26 Introduction of Two Dimensional (2D) Materials
Dr. Anchal Srivastava
Contents of this Unit
1. Introduction
2. Two Dimensional Materials or Quantum Well
3. Particle in 2D Box
3.1 Density of States of Quantum Well
3.2 Population of Band
3.2.1. Conduction Band
3.2.2. Valence Band
3.2.3. Fermi Level
4. Examples of 2D Materials
5. Summary
Learning Outcomes
- After studying this module, you shall be able to
- Know about two dimensional (2D) materials or quantum well structures. Learn calculation of density of states (DOS) for quantum well structures.
- Understand the calculation of occupancy of different bands using density of states. Know various experimentally synthesized 2D materials.
1. Introduction
When the size or the dimension of a material is continuously reduced from a large or macroscopic size, such as a meter or a centimeter, to a very small size, the properties remain the same at first, then small changes begin to occur, until finally when the size drops below 100nm, dramatic changes in properties can occur. If one dimension is reduced to the nanorange while the other two dimensions remain large, then we obtain a structure known as a two dimensional (2D) material or quantum well. If two dimensions are reduced in nanometer range and one remains large, the resulting structure is referred as one dimensional (1D) material or quantum wire. If all the three dimensions are reduced in nanometer range is called zero dimensional (0D) material or quantum dot. The word quantum is associated with these three types of nanostructures because the changes in properties arise from the quantum-mechanical nature of physics in the domain of the ultrasmall. The process of diminishing the size has been illustrated in Fig. 1.
Fig. 1. Figure showing various nansturctures: 3D, 2D, 1D, and 0D.
In recent years, synthesis of 2D NSMs has become a forefront area in materials research, owing to their many low dimensional characteristics different from the bulk properties. In the quest of 2D NSMs, considerable research attention has been focused over the past few years on the development of 2D NSMs. 2D NSMs with certain geometries exhibit unique shape-dependent characteristics and subsequent utilization as building blocks for the key components of nanodevices. In addition, a 2D NSMs are particularly interesting not only for basic understanding of the mechanism of nanostructure growth, but also for investigation and developing novel applications in sensors, photocatalysts, nanocontainers, nanoreactors, and templates for 2D structures of other materials. Keeping in the view of importance of 2D materials in this module we are interested to study two dimensional nanostructured materials (2D NSMs) or quantum well
2. Two Dimensional (2D) material or quantum well:
In bulk metals conduction electrons are delocalized and these electrons are referred as free electrons, but perhaps unconfined electrons would be a better word for them. This is because when the size of a conductor diminishes to the nanoregion, these electrons begin to experience the effects of confinement, meaning that their motion becomes limited by the physical size of the region or domain in which they move. The influence of electrostatic forces becomes more pronounced, and the electrons become restricted by a potential barrier that must be overcome before they can move more freely. More explicitly, the electrons become sequestered in what is called a potential well, an enclosed region of negative energies. A simple model that exhibits the principal characteristics of such a potential well is square well in which the boundary is very sharp or abrupt. Square wells can exist in one, two, three, and higher dimensions; here we describe two-dimensional case.
3. Particle in a Two-Dimensional Box:
Let us now consider a particle in a two-dimensional box, assume that a particle is confined to a square box with dimensions a within the (x, y) plane. The potential is zero inside the box and infinite outside. Solving the Schrödinger equation leads to the following wavefunctions (Equation 1.1):
with corresponding energies (Equation 1.2)
where n2 =nx 2 + ny 2 (nx = 1, 2, 3, . . . ; ny = 1, 2, 3, . . . ) and k2 = kx2 + ky2 (kx = nxπ/a, ky = nyπ/a). Fig. 2(a) list the first 10 energies of the particle and Fig. 2(b) illustrates the first eight energy levels with their degeneracies.
Fig. 2.(a) Energies of a particle in a two-dimensional box, (b) Depiction of the first eight energies of a particle in a two-dimensional box, including degeneracies. The numbers represent the indices nx and ny. The spacing between states is to scale.
At this point, the energy density associated with a unit change in energy is characterized by n. Since E ∝ 2 + 2 , where nx and ny are independent indices, we essentially have circles of constant energy in the (nx, ny) plane with a radius n = √ 2 + 2. This is illustrated in Fig. 3 Thus a small change in radius, n, leads to a slightly larger circle (radius n + n) with a correspondingly larger energy. We are interested in the states encompassed by these two circles, as represented by the area of the resulting annulus. As before, if the dimension of the box, a, becomes large, we may approximate n → dn. We will assume that this holds from here on. Now, the area of the annulus is 2πndn. It represents the total number of states present for a given change dn in energy. Since nx > 0 and ny > 0, we are only interested in the positive quadrant of the circle shown in Fig. 2. As a consequence, the area of specific interest to us is (1/4)(2πn dn). Then if g(E) is our energy density with units of number per unit energy, we can write
Finally, we can define a density of states ρenergy, 2D box = g2Dbox (E)/a2, with units of number per unit enery per unit area by dividing g2Dbox (E) by the physical area of the box. This results in
Fig. 3. Constant-energy perimeters described by the indices nx, ny and the radius n = √ 2 + 2 for a particle in a two-dimensional box.
3.1 Density of states of Quantum Well (2D Material):
Density of states (DOS) of a system in solid-state and condensed matter physics describes the number of states per interval of energy at each energy level available to be occupied. Unlike isolated systems, like atoms or molecules in the gas phase, the density distributions are not discrete like a spectral density but continuous. High DOS at a specific energy level means that there are many states available for occupation. Zero DOS means that no states can be occupied at that energy level. In general we can say a DOS is an average over the space and time domains occupied by the system. Here, we will discuss the DOS for quantum well (2D) structure in detail.
The DOS of a quantum well can be evaluated by considering a circular area (shown in Fig. 4) with radius k = √k2x + k2y. Note that symmetric circular areas are considered, since two degrees of freedom exist within the (x, y) plane. The one direction of confinement that occurs along the z direction is excluded. The perimeters of the circles shown in Fig. 4 therefore represent constant-energy perimeters.
The associated circular area in k-space is then
and encompasses many states having different energies. A given state within this circle occupies an area of
with kx = 2π/Lx and ky = 2π/Ly, as illustrated in Fig. 5. Recall that Lx = Ly = Na, where N is the number of unit cells along a given direction and a represents an interatomic spacing. Thus,
The total number of states encompassed by this circular area is therefore N1 = Ak/ Astate, resulting in
giving the total number of available states for carriers, including spin. At this point, we can define an area density
with units of number per unit energy per unit area. Notice that it is a constant. Notice also that this density of states in (x,y) accompanies states associated with each value of kz (or nz) value is accompanied by a subband and one generally expresses this through
Where nz is the index associated with the confinement energy along the z direction and ϴ(E-Enz) is the Heaviside unit step function, defined by.
Fig. 4. Constant-energy perimeters in k-space. The associated radius is k = √ 2 + 2. Perimeters characterized by smaller radii have correspondingly smaller energies and are denoted by E1, E2, and E3 here.
Fig. 5. Area (shaded) associated with a given state in k-space for a two-dimensional system.
3.2 Population of the Band:
Till now we have calculated the DOS for 2D system. While we have explicitly considered conduction band electrons in these derivation, such calculations also apply to holes in the valence band. Thus, in principle with some slight modifications, one also has the valence band density of states for all systems of interest.
In either case, whether for the electron or the hole, the above DOS just tell us the density of available states. They say nothing about whether or not such states are occupied. For this, we need the probability P(E) that an electron or hole resides in a given state with an energy E. This will therefore be the focus of the current section and is also our first application of the DOS function to find where most carriers reside in a given band. Through this, we will determine both the carrier concentration and the position of the so-called Fermi level in each system.
3.2.1 Conduction Band:
We start with our derived quantum well density of states
Next, recall from the previous bulk example that the concentration of states for a given unit energy difference can be expressed as
The total concentration of electrons in the first subband is therefore the integral over all available energies:
where the lower limit to the integral appears because the energy of the first subband begins at Ene not ??(?) =(1 + ?(?−??)?? )-1 is again our Fermi–Dirac distribution. We
To be instructive, we continue with an analytical evaluation of the integral by assuming that E − EF>> kT . This leads to
which is our desired carrier concentration in the first subband.3.2.2. Valence Band
Let us repeat the above calculation to find the associated hole density of states. As with the conduction band, we need the probability that the hole occupies a given state within the valence band. The required probability distribution is then Ph(E), which can be expressed in terms of the absence of an electron in a given valence band state (i.e., through Ph(E) = 1 − Pe(E)). We obtain
The concentration of states occupied at a given unit energy difference is then
where we consider only the first hole subband (nh = 1) in Equation 15:
Notice that mh is used instead of meff since we are explicitly considering a hole in the valence band. Since the total concentration of holes in the first subband is the integral over all energies, we find that
and since E < EF in the valence band, we alternatively write
Applying the binomial expansion to the term in parentheses
This ultimately results in our desired expression for the valence band hole concentration:
3.2.3. Fermi Level:
Finally, Let us evaluate the Fermi level of the quantum well. Again we assume that the material is intrinsic and that it remains at equilibrium. In this case, nc = nv,
This shows that the Fermi level of an intrinsic two-dimensional material at equilibrium occurs halfway between its conduction band and valence band. The slight temperature dependence means that the fermi level moves closer to the conduction (valence) band with increasing temperature if mh > me (mh < me).
Fig. 6. First subband (shaded) of the conduction and valence bands in a quantum well. For comparison purposes, the dashed lines represent the bulk density of states for both the electron and the hole.
4. Examples of 2D Materials:
Graphene is one of the most alluring two-dimensional materials because of its unique band structure and exceptionally high carrier mobilities. With the discovery that graphene can be controllably grown on metal surfaces by vapor-phase deposition from hydrocarbon precursors, scalable graphene-based technologies started to look more feasible. However, graphene is not the best 2D material for every application—its zero bandgap renders it rather useless as a semiconductor, for example. Fortunately, a large number of other layered materials span the entire spectrum of electronic properties, as shown in Fig. 7. Among them, hexagonal boron nitride (h-BN) is an insulator with a bandgap exceeding 5 eV, and the family of transition metal dichalcogenides (TMDs) includes several semiconductors with bandgaps ranging from 0.5 eV to 3 eV. By exploiting the material properties of the layers in isolation or by mixing and matching them to create new structures with atomically thin heterojunctions, researchers are exploring interesting new physics and engineering new ultrathin devices.
Fig. 7. THE WORLD OF TWO-DIMENSIONAL MATERIALS includes graphene and its analogues, such as hexagonal boron nitride; black phosphorus (BP) and its analogues; the III–VI family of semiconductors; and the transition-metal dichalcogenides (TMDs). Together they span the full range of electronic properties. The four corner diagrams show the material’s cross-sectional structures— most are not strictly planar.
5. Summary:
- In this module, you study
- Basic definition of 2D nanostructures.
- Calculation of DOS for 2D nanostructure.
- Evaluation of the occupation of states in conduction bands by using DOS. Evaluation of the occupation of states in valence bands by using DOS.
- Calculation of Fermi level for 2D nanostructures.
- Various examples of experimentally synthesized 2D materials, which span whole the electronic materials ranging from metal to insulator and even superconductor too.
Suggested Reading
- Text book of “Introductory Nanoscience” by Masaru Kuno.
- Pulickel Ajayan et. al., “Two-dimensional van der Waals materials” Physics Today 69, 9, 38(2016).
- Text book of “Introduction to Nanotechnology” by Charles P. Poole Jr. and Frank J. Owens.
- Text book of “2D materials for nanoelectronics” by Michel Houssa, Athanasios Dimoulas and Alessandro Molle, CRC Press.
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