Quantum Physics Foundations: From Energy Bands to Semiconductors

From Discrete Atomic Levels to Energy Bands in Solids

In an isolated atom, electrons occupy discrete energy levels. In a solid crystal, you have an enormous number of atoms packed close together in a regular pattern. When atoms approach, their electron wavefunctions overlap. Because electrons are quantum objects and must fit into allowed states that respect the crystal’s periodic structure and the exclusion principle, the “same” atomic level from each atom cannot remain exactly identical in energy. Instead, it splits into many extremely closely spaced levels.

If you imagine N identical atoms forming a crystal, a single atomic level typically becomes roughly N distinct allowed energies. For macroscopic N, these energies are so densely packed that they form what we treat as a continuous energy band. Between bands there can be energy ranges with no allowed electron states (for that crystal and momentum): these are forbidden gaps.

Why “allowed” and “forbidden” appear

In a periodic crystal, electron states must be compatible with the repeating potential of the lattice. Only certain wave patterns “fit” consistently across the repeating structure. Other energies would require wave patterns that destructively conflict with the periodicity, producing no stable solutions. The result is a spectrum that naturally organizes into:

• Allowed bands: energies where electron states exist and can be occupied.
• Band gaps: energies where no electron states exist in the crystal.

A useful mental picture: discrete atomic lines broaden into thick “stripes” (bands) as atoms interact; the blank spaces between stripes are gaps.

Valence Band, Conduction Band, and Band Gap (Physical Meaning)

Valence band

The valence band is the highest-energy band that is (at low temperature) mostly filled with electrons. Electrons here are typically involved in bonding. Because most states are occupied, electrons in a completely full band cannot easily change their state in response to an electric field (there are no nearby empty states to move into), so a full valence band by itself does not conduct well.

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Conduction band

The conduction band is the next higher band above the valence band. If electrons occupy states in this band, they can respond to an electric field by shifting into nearby empty states within the same band, producing electrical current. A partially filled conduction band supports conduction efficiently.

Band gap

The band gap E_g is the energy difference between the top of the valence band and the bottom of the conduction band. Physically, it is the minimum energy required to promote an electron from a bound/bonding-like state (valence) into a mobile state (conduction), leaving behind an empty state in the valence band.

Carriers: Electrons and Holes

In band language, electrical conduction is described in terms of carriers:

• Conduction electrons: electrons occupying the conduction band.
• Holes: empty states in the valence band that behave like positively charged mobile carriers.

What a hole really is

A hole is not a particle separate from electrons; it is a convenient description of the motion of missing electrons in an otherwise nearly full valence band. When an electron in the valence band moves to fill a nearby empty state, the “emptiness” effectively moves in the opposite direction. This collective behavior responds to electric fields as if a positive charge were moving.

How temperature changes carrier populations

At nonzero temperature, some electrons gain enough energy to cross the band gap into the conduction band. Each promoted electron creates:

• one conduction electron (in the conduction band), and
• one hole (left behind in the valence band).

As temperature increases, more electrons are thermally excited across the gap, increasing both electron and hole populations and thus increasing conductivity (especially in semiconductors).

Doping: controlling carriers on purpose

Doping means adding a small concentration of impurity atoms to a semiconductor to dramatically change carrier populations.

n-type doping (donors)

Donor impurities contribute extra electrons that are easily promoted into the conduction band. The result is:

• many conduction electrons (majority carriers),
• few holes (minority carriers).

p-type doping (acceptors)

Acceptor impurities create easily formed holes in the valence band (they “accept” electrons from the valence band). The result is:

• many holes (majority carriers),
• few conduction electrons (minority carriers).

Step-by-step: reasoning about carriers in a doped semiconductor

1. Identify the band gap: is it small enough that thermal excitation matters at the operating temperature?
2. Identify dopant type: donors → n-type; acceptors → p-type.
3. Determine majority carriers: n-type → electrons; p-type → holes.
4. Predict temperature trend: higher temperature increases intrinsic electron–hole pairs; at sufficiently high temperature, intrinsic carriers can dominate over dopant-provided carriers.

Band Structure and Conductivity: Metals vs Insulators vs Semiconductors

The key question for conductivity is: Are there available empty states very close in energy that carriers can move into under an electric field? Band structure answers this.

Metals

• Either the conduction band is partially filled, or the valence and conduction bands overlap.
• There are many nearby empty states available.
• Result: high conductivity even at low temperature.

Insulators

• Valence band is full and conduction band is empty.
• Band gap is large, so thermal energy at ordinary temperatures cannot promote many electrons across.
• Result: extremely low conductivity.

Semiconductors

• Valence band is (mostly) full and conduction band is (mostly) empty at low temperature.
• Band gap is moderate, so thermal excitation creates a useful number of electron–hole pairs.
• Doping can strongly increase carriers without changing the crystal structure much.
• Result: conductivity is tunable over many orders of magnitude.

Device Intuition: The p–n Junction

A p–n junction is formed by joining p-type and n-type regions in one crystal. It is the core of diodes and many other devices. The essential physics comes from carrier diffusion and the electric field that builds up in response.

Before contact: separate carrier imbalances

• n-side: many electrons, few holes.
• p-side: many holes, few electrons.

This imbalance sets up a strong tendency for carriers to spread out (diffuse) when the regions touch.

Step-by-step: how the depletion region forms

1. Initial diffusion: electrons diffuse from n → p (where electrons are scarce); holes diffuse from p → n (where holes are scarce).
2. Recombination near the interface: diffusing electrons meet holes and recombine, removing mobile carriers near the junction.
3. Uncovered fixed charges: when mobile carriers leave, they expose ionized dopant atoms that are fixed in the lattice: positive donor ions on the n-side near the junction, negative acceptor ions on the p-side near the junction.
4. Electric field builds: these fixed charges create an electric field pointing from the positive region toward the negative region.
5. Built-in potential: the electric field corresponds to a potential difference that opposes further diffusion.
6. Equilibrium: diffusion (trying to mix carriers) is balanced by drift in the electric field (pushing carriers back). The region depleted of mobile carriers is the depletion region.

Built-in potential (physical meaning)

The built-in potential is not an externally applied voltage; it is an internal energy barrier created by charge separation in the depletion region. It makes it energetically unfavorable for majority carriers to cross the junction at equilibrium.

Why a diode rectifies (conducts one way)

Rectification comes from how an external voltage changes the depletion barrier.

Forward bias (p-side higher potential than n-side)

• The external voltage reduces the built-in barrier.
• The depletion region narrows.
• Majority carriers can cross more easily: electrons injected into p-side and holes injected into n-side.
• Result: large current flows (after a characteristic turn-on related to the barrier).

Reverse bias (p-side lower potential than n-side)

• The external voltage increases the effective barrier.
• The depletion region widens.
• Majority carriers are pulled away from the junction.
• Result: only a small leakage current from minority carriers flows (until breakdown at high reverse voltage).

Band-diagram intuition (no math required)

Think of the band edges as “terrain” for electrons. In forward bias, the terrain barrier at the junction is lowered, so carriers can pass. In reverse bias, the barrier is raised, so carriers are blocked. The depletion region is the area where there are few mobile carriers available to carry current.

Concept Checks (Predict and Explain)

1) Band gap and electrical behavior

Question: Two materials are identical except that Material A has E_g = 0.2 eV and Material B has E_g = 2.0 eV. At the same temperature, which is more conductive if both are undoped, and why?

• Prediction: Material A is more conductive.
• Reasoning path: Smaller E_g means more electrons can be thermally excited into the conduction band, creating more electron–hole pairs and increasing conductivity.

2) Band gap and optical absorption threshold

Question: A photon has energy E = 1.5 eV. Will it be absorbed to create an electron–hole pair in a semiconductor with E_g = 1.1 eV? What about one with E_g = 2.2 eV?

• Prediction: It can be absorbed in the 1.1 eV material (enough energy to cross the gap), but not via band-to-band absorption in the 2.2 eV material (insufficient energy).
• Reasoning path: Band-to-band absorption requires photon energy at least equal to the band gap: E >= E_g.

3) Doping and majority carriers

Question: You dope a semiconductor so that it becomes n-type. What happens to (a) the number of conduction electrons, (b) the number of holes, and (c) the material’s conductivity?

• Prediction: (a) increases strongly, (b) becomes relatively small (minority), (c) increases strongly.
• Reasoning path: Donor dopants provide electrons that populate the conduction band with little thermal cost, creating many mobile carriers.

4) Band gap change: classify the material

Question: If a material’s band gap is gradually reduced while keeping the lattice otherwise similar, what qualitative transition do you expect in electrical behavior?

• Prediction: Large gap → insulator-like; moderate gap → semiconductor-like; very small/zero gap → metal-like (or semimetal-like).
• Reasoning path: As E_g shrinks, it becomes easier for carriers to exist at ordinary temperatures, and eventually bands overlap or become partially filled.