Valence Band

Table of Contents

Introduction

The valence band is one of the two energy bands that together govern how a solid responds to an electric field. Along with the conduction band and the gap that separates them, it determines whether a material behaves as a metal, a semiconductor, or an insulator, and it underlies properties ranging from chemical bonding to optical absorption. This post explains what the valence band is, why it exists, and how its occupation controls electrical conduction through the motion of both electrons and holes.

Why Energy Bands Form in Solids

In an isolated atom, electrons occupy discrete energy levels. When atoms are brought together to form a solid, the Pauli exclusion principle forbids any two electrons from sharing the same quantum state, so each atomic level must split into many slightly different levels — one for every atom that contributes. A macroscopic crystal contains on the order of $10^{23}$ atoms, so each original level splits into an enormous number of levels packed into a narrow energy range. The result is a quasi-continuous band of allowed energies, separated from neighbouring bands by ranges of forbidden energy.

The valence band arises from the outer-shell (valence) orbitals — the same electrons that participate in chemical bonding. The next band up, formed from higher atomic states, is the conduction band.

The Valence Band

The valence band is the highest range of electron energies that is fully occupied at absolute zero. Its electrons are the ones holding the atoms together, so a completely filled valence band carries no net current: for every electron moving one way there is another moving the opposite way, and the contributions cancel. We denote the energy at the top of the valence band by $E_v$ and the energy at the bottom of the conduction band by $E_c$ . The forbidden region between them is the band gap,

$E_g = E_c - E_v$

The size of this gap, together with the position of the highest occupied states, is what distinguishes the three classes of material. In a metal, the highest occupied band is only partially filled (or the valence and conduction bands overlap), so electrons sit right at the top of the occupied states and can move freely under an applied field. In an insulator, the valence band is completely full and the band gap is large (for diamond, about $5.5\ \text{eV}$ ), so electrons cannot easily reach the empty conduction band. A semiconductor has the same full-valence-band structure as an insulator but a much smaller gap (around $0.5$ to $2\ \text{eV}$ ), small enough that thermal energy can lift a useful number of electrons across it.

Holes: Conduction Within the Valence Band

When an electron is excited out of the valence band, it leaves behind an empty state. The remaining electrons in the band can now shift to occupy it, and the most economical way to describe this collective motion is to track the empty state itself, which moves like a particle carrying positive charge $+e$ . This quasiparticle is called a hole. Current in the valence band is therefore carried by holes, while current in the conduction band is carried by electrons; in a semiconductor both contribute. Introducing holes is what lets us treat an almost-full band with the same simple language we use for an almost-empty one.

Fermi Level and Valence Band

The probability that a state of energy $E$ is occupied by an electron is given by the Fermi-Dirac distribution,

$f(E) = \dfrac{1}{1 + e^{(E - E_F)/k_B T}}$

The Fermi level $E_F$ is the energy at which this occupation probability equals exactly $1/2$ . At $T = 0$ the distribution becomes a sharp step — every state below $E_F$ is filled and every state above it is empty — so $E_F$ marks the boundary between occupied and unoccupied states. In an intrinsic (undoped) semiconductor or an insulator, $E_F$ lies within the band gap, near its middle; in a metal it lies inside a partially filled band, which is precisely why metals have electrons available to conduct.

Temperature, Carriers, and the Band Gap

As temperature rises, the step in the Fermi-Dirac distribution softens and some electrons are thermally excited from the valence band into the conduction band, leaving holes behind. In a non-degenerate semiconductor the concentration of holes in the valence band is

$p = N_v\, e^{-(E_F - E_v)/k_B T}, \qquad N_v = 2\left(\dfrac{2\pi m_h^* k_B T}{h^2}\right)^{3/2}$

where $N_v$ is the effective density of states at the valence-band edge, $m_h^*$ is the effective mass of a hole, $k_B$ is the Boltzmann constant, $h$ is the Planck constant, and $T$ is the absolute temperature. Combining the electron and hole populations gives the intrinsic carrier concentration,

$n_i = \sqrt{N_c N_v}\; e^{-E_g/2k_B T}$

The exponential dependence on $E_g$ is the heart of the matter. For silicon, $E_g \approx 1.12\ \text{eV}$ , while at room temperature $k_B T \approx 0.0259\ \text{eV}$ , so the exponent $E_g / 2k_B T \approx 21.6$ suppresses the carrier concentration by a factor of about $10^{-10}$ , leaving $n_i$ on the order of $10^{10}\ \text{cm}^{-3}$ . For diamond, with its much larger gap, the same factor is so tiny that essentially no carriers are thermally generated — which is exactly why one material is a useful semiconductor and the other an insulator.

Conclusion

The valence band is the highest band of occupied electron states, and its behaviour is best understood through the holes left behind when electrons are excited across the band gap. The energy difference between the valence and conduction bands governs whether a material conducts freely, conducts a little, or barely conducts at all, and the exponential sensitivity of carrier concentration to that gap is what makes band engineering so powerful. Controlling the valence band — through composition, temperature, and doping — is the foundation of modern electronic and optoelectronic devices.

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