5 Quantum Confinement of electrons in semiconductor nanostructures

S.S. islam

This module discusses the evolution of quantum size effects and their influence on the electronic structure

of  the  semiconducting  nanostructures. Quantum confinement  effect  has  been  introduced  via  two  approaches: top-down, and bottom-up.

Quantum Confinement

Quantum confinement refers to the spatial confinement of electrons in a nanomaterial rather than the free existence within the bulk materials. The effects of spatial confinement are different for unlike materials and their properties, because it varies with the characteristic length scale of the property that is eventually shaped by a material’s structure and composition. The significant length scale here is the exciton (or electron-hole pair) Bohr radius (a0), a dimension that describes the spatial extent of the excitons in a semiconductor and is in between 2 and 50nm based on the type of material. This spatial extension of the excitons inside the semiconducting nanostructures results in effect of quantum confinement. Appreciating its cause and influence on electronic structure of the semiconductor nanostructure is a vital aspect of the nanoscience. To appreciate the influence of quantum confinement phenomenon on electronic configuration of the given nanomaterial, knowledge of electronic structure of bulk material is a prerequisite.

Electronic Organization of Bulk Semiconductors

Contrary to an unbound (or free) electron moving in vacuum, the electrons within semiconductor encounter the periodic potential produced by atoms of the crystal lattice. For simplicity, we begin with description of a 1D hypothetical crystal, expressed by the Hamiltonian:

here, ?(?) is periodic potential caused by the lattice and ?0 denotes the kinetic energy operator. The potential, ?(?) has periodicity over lattice spacing ?, that is, ?(?)=?(?+?); signifying that Schrödinger equation with translation of a wave function ?(?) by the distance ?(?(?+?)), can be written as:

The wave functions will have similar eigenvalue ?, after translation over ?. This means that wave functions ?(?) and ?(?+?) will only vary by phase-factor. Wave function obeying this condition along with being the eigen-function of eq. (2) may be given as:

?_????ℎ,?(?)=?^???.?_?(?)                                                          (3)

Where, ?=2?/? is the wave vector and ??(?) is a periodic function having the periodicity of lattice constant (??(?)=??(?+?)). This resultant wave function is termed ‘Bloch function’ representing the plane wave (????), modulated with the periodic function (??(?)). In fact, this plane wave signifies phase of wave function, while ??(?) expresses response of valence electrons towards periodic lattice potential caused by lattice. Translating ‘Bloch function’ over ? yields:

?_????ℎ,?(?+?)=?^???.?_????ℎ,?(?)                                              (4)

This relation demonstrates that after translation over , the wavefunction only differs by the phase constant.

Let us consider an unbound and undisturbed electron in vacuum, defined by a plane wave ?(?)=?^???, having continuous energy levels described by dispersion relation (Figure 1):

Figure 1 (a) Dispersion relation of an unbound electron traveling through vacuum or free space, (b) the dispersion curve for confined electron in 1D crystal of lattice spacing ‘a’, and (c) the reduced zone depiction of dispersion curve from (b).

Where m₀ represents the mass of free electron.

The lattice produced periodic potential results significant changes in electronic dispersion relation. The electrons propagating through lattice having wavelength from ‘ ’ get barely disturbed and hence their energy is comparable to the free or unbound electrons. Nonetheless, as electrons’ wavelength is double the lattice constant or else when ?=?/?, these electrons get reflected from the crystal potential. These are nothing but Bragg reflections and produce standing waves, represented with linear combination of ?=−?/? and ?=?/? plane waves. These standing waves ‘?????ℎ+’ and ‘?????ℎ−’ are alike apart from the displacement of ?/2 in ?. This results in a considerable disparity in energy of these standing waves since first wave concentrates electronic probability at ions (thereby reducing potential energy), while second one amasses the probability of electrons in the middle of the ions (thereby raising potential energy). Therefore, because of disparity in electron charge distributions, these standing waves possess dissimilar energies at identical ?-value, resulting in an energy-gap within the dispersion-relation. Lattice periodicity initiates Bragg reflections at many ?-values (where, ? = ??/?; and ? : integer); resulting in the dispersion-relation having several energy gaps (band gaps Figure 1(b)).

Alternatively, Bloch function is periodic in ?: implying that the wave functions having k-values differing by ?2?/? are same, which may be obtained from eq. 4. This can be understood as following: value of phase ( ????) at ions (? = ??; ?: integer) is equal at ? and ?+2 ?/?, thereby portraying similar physical situation. Accordingly, energy dispersion-relation can be limited to the ?-values in −?/? < ? < ?/? window and values of higher ? may be folded in this zone by deducting 2?/? from the ?-value. This resultant interval (in 1D) is first ‘Brillouin zone’ within which energy-dispersion relation may be plotted (Figure 1(C)). The restriction of dispersion relation within the limit: −?/? < ? < ?/?, results in the occurrence of energy bands and each ?-value possess many related energy value (and can be considered as overtones). Gaps occur between different bands at ? = 0 and ? = ?/?. The electronic structure of 3D crystal may be described on similar to the 1D case.

Electron Transitions in Bulk Semiconductors

Valence band (VB) in semiconductors is full, whereas conduction band (CB) remains vacant. An external perturbation can promote an electron from valence- to conduction-band, for example, absorption of a photon with energy equal or greater than the bandgap. The VB to CB transition of an electron can be depicted like a system where CB has one particle while VB, several many particles, is devoid of one particle. This becomes many-body system and can be simplified by expressing many interacting particles with the least possible non-interacting particles termed as ‘quasi-particles’. We start by introducing the quasi-particles in VB, i.e. ‘hole’, that can be associated with an group of electrons present in VB out of which the electron was taken out. Electron in CB can be defined by the charge (?−), electronic spin (?=1/2), and an effective mass (??∗ ), while these variable change to ?+, ?=1/2, and ?ℎ ∗ respectively for charge, spin, and mass. As they possess charge, the electron and hole are described as the charge carriers. Here, effective mass takes into consideration the interactions between the particle and the lattice; and therefore reflects the increase or decrease in mobility of the particle within the semiconductor with reference to that of the free electron in vacuum. A larger effective mass implies such interaction with the lattice which slows the particle, while greater carrier mobility corresponds to lower effective mass. Conversely, the effective masses can point towards the spatial limit of the carrier wave function, because delocalization degree varies inversely with effective mass of charge carriers (meaning that lighter carrier delocalize more).

As electrons and holes are oppositely charged, they interact by the Coulombic potential, creating an electron-hole (e-h) pair which is expressed by the quasi-particle (or ‘exciton’). The creation of exciton involves some minimum energy which can be written as:

?=ħ?=?_?+?_?,???????+?_ℎ,???????                                        (6)

Where, ?? is fundamental band-gap of the semiconductor, and ?_?,??????? and ?_ℎ,??????? are respective kinetic energies of electron and hole. Additionally, momentum conservation requires:

ħ?_??=ħ?_??+ħ?_?ℎ????                                                      (7)

where, ??? and ??? are respective wave vectors of elevated electron in CB and hole in VB. Since the momentum of photons is negligible, the condition: ???=??? must be satisfied. Thus, in absence of external disruptions, the transitions only within same ?-values (direct or vertical transitions) can take place (Figure 2).

Figure 2 Schematic presentation of the transition of an electron from VB to CB within a direct bandgap semiconductor as a result of photon absorption.

The exciton energy is given by the dispersion relation:

The second term in eq. 8 consists Hydrogen-like energy levels where ∗ denotes the exciton Rydberg energy (corresponding to ionization energy of lowest hydrogenic state). 3rd term is kinetic energy of exciton’s centre-of-mass movement with exciton wave-vector . Kinetic energy term looks like dispersion curve of the free electron except owing to the difference that it incorporates effective mass of the exciton. This term stems from the fact that the interacting electrons and holes might be expressed as distinct particles interacting by a Coulombic potential. A Hamiltonian similar to that of a hydrogen atom can be employed to estimate energy of the exciton, wherein the effective mass of exciton substitutes the free electron mass, m₀. Analogous to Hydrogen atom, distance of an electron from hole within the exciton is described as the exciton Bohr radius ( a₀):

here ∗ and ℎ∗ are respective effective masses of electron and hole. is electronic charge and is semiconductor dielectric constant. Exciton Bohr radius gives an important length scale to represent spatial extent of the excitons in semiconductor varying from ~2 – ~50 nm relying on the semiconductor. Note that the bandgap and Bohr radius ( 0) of a semiconductor are correlated, such that the materials with wider bandgaps have smaller 0.

Electronic Structure of the Semiconductor Nanostructures

As shown in Figure 3, electronic structure of semiconductor nanocrystals demonstrates strong dependence on size of the nanocrystal. The effect, called quantum confinement, may be described by two approaches – first approach (the “top-down” approach) treats nanocrystal being a tiny piece of the semiconductor material wherein exciton is confined spatially; while second approach (the “bottom-up” approach) is a molecular or quantum chemical methodology, wherein the nanocrystal is constructed atom-by-atom into a progressively bigger molecular cluster which ultimately grows into the bulk crystal. We will discuss both these approaches in following subsections.

(a) Top-Down Approach

In this method, Bloch functions relating semiconductor bulk properties (comprising band-structure and Brillouin zone) are maintained, except the multiplication with an envelope function taking care of spatial extension of the charge carriers and exciton in the nanocrystal:

?_?????(?)=?_????ℎ(?)._????(?)                                                           (10)

Entire wave function (??????) therefore combines the Bloch function (bulk properties) with the envelope function (????) (confinement) of the charge carriers within nanocrystal. ???? is therefore a solution to Schrödinger equation for ‘particle-in-a-box’ problem. In a 3D box of dimensions ?, the wave function is a product of the sinusoidal functions in ?, ?, and ?-directions. For isotopic confinement across all directions, nanocrystal is characterized as spherical potential box (e.g., quantum dot, QD). The eigenfunctions are expressed as a product of spherical harmonics (?(?,?)) and a radial Bessel function (?(?)):

?_???(?,?,?)=?_?^?(?,?).?(?)                                                            (11)

Envelope function has high similarity to the wave functions depicting electron in hydrogen atom. There, the potential encountered by electron is decided by a positively charged proton (or ? (?)~ 1/?), whereas a QD does not have such positively charged center. Alternatively, electrons come across a spherical potential well with diameter D wherein ?(?)=−?0 for ? < ?/2 while ?(?)=0 at other places. Putting Eq. 11 into Schrödinger equation provides solution for discrete the energy-levels of the spherically confined electron:

here ?∗ = effective mass of electrons/holes; ??? represents solutions of Bessel function and are absolute values relying on (or increasing with) the principal (? = 1, 2, 3,…) and azimuthal (? = 0, 1, 2, 3…, analogous to s, p, d,…, orbitals) quantum numbers (Figure 3).

Figure 3 Schematics of quantum confinement effects on electronic structure of a semiconductor. These arrows correspond to the transition with minimum energy absorption. (a) Bulk semiconductor, (b) Three lowest electron (????) and hole (????) energy-levels in a QD; with broken lines showing the corresponding wave functions, and (c) Semiconductor nanocrystal (QD).

The minimum energy level (described by: ? = 1 and ? = 0) demonstrates symmetry of the 1s orbital of hydrogen atom. An important consequence from the difference in potentials of a QD and hydrogen atom comes out to be that there is no restriction on ? (azimuthal quantum number) with respect to ? (principal quantum number) in a QD whereas in case of hydrogen atom, the two are related by the relation: ? ≤ ? − 1). Thus next energy level in the QD is expressed as ? = ? = 1 (1P level). 3rd level is defined as ? = 1 ??? ? =2 (1D level). Fourth is the 2S level defined by: ? = 2 ??? ? = 0. As the envelope wave function of lowest lying energy-levels is atomic-like, the QDs are frequently regarded as the “artificial atoms”.

Bandgap of a quantum dot is therefore sum total of bulk bandgap (??0) and confinement energy (?????) of electrons and holes:

It is notable that the energy-levels of electron and hole are treated independently in eq. 13, implying that the Coulombic interaction between them is not strong enough to keep them bound as exciton anymore. This approximation works only for a strong confinement system, which is valid for nanocrystal (as its radius, ?<?0, exciton Bohr radius,). In this region, confinement potential exceeds the Coulomb interaction. It may be noted here that the Coulombic interaction is still much larger in a QD than in the bulk crystal, owing to the close confinement of electron and hole. Yet, at ? ≪ ?0, the kinetic energy of charge carriers is even now much greater than their Coulombic interaction. Thus, electron and hole can no longer be interrelated and may be considered individually.

If the nanocrystal radius, ? > ?0 (the case of weak confinement) then the increment in the exciton energy is because of quantization of excitonic center-of-mass movements. The exciton can be expressed as a typical ‘particle-in-a-spherical’ potential, with its discrete energy-levels described by the relation identical to eq. 12, only by substituting electron/hole effective mass with exciton effective mass. Energy level shift in case of weak confinement region (i.e., <100 meV) is very small than in strong confinement case. The quantum confinement effect is generally not apparent for dimensions > 2–3 times of ?0.

If Coulombic interaction is also considered, the bandgap of a quantum dot having radius ? may be given by:

Figure 4 Schematic illustration of quantum confinement effects: the semiconductor bandgap increases with decrease in size. Also, energy levels become discrete at band-edges and energy difference between band-edge levels increase with reducing size. Bottom image shows fluorescence of 5 dispersions of CdSe QDs of differing dimensions, under UV excitation.

Equations 13 and 14 describe two vital outcomes of quantum confinement (Figure 4) – firstly, bandgap of a semiconductor nanocrystal increases with reducing dimensions, varying as −2 for negligible coulomb interaction; and secondly, discrete energy-levels (having different quantum numbers) appear at band-edges of both CB and VB. Consequently, the optical bandgap of QDs can be altered by modifying their size.

It is noteworthy 0 gives a suitable length scale for evaluating the quantum confinement effects on the semiconductor nanocrystal properties. Confinement sets to influence exciton wave functions as soon as size of nanocrystal reaches 0. This implies that quantum confinement begins at different nanocrystal sizes for different semiconductors, as 0 varies broadly within the semiconductor materials. It is to be noted that 0 and bandgap are interrelated, such that the materials having narrower have larger 0 and will therefore feel quantum confinement at bigger nanocrystals. On the contrary, insulators are described by extremely confined excitons and therefore possess very short 0 (usually <1 nm), and thus get influenced by the quantum confinement effects for the sizes already in cluster region (usually less than 20 atoms).

    References & Suggested Reading:

1. phycomp.technion.ac.il/~anastasy/thesis/node10.html.
2. T. Pradeep, A Textbook of “Nanoscience and Nanotechnology”, McGraw Hill Education, 2012.
3. https://en.wikipedia.org/wiki/Potential_well.
4. shodhganga.inflibnet.ac.in/bitstream/10603/23484/3/03.chapter%201.pdf
5. courses.washington.edu/overney/…/Lecture12_Reid_Quantum_Confinement.pdf
6. web.eecs.umich.edu/~peicheng/teaching/…06…/Lecture%2011%20-%20Feb%209.pdf
7. http://www.iue.tuwien.ac.at/phd/ungersboeck/node50.html
8. https://en.wikipedia.org/wiki/Quantum_dot