Abstract
In this chapter, we try to understand the similarities and the differences between the quantum confinements of three-dimensional Bloch waves in one specific \(\mathbf{a}_3\) direction and the quantum confinement of one-dimensional Bloch waves treated in Chap. 4. We prove a basic theorem that is the mathematical basis of the theory in this chapter at the beginning, and then we discuss some consequences of this theorem. Afterwards, we obtain the electronic states in several ideal quantum films of different Bravais lattices by reasonings based on this theorem and physical intuition. It is found that in the simplest cases, there are one surface-like subband and \(N_3-1 \) bulk-like subbands in an ideal film of \(N_3\) layers for each bulk energy band. Some understandings obtained are quite different from what are widely believed in the solid-state physics community. A surface state in a multidimensional crystal, in general, does not have to be in a band gap. A thin film of a semiconductor may have a smaller band gap than the bulk semiconductor.
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Notes
- 1.
It was pointed out in [2] that many properties of the energy band structure of a one-dimensional crystal as summarized in p. 43 are closely related to the fact that a second-order ordinary differential equation cannot have more than two independent solutions. For a partial differential equation, these claims are generally incorrect.
- 2.
Part of the results of this chapter was published in [9].
- 3.
- 4.
For a free-standing film with a boundary at \( x_3 = \tau _3 \), in general, we have neither a reason to require that \(\tau _3\) be a constant nor a reasonable way to assign \(\tau _3(x_1, x_2)\) beforehand. However, since what we are interested in is mainly the quantum confinement effects, in this book it is assumed that the existence of the boundary \(\tau _3\) does not change the two-dimensional space group symmetry of the system, including but not limited to that \(\tau _3 = \tau _3(x_1, x_2)\) must be a periodic function of \(x_1\) and \(x_2\): \( \tau _3 = \tau _3(x_1,x_2) = \tau _3(x_1+1,x_2) = \tau _3(x_1,x_2+1) \).
- 5.
In such a case, \(\hat{\lambda }_n(\hat{\mathbf {k}}; \tau _3) = \varepsilon _{n'}({\mathbf {k}}) = \varepsilon _{n'}(\hat{\mathbf {k}} + k_3 {\mathbf {b}}_3) = \varepsilon _{n'}(\hat{\mathbf {k}} - k_3 {\mathbf {b}}_3)\) is true. Only when either \(k_3 = 0\) or \(k_3 = \pi \) is \(\phi _{n'}(\hat{\mathbf {k}}\pm k_3{\mathbf {b}}_3,{\mathbf {x}})\) one single function and \(\varepsilon _{n'}(\hat{\mathbf {k}}\pm k_3{\mathbf {b}}_3)\) one single eigenvalue.
- 6.
Note that we commented on the zeros of one-dimensional Bloch function \(\phi _n(k, x)\) on p. 48, as a consequence of Theorem 2.8.
- 7.
In one-dimensional cases, it is Theorem 2.7 that warrants that a band-edge Bloch function \(\phi _n(k, x)\) at \(k=0\) or \(k=\frac{\pi }{a}\) (\(\phi _0(0, x)\) excluded) always has zeros.
- 8.
Mathematically, it is the Floquet theory and Theorem 2.8 that limits the energy range of any surface-like state in a one-dimensional finite crystal always in a band gap. This is also related to that in a one-dimensional crystal, each permitted energy band and each band gap always exist alternatively as the energy increases. By [2], the origin is that a second-order ordinary differential equation cannot have more than two independent solutions.
- 9.
In this book, such a film is usually called a film with \(N_3\) layers in the \({\mathbf {a}}_3\) direction, despite the fact that the film may actually have more atomic layers.
- 10.
Unlike in a one-dimensional case there the stationary Bloch state solutions of Eqs. (4.3) and (4.4) such as (4.10) type always exist for any \(\tau \); the author does not have an in-depth and comprehensive understanding on the existence and properties of solutions of periodic partial differential equations mathematically to assert that for what \(\tau _3\) the stationary Bloch state solutions of (5.12) exist. However, he expects that the existence of such type of states is reasonable for a physical film with a physical boundary \(\tau _3\).
- 11.
\(\hat{\varLambda }_{n,j_3}(\hat{\mathbf {k}})\) can never be equal to \(\hat{\varLambda }_n(\hat{\mathbf {k}}; \tau _3)\): \(\hat{\mathbf {k}} +\kappa _3 {\mathbf {b}}_3\) is neither \(\hat{\mathbf {k}}\) nor \(\hat{\mathbf {k}} + \pi {\mathbf {b}}_3\). The equality in (5.4) can be excluded in (5.33).
- 12.
Therefore, the existence of two occupied surface bands above the VBM is because that (5.32) is true for two valence bands, rather than because that the film has two surfaces.
- 13.
Neither any one of the doubly degenerated \(X_{1v}\) states in Si nor the \(X_{1v}\) state in GaAs could have a nodal (110) plane and, thus, the \(X_{1v}\) state in an ideal Si (110) film and the \(X_{1v}\) state in an ideal GaAs (110) film cannot exist. Therefore, we can only have \(\hat{\varLambda }_0(\hat{\mathbf {k}}= 0; \tau _3) = X_{4v}\) for an ideal Si (110) film and \(\hat{\varLambda }_0(\hat{\mathbf {k}}= 0; \tau _3) = X_{5v}\) for an ideal GaAs (110) film. These are two examples of the special cases mentioned on p. 97 and p. 104 in which \(n=0\) while \(n'=2\).
- 14.
Note that in our notations the lowest energy band index \(n= 0\).
- 15.
As a consequence, one has no reason to expect that the surface states of a quantum film can be passivated from only one surface of it.
- 16.
Surface states with energy in the range of a permitted energy band sometimes are referred as bound states in the continuum(BIC), in differentiation of surface states with energy in the gap which are referred as bound states outside the continuum(BOC). See, for example, [17, 18]. We have seen here that the existence of BIC might be quite general in multidimensional crystals, not only in some special systems.
- 17.
Conduction band minima in Ge are located at the boundary of the Brillouin zone in four equivalent [111] directions, the use of EMA is not justified for the (001) or (110) films investigated here.
- 18.
- 19.
A detailed numerical verification of the corresponding theoretical results in a two-dimensional case was published in [23], where Ajoy and Karmalkar tried to verify the predictions of the analytical theory by numerically computing the subbands of zigzag ribbons and the corresponding bulk band structure in an artificial graphene. It was concluded that in many interesting cases, the analytical theory “predicts all the important subbands in these ribbons, and provides additional insight into the nature of their wavefunctions.” Interesting investigations can also be seen in [24].
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Ren, S.Y. (2017). Electronic States in Ideal Quantum Films. In: Electronic States in Crystals of Finite Size. Springer Tracts in Modern Physics, vol 270. Springer, Singapore. https://doi.org/10.1007/978-981-10-4718-3_5
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