Quantum Wells, Wires, and Dots

The Sommerfeld model treated a metal's electrons as free particles filling a continuum of $k$ states. That continuum is an approximation valid only when the crystal is large compared with the electron's de Broglie wavelength. Modern epitaxy grows semiconductor layers a few nanometers thick, comparable to the wavelength $λ_{F} = 2 π / k_{F}$ of a conduction electron. Once a sample's size in some direction drops to that scale, the wavevector along that direction is quantized into a discrete ladder, and the smooth bulk bands split into subbands. The number of directions left unconfined sets the effective dimensionality: a quantum well is two-dimensional, a quantum wire one-dimensional, and a quantum dot zero-dimensional. The density of states $g (E)$ — the count of single-electron levels per unit energy — takes a qualitatively different form in each, and every optical and transport property of a nanostructure follows from that shape.

The confinement energy scale

Confine a free electron of effective mass $m^{*}$ to a length $L$ in one direction with hard walls. The allowed wavevectors are $k_{n} = nπ / L$ , and the quantized energies are

E_{n} = \frac{ℏ ^{2} k _{n}^{2}}{2 m ^{*}} = \frac{ℏ ^{2} π ^{2}}{2 m ^{*} L ^{2}} n^{2}, n = 1, 2, 3, \dots

The spacing between the lowest levels is of order

Δ E \sim \frac{ℏ ^{2} π ^{2}}{2 m ^{*} L ^{2}} .

Confinement matters when this spacing exceeds the thermal energy $k_{B} T$ , so that electrons cannot be smeared across many levels. For GaAs, whose conduction-band effective mass is $m^{*} = 0.067 m_{e}$ , a well of width $L = 10 nm$ gives

Δ E = \frac{( 1.055 \times 1 0 ^{- 34} J\cdots ) ^{2} π ^{2}}{2 ( 0.067 ) ( 9.11 \times 1 0 ^{- 31} kg ) ( 1 0 ^{- 8} m ) ^{2}} = 9.0 \times 1 0^{- 21} J = 56 meV .

This is more than twice $k_{B} T = 25 meV$ at room temperature, so a 10-nm GaAs well is firmly in the quantum regime. The small effective mass is what makes semiconductor nanostructures accessible: the same calculation for a free electron ( $m^{*} = m_{e}$ ) gives only $3.8 meV$ , quantized only at cryogenic temperatures.

The quantum well: two dimensions

A quantum well is a thin layer of a narrow-gap semiconductor (GaAs) sandwiched between a wider-gap material (Al $_{x}$ Ga $_{1 - x}$ As). The conduction-band edge of the barrier sits above that of the well, so an electron in the well is trapped in a potential of depth equal to the conduction-band offset $Δ E_{c}$ , typically a few tenths of an eV. Motion along $z$ (the growth direction) is quantized into levels $E_{n}$ ; motion in the $x$ – $y$ plane is free. The total energy is

E = E_{n} + \frac{ℏ ^{2}}{2 m ^{*}} (k_{x}^{2} + k_{y}^{2}) .

A GaAs layer between AlGaAs barriers confines electrons in a square well of depth equal to the conduction-band offset; the transverse motion is quantized into subbands E1, E2, E3 whose wavefunctions are standing waves.

The density of states counts the plane-wave states in the $x$ – $y$ plane. In a sample of area $A = L_{x} L_{y}$ , periodic boundary conditions space the allowed $(k_{x}, k_{y})$ on a grid of cell area $(2 π)^{2} / A$ . Including a factor of two for spin, the number of states with wavevector below $k$ is

N (k) = 2 \frac{π k ^{2}}{( 2 π ) ^{2} / A} = \frac{A k ^{2}}{2 π} .

Writing $ε = ℏ^{2} k^{2} /2 m^{*}$ for the in-plane kinetic energy, so that $k^{2} = 2 m^{*} ε / ℏ^{2}$ , the areal density $n = N / A$ becomes $n = m^{*} ε / π ℏ^{2}$ , and its derivative is a constant:

g_{2D} (ε) = \frac{d n}{d ε} = \frac{m ^{*}}{π ℏ ^{2}} .

Each subband contributes this same constant once $E$ exceeds its threshold $E_{n}$ . Summing over subbands gives a staircase,

g_{2D} (E) = \frac{m ^{*}}{π ℏ ^{2}} n \sum Θ (E - E_{n}),

with $Θ$ the unit step. Every time $E$ crosses a new subband edge the density of states jumps by one riser of height $m^{*} / π ℏ^{2}$ .

The quantum wire: one dimension

Confining a second direction leaves only $x$ free. The transverse motion is now quantized in both $y$ and $z$ , so each subband is labeled by a pair $(n_{y}, n_{z})$ with threshold energy $E_{n_{y} n_{z}}$ . Along the wire the states form a one-dimensional gas. Counting states with $∣ k_{x} ∣ < k$ in a wire of length $L$ gives $N (k) = 2 \cdot (2 k) / (2 π / L) = 2 k L / π$ , so the linear density is $n = (2/ π) 2 m^{*} ε /ℏ$ and

g_{1D} (ε) = \frac{d n}{d ε} = \frac{1}{π} \frac{2 m ^{*}}{ℏ ^{2}} \frac{1}{ε} .

Measured from each subband edge, the density of states diverges as $(E - E_{n_{y} n_{z}})^{- 1/2}$ . These inverse-square-root spikes are the one-dimensional van Hove singularities; the wire's density of states is a row of decaying teeth, one at each subband threshold.

The quantum dot: zero dimensions

Confining all three directions removes every continuum. The energy spectrum is fully discrete,

E_{n_{x} n_{y} n_{z}} = \frac{ℏ ^{2} π ^{2}}{2 m ^{*}} (\frac{n _{x}^{2}}{L _{x}^{2}} + \frac{n _{y}^{2}}{L _{y}^{2}} + \frac{n _{z}^{2}}{L _{z}^{2}}),

and the density of states collapses to a set of delta functions,

g_{0D} (E) = 2 n_{x} n_{y} n_{z} \sum δ (E - E_{n_{x} n_{y} n_{z}}) .

A quantum dot is an artificial atom: a box of a few thousand atoms whose electronic states are sharp, discrete levels set by the box size and shape rather than by a nuclear Coulomb field. The four cases stack into a single progression as dimensionality falls.

Density of states versus energy as confinement is added. Bulk (3D) rises as the square root of energy; the well (2D) is a staircase of equal risers; the wire (1D) has inverse-square-root teeth at each subband edge; the dot (0D) is a comb of sharp levels.

System	Confined directions	Density of states	Behavior at a subband edge
Bulk (3D)	none	$g \propto E$	continuous rise from zero
Well (2D)	one	$g = \frac{m ^{*}}{π ℏ ^{2}}$ per subband	finite step
Wire (1D)	two	$g \propto (E - E_{n})^{- 1/2}$	inverse-square-root divergence
Dot (0D)	three	$g = 2 \sum_{n} δ (E - E_{n})$	discrete delta functions

Size-tunable optical gaps

The sharpest signature of a quantum dot is that its optical gap depends on its size. An electron and hole confined to a sphere of radius $R$ each acquire a confinement energy of order $ℏ^{2} π^{2} /2 m^{*} R^{2}$ , so the effective gap exceeds the bulk value by an amount that grows as $R$ shrinks. To leading order,

E_{g} (R) = E_{g}^{bulk} + \frac{ℏ ^{2} π ^{2}}{2 R ^{2}} (\frac{1}{m _{e}^{*}} + \frac{1}{m _{h}^{*}}) - \frac{1.8 e ^{2}}{4 π εR},

where the last term is the electron–hole Coulomb attraction. The confinement term dominates for small dots and scales as $1/ R^{2}$ , so a smaller dot emits a higher-energy (bluer) photon. In CdSe, dots near $2 nm$ across fluoresce blue while $6 nm$ dots fluoresce red, spanning the visible spectrum from a single material by size alone.

The effective gap of a quantum dot rises as its radius falls, blue- shifting the emission; three dots of increasing radius emit from blue through green to red.

Coulomb blockade and the single-electron transistor

A dot small enough to quantize its levels is also small enough that adding a single electron changes its electrostatic energy appreciably. Model the dot as a conductor of total capacitance $C$ . Placing $N$ electrons on it costs electrostatic energy $Q^{2} /2 C = (N e)^{2} /2 C$ , so the cost of adding the $(N + 1)$ -th electron above the $N$ -th is the charging energy

E_{C} = \frac{( N + 1 ) ^{2} e ^{2} - N ^{2} e ^{2}}{2 C} - e V_{g} = \frac{e ^{2}}{2 C} (2 N + 1) - e V_{g},

where a gate voltage $V_{g}$ shifts the electrostatic potential of the dot. When $e^{2} /2 C ≫ k_{B} T$ , thermal fluctuations cannot supply this energy, and no current flows through a dot weakly coupled to two leads: transport is frozen by the Coulomb blockade. The blockade lifts only at the discrete gate voltages where the charge states $N$ and $N + 1$ become degenerate; there a single electron can hop on and off, and the conductance shows a sharp peak. Sweeping the gate produces a periodic train of conductance peaks spaced by $Δ V_{g} = e / C_{g}$ , one electron per peak. Sweeping the bias voltage at fixed gate produces a Coulomb staircase: the current jumps up each time the window $e V$ opens a new charge transition.

In a single-electron transistor, current is blocked until the bias opens a charge transition; each new accessible charge state adds a step, giving the Coulomb staircase (left) and periodic conductance peaks in gate voltage (right).

Devices

The engineered density of states is the reason nanostructures dominate optoelectronics. In a quantum-well laser the gain medium is one or a few wells rather than bulk material; the step density of states concentrates carriers in a narrow energy range at the band edge, which sharpens the gain spectrum and lowers the threshold current density by roughly an order of magnitude compared with a bulk double-heterostructure laser. Quantum-dot displays use the size-tuned emission directly: a film of CdSe or InP dots, each size chosen for its color, converts blue backlight into saturated red and green with narrow linewidths set by the discrete 0D levels. The semiconductor heterostructures that make wells and dots possible are the same band-offset junctions that build diodes and transistors; the difference is only that here one dimension is thin enough to quantize.

Confinement in a magnetic field produces a different and more rigid quantization. The next lesson takes a two-dimensional electron gas, applies a strong perpendicular field, and finds that the continuous 2D staircase collapses into the massively degenerate Landau levels behind the integer quantum Hall effect.