Abstract
After a brief introduction, we prove the major results of this chapter: In an ideal one-dimensional crystal bounded at \(\tau \) and \(\tau + L \) where \(L = N a\), a is the period and N is a positive integer, for each bulk energy band, there are two different types of electronic states. There are \(N-1\) stationary Bloch states in the energy band, whose energies depend on the crystal length L but not on the crystal boundary \(\tau \); There is one and only one state in the band gap above the energy band, whose energy depends on the crystal boundary \(\tau \) but not on the crystal length L. This \(\tau \)-dependent state is either a surface state or a confined band-edge state. The very existence of the \(\tau \) -dependent states is a fundamental distinction of the quantum confinement of Bloch waves. After giving further discussions on the two different types of electronic states, we treat one-dimensional symmetric finite crystals and provide comments on several related problems such as effective mass approximations, surface states, etc.
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Notes
- 1.
- 2.
Otherwise
$$ c_1 p_1(\tau , \varLambda ) = 0, ~~\text {and}~~ c_2 p_2( \tau , \varLambda ) = 0. \quad (A.2) $$It was pointed out on p. 48 that, in general, a one-dimensional Bloch function \(\phi _n(k, x)\) does not have a zero except \(k = 0\) where \(D(\lambda ) = 2\) or \(k = \frac{\pi }{a}\) where \(D(\lambda ) = -2\). Thus, neither \(p_1(\tau , \varLambda )\) nor \(p_2(\tau , \varLambda )\) in (A.2) can be zero. (A.2) leads to that \(c_1 = c_2 = 0\) and no nontrivial solution of (4.5) and (4.6) from (A.2) exists.
- 3.
Remember that the zeros of \(\phi _{2m+1}(0,x)\) and \(\phi _{2m+2}(0,x)\) are distributed alternatively.
- 4.
Any point \(\ell a\) (\(\ell \): an integer) away from an inversion symmetry center of a periodic potential is also an inversion symmetry center of the periodic potential.
- 5.
This is true only in the cases where the interested band edge is located either at the center or at the boundary of the Brillouin zone. It may not be true in the low-dimensional systems or finite crystals investigated in Part III.
- 6.
All of these discussions here are for ideal one-dimensional finite crystals defined by (4.3) and (4.4). An investigation of the electronic states in one-dimensional symmetric finite crystals with relaxed boundary conditions \((\psi '/ \psi )_{x=\tau } = -(\psi '/ \psi )_{x= L+ \tau } = \sigma \) for finite \(\sigma \) can be found in Appendix B.
- 7.
By the theory in this book, the q states per unit cell give q permitted bands thus q \(\tau \)-dependent states. It is the tight-binding approximation used by Hatsugai that gives \( q - 1 \) gaps thus only \( q - 1 \) edge states. It is well known that [18] in a tight-binding formalism with a single state per unit cell, a linear finite chain does not have a surface state. The reason is quite simple—there is no band gap (\( q = 1\)) in the band structure in that formalism.
- 8.
A near-zero band gap makes the largest possible numerator in (4.14) or (4.14a) small; therefore, the largest decay factor \(\beta _{max}\) for a surface state in the band gap is also small. On the contrary, a narrow permitted band width makes \(|D'(\lambda )|\) at its band edge large and thus makes the largest possible numerator in (4.14) or (4.14a) large for the band gap, which leads to a large \(\beta _{max}\) in the band gap.
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Ren, S.Y. (2017). Electronic States in Ideal One-Dimensional Crystals of Finite Length. 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_4
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