to a solution of (1.34) of the form

$$\begin{eqnarray}&&{{\psi }_{n,l,{{m}_{l}}}}\left( r,\theta ,\varphi \right)={{R}_{n,l}}\left( r \right){{{\rm \ \Theta\ }}_{l,{{m}_{l}}}}\left( \theta \right){{{\rm \ \Phi\ }}_{{{m}_{l}}}}\left( \varphi \right)\end{eqnarray} \tag{ 1.41 }$$

where R _n,l controls the radial features of the wavefunction, and ${{{\rm \ \Theta\ }}_{l,{{m}_{l}}}}$ and ${{{\rm \ \Phi\ }}_{{{m}_{l}}}}$ the angular distribution. It is common to refer to the combination of ${{{\rm \ \Theta\ }}_{l,{{m}_{l}}}}$ and ${{{\rm \ \Phi\ }}_{{{m}_{l}}}}$ as spherical harmonics. Figure 1.7 shows a schematic for the probability distribution of the electron in the hydrogen atom occupying different states as well as some of the functional forms for the wavefunction of different electronics states.

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Since an orbital is analogous to the density plot of $\mid {{\psi }_{n,l,{{m}_{l}}}}{{\mid }^{2}}$, it is common to employ an orbital notation to refer to the electronic states. These notations are summarized in table 1.1.

Table 1.1.  The orbital notation used to describe electronic states.

Orbital notation	Quantum number ${\boldsymbol{n}}$
s	0
p	1
d	2
f	3
and so on alphabetically

It is also customary to label orbitals in groups. Orbitals that share the same quantum number n are known as electronic shells. The orbital notation is summarized in table 1.2. There is another subgroup called a subshell. A subshell is the group of orbitals with the same quantum number n but different values of the azimuthal quantum number l. These are summarized in figure 1.8.

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In analogy to the particle in a box problem, it was proved that knowledge of the wavefunction leads to the calculation of the energy values corresponding to an electron in a state described by ${{\psi }_{n,l,{{m}_{l}}}}\left( r,\theta ,\varphi \right)$. For the hydrogen atom, these energies ⁸ are given by

$$\begin{eqnarray}&&{{E}_{n}}=-\frac{13.6\;{\rm eV}}{{{n}^{2}}}.\end{eqnarray} \tag{ 1.42 }$$

The energy levels are summarized in the energy diagram of figure 1.9(a). Unlike the case in the particle in a box problem, the energies describing the electronic states in the hydrogen atom are negative. The electronic states with such energies are referred to as 'bound' states (i.e. the electron is bounded to a localized region in space due to the electric potential it is subject to) and are all discrete (see figure 1.9(b)). Energies above zero exist and are referred to as the continuum spectrum. These energies are interpreted as the kinetic energy of the electron as the electron can move freely once removed from the atom. The minimum energy required to remove an electron from an atom is known as the ionization energy. For an electron in the ground state of the hydrogen atom this energy is 13.6 eV.

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In contrast to the particle in a box problem, for the hydrogen atom some transitions between energy levels are highly improbable (or 'forbidden'). It is possible to determine which transitions are not allowed using the selection rules. The selection rules is a set of rules that provide information about the electronic transitions that are allowed in the system. For hydrogenic atoms (atoms that only have one valence electron) these are:

$$\begin{eqnarray}&&\ \ {\rm \Delta }J=0,\pm 1\end{eqnarray} \tag{ 1.43 }$$

$$\begin{eqnarray}&&{\rm \Delta }{{m}_{J}}=0,\pm 1\end{eqnarray} \tag{ 1.44 }$$

where a transition with $j=0$ to another with $j=0$ is excluded. Here J refers to the total angular momentum quantum number and m_J is the secondary total angular momentum quantum number. The value of J dependa on the orbital the electron is occupying (or the state in which the electron is found) and the electron spin. A summary of the selection rules is shown in table 1.3. The reader must keep in mind that these selection rules may vary depending on which technique is employed to observed the electronic transition. In the case of photoemission, it is customary to refer to the 'photoemission selection rules' which provide information as to which transitions are allowed and not allowed based on the interactions on the experimental set up (i.e. beam polarity, geometry of the system, etc).

Table 1.3.  The selection rules for the electronic transitions in hydrogenic atoms.

Electric dipole (allowed)	Magnetic dipole (forbidden)	Electric quadrupole (forbidden)
Δ J = 0, ±1	ΔJ = 0, ±1	ΔJ = 0, ±1, ±2
(0 ↔ 0)	(0 ↔ 0)	$\left( 0\leftrightarrow 0,\;\frac{1}{2}\leftrightarrow \frac{1}{2},\;0\leftrightarrow 1 \right)$
Δ M = 0, ±1	ΔM = 0, ±1	ΔM = 0, ±1, ±2
Parity exchange	No parity exchange	No parity exchange
One electron jump	No electron jump	One or no electron jump
Δ l = ±1	Δl = 0	Δl = 0, ±2
For L–S coupling	Δn = 0
Δ S = 0	ΔS = 0	ΔS = 0
Δ L = 0, ±1	ΔL = 0	ΔL = 0, ±1, ±2
(0 ↔ 0)		(0 ↔ 0, 0 ↔ 1)

In summary, an analogy with the particle in a box problem was established to show the descriptive power of the Schrödinger equation in a real quantum mechanical system, such as the hydrogen atom. As previously suggested, it is strongly recommended that the reader comprehend the following points.

• 1.  
  Electronic states that the electron can occupy with negative energy are referred to as bound states. These are states that are bounded to the atom (highly localized in a region near the atom) due to the electrostatic attraction to the nucleus of the atom.
• 2.  
  If an electron in an orbital gains enough energy, it can be removed from the atom. The energy needed for that to occur is called the ionization energy.
• 3.  
  To describe the region where the electrons are likely to be found (orbitals), a set of notations was introduced.
• 4.  
  Unlike the particle in a box case, some of the transitions between energy levels are forbidden in the hydrogen atom. The outcomes of an atomic transition between energy levels are dictated by the selection rules (the photoemission process also has selection rules of its own that depend entirely on the symmetry of the specimen as well as the geometry of the experimental set up).

This concludes the discussion of the electron behavior in the hydrogen atom. However, in practice, one mostly deals with materials that are made out of two or more atoms (molecules, compounds, etc). For those materials to be formed there must be a force that keeps them together. It is the purpose of section 1.1.4 to provide a basic understanding of how that is possible. Along this line of thought, an interesting problem is what happens to the electrons in the hydrogen atoms if another hydrogen atom is placed in the vicinity of the primary atom. This is the topic to be discussed next.

Let us extend our discussion on quantum mechanical systems to a molecule with the purpose of studying the effects of bonding. To begin, consider the H₂ cation, which provides the simplest discussion for a molecule. This system consists of two hydrogen nuclei separated by a distance $R$ and one electron at a distance ${{r}_{{\rm A}}}$ from atom A, ${{r}_{{\rm B}}}$ from atom B and r from the center of mass reference frame O as shown in figure 1.10. The origin is conveniently placed at the midpoint of the inter-nuclear position vector to facilitate the mathematical formulation ⁹ .

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The Hamiltonian that correspond to this system has the form

$$\begin{eqnarray}&&{\hat{H}}\left( {\boldsymbol{r}} ,{\boldsymbol{R}} \right)=-\frac{{{\hbar }^{2}}}{2{{\mu }_{pp}}}\nabla _{R}^{2}-\frac{{{\hbar }^{2}}}{2{{m}_{e}}}\nabla _{r}^{2}+\frac{{{e}^{2}}}{4\pi {{\varepsilon }_{0}}}\left( \frac{1}{{\boldsymbol{R}} }-\frac{1}{{{{\boldsymbol{r}} }_{{\rm A}}}}-\frac{1}{{{{\boldsymbol{r}} }_{{\rm B}}}} \right)\end{eqnarray} \tag{ 1.45 }$$

where ${{\mu }_{pp}}$ is the reduced mass between the two nuclei of the system. The potential energy term arises from the electrostatic interaction between the nuclei and electron and between the nuclei themselves. Since the nuclei and the electron are subject to the same force magnitude, the motion of the nuclei is negligible as they are much more massive than the electron. This assumption allows us to neglect the first term of the Hamiltonian, reducing it to the electronic Hamiltonian (i.e. the Hamiltonian that describes the electron behavior in the molecule). Unlike the other cases seen previously, a Hamiltonian with such a functional form needs to be solved using an approximation method [7]. Since our interest lies in understanding how the electron energy will be affected by bringing another atom to its vicinity, one can use the variational method ¹⁰ . This technique will provide information on the ground state of the electron with enough accuracy to deliver a close description of the electron behavior in this configuration. The mathematical formulation of the method consists of calculating

$$\begin{eqnarray}&&E\approx \frac{\int {{{\rm \Psi}}^{*}}\left( {\boldsymbol{r}} \right){{H}_{e}}{\rm \Psi}\left( {\boldsymbol{r}} \right){\rm d}{\boldsymbol{r}} }{\int \mid {\rm \Psi}\left( {\boldsymbol{r}} \right){{\mid }^{2}}{\rm d}{\boldsymbol{r}} }\end{eqnarray} \tag{ 1.46 }$$

where ${{H}_{e}}$ is the electronic Hamiltonian and ${\rm \Psi}\left( {\boldsymbol{r}} \right)$ is a trial wavefunction describing the system. The idea behind the method is to construct a wavefunction that is as accurate as possible, and then use it to solve (1.45) to find the ground state energy of the electron. As an educated guess, one can choose the wavefunction that describes the electron ground state in the hydrogen atom, this is

$$\begin{eqnarray}&&{{\psi }_{1,0,0}}\left( {\boldsymbol{r}} \right)=\psi \left( {\boldsymbol{r}} \right)=\frac{1}{\sqrt{\pi }}{{e}^{-\frac{{\boldsymbol{r}} }{{{a}_{0}}}}}\end{eqnarray} \tag{ 1.47 }$$

where ${{a}_{0}}$ is the Bohr radius and r the distance between the electron and the nucleus. While one could claim that this choice poorly represents the system at close internuclear distances R (which it does), at large enough R, the electron is expected to resemble the same behavior as in the hydrogen atom (it must be attached to one of the nuclei), establishing the legitimacy of the chosen wavefunction. In any case, one must keep in mind that this is just an approximation ¹¹ . The main issue inherent in the use of (1.46) is that the wavefunction does not fulfill the symmetry requirements for the center of mass frame of reference. This is because such a wavefunction assumes prior knowledge as to which nucleus the electron is attached to and, due to the quantum mechanical nature of the problem, that information is not known. In reality, there is no way to distinguish whether an electron is attached to one nucleus or the other, furthermore, it is not possible to identify which hydrogen atom the electron was attached to initially, before the cation was formed. All that quantum mechanics allows us to do is to describe the probability of that event happening. Nonetheless, all hope is not lost; while (1.46) is not a solution for this particular system, it is possible to construct a wavefunction that obeys the required symmetry by using a linear combination of (1.46). In such a case the total wavefunction can be written as

$$\begin{eqnarray}&&{{{\rm \Psi}}_{{\rm even}}}\left( {\boldsymbol{r}} \right)=\frac{1}{\sqrt{2}}\left[ {{\psi }_{{\rm A}}}\left( {{{\boldsymbol{r}} }_{{\rm A}}} \right)+{{\psi }_{{\rm A}}}\left( {{{\boldsymbol{r}} }_{{\rm B}}} \right) \right]\end{eqnarray} \tag{ 1.48 }$$

$$\begin{eqnarray}&&{{{\rm \Psi}}_{{\rm odd}}}\left( {\boldsymbol{r}} \right)=\frac{1}{\sqrt{2}}\left[ {{\psi }_{{\rm A}}}\left( {{{\boldsymbol{r}} }_{{\rm A}}} \right)-{{\psi }_{{\rm B}}}\left( {{{\boldsymbol{r}} }_{{\rm B}}} \right) \right]\end{eqnarray} \tag{ 1.49 }$$

where both equations are normalized and the even and odd subscripts denote the symmetry inherent in the functional form of the wavefunction. Note that both wavefunctions satisfy the Schrödinger equation, meaning that the electron can be found in either of the two states. Let us then calculate the energies corresponding to these states by placing (1.47) and (1.48) into (1.45). One obtains

$$\begin{eqnarray}&&E(R)\approx \frac{{{E}_{1}}\left( 1+I\left( R \right) \right)\pm \frac{{{e}^{2}}}{4\pi {{\varepsilon }_{0}}}\frac{1}{\pi }\left[ \frac{1}{R}\left( 1+I\left( R \right) \right)-\;J\left( R \right)-X\left( R \right) \right]}{1\pm J\left( R \right)}\end{eqnarray} \tag{ 1.50 }$$

where the plus sign corresponds to the energy of the electron if found in a state described by ${{{\rm \Psi}}_{{\rm even}}}\left( {\boldsymbol{r}} \right)$ and the minus sign to ${{{\rm \Psi}}_{{\rm odd}}}\left( {\boldsymbol{r}} \right)$. As (1.50) dictates, the energy levels depend on the interatomic distance R. Let us plot the energy of the electron as a function of the inter-nuclear distance R to reflect on its behavior.

The implications of (1.50) are remarkable. At a large interatomic distance R, the energy of the electron tends to that of the hydrogen atom. While this is not surprising, it suggests that our trial wavefunction at least predicts the correct energies at large R. As the nuclei are brought together, the energy levels split in two, one corresponding to an electron described by an even wavefunction and the other by an odd wavefunction. Of particular importance, the energy of the electron described by ${{{\rm \Psi}}_{{\rm even}}}\left( {\boldsymbol{r}} \right)$ has a minimum energy positioned at 2.2 a ₀, suggesting a stable configuration. The energy for an electron described by ${{{\rm \Psi}}_{{\rm odd}}}\left( {\boldsymbol{r}} \right)$ has no energy minimum at any R. As nature dictates, a quantum mechanical system will always tend to rearrange itself in a configuration such as to minimize the energy. This means that the electron in this configuration will establish at a distance of 2.2 a ₀, as shown in figure 1.11.

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Let us inspect the wavefunction describing the possible electronic states of the electron in the H₂ cation to gain information about the system. Figure 1.12 shows a one-dimensional plot of the probability distribution of the electron as a function of spatial coordinates. Notice that for $\mid {{{\rm \Psi}}_{{\rm even}}}\left( {\boldsymbol{r}} \right){{\mid }^{2}}$ the probability distribution between the nuclei is enhanced, suggesting that it is highly probable to find the electron within that region. Since a molecule configuration as such is favorable in terms of energy (requiring the least energy), one can call this state the bonding state and the density plot of it the bonding orbital. For $\mid {{{\rm \Psi}}_{{\rm odd}}}\left( {\boldsymbol{r}} \right){{\mid }^{2}}$, the probability distribution decreases within the nuclei, while it increases away from the nuclei. Such a configuration will result in an unstable molecule as the electrons will not bond. This is clearly shown in the density plot.

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In the case of the hydrogen atom, an electron in the ground state was described with a single energy level in terms of energy. In the case of the hydrogen cation, the ground state energy of the electron is split into two distinct energy levels, one corresponding to a bonding configuration and the other corresponding to an antibonding configuration as shown in figure 1.13. Since the bonding orbital has the lowest energy, the electron will be found in that state.

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Let us summarize. When two hydrogen atoms come into close contact and form a cation molecule:

• 1.  
  The ground state electron can be described by a symmetric and an antisymmetric wavefunction. This lead to the formation of a bonding orbital, which is a configuration in which the electron is shared between the nuclei and an antibonding orbital, which is an unstable configuration.
• 2.  
  As the nuclei come together, the ground state energy level of the electron will split in two. The level with the lowest energy corresponds to the bonding orbital, while the level with the highest energy corresponds to the antibonding orbital.

While most molecules are more complex than the example provided in this section, the main ideas are the same. There will always be a bonding orbital which the electron can occupy without the need to supply energy and an antibonding orbital, which the electron can occupy if energy is invested.

In terms of photoemission, it is often important to separate even and odd wavefunctions, especially with polarized light as the geometry of the photoemission will result in different photoemission cross sections (symmetry selection rules) for even and odd states. But these issues are for readers who are interested in valence band photoemission mechanically the same as x-ray photoemission spectroscopy but with a light source with energy in the ultraviolet range), here, core level photoemission is the emphasis. Let us now describe what happens to an electron when placed in a periodic arrangement of atoms, which is the most pertinent example for the description of the photoemission process.

A crystalline solid is a material composed of a periodic arrangement of atoms. The calculation of a wavefunction in a solid is very complex and even the simplest model (the free electron model) would require the introduction of concepts not discussed in this work. Therefore, let us emphasize the properties of the electrons' energy levels inside a solid as this is the information pertinent to the photoemission process. To this purpose, one can assume that the solid material behaves like a giant molecule. Qualitatively speaking, this sharing of electrons can be pictured as in the di-hydrogen cation case. For instance, if two atoms each share one electron, the energy levels will look like those in figure 1.14 (left panel). Since the Pauli principle forbids two electrons from occupying the same state, one electron will have a spin up and the other a spin down. If another atom comes close, one might argue that the energies should be the same, however, once again, the Pauli principle forbids two electrons from occupying the same state, hence the energy will be slightly different, see figure 1.14 (middle panel). As the number of atoms sharing electrons increases, so do the energy levels. Even though the separation of these energy levels is discrete, there are so many atoms in a solid (~10²³), that the energy levels can be thought of as a continuum of energy as shown in figure 1.14 (right panel). That energy continuum is called an energy band.

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As in the case of a molecule, the atoms in the crystal will interact with each other until their equilibrium position is reached at a distance a ₀. If one maps the line at point a ₀ in terms of energy, the energy diagram for a solid can be obtained as shown in figure 1.15.

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In this type of schematic, the wavevector dependence (momentum dependence of the bound state) can be ignored. Notice that the electronic band structure of a solid is composed of two continuous sets of bands which may be divided by a forbidden gap, depending on the material's electrical properties. The bottom band is called the valence band and represents all the states that are filled by electrons. The top band is called the conduction band and represents the states that electrons can occupy if external energy is supplied. The gap is known as the electronic band gap and represents a range of energies where there are no allowed states. It is this region of forbidden states that is used to provide us with a more phenomenological distinction between metals and insulators.

In terms of the classical model, the electric properties of a solid material can be classified into three categories: conductor, semiconductor and insulator (figure 1.16). Metals (conductors) are materials with the ability to conduct vast quantities of electrons (around 10⁴–10⁵ S cm⁻¹). On the opposite side are insulators, which are materials in which the conduction of electrons is very small and usually negligible, ranging around 10⁻² S cm⁻¹ or less. Between these two lie semiconducting materials, which partly conduct current (10⁻²–10⁴ S cm⁻¹).

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The classification of the electrical properties of materials using the classical model is rather cumbersome. The band structures (or energy model) allow for a clearer distinction of the material's electric properties. In terms of their electronic band structure, the electrical properties of a solid material can be classified within the same three categories by means of their band gap. Under this model a metal will be defined as a material with no energy band gap (i.e. the conduction and valence band overlap). Due to the continuity of the bands, electrons from the valence band can move to the conduction band with little energy cost, which results in large numbers of carriers that can move freely through the material. In the cases of an insulator and a semiconductor, the distinction becomes a bit more complicated. An insulator is a material with a wide electronic band gap and a semiconductor a material with a smaller band gap, but what is wider or smaller? From a standard textbook point of view, it is customary to assign a band gap of 2–3 eV or smaller to semiconductors and 3 eV or greater to insulators. However, one must keep in mind that this distinction may not be adequate for all cases, given the fact that this definition applies to a material that is entirely periodic with no defects of any kind. In reality, an insulator can have enough free carriers to behave as a semiconductor and sometimes even a metal [8]. This is usually due to defects introduced in the material lattice structure that can provide free carriers at low energy cost.

It is customary (and necessary) to define a reference energy level to describe the electronic band structure. For this purpose the Fermi level is defined. The Fermi level is the highest electronic occupied orbital at 0 K. To define this energy, it is necessary to cool down the solid material to 0 K. This will 'freeze' all the electrons in the solid material and allows their classification by energy. Due to the Pauli exclusion principle, the electrons will occupy the states with the lowest energy possible. Hence, the last electron or the electron in the highest occupied state will have energy equal to the Fermi energy. The Fermi level is then the energy reference from which any energy below this level corresponds to the energies electrons will have when in an occupied state. There is a slight problem with this definition as the Fermi level is defined only at a temperature of 0 K. To describe the system as a function of temperature one must use the chemical potential formalism. At room temperature the existence of a metallic Fermi level is even more complicated as thermal excitations are involved and not all electronic states will be occupied. However, for the purposes of the photoemission process, one can approximate the Fermi level to be equal to the chemical potential at room temperature. For semiconductors and insulators, assigning the Fermi level is more complex and its definition is much more ambiguous. Therefore, caution must be taken. This will be discussed in more detail in chapter 2.

For the sake of completeness, the reader must keep in mind that the band structure model discussed above is far too simplistic. A more realistic band structure is shown in figure 1.17 in which the wavevector dependence is taken into account.

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At this point it is sufficient for the reader to realize that in this theoretical band structure the calculation of a semiconductor still has an electronic band gap and all electronic states below the Fermi level are occupied by electrons and all states above the Fermi level are unoccupied. However, do not be deceived by the simplicity of the previous argument, as there are very important details inherent in the calculation process and differences between the calculation and experimental results that have not been mentioned yet.