3 Quantum Confinement Effects

S.S. islam

epgp books

 

 

 

After discussing the length and energy scales relevant in nanoscience and nanotechnology, we will now discuss the essential structures to understand the properties of nanostructures.

 

In this module we will learn –

–          Various types of semiconductor nanostructures

–          Quantum confinement

–          Potential Well

–          Energy and wavefunction of a free particle

 

1.   Semiconductor Nanostructures

 

Among the semiconductor systems, the three important generic types of nanostructures are referred as:

  • Quantum well
  • Quantum wire or nanowire
  • Quantum dot or nanocrystal, colloidal quantum dots, etc.

 Alternatively, these structures are also referred as “low-dimensional” systems and possess a given dimensionality. Specifically, they are called:

  • Two-dimensional (2D) systems (quantum wells)
  • One-dimensional (1D) systems (quantum wires)
  • Zero-dimensional (0D) systems (quantum dots)

   where the stated dimensionality reflects the degrees of freedom of carriers (i.e., electrons and holes) in the material.

 

Quantum well is a thin slab of semiconductor sandwiched by an insulator on either side. When its physical thickness becomes narrow such that ‘L’ depicted along z direction of a Cartesian coordinate system, is shorter than the characteristic length of electrons and holes in the material, carrier confinement occurs. At the same time, since the dimensions of the material along x and y are much larger than the critical length, carriers experience no additional confinement effects. They therefore possess two degrees of freedom. Thus, electrons and holes in quantum wells experience one degree of confinement and two degrees of freedom.

 

A quantum wire or nanowire is a semiconductor elongated along a single direction with two associated orthogonal dimensions that are much narrower. Carriers in quantum wire possess one degree of freedom and two degrees of confinement.

 

Quantum dots structure is narrow along all the three directions of a Cartesian coordinate system. Quantum dot is a small semiconductor particle having its radius smaller than the characteristic de Broglie wavelength or Bohr radius of carriers in the material and as such, carriers do not possess any degree of freedom.

 

2.   Quantum Confinement

 

Charge carriers (electrons and/or holes) in a bulk (0D confinement) semiconductor can freely move in all the three spatial directions and behave as free particles with continuous energy levels and density of states (DOS). If any of the structural dimensions is squeezed such that it becomes comparable to the de Broglie wavelength of the carrier or the exciton Bohr radius, the motion of carriers becomes restricted in the corresponding direction, leading to the quantum confinement effect where the carrier energy levels and DOS are strongly related to the structural size, and entirely different from the bulk form.

 

Low-dimensional or quantum structures are often categorized by confinement dimension(s) into 3 types: 1D, 2D, and 3D confinements which are respectively described as quantum well (QW), quantum wire, and quantum dot (QD). The carrier wave functions in these structures can be determined by the solution of time-independent Schrodinger equation with effective mass approximation. From this, the carrier energy levels and the corresponding DOS are obtained.

 

An advantage of using the low-dimensional structures is that the energy levels and DOS can be tailored by controlling the structural size of semiconductors. Therefore, the electronic, optical as well as transport properties can be engineered to meet the requirements of devices or improve their efficiency. For example,

 

(1) Semiconductor QD laser needs rather low threshold current because of the strong carrier confinement within the QD.

 

(2) The emission wavelength of semiconductor QD laser and LED can be tuned by adjusting size of the quantum dot because the carrier energy level and thus emission wavelength strongly depend on the structural size.

 

2.1 Potential Well

 

The region surrounding a local minimum of potential energy is known as the potential well. The energy entrapped inside the potential well cannot convert to any other energy since it is captured within the local minimum of the potential well. This leads to a capturing or confinement effect, and the body cannot release energy. Thus, the body does not strive towards minimizing its potential energy, as mandated by entropy.

 

To release energy from the potential well the local maximum must be surmounted. This can be achieved by adding sufficient energy to the system. In quantum physics, owing to the probabilistic characteristics of quantum particles, the potential energy may also escape the potential well without adding additional energy to the system. In this case, the particle may tunnel through the walls of the potential well.

 

A 2D potential energy function is the potential energy surface and can be considered as the surface of the earth consisting of hills as well as valleys. In this analogy, potential well may be represented by a valley surrounded on all sides by a higher terrain. This well can be filled with water. Without any passage for outflow, this water will remain entrapped in this low lying area and will not move toward the global minimum, that is, the sea level.

 

The opposite of a potential well is represented by a potential hill and is described as a region surrounding the local maximum.

 

2.2 Quantum Confinement

 

If the diameter of a molecule is reduced such that it becomes comparable to the de Broglie wavelength of electron wavefunction, it experiences quantum confinement effects. The electronic as well as optical properties of such small materials considerably differ from the properties of bulk materials.

 

Figure 2 Quantum confinement causes an increase in the energy difference between energy states as well as the band gap. Both these effects strongly influence the optical and electronic properties of the materials at nanoscale.

 

If the dimensions of the confining structure are very large in comparison to the de Broglie wavelength of the particle, the particle behaves like a free particle. In this state, the energy states are continuous and the bandgap is at its original position. Nonetheless, when the dimensions of confining structure are decreased down to the nanoscale, the energy spectrum does not remain continuous and becomes discrete. Consequently, the bandgap exhibits size dependence, and eventually causes a blue shift in the emitted light as the particle’s size is decreased.

 

In particular, this effect demonstrates the consequences of confining the electrons and electron-hole (or the excitons) within a dimension which approaches the critical quantum limit, often termed as the exciton Bohr radius. In this view, a quantum dot confines in all the three dimensions; a quantum wire (nanowire) confines in two dimensions; and a quantum well confines only in one dimension. The corresponding structures are also termed as zero dimensional (0D), one dimensional (1D), and two dimensional (2D) potential wells, respectively, with regard to the number of dimensions in which the confined particle has freedom of movement.

 

3. Understanding through Quantum Mechanics

 

Nanostructured materials demonstrate electronic and optical properties which are strongly influenced by their size as well as shape. The theoretical simulations have been used to investigate the size and shape dependency of the material’s properties. This has led to the developments of various novel structures such as quantum dots which exhibit strong quantum confinement effects. These theoretical evaluations are based on the fact that an exciton behaves increasingly as an atom when its surrounding space become smaller and smaller. An excellent approximation of the behavior of an exciton is three dimensional model of a particle in a box. The solution to this problem suggests that the energy states and the dimensions of the space are related by a pure mathematical relation. The energy of states increases as the volume or the dimensions of the available space are reduced. Figure 2 shows the differences in the electron energy levels and the bandgap of a nanomaterial and its bulk counterpart. The relation between energy level and dimension spacing is given by:

 

 

The variations in the properties of materials at nanoscale can also be understood as follows: in bulk materials, the surface controls most of the macroscopic properties of the material. Whereas, in nanostructured materials, the surface atoms or molecules do not follow the expected configuration in space. Consequently, the surface tension is remarkably changed.

 

Classical Mechanics View

 

The Young-Laplace relation can be used to understand the scale of forces applicable on the surface molecules:

 

 

 

 

 

 

If the shape of the confining structure is assumed to be spherical, we obtain R1=R2=R. Now by solving the Young-Laplace equation for the new radii R (nm), new ΔP (GPa) can be estimated. As can be observed from the above relation, ΔP scales inversely with R. As a consequence, smaller the R, greater is the applied pressure. The increased pressure at the nanoscale dimensions causes strong forces towards the interior of the particle. As a result, the molecular structure of the particle at nanoscale greatly differs from that of the bulk counterparts. This effect is more pronounced at the surface. Such aberrations at the surface result in modified interatomic interactions and bandgap at nano-dimensions.

Figure 3 The classical mechanic explanation employs the Young-Laplace law to provide evidence on how pressure drop advances from scale to scale.

 

Quantum mechanics is used to solve the model systems that illustrate the confinement of a particles (e.g., an electron) in one, two, or three dimensions. These are referred to as:

  • The particle in a box
  • The particle in a cylinder
  • The particle in a sphere

    The wavefunctions and energies of confined particles can be obtained by solving Schrodinger equation. The difference between these systems lies in their boundary conditions. These constraints dictate where the particle can and cannot be and reflect the underlying physical geometry and restrictions of the system. This is why one often refers to a box, a cylinder, and a sphere while modeling quantum wells, quantum wires, and quantum dots, since these are their natural geometries.

 

Free Particle

 

In one dimension, the free particle is described by a plane wave, = . Multiplication of with yields a travelling wave.

 

Time-independent Schrodinger equation is

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

where ? and ? are constant coefficients to be determined

 

As the plane wave propagates, it repeats and thus, the only constraint on it is periodicity (i.e., it has periodic boundary conditions, also called Born-von Karman boundary conditions). The following relation applies:

 

?(?)=?(?+?)

 

where   is the wavelength or the period of the wave.

 

Using our previous solution for ?, we then have the following relation for the first term:

 

?????=????(?+?)=?????????

 

A similar equality applies to the second term. Thus, for either to be true, ????=1, since the Euler relation says that ????=cos??+???? ??, we have:

 

??=2??

 

where ?=0,±1,±2,±3,… We thus find:

 

?=2??/?

 

as constraints on ? tgat arise due to the periodic boundary conditions of the system.

 

Wavefunctions

 

Since there no constraints on or , we are free to choose a solution. For convenience, let B= 0. This gives:

 

?(?)=?????

 

as the free particle wavefunction.

 

Energies

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4 Parabolic dependence of the free particle energy with

 

you can view video on Quantum Confinement Effects

 

References:

 

Questions

 

1. Find examples in literature of the following nanostructures:

a. A “self-assembled” quantum dot

    b. A colloidal quantum dot

Explain whether the approaches used to make these quantum dots are bottom-up or top-down.

 

References & Suggested Reading:

  1. https://en.wikipedia.org/wiki/Potential_well.
  2. https://nanoyou.eu/.
  3. Paul Harrison, Alex Valavanis, “Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures”, 4th Edition, Wiley, 2016.
  4. M. Cahay (2001). Quantum Confinement VI: Nanostructured Materials and Devices: Proceedings of the International Symposium. The Electrochemical Society.
  5. Hartmut Haug; Stephan W. Koch (1994). Quantum Theory of the Optical and Electronic Properties of Semiconductors. World Scientific.
  6. Norris, DJ; Bawendi, MG (1996). “Measurement and assignment of the size-dependent optical spectrum in CdSe quantum dots”. Physical Review B. 53 (24): 16338–16346.
  7. Brus, L. E. (1983). “A simple model for the ionization potential, electron affinity, and aqueous redox potentials of small semiconductor crystallites”. The Journal of Chemical Physics. 79 (11): 5566.
  8. Kunz, A B; Weidman, R S; Collins, T C (1981). “Pressure-induced modifications of the energy band structure of crystalline CdS”. Journal of Physics C: Solid State Physics. 14 (20): L581.
  9. H. Kurisu; T. Tanaka; T. Karasawa; T. Komatsu (1993). “Pressure induced quantum confined excitons in layered metal triiodide crystals”. Jpn. J. Appl. Phys. 32 (Supplement 32–1): 285–287.
  10. Lee, Chieh-Ju; Mizel, Ari; Banin, Uri; Cohen, Marvin L.; Alivisatos, A. Paul (2000). “Observation of pressure-induced direct-to-indirect band gap transition in InP nanocrystals”. The Journal of Chemical Physics. 113 (5): 2016.
  11. Maseru Kuno, Introductory Nanoscience: Physical and Chemical concepts. Garland Science, Taylor & Francis Group, 2012.