**10. Ballistic electrons in 2DESs**

As already mentioned, the 2DES, formed at the interface of two semiconductors might play a central role in mesoscopic physics. The thin 2D conduction layer formed in the GaAs/AlGaAs heterojunction may reach a carrier concentration of 2 <sup>×</sup> <sup>10</sup><sup>12</sup> *cm*−<sup>2</sup> and can be depleted by applying a negative voltage to a metalic gate deposited on the surface [Datta]. The mobility can be as high as 106*cm*2/*Vs*, two order of magnitude higher than in bulk semiconductors. The Fermi wavelength *λ<sup>M</sup>* is about 35 nm, and the electron mean free path may be as long as *λ<sup>m</sup>* = 30*μ*m− the same order of magnitude as the liniar dimension of the sample; the ballistic regime of electrons is therefore easily reached.

At low temperature, the conduction in mesoscopic semiconductor is mainly due to electrons in the conduction band. Their dynamics can be described by an equation of the form:

$$\left[\mathcal{E}\_c + \frac{\left(i\hbar\nabla + e\mathbf{A}\right)^2}{2m} + \mathcal{U}\left(\mathbf{r}\right)\right]\Psi\left(\mathbf{r}\right) = \mathcal{E}\Psi\left(\mathbf{r}\right)\tag{129}$$

where E*<sup>c</sup>* referrs to the conduction band energy, *U* (**r**) is the potential energy due to space-charge etc., **A** is the vector potential and *m* is the effective mass. Any band discontinuity ΔE*<sup>c</sup>* at heterojunctions is incorporated by letting E*<sup>c</sup>* be position-dependent [11].

In the case of a homogenous semiconductor, *U* (**r**) = 0, assuming **A** = 0 and E*<sup>c</sup>* = *const*., the solution of (129) is given by plane waves, exp (*i***k** · **r**), and not by Bloch functions, *uk* (**r**) · exp (*i***k** · **r**). So, the solutions of (129) are not true wavefunctions, but wavefunctions smoothed out over a mesoscopic distance, so any rapid variation at atomic scale is suppressed; eq. (129) is called single-band effective mass equation.

Let us consider a 2DES contained mainly in the *xy* plane. This means that, in the absence of any external potential, the electrons can move freely in the *xy* plane, but they are confined in the *z*-direction by some potential *V* (*z*), so their wavefunctions will have the form:

$$\Psi\left(\mathbf{r}\right) = \phi\_n\left(z\right) \exp\left(i k\_x \mathbf{x}\right) \exp\left(i k\_y y\right) \tag{130}$$

The quantization on the *z*−direction, expressed by the functions *φ<sup>n</sup>* (*z*), generate several subbands, with cut-off energy *εn*. At low temperature, only the first subband, corresponding to *n* = 1, is occupied, so, instead of (129), the electrons of the 2DES are described by:

$$\left[\mathcal{E}\_s + \frac{\left(i\hbar\nabla + e\mathbf{A}\right)^2}{2m} + \mathcal{U}\left(\mathbf{r}\right)\right]\Psi\left(\mathbf{r}\right) = \mathcal{E}\Psi\left(\mathbf{r}\right) \tag{131}$$

where the subband energy is E*<sup>s</sup>* = E*<sup>c</sup>* + *ε*1; so, the ‡−dimension enters in this equation only through *ε*1, which depends on the confining potential *V* (*z*). The eq. (131) correctly describes the 2DESs formed in semiconductor heterostructures, but is inappropriate for metallic thin films, where the electron density is much higher, and even at nanoscopic scale, there are tens of occupied bands; so, the system is merely 3D. Consequently, the dimensionality of a system depends not only on its geometry, but also on its electron concentration. Let us remind that the conductive / dielectric properties of a sample depends on frequency of electromagnetic waves: so even basic classification of materials is not necessarely intrinsec, but it might depend of the value of some parameters.

## **11. Transverse modes (or magneto-electric subbands)**

We shall discuss now the concept of transverse modes or subbands, which are analogous to the transverse modes of electromagnetic waveguides [11]. In narrow conductors, the different transverse modes are well separated in energy, and such conductors are often called electron waveguides.

We consider a rectangular conductor that is uniform in the *x*−direction and has some transverse confining potential *U*(*y*). The motion of electrons in such a conductor is described by the effective mass eq (131):

$$\left[\mathcal{E}\_s + \frac{\left(i\hbar\nabla + e\mathbf{A}\right)^2}{2m} + \mathcal{U}\left(y\right)\right]\Psi\left(\mathbf{x}, y\right) = \mathcal{E}\Psi\left(\mathbf{x}, y\right) \tag{132}$$

We assume a constant magnetic field *B* in the *z*−direction, perpendicular to the plane of the conductor, which can be represented by a vector potential defined by:

$$A\_X = -By, \; A\_Y = 0 \tag{133}$$

so that the effective-mass equationcan be rewritten as:

$$\left[\mathcal{E}\_{\rm s} + \frac{(p\_{\rm x} + eBy)^2}{2m} + \frac{p\_y^2}{2m} + \mathcal{U}\left(y\right)\right] \Psi\left(x, y\right) = \mathcal{E}\Psi\left(x, y\right) \tag{134}$$

Writing

18 Will-be-set-by-IN-TECH

the same (Helmholtz) equation. So, if the dynamics of a particle, given by the Schrodinger equation, can be considered as the central aspect of quantum mechanics, the scattering of light

least when the Maxwell equations can be reduced to a Helmholtz equation. Remembering Goethe's opinion, that the "Urphänomenon" of light science is the scattering of light on a "turbid" medium, one could remark that his theory of colours is not always as unrealistic as it

As already mentioned, the 2DES, formed at the interface of two semiconductors might play a central role in mesoscopic physics. The thin 2D conduction layer formed in the GaAs/AlGaAs heterojunction may reach a carrier concentration of 2 <sup>×</sup> <sup>10</sup><sup>12</sup> *cm*−<sup>2</sup> and can be depleted by applying a negative voltage to a metalic gate deposited on the surface [Datta]. The mobility can be as high as 106*cm*2/*Vs*, two order of magnitude higher than in bulk semiconductors. The Fermi wavelength *λ<sup>M</sup>* is about 35 nm, and the electron mean free path may be as long as *λ<sup>m</sup>* = 30*μ*m− the same order of magnitude as the liniar dimension of the sample; the ballistic

At low temperature, the conduction in mesoscopic semiconductor is mainly due to electrons

+ *U* (**r**)

where E*<sup>c</sup>* referrs to the conduction band energy, *U* (**r**) is the potential energy due to space-charge etc., **A** is the vector potential and *m* is the effective mass. Any band discontinuity

In the case of a homogenous semiconductor, *U* (**r**) = 0, assuming **A** = 0 and E*<sup>c</sup>* = *const*., the solution of (129) is given by plane waves, exp (*i***k** · **r**), and not by Bloch functions, *uk* (**r**) · exp (*i***k** · **r**). So, the solutions of (129) are not true wavefunctions, but wavefunctions smoothed out over a mesoscopic distance, so any rapid variation at atomic scale is suppressed; eq. (129)

Let us consider a 2DES contained mainly in the *xy* plane. This means that, in the absence of any external potential, the electrons can move freely in the *xy* plane, but they are confined in

The quantization on the *z*−direction, expressed by the functions *φ<sup>n</sup>* (*z*), generate several subbands, with cut-off energy *εn*. At low temperature, only the first subband, corresponding

*ikyy*

the *z*-direction by some potential *V* (*z*), so their wavefunctions will have the form:

to *n* = 1, is occupied, so, instead of (129), the electrons of the 2DES are described by:

+ *U* (**r**)

where the subband energy is E*<sup>s</sup>* = E*<sup>c</sup>* + *ε*1; so, the ‡−dimension enters in this equation only through *ε*1, which depends on the confining potential *V* (*z*). The eq. (131) correctly describes

2

<sup>Ψ</sup> (**r**) <sup>=</sup> *<sup>φ</sup><sup>n</sup>* (*z*) exp (*ikx <sup>x</sup>*) exp

in the conduction band. Their dynamics can be described by an equation of the form:

2

<sup>E</sup>*<sup>c</sup>* <sup>+</sup> (*ih*¯ <sup>∇</sup> <sup>+</sup> *<sup>e</sup>***A**)

2*m*

ΔE*<sup>c</sup>* at heterojunctions is incorporated by letting E*<sup>c</sup>* be position-dependent [11].

can be considered as the central aspect of optics, at

Ψ (**r**) = EΨ (**r**) (129)

(130)

Ψ (**r**) = EΨ (**r**) (102) (131)

−→*r* 

by a medium with refractive index *n*

was generally considered. [18]

**10. Ballistic electrons in 2DESs**

regime of electrons is therefore easily reached.

is called single-band effective mass equation.

<sup>E</sup>*<sup>s</sup>* <sup>+</sup> (*ih*¯ <sup>∇</sup> <sup>+</sup> *<sup>e</sup>***A**)

2*m*

$$\Psi\left(\mathbf{x},\mathbf{y}\right) = \frac{1}{\sqrt{L}} \exp\left(i\mathbf{k}\mathbf{x}\right)\chi\left(\mathbf{y}\right) \tag{135}$$

we get for the transverse function the equation:

$$\left[\mathcal{E}\_s + \frac{(\hbar k + \varepsilon By)^2}{2m} + \frac{p\_y^2}{2m} + \mathcal{U}\left(y\right)\right] \chi\left(y\right) = \mathcal{E}\chi\left(y\right) \tag{136}$$

We are interested in the nature of the transverse eigenfunctions and eigenenergies for different combinations of the confining potential *U*and the magnetic field *B*. A parabolic potential

$$
\mathcal{U}\left(y\right) = \frac{1}{2}m\omega\_0^2 y^2\tag{137}
$$

is often a good description of the actual potential in many electron wave guides.

with *<sup>n</sup>* <sup>−</sup> the unit vector normal to the interface. Analogously, the boundary conditions for an electromagnetic waves at an interface between two dielectrics require the continuity of tangential components *Et*, *Ht* across the interface. Based on these considerations, it is

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 109

For bulk propagation in a homogenous medium, an exact analogy can be drawn between Φ

<sup>∇</sup>2<sup>Φ</sup> <sup>=</sup> <sup>−</sup>*k*2Φ, *<sup>k</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*<sup>m</sup>* (*<sup>E</sup>* <sup>−</sup> *<sup>V</sup>*)

The wave equation (148) is exactly analogous to the Helmholtz equation for an electromagnetic wave propagating in a homogenous dielectric of permittivity *�* and

So, an exact analogy can be drawn between Φ and both *E* or *H*. With these analogies, one can

(E − *<sup>V</sup>*)*<sup>r</sup>* <sup>=</sup> E − *<sup>V</sup>*

is the relative kinetic energy, where *mref* , *Vref* are the effective mass and the potential energy in a reference region. This electron wave phase-refractive index is analogous to the

With these results, phase-propagation effects, such as interference, can be analyzed using

all the Hamiltonians (143). For the electron wave amplitude, the index of refraction is defined

These expressions are exactly the same as the anagous electromagnetic expressions for the reflection/refraction of an electromagnetic wave from an interface between two dielectrics

The theory outlined in this section can be extended for 1D or 2D inhomogenous materials, but

Dragoman and Dragoman [22], [23], [24] obtained a quantum-mechanical - electromagnetic analogy, similar to [19], in the sense that the electronic wave function does not correspond to

*nEM*

*amp*,*<sup>l</sup>* <sup>=</sup> *<sup>m</sup>β*+1/2

*nEW*

*<sup>r</sup>* (*E* − *V*)

E − *Vref*

*ph* is replaced by <sup>Φ</sup>, *<sup>n</sup>EW*

1/2

*<sup>m</sup>αψ* <sup>→</sup> *<sup>A</sup>* (TM wave) or (*ε*/*μ*) (TE wave) (153)

*<sup>r</sup>*,*<sup>l</sup>* (E − *V*)

1/2

and both *E* or *H*. In this case, eq. (143) reduces to a Helmholtz equation of the form:

Φ = *mαψ* (146)

*k*<sup>2</sup> = *ω*2*μ�* (148)

*ph* <sup>=</sup> <sup>√</sup>*μr�<sup>r</sup>* (151)

*<sup>h</sup>*¯ <sup>2</sup> (147)

*<sup>r</sup>* (149)

*ph* . These results are valid for

*<sup>r</sup>*,*<sup>l</sup>* (152)

(150)

reasonable to look for analogies between

permeability *μ*, with Φ replaced by *E* or *H*, and

define a phase-refractive index for electron waves as:

where *mr* = *m*/*mref* is the relative effective mass and

phase-refracting index for electromagnetic waves,

standard em results, where *E* (or *H*), *nEM*

not for three dimensions [19].

the fields, but to the vector potential:

as

[3].

*nEW ph* <sup>=</sup> *<sup>m</sup>*1/2

and either *E* or *H*.

Let us consider the case of confined electrons (*U* �= 0) in zero magnetic field (*B* = 0). Eq. (134) becomes:

$$\left[\mathcal{E}\_s + \frac{\hbar^2 k^2}{2m} + \frac{p\_y^2}{2m} + \frac{1}{2}m\omega\_0^2 y^2\right] \chi(y) = \mathcal{E}\chi(y) \tag{138}$$

with solutions:

$$
\chi\_{n,k}\left(y\right) = u\_n\left(q\right), \ q = \sqrt{\frac{m\omega\_0}{\hbar}}y \tag{139}
$$

$$E\left(n,k\right) = E\_{\rm s} + \frac{\left(\hbar k\right)^2}{2m} + \left(n + \frac{1}{2}\right)\hbar\omega\_0\tag{140}$$

where:

$$u\_n\left(q\right) = \exp\left(-\frac{1}{2}q^2\right)H\_n\left(q\right) \tag{141}$$

with *Hn* - Hermite polynomials. The velocity is obtained from the slope of the dispersion curve:

$$\operatorname{tr}\left(n,k\right) = \frac{1}{\hbar} \frac{\partial E\left(n,k\right)}{\partial k} = \frac{\hbar k}{m} \tag{142}$$

States with different index *n* are said to belong to different subbands; the situation is similar to that described in Sect. 8, where we have discussed the confinement due to the potential *V* (*z*). However, the confinement in the *y*−direction is somewhat weaker, and several subbands are now normally occupied. The subbands are often referred to as *transverse modes*, in analogy with the modes of an electromagnetic waveguide [11].

#### **12. Effective mass approximation revisited**

A more attentive investigation of the effective-mass approximation for electrons in a semiconductor, introduced in a simplified form in Sect.10, will allow quantitative analogies between propagation ballistic electrons and guided electromagnetic waves past abrupt interfaces [19].

According to Morrow and Brownstein [20], out of the general class of Hamiltonians suggested by von Ross [21], only those of the form:

$$H\psi = -\frac{\hbar^2}{2}\left\{ m\left(\overrightarrow{r'}\right)^a \nabla \cdot \left[ m\left(\overrightarrow{r'}\right)^\beta \nabla \left( m\left(\overrightarrow{r'}\right)^a \psi \right) \right] \right\} + V\left(\overrightarrow{r'}\right)\psi = \mathcal{E}\psi \tag{143}$$

with the constraint

$$2\alpha + \beta = -1\tag{144}$$

(where *α* and *β* have specific values for specific substances) can be used in the study of refraction of ballistic electrons at the interface between dissimilar semiconductors.

For the Hamiltonian (143), the boundary conditions for an electron wave at an interface are:

$$m^\mu \psi = \text{continuous} \tag{145}$$

and

$$m^{a+\beta}\nabla\psi \cdot \hat{n} = \text{continuous} \tag{146}$$

with *<sup>n</sup>* <sup>−</sup> the unit vector normal to the interface. Analogously, the boundary conditions for an electromagnetic waves at an interface between two dielectrics require the continuity of tangential components *Et*, *Ht* across the interface. Based on these considerations, it is reasonable to look for analogies between

$$
\Phi = m^\mu \psi \tag{146}
$$

and either *E* or *H*.

20 Will-be-set-by-IN-TECH

Let us consider the case of confined electrons (*U* �= 0) in zero magnetic field (*B* = 0). Eq.

2 2*m* + *n* + 1 2 

 −1 2 *q*2 

with *Hn* - Hermite polynomials. The velocity is obtained from the slope of the dispersion

States with different index *n* are said to belong to different subbands; the situation is similar to that described in Sect. 8, where we have discussed the confinement due to the potential *V* (*z*). However, the confinement in the *y*−direction is somewhat weaker, and several subbands are now normally occupied. The subbands are often referred to as *transverse modes*, in analogy

A more attentive investigation of the effective-mass approximation for electrons in a semiconductor, introduced in a simplified form in Sect.10, will allow quantitative analogies between propagation ballistic electrons and guided electromagnetic waves past abrupt

According to Morrow and Brownstein [20], out of the general class of Hamiltonians suggested

*<sup>β</sup>* <sup>∇</sup> *m* −→*r*

(where *α* and *β* have specific values for specific substances) can be used in the study of

For the Hamiltonian (143), the boundary conditions for an electron wave at an interface are:

refraction of ballistic electrons at the interface between dissimilar semiconductors.

*<sup>α</sup> ψ* 

+ *V* −→*r* 

2*α* + *β* = −1 (144)

*mαψ* = *continous* (145)

*<sup>m</sup>α*+*β*∇*<sup>ψ</sup>* · *<sup>n</sup>* <sup>=</sup> *continuous* (146)

*ψ* = E*ψ* (143)

*∂E* (*n*, *k*)

*<sup>∂</sup><sup>k</sup>* <sup>=</sup> *hk*¯

*h*¯

*mω*<sup>0</sup>

*χ* (*y*) = E*χ* (*y*) (138)

*<sup>h</sup>*¯ *<sup>y</sup>* (139)

*Hn* (*q*) (141)

*<sup>m</sup>* (142)

*h*¯ *ω*<sup>0</sup> (140)

(134) becomes:

with solutions:

where:

curve:

interfaces [19].

with the constraint

and

 E*<sup>s</sup>* +

with the modes of an electromagnetic waveguide [11].

**12. Effective mass approximation revisited**

by von Ross [21], only those of the form:

2 *m* −→*r*

*<sup>α</sup>* ∇ · *m* −→*r*

*<sup>H</sup><sup>ψ</sup>* <sup>=</sup> <sup>−</sup> *<sup>h</sup>*¯ <sup>2</sup>

*h*¯ <sup>2</sup>*k*<sup>2</sup> 2*m*

<sup>+</sup> *<sup>p</sup>*<sup>2</sup> *y* 2*m* + 1 2 *mω*<sup>2</sup> 0*y*2 

*<sup>E</sup>* (*n*, *<sup>k</sup>*) <sup>=</sup> *Es* <sup>+</sup> (*hk*¯ )

*un* (*q*) = exp

*<sup>v</sup>* (*n*, *<sup>k</sup>*) <sup>=</sup> <sup>1</sup>

*χn*,*<sup>k</sup>* (*y*) = *un* (*q*), *q* =

For bulk propagation in a homogenous medium, an exact analogy can be drawn between Φ and both *E* or *H*. In this case, eq. (143) reduces to a Helmholtz equation of the form:

$$
\nabla^2 \Phi = -k^2 \Phi, \; k^2 = \frac{2m \ (E - V)}{\hbar^2} \tag{147}
$$

The wave equation (148) is exactly analogous to the Helmholtz equation for an electromagnetic wave propagating in a homogenous dielectric of permittivity *�* and permeability *μ*, with Φ replaced by *E* or *H*, and

$$k^2 = \omega^2 \mu \epsilon \tag{148}$$

So, an exact analogy can be drawn between Φ and both *E* or *H*. With these analogies, one can define a phase-refractive index for electron waves as:

$$m\_{ph}^{EW} = m\_r^{1/2} \left( E - V \right)\_r^{1/2} \tag{149}$$

where *mr* = *m*/*mref* is the relative effective mass and

$$(\mathcal{E} - V)\_r = \frac{\mathcal{E} - V}{\mathcal{E} - V\_{ref}} \tag{150}$$

is the relative kinetic energy, where *mref* , *Vref* are the effective mass and the potential energy in a reference region. This electron wave phase-refractive index is analogous to the phase-refracting index for electromagnetic waves,

$$m\_{\rm pli}^{EM} = \sqrt{\mu\_r \epsilon\_r} \tag{151}$$

With these results, phase-propagation effects, such as interference, can be analyzed using standard em results, where *E* (or *H*), *nEM ph* is replaced by <sup>Φ</sup>, *<sup>n</sup>EW ph* . These results are valid for all the Hamiltonians (143). For the electron wave amplitude, the index of refraction is defined as

$$m\_{amp,l}^{EW} = m\_{r,l}^{\mathcal{S}+1/2} \left(\mathcal{E} - V\right)\_{r,l}^{1/2} \tag{152}$$

These expressions are exactly the same as the anagous electromagnetic expressions for the reflection/refraction of an electromagnetic wave from an interface between two dielectrics [3].

The theory outlined in this section can be extended for 1D or 2D inhomogenous materials, but not for three dimensions [19].

Dragoman and Dragoman [22], [23], [24] obtained a quantum-mechanical - electromagnetic analogy, similar to [19], in the sense that the electronic wave function does not correspond to the fields, but to the vector potential:

$$m^\mu \psi \to A \text{ (TM wave)} \text{ or } (\varepsilon/\mu) \text{ (TE wave)}\tag{153}$$

**14. Conclusions**

**15. Acknowledgement**

McGraw-Hill, 2002

2433 (1990)

2005

**16. References**

ANCS, received during the elaboration of this chapter.

[2] J.W. Goethe, The Teory of Colours, MIT Press, 1982

[5] K. C. Kao, G. A. Hockham, Proc. IEEE, vol. 113, p. 1151 (1966)

equations and Schrodinger equation, John Wiley & Sons, 2001

[19] G.N. Henderson, T.K. Gaylord, E.N. Glytsis, Phys.Rev. B45, 8404 (1992)

[17] S. Flugge, Practical quantum mechanics, Springer, 1971 [18] A. Zajonc: Catching the light, Oxford University Press, 1993

[20] R.A. Morrow, K.R. Brownstein, Phys.Rev. B30, 687 (1984)

[4] C.K. Kao, Nobel Lecture, December 8, 2009

[7] P. K. Tien, Rev.Mod.Phys. 49, 361 (1977)

[10] V. S. Tsoi, JETP Lett. 19, 70 (1974)

[9] Yu. V. Sharvin, Sov.Phys.JETP 21, 655 (1965)

[13] A. Messiah, Quantum mechanics, Dover (1999) [14] A. Hondros, P. Debye, Ann. Phys. 32, 465 (1910)

[21] O. von Ross, Phys.Rev. B27, 7547 (1983)

Several analogies between electromagnetic and quantum-mechanical phenomena have been analyzed. They rely upon the fact that both wave equation for electromagnetic and electric field with well defined frequency, obtained from the Maxwell equations, and the time-independent Schrodinger equation, have the same form - which is a Helmholtz equation. However, the description of these analogies is by no way a simple dictionary between two formalisms. On the contrary, their physical basis has been discussed in detail, and they have been developed for very modern domains of physics - optical fibers, 2DESs, electron waveguides, electronic transport in mesoscopic and nanoscopic regime. So, the analogies examined in this chapter offers the opportunity of reviewing some very exciting, new and rapidely developing fields of physics, interesting from both the applicative and fundamental perspective. It has been stressed that the analogies are not simple curiosities, but they bear a significant cognitive potential, which can stimultate both scientific understanding and

Waveguides, Resonant Cavities, Optical Fibers and Their Quantum Counterparts 111

technological progress in fields like waveguides, optical fibers, nanoscopic transport.

[3] J.D. Jackson, Classical Electrodynamics, 3rd edition, John Wiley & Sons, 1999

The author acknowledges the financial support of the PN 09 37 01 06 project, granted by

[1] R.J. Black, A. Ankiewicz: Fiber-optic analogies with mechnics, Am.J.Phys. 53, 554 (1985)

[6] T. G. Brown, in: M. Bass, E. W. Van Stryland (eds.): Fiber Optics Handbook,

[8] J. Spector, H.L. Stormer, K.W. Baldwin, L.N. Pfeiffer, K.W. West, Appl. Phys. Lett. 56,

[11] S. Datta, Electronic transport in mesoscopic systems, Cambridge University Press, 1995 [12] H. J. Pain, The Physics of Vibrations and Waves, John Wiley & Sons, Ltd, 6th edition,

[15] A. W. Snyder, J. D. Love: Optical Waveguide Theory, Chapman and Hall, London, 1983 [16] K. Kawano, T. Kioth: Introduction to optical waveguide analysis: solving Maxwell's

#### **13. Optics experiments with ballistic electrons**

In 2DESs, there exists a unique oportunity to control ballistic carriers via electrostatic gates, which can act as refractive elements for the electron path, in complete analogy to refractive elements in geometrical optics. [8], [25], [26]. The refraction of a beam of ballistic electrons can be simply described, using elementary considerations. If in the "left" half-space, the potential has a constant value *V*, and in the "right" one, a different (but also constant) one, *V* + Δ*V*, an incident electron arriving with an incident angle *θ* and kinetic energy

$$\mathcal{E} = \frac{\hbar^2 k^2}{2m^\*} \tag{154}$$

emerges in the "right" half-plane under the angle *θ*� , with the kinetic energy

$$\mathcal{E}' = \mathcal{E} + \varepsilon \Delta V = \frac{\hbar^2 k'^2}{2m^\*} \tag{155}$$

Translational invariance along the interface preserves the parallel component of electron momentum and thus

$$k\sin\theta = k'\sin\theta'\tag{156}$$

or

$$\frac{\sin \theta}{\sin \theta'} = \frac{k'}{k} = \sqrt{\frac{\mathcal{E'}}{\mathcal{E}}} \tag{157}$$

Considering that the energies E, E� are Fermi energies, proportional (in a 2D system) to the electron densities *nel* (not to be confused with refractive index!), the Snell's law takes the form:

$$\frac{\sin \theta}{\sin \theta'} = \sqrt{\frac{n\_{el}'}{n\_{el}}} \tag{158}$$

An electrostatic lens for ballistic electrons was set up in [8] and its focusing action was demonstrated in the GaAs / AlGaAs heterostructure. In this way, the close analogy between the propagation of ballistic electrons and geometrical optics has been put in evidence.

Another nice experiment used a refractive electronic prism to switch a beam of ballistic electrons between different collectors in the same 2DES [25]

The quantum character of ballistic electrons is clearly present, even they are regarded as beams of particles. Transmission and reflection of electrons on a sharp (compared to *λF*), rectangular barrier, induced via a surface gate at *T* = 0.5*K*, follow the laws obtained in quantum mechanics, for instance the following expression for the transmission coefficient:

$$T = \frac{1}{1 + 0.25\left(\frac{k}{k\_0} - \frac{k\_0}{k}\right)^2 \sin^2\left(ka\right)}$$

with *k*, *k*<sup>0</sup> - the wavevectors inside and outside the barrier, and *a* - its width (compare the previous formula with the results of III.I.7 of [13]). [27]
