Technical Article

# Understanding Atomic Energy Levels in Solid-state Electronic Devices

May 11, 2021 by Lorenzo Mari

## Learn about the shell structure of atomic energy levels.

Understanding solid-state electronic devices requires a knowledge of the physical principles on which they operate. Such knowledge should allow the engineer to employ the many appliances that the future will bring effectively. The electron’s energy levels depend on the quantum numbers n and l – the energy and the angular momentum of the orbital motion – and the symbol nl identifies each electronic state.

## Stationary States, Ground State, and Binding Energy

Stationary states correspond to the allowed energies for electrons, and the ground state is the one with the lowest energy. In the ground state, each electron occupies its lowest energy orbit – this is a stable configuration.

The energy of the stationary states for one-electron atoms (hydrogen H, Z = 1; singly-ionized helium He+, Z = 2; doubly-ionized lithium Li2+, Z = 3) follows the expression

En=-constant x (Z2/n2)

where

n = 1,2,3,… denotes the state and Constant = 13.6 eV.

This equation, developed from the Bohr theory, serves as a good approximation in quantum theory.

Hypothetically, the number of levels that exist for each atom is infinite. Table 1 and figure 1 show some energy levels for one-electron atoms.

 State(n) H He+ Li2+ 1 -13.61 -54.42 -122.45 2 -3.40 -13.61 -30.61 3 -1.51 -6.05 -13.61 4 -0.85 -3.40 -7.65 5 -2.18 -4.90 6 -1.51 -3.40 7 -1.11 -2.50 8 -1.91 9 -1.51 10 -1.22
##### Figure 1. Some energy levels for one-electron atoms.

In figure 1, the energy’s reference level (E = 0) corresponds to n = ∞ and describes the situation when the distance between the electron and proton (nucleus) is infinite.

The ground state – lowest energy state – is n = 1. When the atom is in the lowest energy state, the electron moves around the proton in a circle of radius r1 – the Bohr radius.

The atom’s stationary states are also known as excited, radiating, critical, or resonance levels. The ground state is also called the normal level. The stationary states’ energy values are typically expressed in electron volts (eV) rather than in Joules.

As the electron receives increasing amounts of energy, it moves into stationary states farther and farther away from the nucleus. When the electron’s energy is enough to carry the electron out of the nucleus’s influence, it becomes detached from the atom. The binding energy (Eb) of an electron is the energy magnitude that, when absorbed by the electron, removes it from the atom altogether.

The atom deprived of one or more electrons becomes positively charged – an ion. The minimum energy required to ionize an atom in the ground state is E1 – the ionization energy. This value is 13.6 eV for hydrogen.

The binding energy depends on the kind of atom and the shell from which the electron is removed. Table 2 shows examples of approximate electron binding energies.

 Element Binding Energy (eV) H 13.6 C 11.26 Si 8.15 Ge 7.88 Li 5.39
##### Table 2. Electron binding energies.

The binding energy of the inner-filled shells is much higher than that of the outermost shell.

## The Angular Momentum of an Electron

The presence of stationary states of one-electron atoms requires an electron’s angular momentum with values L1, L2, L3,…, Ln,… – i.e., the angular momentum of the electronic motion is quantized.

In quantum mechanics, each energy level established by n is associated with some electron angular-momentum states determined by l = 0,1, 2, 3, 4,…, (n-1)   (n values).

The letters shown in table 3 designate these angular-momentum states.

 Angular Momentum (I) Letter Maximum Number of Electrons 0 s 2 1 p 6 2 d 10 3 f 14 4 g 18

## Electron Shells and Subshells

The different electron orbits are organized in groups called electron shells. The innermost is the K-shell, followed by the L-shell, then the M-shell, and continues farther and farther from the nucleus. These letters designate the states for which n = 1,2,3, . . . .

Each shell accommodates a limited number of electrons. In general, the nth shell can take up to 2n2 electrons. A shell filled is a closed shell – a very stable configuration.

It was initially thought that all electrons in a shell occupied the same energy level. But further investigation revealed that electrons in shells are split into several groups, forming subshells. As shown in table 3, the letters s,p,d,f,g, . . . designate the subshells for which l = 0,1,2,3,4, ….

For example, the 3p state has the quantum numbers n = 3 and l = 1; likewise, the 2s state has the quantum numbers n = 2 and l = 0.

Pauli’s exclusion principle establishes that no two electrons in an atom may have the same set of quantum numbers. A state nl can contain a maximum number of 2∙(2∙l+1) electrons (table 3) to comply with Pauli’s exclusion principle.

The electronic structure of complex atoms is a series of filled levels rising in energy. Electrons first occupy the lowest-lying energy states to the maximum number allowed, and the next electron goes into the next lowest-energy vacant subshell. If the atom were not in the lowest available energy state, it would radiate energy until reaching it.

Table 4 shows the electron distribution for a few selected chemical elements.

 Symbol Z Electron Shell K (n=1) L (n=2) M (n=3) N (n=4) 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f H 1 1 He 2 2 Li 3 2 1 Be 4 2 2 B 5 2 2 1 C 6 2 2 2 N 7 2 2 3 O 8 2 2 4 F 9 2 2 5 Ne 10 2 2 6 Na 11 2 2 6 1 Mg 12 2 2 6 2 Al 13 2 2 6 2 1 Si 14 2 2 6 2 2 P 15 2 2 6 2 3 S 16 2 2 6 2 4 Cl 17 2 2 6 2 5 A 18 2 2 6 2 6 Ni 28 2 2 6 2 6 8 2 Ge 32 2 2 6 2 6 10 2 2
##### Table 4. Distribution of electrons per nl state.

Figure 2 shows the shell structure of the atomic energy levels.

##### Figure 2. Shell structure of energy levels.

The separation between lines indicates the differences between energy levels. The energy differences in certain regions is considerably larger than others. These regions are energy gaps emerging between 1s and 2s, 2p and 3s, 3p and 4s, 4p and 5s, 5p and 6s, and 6p and 7s. A shell is a set of energy levels between two energy gaps. Each nl level composing a shell is a subshell.

It should be highlighted that there are some overlaps in the energy levels of successive subshells and shells. Because of these overlaps, a subshell of an upper shell can receive electrons before a lower shell completely fills up. This is due to the fact that electrons are most stable when they have the lowest energy. For example, nickel (Z=28) has its 4s subshell – lower energy state – filled before completing the ten electrons in the 3d subshell.

The electrons in the outermost shell are the valence electrons. The valence electrons determine the element’s chemical activity.

Figure 2 also shows those levels where the s and p subshells of the outermost shell are completely filled. These subshells close at the inert or noble gases – Z = 2,10,18,36,54, 86, (and 118).

As an example, we can see, comparing figure 2 and table 4, that Helium (Z = 2) has the subshell 1s closed (shell K), Neon (Z = 10) has the subshells 1s, 2s, and 2p closed (shells K and L), and Argon (Z = 18) have the subshells 1s, 2s, 2p, 3s, and 3p closed (shells K, L, and M). Subshells 2s and 2p close after the first energy gap, subshells 3s and 3p close after the second energy gap, and so on.

The inert gases are in a stable state and have no free electrons in their valence shell – the outer subshells s and p are closed. For this reason, their valence is zero, and their ability to react with other elements is meager, except under exceptional conditions.

## Electronic Notation

The energy state of an electron depends on n and l, and the symbol nl identifies each electronic state – i.e., 1s,2s,2p,3s,3p,3d,… nl˟ indicates x electrons in the nl state.

The notation s designates the number of electrons in each shell with the lowest energy levels. Consequently, 1s² indicates that two electrons are in the first shell’s (K) most down energy position. Similarly, 2s² specifies two electrons located in the second shell’s (L) lowest energy level. The maximum number of electrons in the s subshell is two, as shown in table 3.

The number of electrons in each nl state specifies the complete state of the atom – the configuration.

The ground state of helium (Z = 2) with two electrons in n = 1 has the configuration 1s2. Exciting one electron to the state 2s changes the configuration to 1s2s.

Neon (Z = 10), with its L-shell closed, has the configuration 1s²2s²2p⁶, showing two electrons in the first shell (K) and eight in the second (L) – two in the s-subshell and six in the p-subshell.

## The Tetravalent Atoms

The tetravalent atoms – they make four stable bonds to other atoms – are of substantial importance to the semiconductor industry. Table 4 shows that the simplest tetravalent atom is carbon, with six electrons and six protons (Z = 6). The two electrons nearest the nucleus close the K-shell because only two electrons can be accommodated at that energy level.  The outer level requires four additional electrons to fill the L-shell.

Silicon has fourteen electrons and fourteen protons (Z = 14). Like carbon, it has four valence electrons but has a filled shell (L) between the inner shell (K) and the valence shell (M). Silicon and carbon’s chemical properties are similar.

Another element in this group is germanium, with thirty-two protons and thirty-two electrons (Z = 32), including four valence electrons. Germanium has two filled shells (L and M) between the valence electrons and the inner shell (K).

In steady-state conditions, the electrons move around the nucleus in allowed stable orbits. Each orbit corresponds to a discrete energy level for the atom. Electrons can exist only in this set of discrete states with discrete energies.

The allowed energies for electrons are the stationary states. The configuration is stable when each electron occupies its lowest energy orbit – the ground state.

The binding energy is the energy needed to remove an electron from the atom. The binding energy of electrons filling the inner shells are much higher than that of the valence electrons.

The electron orbits are bunched in individual shells, each having a finite capacity. The states with the same n form a shell. The states with the same n and l form a subshell. A shell is closed when filled.

Pauli’s exclusion principle guides electrons’ assignment to the states in a multielectron atom – no two electrons can have the same set of quantum numbers.

The symbol nl identifies the energy state of an electron.

The tetravalent atoms make four bonds to other atoms.  Germanium, silicon, and gallium arsenide are tetravalent atoms widely used in semiconductor technology. 