---
title: Quantum Wells, Wires, and Dots
module: Nanostructures
moduleNumber: 11
lessonNumber: 1
order: 1101
summary: >
  When a crystal is shrunk until one or more of its dimensions approaches the
  electron wavelength, the continuous bands of the bulk break into discrete
  subbands. Confining in one direction gives a quantum well with a step-like
  density of states, in two directions a quantum wire with inverse-square-root
  singularities, and in all three a quantum dot whose levels are sharp like an
  atom's. This lesson derives the density of states in each case and applies it
  to size-tunable dot emission and the Coulomb blockade of a single-electron
  transistor.
topics: [Nanostructures]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 18 — Nanostructures"
  - book: Simon
    ref: "Ch. 4 — Sommerfeld (Free Electron) Theory of Metals; confinement"
  - book: Ashcroft & Mermin
    ref: "Ch. 2 — The Sommerfeld Theory of Metals (density of states background)"
---

The [Sommerfeld model](/condensed-matter/free-electron-fermi-gas/sommerfeld-model-and-heat-capacity)
treated a metal's electrons as free particles filling a continuum of $\vec k$
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
$\lambda_F = 2\pi/k_F$ 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 $m^\ast$ to a length $L$ in one
direction with hard walls. The allowed wavevectors are $k_n = n\pi/L$, and the
quantized energies are

$$
E_n = \frac{\hbar^2 k_n^2}{2m^\ast} = \frac{\hbar^2 \pi^2}{2m^\ast L^2}\,n^2,
\qquad n = 1, 2, 3, \dots
$$

The spacing between the lowest levels is of order

$$
\Delta E \sim \frac{\hbar^2\pi^2}{2m^\ast L^2}.
$$

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 $m^\ast = 0.067\,m_e$, a well of width $L = 10\ \text{nm}$ gives

$$
\Delta E = \frac{(1.055\times10^{-34}\ \text{J·s})^2\,\pi^2}
{2\,(0.067)(9.11\times10^{-31}\ \text{kg})(10^{-8}\ \text{m})^2}
= 9.0\times10^{-21}\ \text{J} = 56\ \text{meV}.
$$

This is more than twice $k_B T = 25\ \text{meV}$ 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 ($m^\ast = m_e$) gives only $3.8\ \text{meV}$, quantized only at
cryogenic temperatures.

> **Definition (Subband).** In a structure confined along one or more
> directions, each quantized transverse level $E_n$ carries with it a continuum
> (or discrete set) of states from the unconfined directions. The family of
> states built on a single confinement level $E_n$ is a **subband**, and the
> total energy is the confinement energy plus the free kinetic energy of the
> remaining directions.

## 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 $\Delta E_c$,
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

$$
E = E_n + \frac{\hbar^2}{2m^\ast}\bigl(k_x^2 + k_y^2\bigr).
$$

$$
% caption: A GaAs layer between AlGaAs barriers confines electrons in a square
% well of depth equal to the conduction-band offset; the transverse motion is
% quantized into subbands E1, E2, E3 whose wavefunctions are standing waves.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % barrier / well / barrier conduction band edge
  \draw[black, ->] (-0.4,0) -- (7.6,0) node[below] {growth direction $z$};
  \draw[black, ->] (-0.4,0) -- (-0.4,3.6) node[left] {energy};
  % well profile
  \draw[black, very thick] (0.0,2.6) -- (2.2,2.6) -- (2.2,0.2) -- (5.0,0.2)
    -- (5.0,2.6) -- (7.2,2.6);
  \node[black, anchor=south] at (1.0,2.62) {AlGaAs};
  \node[black, anchor=south] at (6.2,2.62) {AlGaAs};
  \node[black, anchor=south] at (3.6,2.62) {GaAs well};
  % offset arrow
  \draw[black, <->] (5.4,0.2) -- (5.4,2.6);
  \node[black, anchor=west] at (5.5,1.4) {band of\/fset};
  % subband levels inside the well
  \draw[acc, thick] (2.2,0.7) -- (5.0,0.7);
  \node[acc, anchor=east] at (2.15,0.7) {$E_1$};
  \draw[acc, thick] (2.2,1.35) -- (5.0,1.35);
  \node[acc, anchor=east] at (2.15,1.35) {$E_2$};
  \draw[acc, thick] (2.2,2.05) -- (5.0,2.05);
  \node[acc, anchor=east] at (2.15,2.05) {$E_3$};
  % ground-state standing wave on E1
  \draw[acc, densely dotted, domain=2.2:5.0, samples=40, variable=\x]
    plot ({\x},{0.7 + 0.45*sin(pi*(\x-2.2)/2.8 r)});
\end{tikzpicture}
$$

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 $(2\pi)^2/A$. Including a factor of two for
spin, the number of states with wavevector below $k$ is

$$
N(k) = 2\,\frac{\pi k^2}{(2\pi)^2/A} = \frac{A k^2}{2\pi}.
$$

Writing $\varepsilon = \hbar^2 k^2/2m^\ast$ for the in-plane kinetic energy, so
that $k^2 = 2m^\ast \varepsilon/\hbar^2$, the areal density $n = N/A$ becomes
$n = m^\ast \varepsilon/\pi\hbar^2$, and its derivative is a constant:

$$
g_{\text{2D}}(\varepsilon) = \frac{\d n}{\d\varepsilon} = \frac{m^\ast}{\pi\hbar^2}.
$$

Each subband contributes this same constant once $E$ exceeds its threshold $E_n$.
Summing over subbands gives a **staircase**,

$$
g_{\text{2D}}(E) = \frac{m^\ast}{\pi\hbar^2}\sum_n \Theta(E - E_n),
$$

with $\Theta$ the unit step. Every time $E$ crosses a new subband edge the
density of states jumps by one riser of height $m^\ast/\pi\hbar^2$.

## 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 $N(k) = 2\cdot(2k)/(2\pi/L) = 2kL/\pi$, so the linear density is
$n = (2/\pi)\sqrt{2m^\ast \varepsilon}/\hbar$ and

$$
g_{\text{1D}}(\varepsilon) = \frac{\d n}{\d\varepsilon}
= \frac{1}{\pi}\sqrt{\frac{2m^\ast}{\hbar^2}}\,\frac{1}{\sqrt{\varepsilon}}.
$$

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,

$$
E_{n_x n_y n_z} = \frac{\hbar^2\pi^2}{2m^\ast}
\left(\frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2}\right),
$$

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

$$
g_{\text{0D}}(E) = 2\sum_{n_x n_y n_z}\delta\!\left(E - E_{n_x n_y n_z}\right).
$$

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.

$$
% caption: 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.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- 3D bulk ----
  \begin{scope}
    \draw[black, ->] (0,0) -- (2.4,0) node[below] {$E$};
    \draw[black, ->] (0,0) -- (0,2.6) node[left] {$g$};
    \draw[acc, very thick, domain=0.0:2.1, samples=50, variable=\x]
      plot ({\x},{1.5*sqrt(\x)});
    \node[black, anchor=south] at (1.2,2.55) {bulk 3D};
  \end{scope}
  % ---- 2D well ----
  \begin{scope}[xshift=3.4cm]
    \draw[black, ->] (0,0) -- (2.4,0) node[below] {$E$};
    \draw[black, ->] (0,0) -- (0,2.6) node[left] {$g$};
    \draw[acc, very thick]
      (0.4,0) -- (0.4,0.7) -- (1.1,0.7) -- (1.1,1.4)
      -- (1.8,1.4) -- (1.8,2.1) -- (2.2,2.1);
    \node[black, anchor=south] at (1.2,2.55) {well 2D};
  \end{scope}
  % ---- 1D wire ----
  \begin{scope}[xshift=6.8cm]
    \draw[black, ->] (0,0) -- (2.4,0) node[below] {$E$};
    \draw[black, ->] (0,0) -- (0,2.6) node[left] {$g$};
    \draw[acc, very thick, domain=0.42:1.05, samples=40, variable=\x]
      plot ({\x},{0.33/sqrt(\x-0.4)});
    \draw[acc, very thick, domain=1.12:1.75, samples=40, variable=\x]
      plot ({\x},{0.33/sqrt(\x-1.1)});
    \draw[acc, very thick, domain=1.82:2.2, samples=30, variable=\x]
      plot ({\x},{0.33/sqrt(\x-1.8)});
    \node[black, anchor=south] at (1.2,2.55) {wire 1D};
  \end{scope}
  % ---- 0D dot ----
  \begin{scope}[xshift=10.2cm]
    \draw[black, ->] (0,0) -- (2.4,0) node[below] {$E$};
    \draw[black, ->] (0,0) -- (0,2.6) node[left] {$g$};
    \draw[acc, very thick, ->] (0.5,0) -- (0.5,1.4);
    \draw[acc, very thick, ->] (1.1,0) -- (1.1,1.9);
    \draw[acc, very thick, ->] (1.7,0) -- (1.7,1.2);
    \draw[acc, very thick, ->] (2.05,0) -- (2.05,1.6);
    \node[black, anchor=south] at (1.2,2.55) {dot 0D};
  \end{scope}
\end{tikzpicture}
$$

| System | Confined directions | Density of states | Behavior at a subband edge |
| --- | --- | --- | --- |
| Bulk (3D) | none | $g \propto \sqrt{E}$ | continuous rise from zero |
| Well (2D) | one | $g = \dfrac{m^\ast}{\pi\hbar^2}$ per subband | finite step |
| Wire (1D) | two | $g \propto (E-E_n)^{-1/2}$ | inverse-square-root divergence |
| Dot (0D) | three | $g = 2\sum_n \delta(E-E_n)$ | 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 $\hbar^2\pi^2/2m^\ast R^2$, so the effective gap
exceeds the bulk value by an amount that grows as $R$ shrinks. To leading order,

$$
E_g(R) = E_g^{\text{bulk}} + \frac{\hbar^2\pi^2}{2R^2}
\left(\frac{1}{m_e^\ast} + \frac{1}{m_h^\ast}\right)
- \frac{1.8\,e^2}{4\pi\varepsilon R},
$$

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 $2\ \text{nm}$ across fluoresce
blue while $6\ \text{nm}$ dots fluoresce red, spanning the visible spectrum from a
single material by size alone.

$$
% caption: The effective gap of a quantum dot rises as its radius falls, blue-
% shifting the emission; three dots of increasing radius emit from blue through
% green to red.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (0,0) -- (7.4,0) node[below] {dot radius $R$};
  \draw[black, ->] (0,0) -- (0,3.4) node[left] {emission energy $E_g$};
  % gap curve: bulk + C/R^2 (overflow-safe on domain)
  \draw[acc, very thick, domain=1.5:6.6, samples=90, variable=\x]
    plot ({\x},{0.7 + 5.2/(\x*\x)});
  \draw[black, dashed] (0,0.7) -- (6.8,0.7);
  \node[black, anchor=west] at (5.4,0.55) {bulk gap};
  % three dots of increasing size
  \fill[acc] (1.6,2.7) circle (3pt);
  \node[black, anchor=south] at (1.6,2.85) {small: blue};
  \fill[acc] (3.2,1.4) circle (5pt);
  \node[black, anchor=south] at (3.4,1.6) {medium: green};
  \fill[acc] (5.2,0.95) circle (7pt);
  \node[black, anchor=south] at (5.6,1.15) {large: red};
\end{tikzpicture}
$$

## 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 $Q^2/2C = (Ne)^2/2C$, so the cost of adding the
$(N+1)$-th electron above the $N$-th is the **charging energy**

$$
E_C = \frac{(N+1)^2 e^2 - N^2 e^2}{2C} - eV_g
= \frac{e^2}{2C}(2N+1) - eV_g,
$$

where a gate voltage $V_g$ shifts the electrostatic potential of the dot. When
$e^2/2C \gg k_B T$, 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 $\Delta V_g = e/C_g$,
one electron per peak. Sweeping the bias voltage at fixed gate produces a
**Coulomb staircase**: the current jumps up each time the window $eV$ opens a new
charge transition.

$$
% caption: In a single-electron transistor, current is blocked until the bias
% opens a charge transition; each new accessible charge state adds a step,
% giving the Coulomb staircase (left) and periodic conductance peaks in gate
% voltage (right).
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % ---- I-V staircase ----
  \begin{scope}
    \draw[black, ->] (0,0) -- (4.6,0) node[below] {bias voltage $V$};
    \draw[black, ->] (0,0) -- (0,3.0) node[left] {current $I$};
    \draw[acc, very thick]
      (0,0) -- (1.0,0) -- (1.0,0.7) -- (2.0,0.7) -- (2.0,1.5)
      -- (3.0,1.5) -- (3.0,2.3) -- (4.0,2.3);
    \node[black, anchor=north] at (0.5,-0.05) {blockade};
  \end{scope}
  % ---- conductance peaks ----
  \begin{scope}[xshift=6.0cm]
    \draw[black, ->] (0,0) -- (4.6,0) node[below] {gate voltage $V_g$};
    \draw[black, ->] (0,0) -- (0,3.0) node[left] {conductance};
    \foreach \c in {0.8,1.9,3.0}{
      \draw[acc, very thick, domain=-0.55:0.55, samples=40, variable=\x]
        plot ({\c+\x},{2.2*exp(-(\x*\x)/0.03)});
    }
    \draw[black, <->] (0.8,-0.35) -- (1.9,-0.35);
    \node[black, anchor=north] at (1.35,-0.35) {one electron};
  \end{scope}
\end{tikzpicture}
$$

## 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](/condensed-matter/semiconductors/semiconductor-bands-and-junctions)
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](/condensed-matter/nanostructures/integer-quantum-hall-effect)
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.