Quantum confinement

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Quantum confinement is a physics effect unique to quantum mechanics. Much of our current understanding of quantum confinement comes from the well-known Particle in a box model in quantum mechanics. The model treated quantum confinement of plane waves(see, for example, Particle in a box and Potential well).

Bloch's theorem is the very basis of modern solid-state physics. The theory's central concept --- the Bloch wave --- is a more general type of wave than the plane wave(see Particle in a box). However, based on Bloch's theorem, the conventional theory of electronic states in crystals could not explain genuine crystals' boundary and size effects since Bloch's theorem is based on periodicity with no boundary, see Bloch's theorem. Both theories are essential concepts in quantum mechanics and have been well-known for almost a century.

About half a century ago, the mathematical theory of periodic differential equations made significant progress. The new mathematical progress provided a basis to combine the physics core of Bloch theorem --- the potential periodicity --- and the physical core of Particle in a box model --- the very existence of a specific boundary and a specific finite size---to develop a more general theory (QCBW theory).

Primary theoretical results Rather than the two extremes: either all electronic states are Bloch waves in a perfect crystal as in conventional solid-state physics (see, for example, Bloch's theorem and Particle in a one-dimensional lattice) or all electronic states are stationary confined plane waves (see, for example, Particle in a box and Potential well), the QCBW theory found that: quantum confinement of Bloch waves in one specific dimension may lead to two different types of electronic states: size-dependent or boundary-dependent in some simple and essential cases. The size-dependent states are stationary Bloch waves due to the finite size of the system, with properties and numbers are determined by the size. The boundary-dependent states are electronic states of a different type whose properties are due to the very existence of the boundary and its position, usually localized in at least one dimension (including but not limited to surface states). In multi-dimensional crystals they can be in the range of permitted energy bands.

New physics revealed or predicted The existence of the boundary-dependent states --- which exist in neither the Bloch theorem nor the Particle in a box model --- and their general properties understood in the QCBW theory lead to some conclusions that significantly differ from understandings traditionally accepted in the solid-state physics community. Such as:

• For treating physics problems in low-dimensional systems, a widely used approximation in the solid-state physics community is the Effective Mass Approximation(EMA); see, for example, . The QCBW theory concluded that the EMA treats the quantum confinement of Bloch waves as the quantum confinement of plane waves, thus missing a fundamental distinction of the quantum confinement of Bloch waves.

• The QCBW theory found that the usually localized boundary-dependent states may be in the range of permitted energy band in multi-dimensional cases. Therefore, a Bound state in the continuum (BIC) is normal in multi-dimensional cases. The atypical cases of "localization does not require a band gap" described in might be general in multi-dimensional cases.

• The QCBW theory concluded that an ideal one-dimensional crystal of finite length $Na$ could have at most one surface state --- localized near either of the two ends --- in each band gap. It is different from a well-accepted concept in the solid state physics community proposed first by Fowler in 1933, then accepted in Seitz's classic book that "in a finite one-dimensional crystal the surface states occur in pairs, one state being associated with each end of the crystal." This concept seemly has never been questioned since then for many years. See, for example, in . The QCBW theory analyzed this puzzle: it might likely relate to a mathematics issue in Seitz's book.

• Since the gap between the boundary-dependent occupied states and the size-dependent vacant states in a semiconductor low-dimensional system is always smaller than the band gap of the bulk semiconductor, the QCBW theory predicted that as the size of the system decreases, the boundary-dependent states play a more significant role, the fundamental differences between an ideal semiconductor and an ideal metal might become blurred. An ideal low-dimensional semiconductor system might have a metallic electrical conductivity.

The existence of

One-dimensional cases

The Schrödinger differential equation for a one-dimensional periodic potential $v(x)$ with a period $a$ can be written as:

$$-\frac{d^2}{d{x}^2}y(x)+[v(x)-\lambda ]y(x)=0,-\mathrm{\infty }<x<\mathrm{\infty },$$

(1)

The solutions are one-dimensional Bloch waves with a wave-vector $k$ in the Brillouin zone $(-\frac{\pi }{a},\frac{\pi }{a}]$. Each energy band ${ϵ}_n(k)$ ( $n=0,1,2,...$) is monotonical in the half Brillouin zone $[0,\frac{\pi }{a}]$, and the band edges can be ordered as(for general cases where a band gap is between two consecutive energy bands):

$${ϵ}_0(0)<{ϵ}_0(\frac{\pi }{a})<{ϵ}_1(\frac{\pi }{a})<{ϵ}_1(0)<{ϵ}_2(0)<{ϵ}_2(\frac{\pi }{a})<{ϵ}_3(\frac{\pi }{a})<{ϵ}_3(0)\cdots .$$

( 2)

Mathematically, the problem of quantum confinement of one-dimensional Bloch waves --- electronic states in an ideal one-dimensional crystal of finite length $L=Na$ ($N$ is a positive integer) is that: Under the assumption that Eq.（1）is solved, one tries to find the solutions of the Schrödinger equation for the confined system:

$$\{\begin{array}{l}-\frac{d^2}{d{x}^2}y(x,\lambda )+[v(x)-\lambda ]y(x,\lambda )=0,\tau <x<\tau +Na,\\ y(x,\lambda )=0,x\le \tau \text{or}x\ge \tau +Na.& \end{array}$$

(3)

The problem can be rigorously solved analytically. The theorem that a second-order linearly ordinary differential equation can only have two linearly independent solutions and the theorems on zeros of solutions of second-order periodic ordinary differential equations played an essential role.

Depending on whether the value of $\lambda$ is inside a permitted band, there are two different cases:

1. $\lambda$ is inside a permitted band. It was found that inside each permitted energy band ${ϵ}_n(k)$, there are $N-1$ solutions of Eq.(2), with the eigenvalues

$${\mathrm{\Lambda }}_{n,j}={ϵ}_n(\frac{j\pi }{Na}).j=1,2,3,\cdots ,N-1$$

(4)

They depend on $N$ only but not $\tau$. These states are stationary Bloch wave states: each stationary Bloch state is made of two bulk Bloch wave states with opposite wave vectors inside the band, formed due to the multiple reflections of Bloch waves at the two ends $\tau$ and $\tau +Na$ of the confined range. Their energies ${\mathrm{\Lambda }}_{n,j}$ can be obtained from the bulk energy band ${ϵ}_n(k)$ and map the energy band ${ϵ}_n(k)$ precisely. They can be called bulk-like states.

2. $\lambda$ is not inside a permitted band. Simple mathematics based on the differential equation theory and Theorem 3.1.3 in leads to the conclusion that there is always one and only one state for each band gap, with energy ${\mathrm{\Lambda }}_{n,\tau }$ depending on $\tau$ only but not $N$. Such a state is either a band edge state or a state inside the band gap, localized near either the end $\tau$ or the end $\tau +Na$. Each such state can be called a surface-like state.

Consequently, the energy of a $\tau$-dependent state is always higher than the energy of each corresponding bulk-like $N$-dependent state:

$${\mathrm{\Lambda }}_{n,\tau }>{\mathrm{\Lambda }}_{n,j}.$$

( 5)

Since, by Eq.（2), the bandgap between ${ϵ}_n(k)$ and ${ϵ}_{n+1}(k)$ is always above the energy band ${ϵ}_n(k)$.

Therefore, QCBW theory found that the quantum confinement in the range $[\tau ,\tau +Na]$ of one-dimensional Bloch waves leads to the following:

• Inside each permitted band of Eq.(1) are $N-1$ bulk-like Bloch stationary states; their energies depend on the size $N$ but not the boundary $\tau$. These states are similar to those in the Particle in a box model.

• The existence of the $1$ localized or band edge state in each band gap --- whose energy and properties depend on the boundary $\tau$ but not size $N$ --- is a fundamental distinction of quantum confinement of one-dimensional Bloch waves. Each surface-like boundary-dependent state is always energetically higher than each of its relevant bulk-like size-dependent states.

Numerical calculations confirmed these conclusions. Moreover, such behaviors have been observed in theoretical and experimental investigations on various one-dimensional systems, such as in .

Multi-dimensional cases

Unlike the one-dimensional cases that can be understood rigorously based on mathematical theorems, understanding multi-dimensional problems is more difficult --- since for periodic second-order differential equations, "No general theorems on the nature of the spectrum in the whole space problem in more than one dimension seem to be known." So Titchmarsh wrote in 1958. Eastham's work made significant progress 15 years later. However, Eastham wrote, "My feeling is, nevertheless, that the mathematical theory of the periodic Schrödinger equation is far from complete at present." Nonetheless, his progress was essential to the understanding that the QCBW theory obtained.

The Schrödinger differential equation for a multi-dimensional periodic potential can be written as

$$-{\mathrm{\nabla }}^{2}y(\mathbf{𝐱})+[v(\mathbf{𝐱})-\lambda ]y(\mathbf{𝐱})=0,$$

( 6)

where $v(\mathbf{𝐱}+{\mathbf{𝐚}}_1)=v(\mathbf{𝐱}+{\mathbf{𝐚}}_2)=v(\mathbf{𝐱}+{\mathbf{𝐚}}_3)=v(\mathbf{𝐱})$ is the periodic potential, $\mathbf{𝐱}$ is the position, and ${\mathbf{𝐚}}_i$ is the primitive lattice vector in the $i$ direction.

The solutions' eigenvalues $\lambda$ form energy bands ${ϵ}_n(\mathbf{𝐤})$, with wave vectors $\mathbf{𝐤}$ in the three-dimensional Brillouin zone defined by ${\mathbf{b}}_{\mathbf{𝟏}},{\mathbf{b}}_{\mathbf{𝟐}}$, and ${\mathbf{b}}_{\mathbf{𝟑}}$, the reciprocal primitive vectors (see Reciprocal lattice). The energy bands ${ϵ}_n(\mathbf{𝐤})$ are ordered as

$${ϵ}_0(\mathbf{𝐤})\le {ϵ}_1(\mathbf{𝐤})\le {ϵ}_2(\mathbf{𝐤})\le {ϵ}_3(\mathbf{𝐤})\le \cdots .$$

(7)

The corresponding eigenfunctions are written as ${\varphi }_n(\mathbf{𝐤},\mathbf{𝐱}),n=0,1,2,3\cdots .$

The fundamental difference between the quantum confinement of a multi-dimensional Bloch wave and a one-dimension Bloch wave is that the Schrödinger equation for multi-dimensional Bloch waves is a partial differential equation with no limitation to the number of linearly-independent solutions. Thus the energy band structures are significantly different: The permitted energy bands in a multi-dimensional crystal are often overlapped. The number of band gaps in a multi-dimensional crystal is always finite. There are even no band gaps if the potential is minimal.

Quantum confinement in one specific dimension of three-dimensional Bloch waves

For the quantum confinement of Bloch waves ${\varphi }_n(\mathbf{𝐤},\mathbf{𝐱})$ of solutions of Eq. (4) in a specific ${\mathbf{𝐚}}_3$ direction, the investigation treated a quantum film with its bottom face defined by $x_3={\tau }_3$ and its thickness defined by a positive integer $N_3$. The corresponding Schrödinger equation in such a quantum film is:

$$\{\begin{array}{l}-{\mathrm{\nabla }}^{2}y(\mathbf{𝐱})+[v(\mathbf{𝐱})-\lambda ]y(\mathbf{𝐱})=0,{\tau }_3<x<{\tau }_3+N_3,\\ y(\mathbf{𝐱})=0,x_3\le {\tau }_3\text{or}{x}_3\ge {\tau }_3+N_3.& \end{array}$$

(8)

Solutions of Eq. (8) are two-dimensional Bloch waves with wave vector $\widehat{\mathbf{𝐤}}$ in the plane defined by ${\mathbf{𝐚}}_1$ and ${\mathbf{𝐚}}_2$. They can be written as wavefunctions ${\hat{\psi }}_n(\widehat{\mathbf{𝐤}},\mathbf{𝐱})$ and eigenvalues ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}})$ with different indices to specify their properties further. This equation looks similar to Eq.(3) but is a much less known periodic partial differential equation. The theory on periodic partial differential equations in Eastham's book, sheds light on understanding the solutions of Eq. (8).

Based on the theory in and with help from physics intuitions, The QCBW theory found that in some simple but essential cases (the simplest is the crystal has a simple cubic Bravais lattice), there are two different types of solutions of Eq. (6): size($N_3$)-dependent states and boundary(${\tau }_3$)-dependent states. For each bulk energy band $n$ and each two-dimensional Bloch waves vector $\widehat{\mathbf{𝐤}}$, there are $N_3-1$ size-dependent states with energy ${\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}})$ depending on $N_3$ but not ${\tau }_3$, and one boundary-dependent state with energy ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)$ depending on ${\tau }_3$ but not $N_3$.

The size-dependent states are Bloch waves in the ${\mathbf{𝐚}}_1$ and ${\mathbf{𝐚}}_2$ plane but are Bloch stationary states formed due to multiple reflections of the three-dimensional Bloch waves ${\varphi }_n(\mathbf{𝐤},\mathbf{𝐱})$ at the bottom face $x_3={\tau }_3$ and the top face $x_3={\tau }_3+N_3$ of the film in the ${\mathbf{𝐚}}_3$ direction. Their energy ${\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}})$ can be obtained from the three-dimensional bulk energy band ${ϵ}_n(\mathbf{𝐤})$:

$${\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}})={ϵ}_n(\widehat{\mathbf{𝐤}}+\frac{j_3}{2N_3}{\mathbf{𝐛}}_3),j_3=1,2,3,\cdots ,N_3-1.$$

( 9)

Eq. (9) is similar to Eq. (4) in one-dimensional cases. These size-dependent states can be called bulk-like states.

Each boundary-dependent state is a Bloch wave in the ${\mathbf{𝐚}}_1$ and ${\mathbf{𝐚}}_2$ plane but localized near either the top face $x_3={\tau }_3+N_3$ or the bottom face $x_3={\tau }_3$ of the film, so can be called a surface-like state in the ${\mathbf{𝐚}}_3$ direction. Due to a theorem in (an extension of Theorem 6.3.1 in ), each boundary-dependent surface-like state's energy ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)$ is always above the energy of each relevant bulk-like size-dependent state:

$${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)>{\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}}).$$

( 10)

Eq. (10) is similar to Eq. (5) in the one-dimensional cases, but ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)$ does not have to be in a band gap: There is not always a band gap between ${ϵ}_n(\mathbf{𝐤})$ and ${ϵ}_{n+1}(\mathbf{𝐤})$ in a multi-dimensional case.

Therefore, the QCBW theory found that, as solutions of Eq.(8), in some simple and essential cases, the quantum confinement of three-dimensional Bloch wave ${\varphi }_n(\mathbf{𝐤},\mathbf{𝐱})$ in the specific ${\mathbf{𝐚}}_3$ direction will lead to two types of two-dimensional Bloch waves with each vector $\widehat{\mathbf{𝐤}}$:

• There are $N_3-1$ bulk-like Bloch stationary states with energies ${\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}})$ and properties depending on the size $N_3$ but not the boundary ${\tau }_3$ of the film.

• The existence of the $1$ surface-like localized state with energy ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)$ --- whose energy and properties depend on the boundary ${\tau }_3$ but not size $N_3$ --- is a fundamental distinction of quantum confinement of three-dimensional Bloch waves. Such a state is localized near either the top or bottom face of the film with energy always above its relevant bulk-like Bloch stationary states ${\widehat{ϵ}}_{n,j}(\widehat{\mathbf{𝐤}})$.

Those are similar to the one-dimensional cases. But the surface-like states ${\widehat{ϵ}}_n(\widehat{\mathbf{𝐤}},{\tau }_3)$ do not have to be in a band gap. Therefore, “localized states in the ranges of permitted bands” are general in multi-dimensional cases.

The analytical theory provided a basis for understanding many unexpected or uncomprehended results from previous relevant numerical calculations, for example , on semiconductor quantum films: The so-called "zero-confinement" states therein is a particular case of the boundary-dependent states the analytical theory obtained; the bulk energy band-mapping states are the stationary Bloch states in the analytical theory.

An article aiming to verify the analytical theory by numerical calculations concluded that it "predicts all the important subbands $\cdots$ and provides additional insight into the nature of their wavefunctions."

Quantum confinement in three-dimensions of three-dimensional Bloch waves

Further quantum confinements in the ${\mathbf{𝐚}}_2$ direction and then further in the ${\mathbf{𝐚}}_1$ direction are similarly analyzed in . In many simple but essential cases, the QCBW theory found that each quantum confinement of multi-dimensional Bloch waves in one specific direction $i$ between ${\tau }_i$ and $N_i{a}_i+{\tau }_i$ ($a_i$; the period, $N_i$: the positive integer indicating the size in the $i$ direction) could lead to two types of states: $N_i-1$ stationary Bloch states whose energy and properties depend on $N_i$ but not ${\tau }_i$, and $1$ boundary-dependent state whose energy and properties depend on ${\tau }_i$ but not $N_i$. The energy of the ${\tau }_i$-dependent state is always above its relevant $N_i$-dependent stationary Bloch states.

Therefore, in a crystal of simple cubic structure with a rectangular cuboid shape having $N_1,N_2,\text{and}{N}_3$ periods separately in three perpendicular dimensions, for each bulk energy band, there are

• one vertex-like state,
• ($N_1-1$) $+$ ($N_2-1$) $+$ ($N_3-1$) edge-like states,
• ($N_1-1$) ($N_2-1$) $+$ ($N_2-1$) ($N_3-1$) $+$ ($N_3-1$) ($N_1-1$) surface-like states,
• ($N_1-1$) ($N_2-1$) ($N_3-1$) bulk-like states.

And,

• The properties and energy of the vertex-like state depend on three ${\tau }_i$, but neither one $N_i$;
• The properties and energy of each edge-like state depend on two ${\tau }_i$ and the other $N_i$, but neither one of the corresponding two $N_i$ nor the other ${\tau }_i$;
• The properties and energy of each surface-like state depend on two $N_i$ and the other ${\tau }_i$, but neither one of the corresponding two ${\tau }_i$ nor the other $N_i$;
• The properties and energy of each bulk-like state depend on three $N_i$, but neither one ${\tau }_i$.

Among the above states from the same bulk energy band, the following general relations exist

The energy of the vertex-like state is above

the energy of every edge-like state, which is above

the energy of every relevant surface-like state, which is above

the energy of every relevant bulk-like state.

The QCBW theory found that in some practically more interesting but more complicated structures, the numbers of edge-like states, surface-like states, and bulk-like states might be different; however, the general properties of the various states and the general relationship between different types of states remain unchanged.

Summary

The mathematics progress about fifty years ago led to a preliminary understanding of the quantum confinement of Bloch waves. Some unexpected conclusions of the analytical theory have been confirmand by numerical calculations or/and theoretical or/and experimental investigations in related systems.

The very existence and properties of the boundary-dependent states --- which exist in neither Particle in a box model nor Bloch's theorem lead to fruitful new physics.

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