Semiconductor Physics and Computational Methods - Module 5: Density of States & Low Dimensional Systems

Density of States

• Definition: Density of states describes the number of available energy states within a system. It's crucial for determining carrier concentrations and energy distributions within a semiconductor.
• In semiconductors, carrier motion is confined to two, one, or zero spatial dimensions. Therefore, density of states in quantum wells (2D), quantum wires (1D), and quantum dots (0D) must be known for accurate semiconductor statistics.

Density of States in 3D (Bulk Material)

• Defined as the number of allowed electronic energy states per unit energy range per unit volume.

Density of States in Lower-Dimensional Systems

• Challenges with 3D Materials:
  • Doped semiconductors are not ideal for studying quantum effects due to disorder from ionized impurities.
  • Quantum effects are more pronounced in lower dimensions.
• Benefits of Reduced Dimensionality:
  • Leads to new electronic and photonic device designs.
  • Devices operate based on quantum mechanics with rapidly changing potentials.
• Low-dimensional systems critically influence material properties due to varying electron interactions in 2D, 1D, and 0D structures.
• Charge carriers are confined differently in quantum wells (2D), quantum wires (1D), and quantum dots (0D).
• High confinement leads to technological applications and plays a major role as semiconductor size approaches nanoscale.
• Applications include:
  • Double heterostructure lasers with low threshold at room temperature.
  • High-efficiency LEDs.
  • Bipolar transistors.
  • p-n-p-n switching devices.
  • High electron mobility transistors (HEMTs).
  • Optoelectronic devices.

Density of States in 1D

• Quantum effects occur when electrons are confined to regions comparable to their de Broglie wavelength.
• 1D systems are created when confinement occurs in two dimensions (e.g., z and y directions), with free motion in the x-direction.
• Electrons are free to move in one direction but confined in two directions (Quantum wire).
• In 1D, two of the k-components are fixed, so the area of k-space becomes a length, and the area of the annulus becomes a line.
• For a 1D box of width L, eigenenergies are given by:
  $E<em>n = n^2 E</em>1$
  where $E<em>1$ is given by: $E</em>1 = {h^2 \over 8mL^2}$
• The density of states g(E)dE gives the number of energy states in the interval E and E+dE per unit volume.
  $g<em>1(E) = {1 \over L} {dn \over dE}$ Using the equation above: $g</em>1(E) = {\sqrt{2m} \over h \sqrt{E}}$
• Multiply the expression by a factor of 2 to account for spin-up and spin-down electrons.
• The final expression for the density of states in 1D:
  $g_1(E) = {2\sqrt{2m} \over h \sqrt{E}}$
• For an electron confined to a 1D box, the density of states is a decreasing function of E and vanishes in the limit of large energy.

Density of States in 2D

• Quantum effects arise in systems where electrons are confined in one dimension (e.g., the z-direction), allowing free motion in the x- and y-directions, creating a two-dimensional system.

• Consider a slab of material with macroscopic dimensions in x- and y-directions but small thickness (nanometer range - Quantum Well).

• The two-dimensional square-box has energy eigenvalue equation:
  $E<em>{n</em>1,n<em>2} = E</em>1(n<em>1^2 + n</em>2^2)$

• In Cartesian $n<em>1n</em>2$ space, there is an energy eigenstate at every point $(n<em>1, n</em>2)$.

• Let $n = \sqrt{n<em>1^2 + n</em>2^2}$ denote the radius vector in this space. The area of this strip is given by:
  $dA = {1 \over 4} 2 \pi n dn$
  The factor (1/4) is necessary since the quantum numbers $n<em>1$ and $n</em>2$ can only take positive values.

• Since each quantum state occupies one unit of area in $n<em>1n</em>2$ space, the number of states available in the strip dN is equal to dA.
  $dN = {1 \over 4} \pi n dn$

• Multiply the above expression by a factor of 2 because in each state, two electrons, one with spin-up and one with spin-down can be accommodated.
  $dN = {\pi \over 2} n dn$

• The equation above can be written as
  $E = n^2 E<em>1$ Taking differential on both sides we can find that $n dn = {dE \over 2E</em>1}$
  Substituting the above expression in Eq (3), we get
  $dN = {\pi \over 2} {dE \over 2E<em>1} = {\pi \over 4} {dE \over E</em>1} = {\pi \over 4} {dE \over {h^2 \over 8mL^2}} = {4\pi mL^2 \over h^2} dE$

• The density of states in two dimension is defined as

  $g<em>2(E) = {1 \over L^2} {dN \over dE}$ Hence $g</em>2(E) = {4\pi m \over h^2}$

• In two-dimensions, the density of states is constant, independent of energy.

• The 2D density of states is independent of energy, but DOS depends on the number of levels and is a sum of contributions from discrete levels due to quantization.

Density of States in 0D

• Electrons are confined in all three dimensions in a dot.

• Analogous to a hydrogen atom, only discrete energy levels are possible for electrons trapped by a zero-dimensional potential.

• The spacing of these levels depends on the precise shape of the potential.

• When considering the density of states for a 0D structure (Quantum dot), no k – space is available as all dimensions are reduced.

• Therefore DOS of 0D can be expressed as a delta function

  $g(E)<em>{0D} = 2\delta(E - E</em>i)$

Conclusion

Introduction to Low Dimensional Systems

Nanoscience & Nanotechnology

• Studies phenomena at very small length scales.

Nano Scale

• Nano is a prefix meaning very, very small.
• One billionth of something, or $10^{-9}$.

Nanoscience Size and shape dependent properties

• Nanometer scale: The length scale where corresponding property is size & shape dependent.
• Actual physical dimensions relevant to Nanosystem.

Surface to Volume Ratio

• As surface to volume ratio increases:
  • A greater amount of a substance comes in contact with surrounding material.
  • Results in better catalysts since a greater proportion of the material is exposed for potential reaction.

Properties at the Nanoscale

• Nano-sized particles exhibit different properties than larger particles of the same substance.
• Nano-sized particles exhibit size & shape dependent properties.

Optical Properties: Colour of Gold

• Bulk gold appears yellow.
• Nano-sized gold appears red.
  • Electrons in nano gold are not free to move, causing different reactions with light.

Nanoscience: Nanometer scale science

• A part of science that studies small stuff.
• It involves all sciences: Biology, Physics, Chemistry, Engineering.

Interdisciplinary Nature

• Physicists: Study physical forces between individual atoms.
• Chemists: Study the interaction of different molecules governed by chemical forces.
• Biologists: creation of small devices (encoding informations in DNA to perform multitasks
• Computer Scientists: Steady miniaturization, Moore’s Law.
• Electrical Engineers: Control of electric signals, power supply.
• Mechanical Engineers: Study nanolevel issues like load bearing, wear, material fatigue, and lubrication.

What makes the nanoscale special?

1. High density of structures is possible with small size.
2. Physical and chemical properties can be different at the nano-scale (e.g. electronic, optical, mechanical, thermal, chemical).
3. The physical behavior of material can be different in the nano-regime because of the different ways physical properties scale with dimension (e.g. area vs. volume).

Dr. Richard P. Feynman