External Fields

Abstract

The effects of external electric and magnetic fields on the electronic and optical properties of bulk materials and quantum wells are discussed including the Stark effect and quantum-confined Stark effect, the Hall effect and Quantized Hall Effects. The energy levels of the solid and its optical and electronic properties depend on external electric and magnetic fields. In high magnetic fields and at low temperatures the quantum Hall effects give evidence for new states of matter in many-body systems.

1 Electric Fields

1.1 Bulk Material

The center-of-mass motion of the exciton is not influenced by a homogeneous electric field. The Hamilton operator for the relative motion of an electron-hole pair of reduced mass $\mu$ along z in the presence of an electric field E along the z direction is

$$\begin{aligned} \hat{H}=-\frac{\hbar ^2}{2\mu } \, \Delta -e\, E \,z \;. \end{aligned}$$

(15.1)

Here, the Coulomb interaction, leading to the formation of bound exciton states, is neglected. In the plane perpendicular to the field (here the z direction) the solutions for the relative motion are just plane waves.

In the electric field the bands are tilted (Fig. 15.1), i.e. there is no longer an overall band gap. Accordingly, the wavefunctions are modified and have exponential tails in the energy gap.

Fig. 15.1

Impact of an electric field on a bulk material (tilt of bands) and b a quantum well (quantum confined Stark effect, QCSE)

After separation of the motion in the (x,y) plane the Schrödinger equation for the motion in the z direction is

$$\begin{aligned} \left( -\frac{\hbar ^2}{2\mu } \, \frac{\mathrm {d}^2}{\mathrm {d} z^2} - e \, E \, z - E_z \right) \phi (z)=0 \;, \end{aligned}$$

(15.2)

which is of the type

$$\begin{aligned} \frac{\mathrm {d}^2f(\xi )}{\mathrm {d} \xi ^2}-\xi \, f(\xi )=0 \;, \end{aligned}$$

(15.3)

with $\xi =\frac{E_z}{\Theta }-z\left( \frac{2\mu }{\hbar ^2}\,e\,E\right) ^{1/3}$ and the optoelectronic energy $\Theta =\left( \frac{e^2E^2\hbar ^2}{2\mu }\right) ^{1/3}$. The solution of (15.3) is given by the Airy function Ai (cf. Fig. 15.2):

$$\begin{aligned} \phi _{E_z}(\xi )=\frac{\sqrt{e \, E}}{\Theta } \, Ai(\xi ) \;. \end{aligned}$$

(15.4)

The pre-factor guaranties the orthonormality (with regard to the $E_z$). The absorption spectrum is then given by

$$\begin{aligned} \alpha (\omega ,E) \propto \frac{1}{\omega } \, \sqrt{\Theta } \, \pi \left[ Ai^{'2}(\eta )-\eta \, Ai^2(\eta )\right] \;, \end{aligned}$$

(15.5)

with $\eta =(E_{\mathrm {g}}-E)/ \Theta$ and $Ai'(x)=\mathrm {d}Ai(x)/\mathrm {d}x$.

Fig. 15.2

a Airy function Ai(x), b Ai’(x)

Optical transitions below the band gap become possible that are photon-assisted tunneling processes. The below-bandgap transitions have the form of an exponential tail. Additionally, oscillations develop above the band gap, the so-called Franz-Keldysh oscillations (FKO) [1417, 1418] (Fig. 15.3a).

The absorption spectrum scales with the optoelectronic energy $\Theta$. The energy position of the FKO peaks $E_n$ is periodic with ($\nu \sim 0.5$)

$$\begin{aligned} \left( E_n-E_{\mathrm {g}} \right) ^{3/2} \propto (n-\nu )\, E \, \sqrt{\mu } \;. \end{aligned}$$

(15.6)

A more detailed theory including interband coupling and excitonic effects can be found in [1419]. The effect is best determined with modulation spectroscopy can measuring the difference of absorption with and without field [1420] since it exhibits more defined features (Fig. 15.3a, lower panel).

A nonperiodicity can indicate a nonparabolicity of the mass. Also, the contributions of heavy and light holes merge. For a given mass the electric field strength can be determined. Well-pronounced oscillations are only present for homogeneous fields.

Fig. 15.3

Adapted from [1421]

a Theoretical absorption (top panel) with (solid line) and without (dash-dotted line) electric field for a volume semiconductor (without Coulomb interaction) and theoretical change of absorption (bottom panel). b Experimental absorption spectra of (In,Ga)As on InP at $T=15$ K for various applied voltages as labeled.

Experimental spectra show additionally the peaks due to excitonic correlation (Fig. 15.3b) at low field strength. At higher fields the FKO evolve and the amplitude of the excitonic peaks decreases because the excitons are ionized in the field.

1.2 Quantum Wells

 

In a quantum well an electric field along the confinement direction (z direction) causes electrons and holes to shift their mean position to opposite interfaces (Fig. 15.1b). However, excitons are not ionized due to the electric field. With increasing field (for both field directions) the energy position of the absorption edge and the recombination energy is reduced. This is the quantum confined Stark effect (QCSE). Corresponding experimental data are shown in Fig. 15.4i–v. The shift depends quadratically on the electric field since the exciton has no permanent dipole moment (mirror symmetry of the quantum well). Thus, only the second-order Stark effect is present (as for the hydrogen atom) in which the field first induces a dipole $\mathbf{p}=\alpha \mathbf{E}$. This dipole interacts with the field with an energy $E=-\mathbf{p}\cdot \mathbf{E}=-\alpha \mathbf{E}^2$. The carrier separation in opposite sides of the quantum well (Fig. 15.4b) leads to a reduced overlap of the electron and hole wavefunctions and subsequently to an increased recombination lifetime (see Fig. 12.43).

Fig. 15.4

Adapted from [1422]

Impact of electric fields on the absorption spectrum of $n\times (9.5$ nm GaAs/9.8 nm Al$_{0.32}$Ga$_{0.68}$As) multiple quantum well structures. a Electric field along the [001] growth direction ($n=50$), (i)–(v) $E=0$, 0.6, 1.1, 1.5, and $2\times 10^5$ V/cm. b Electric field within the interface plane (n $=$ 60), (i, ii, iii) $E=0$, 1.1, and 2$\times 10^5$ V/cm.

If the field is within the quantum-well interface plane, the field leads to the ionization of excitons without shift of the energy position. The loss of the excitonic peak is visualized in the spectra in Fig. 15.4a–c.

2 Magnetic Fields

 

In magnetic fields, electrons (or holes) perform a cyclotron motion with frequency $\omega _{\mathrm {c}}=eB/m^*$, i.e. a motion perpendicular to the magnetic field on a line of constant energy in $\mathbf{k}$-space. This line is the intersection of a plane perpendicular to the magnetic field and the respective isoenergy surface in $\mathbf{k}$-space. For semiconductors with anisotropic mass, such as Si and Ge, the quantum theory of cyclotron resonance has been given in [1423]. The physics of semiconductors in magnetic fields is covered in detail in [1424].

The ballistic cyclotron motion can only occur between two scattering events. Thus, a significantly long path along the cyclotron trajectory (classically speaking) and the connected magnetotransport properties are only possible when

• $\omega _{\mathrm {c}} \tau \gg 1$, i.e. when the average scattering time $\tau$ is sufficiently large. This requires high mobility.

• the magnetic field is sufficiently strong and the temperature sufficiently low, i.e. $\hbar \omega _{\mathrm {c}} \gg kT$, such that thermal excitations do not scatter electrons between different Landau levels.

• the cyclotron path is free of geometric obstructions.

An external magnetic field also produces a Zeeman-like splitting of the spin states. For the electron, the energy splitting $\Delta E$ is given by

$$\begin{aligned} \Delta E = g_{\mathrm {e}}^* \, \mu _{\mathrm {B}} \, B \;, \end{aligned}$$

(15.7)

where B is the magnetic-field amplitude and $g_{\mathrm {e}}^*$ the (effective) electron g-factor. This value differs from the free-electron value in vacuum of $g_{\mathrm {e}}=2.0023$ due to the presence of spin-orbit interaction (see Sect. 15.2.3). Values for $g_{\mathrm {e}}^*$ at low carrier density and low temperatures are 2 for Si, 1.2 for InP and ZnSe, $-1.65$ for CdTe, $-0.44$ for GaAs, $-15$ in InAs, and $-50$ for InSb. In [1425] the temperature dependence of $g_{\mathrm {e}}^*$ in GaAs, InP and CdTe is also measured and discussed. The electron g-factor increases in thin GaAs/(Al,Ga)As quantum wells [1426].

2.1 Classical Hall Effect

 

An electrical current along the x (longitudinal) direction in a perpendicular magnetic field $\mathbf{B}=(0,0,B)$ along z causes an electric field $E_y$ along the transverse (y) direction (Fig. 15.5). The charge accumulation is due to the Lorentz force. The related transverse voltage is called the Hall voltage and the resistivity $\rho _{xy}=E_y/j_x$ the Hall resistivity [28, 31, 32]. Many aspects of the Hall effect are discussed in [1427]. For thin-film samples typically Hall bars [1428] (see Fig. 15.6 and also Fig. 15.19, for a reasonable measurement of the Hall voltage the ratio of length and width of the Hall bar should be at least 3) or the van-der-Pauw geometry (Fig. 15.7) and method are used [1429–1431].

Fig. 15.5

Scheme of the Hall-effect geometry. The movement of one electron in the longitudinal electric current I is shown schematically. The coordinate system (x, y, z) and the directions of the magnetic field $\mathbf{B}$, the drift velocity of an electron $\mathbf{v}$ and the resulting Lorentz force $\mathbf{F}$ are given. The transverse field $E_y$ is given by $V_{\mathrm {H}}/w$

Fig. 15.6

Hallbar geometry. The current I is driven by a source from contact 1 to 2. The Hall voltage is measured between contacts 3 and 4, the longitudinal voltage between 3 and 5

Fig. 15.7

a–d Geometry for van-der-Pauw Hall measurements. a Best geometry (cloverleaf), b acceptable square geometry with small contacts on the corners, c,d not recommended geometries with contacts on the edge centers or inside the square, respectively. e Current distribution, as visualized by lock-in thermography [1432], in epitaxial ZnO layer on sapphire with Hall geometry as in part b. Grey dashed line indicates the outline of the $10 \times 10$ mm$^2$ substrate, grey areas indicate gold ohmic contacts

For band transport in the relaxation time approximation (Chap. 8.2), the steady-state equation of motion is (in the vicinity of an isotropic extremum)

$$\begin{aligned} m^* \frac{\mathbf{v}}{\tau } = q \, \left( \mathcal {\mathbf{E}} + \mathbf{v} \times \mathbf{B} \right) \;. \end{aligned}$$

(15.8)

We note that this equation of motion is also valid for holes, given the convention of Sect. 6.10.1, i.e. positive effective mass and charge. With the cyclotron frequency $\omega _{\mathrm {c}}=qB/m^*$ the conductivity tensor is ($\mathbf{j}=qn\mathbf{v}=\varvec{\sigma }\,\mathbf{E}$)

$$\begin{aligned} \varvec{\sigma }= & {} \left( \begin{array}{ccc} \sigma _{xx} &{} \sigma _{xy} &{} 0 \\ \sigma _{yx} &{} \sigma _{yy} &{} 0 \\ 0 &{} 0 &{} \sigma _{zz} \end{array} \right) \end{aligned}$$

(15.9a)

$$\begin{aligned} \sigma _{xx}=\sigma _{yy}= & {} \sigma _0 \,\frac{1}{1+\omega _{\mathrm {c}}^2 \, \tau ^2}=\sigma _0 \, \frac{1}{1+\mu ^2 \, B^2} \end{aligned}$$

(15.9b)

$$\begin{aligned} \sigma _{xy}=-\sigma _{yx}= & {} \sigma _0 \, \frac{\omega _{\mathrm {c}}\tau }{1+\omega _{\mathrm {c}}^2 \, \tau ^2}=\sigma _0 \, \frac{\mu \, B}{1+\mu ^2 \, B^2} \end{aligned}$$

(15.9c)

$$\begin{aligned} \sigma _{zz}= & {} \sigma _0=\frac{q^2 \,n \,\tau }{m^*}=q \,n \,\mu \;. \end{aligned}$$

(15.9d)

Perpendicular to the magnetic field, the conductivity ($\sigma _{zz}$) is given by (8.5). If only one type of carrier (charge q, density n) is considered, the condition $j_y=0$ leads to $E_y=\mu B\,E_x$ and $j_x=\sigma _0\,E_x$. The Hall coefficient is defined as $R_{\mathrm {H}}=E_y/(j_x\,B)$ or more precisely as

$$\begin{aligned} R_{\mathrm {H}}=\frac{\rho _{xy}}{B} \;, \end{aligned}$$

(15.10)

where the resistivity tensor $\varvec{\rho }$ is the inverse of the conductivity tensor $\varvec{\sigma }$,

$$\begin{aligned} \varvec{\rho }=\varvec{\sigma }^{-1}= & {} \left( \begin{array}{ccc} \rho _{xx} &{} \rho _{xy} &{} 0 \\ \rho _{yx} &{} \rho _{yy} &{} 0 \\ 0 &{} 0 &{} \rho _{zz} \end{array} \right) \end{aligned}$$

(15.11a)

$$\begin{aligned} \rho _{xx}=\rho _{yy}= & {} \frac{\sigma _{xx}}{\sigma _{xx}^2+\sigma _{xy}^2} \end{aligned}$$

(15.11b)

$$\begin{aligned} \rho _{xy}=-\rho _{yx}= & {} \frac{\sigma _{xy}}{\sigma _{xx}^2+\sigma _{xy}^2} \end{aligned}$$

(15.11c)

$$\begin{aligned} \rho _{zz}= & {} \frac{1}{\sigma _{zz}} = \frac{1}{\sigma _0} \;. \end{aligned}$$

(15.11d)

For a single type of carriers, the Hall coefficient is therefore given by

$$\begin{aligned} R_{\mathrm {H}}=\frac{\mu }{\sigma _0} = \frac{1}{q \, n} \;. \end{aligned}$$

(15.12)

It is negative (positive) for electron (hole) conduction. We note that electrons and holes are deflected in the same y-direction by the magnetic field and collect at the same electrode. Thus the Hall effect allows the determination of the carrier type and the carrier density.¹

If both types of carriers are present simultaneously, the conductivity (two-band conduction) is given by the sum of electron and hole conductivity (8.11),

$$\begin{aligned} \varvec{\sigma } = \varvec{\sigma }_{\mathrm {e}} + \varvec{\sigma }_{\mathrm {h}}\;. \end{aligned}$$

(15.13)

The Hall constant (15.10) is then

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{e} \, \frac{-n \, \mu _{\mathrm {e}}^2\, (1+\mu _{\mathrm {h}}^2\, B^2)+p \, \mu _{\mathrm {h}}^2\, (1+\mu _{\mathrm {e}}^2\, B^2)}{n^2 \, \mu _{\mathrm {e}}^2\, (1+\mu _{\mathrm {h}}^2\, B^2)-2\, n\, p \, \mu _{\mathrm {e}} \, \mu _{\mathrm {h}}\, (1+\mu _{\mathrm {e}} \, \mu _{\mathrm {h}} \, B^2)+p^2 \, \mu _{\mathrm {h}}^2\, (1+\mu _{\mathrm {e}}^2\, B^2)} \;. \end{aligned}$$

(15.14)

Under the assumption of small magnetic fields,² i.e. $\mu B\ll 1$, the Hall coefficient is

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{e} \, \left[ \frac{-n \, \mu _{\mathrm {e}}^2+p\, \mu _{\mathrm {h}}^2}{(-n \, \mu _{\mathrm {e}}+p \, \mu _{\mathrm {h}})^2}+\frac{n \, p \, (-n+p) \, \mu _{\mathrm {e}}^2 \, \mu _{\mathrm {h}}^2\, (\mu _{\mathrm {e}}-\mu _{\mathrm {h}})^2}{(-n \, \mu _{\mathrm {e}}+p \, \mu _{\mathrm {h}})^4}\;B^2 + \ldots \right] \;. \end{aligned}$$

(15.15)

For small magnetic field this can be written as

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{e} \, \frac{p-n \, \beta ^2}{(p-n \, \beta )^2} \;, \end{aligned}$$

(15.16)

with $\beta =\mu _{\mathrm {e}}/\mu _{\mathrm {h}}<0$. For large magnetic fields, i.e. $\mu B \gg 1$, the Hall coefficient is given by

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{e} \, \frac{1}{p-n} \;. \end{aligned}$$

(15.17)

In Fig. 15.8, the absolute value of the Hall coefficient for InSb samples with different doping concentrations is shown. The p-doped samples exhibit a reverse of the sign of the Hall coefficient upon increase of temperature when intrinsic electrons contribute to the conductivity. The zero in $R_{\mathrm {H}}$ occurs for $n=p\,\mu _{\mathrm {h}}^2/\mu _{\mathrm {e}}^2=n_{\mathrm {i}}/|\beta |$. For high temperatures, the Hall coefficient for n- and p-doped samples is dominated by the electrons that have much higher mobility (Table 8.2).

Fig. 15.8

Adapted from [1433]

a Conductivity and b absolute value of the Hall coefficient versus inverse temperature for four p-doped (A–D) and two n-doped (E, F) InSb samples. The doping levels are given in (a).

The simultaneous conduction in a band and an impurity band (cmp. Sect. 8.6) has been separated with a suitable model assuming two conduction channels for holes [1434] (Fig. 15.9).

Fig. 15.9

Adapted from [1434]

Carrier densities in valence band (circles) and impurity band (triangles) from evaluating the Hall effect on GaAs doped with lithium (and annealed) taking into account two conduction channels.

In (15.8) an isotropic mass was considered. For semiconductors with multiple valleys and anisotropic extrema, in particular the conduction bands of Si and Ge (cf. Sect. 6.9.2), the Hall coefficient has been derived in [1435, 1436]. With $K=m_{\mathrm {l}}/ m_{\mathrm {t}}$ being the mass anisotropy (cmp. Table 6.5), the Hall coefficient (15.12) changes to,

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{q \, n} \, \frac{3\,K\,(K+2)}{(2K+1)^2} \;. \end{aligned}$$

(15.18)

In the derivation of the (unipolar) Hall coefficient we had assumed that all carriers involved in the transport have the same properties, in particular that they are subject to the same scattering time. This assumption is generally not the case (cmp. Chap. J) and we need to operate with the ensemble average of the discussed quantities. The ensemble average of an energy-dependent quantity $\zeta (E)$ over the (electron) distribution function f(E) is denoted as $\langle \zeta \rangle$ and is given as³

$$\begin{aligned} \langle \zeta \rangle = \frac{\int \zeta (e) \, f(E) \, \mathrm {d}E}{\int f(E) \, \mathrm {d}E} \;. \end{aligned}$$

(15.19)

In particular, the average $\langle \tau \rangle ^2$ is now different from $\langle \tau ^2\rangle$. Considering the equation $\langle \mathbf{j}\rangle =\langle \varvec{\sigma }\rangle \,\mathbf{E}$ for the ensemble-averaged current density we find (for one type of carrier, cf. (15.12))

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{q \, n} \, r_{\mathrm {H}} \;, \end{aligned}$$

(15.20)

with the so-called Hall factor $r_{\mathrm {H}}$ given by

$$\begin{aligned} r_{\mathrm {H}}= & {} \frac{\gamma }{\alpha ^2 + \omega _{\mathrm {c}}^2 \, \gamma ^2} \;,\nonumber \\ \alpha= & {} \left\langle \frac{\tau }{1+\omega _{\mathrm {c}}^2 \, \tau ^2} \right\rangle \;,\quad \gamma = \left\langle \frac{\tau ^2}{1+\omega _{\mathrm {c}}^2 \, \tau ^2} \right\rangle \;. \end{aligned}$$

(15.21)

The Hall factor depends on the scattering mechanisms and is of the order of 1. For large magnetic fields the Hall factor approaches 1. For small magnetic fields we have

$$\begin{aligned} R_{\mathrm {H}}=\frac{1}{q \, n}\,\frac{\langle \tau ^2 \rangle }{\langle \tau \rangle ^2} \;. \end{aligned}$$

(15.22)

The mobility calculated from (cf. (15.9d)) $\sigma _0 R_{\mathrm {H}}$ is called the Hall mobility $\mu _{\mathrm {H}}$ and is related to the mobility via

$$\begin{aligned} \mu _{\mathrm {H}}= r_{\mathrm {H}} \, \mu \;. \end{aligned}$$

(15.23)

It is assumed so far that the free carrier density and mobility are homogeneous within the volume of current transport. Multi-layer models can be fitted to experimental Hall data in order to account for different conduction channels in different layered parts of the sample [1437]. E.g., in a two-layer model, contributions from bulk and surface/interface conduction can be separated [1438–1440].

The magnetic field dependence of $\varvec{\sigma }$ can be used in a general case to separate contributions of carriers with different density and mobility (including its sign) without assumptions and obtain the mobility spectrum $s(\mu )$ (MSA, mobility spectral analysis),

$$\begin{aligned} \sigma _{xx}= & {} \int \limits _{-\infty }^{\infty } s(\mu ) \, \frac{1}{1+\mu ^2 B^2} \, \mathrm {d}\mu \end{aligned}$$

(15.24a)

$$\begin{aligned} \sigma _{xy}= & {} \int \limits _{-\infty }^{\infty } s(\mu ) \, \frac{\mu B}{1+\mu ^2 B^2} \, \mathrm {d}\mu \;, \end{aligned}$$

(15.24b)

as a generalization of (15.13), (15.9b) and (15.9c) [1441–1443]. Examples are the separation of electron conductivity in (GaAs-) $\Gamma$- and (InAs-) X-Minima in a GaAs/(Al,Ga)As/InAs double quantum well structures [1441] (Fig. 15.10a), substrate and 2DEG electron conductivity in (Al,Ga)N/GaN heterostructures [1444], and electrons and holes in an InAs/GaSb quantum well [1442] (Fig. 15.10b).

Fig. 15.10

Adapted from [1442]

a Mobility spectrum of GaAs/(Al,Ga)As/InAs double quantum well structure. Adapted from [1441]. b Mobility spectrum of InAs/GaSb quantum well structure.

In the case of hopping conduction (Chap. 8.8), the theory of Hall effect is more involved [1445, 1446]. If carriers are transported by hopping, generally, they may not be free to move as expected by (15.8) in response to the applied magnetic field and the Lorentz force. A Hall effect occurs only at the junction of three (or more) hopping sites [1445]. The sign of the experimentally determined Hall coefficient is often opposite to the one expected from the carrier type according to (15.12), e.g., as studied in a-Si [1447, 1448]. The sign anomaly depends on intricacies such as the local site geometry and interference processes taking place among bonding and antibonding orbitals of various numbers as summarized in [1449].

2.2 Free-Carrier Absorption

 

The absorption of free carriers was treated in Sect. 9.9.1 without the presence of a static magnetic field. Solving (15.8) for a static magnetic field $\mathbf{B}=\mu _0\,\mathbf{H}$ with $\mathbf{H}=H\,(h_x,h_y,h_z)$ and a harmonic electric field $\mathbf{E}\propto \exp (-i\omega t)$ yields for the dielectric tensor (cf. (9.72))

$$\begin{aligned} \varvec{\epsilon }=\frac{i}{\epsilon _0 \, \omega } \, \varvec{\sigma } \;, \end{aligned}$$

(15.25)

and by comparing to $\mathbf{j}= \varvec{\sigma } \, \mathbf{E} = q \, n \, \mathbf{v}$,

$$\begin{aligned} \varvec{\epsilon }(\omega )=- \omega _{\mathrm {p}}^{*2} \left[ (\omega ^2 + i\, \omega \, \gamma ) \, \mathbf{1} -i\, \omega _{\mathrm {c}}^2 \left( \begin{array}{ccc} 0 &{} -h_z &{} h_y \\ h_z &{} 0 &{} -h_x \\ -h_y &{} h_x &{} 0 \end{array} \right) \right] ^{-1} \;, \end{aligned}$$

(15.26)

where $\mathbf{1}$ denotes the (3 by 3) unity matrix and $\gamma =1/\tau =q/(m^*\mu )$ is the damping parameter with $\mu$ representing the optical carrier mobility (in the non-isotropic case a tensor $\varvec{\gamma }$ needs to be used). The (unscreened) plasma frequency is given by (cmp. (9.77))

$$\begin{aligned} \omega _{\mathrm {p}}^{*} = \sqrt{n \, \frac{e^2}{\epsilon _0 \, m^*}} \;. \end{aligned}$$

(15.27)

The free-carrier cyclotron frequency is

$$\begin{aligned} \omega _{\mathrm {c}} = e \, \frac{\mu _0\,H}{m^*} \;. \end{aligned}$$

(15.28)

If the effective mass is treated as a tensor, $1/m^*$ is replaced by $\mathbf{m^*}^{-1}$ in (15.27) and (15.28). For zero magnetic field the classical Drude theory for one carrier species is recovered (cf. (9.74a))

$$\begin{aligned} \mathbf{\epsilon }(\omega )=- \frac{\omega _{\mathrm {p}}^{*2}}{\omega \, (\omega + i\gamma )} \;. \end{aligned}$$

(15.29)

With the magnetic field perpendicular to the sample surface, i.e. $\mathbf{B}=\mu _0 \, (0,0,H)$ the magneto-optic dielectric tensor simplifies to (cf. (15.9d))

$$\begin{aligned} \mathbf{\epsilon }(\omega )= & {} - \frac{\omega _{\mathrm {p}}^{*2}}{\omega ^2} \left( \begin{array}{ccc} \tilde{\epsilon }_{xx} &{} i\tilde{\epsilon }_{xy} &{} 0 \\ -i\tilde{\epsilon }_{xy} &{} \tilde{\epsilon }_{xx} &{} 0 \\ 0 &{} 0 &{} \tilde{\epsilon }_{zz} \end{array} \right) \end{aligned}$$

(15.30a)

$$\begin{aligned} \tilde{\epsilon }_{xx}= & {} \frac{1+i\gamma /\omega }{(1+i\gamma /\omega )^2-(\omega _{\mathrm {c}}/\omega )^2}\end{aligned}$$

(15.30b)

$$\begin{aligned} \tilde{\epsilon }_{zz}= & {} \frac{1}{(1+i\gamma /\omega )}\end{aligned}$$

(15.30c)

$$\begin{aligned} \tilde{\epsilon }_{xy}= & {} \frac{\omega _{\mathrm {c}}/\omega }{(1+i\gamma /\omega )^2-(\omega _{\mathrm {c}}/\omega )^2} \;. \end{aligned}$$

(15.30d)

The in-plane component $\epsilon _{xx}$ provides information about $\omega _{\mathrm {p}}^*$ and $\gamma$, i.e. two of the three parameters n, $\mu$ and $m^*$ are known. Additionally, the antisymmetric tensor component $\epsilon _{xy}$ is linear in the cyclotron frequency and provides $q/m^*$. This subtle but finite birefringence depends on the strength (and orientation) of the magnetic field and can be experimentally determined in the infrared using magneto-ellipsometry [1450, 1451]. Such ‘optical Hall effect’ experiment allows the determination of the carrier density n, the mobility $\mu$, the carrier mass⁴  $m^*$ and the sign of the carrier charge $\mathrm {sgn}(q)$ with optical means. The electrical Hall effect (Sect. 15.2.1) can reveal n, $\mu$ and $\mathrm {sgn}(q)$ but cannot reveal the carrier mass.

2.3 Energy Levels in Bulk Crystals

In a 3D electron gas (the magnetic field is along z, i.e. $\mathbf{B}=B\,(0,0,1)$) the motion in the (x, y) plane is described by Landau levels. Quantum mechanically they correspond to levels of a harmonic oscillator. The magnetic field has no impact on the motion of electrons along z, such that in this direction a free dispersion relation $\propto k_z^2$ is present. The energy levels are given as

$$\begin{aligned} E_{nk_z}=\left( n+\frac{1}{2} \right) \hbar \omega _{\mathrm {c}}+\frac{\hbar ^2}{2m} \, k_z^2 \;. \end{aligned}$$

(15.31)

Thus, the states are on concentric cylinders in $\mathbf{k}$-space (Fig. 15.11a). The populated states of the 3D electron gas (at 0 K) lie within the Fermi vector of length $k_{\mathrm {F}}$. For the 3D system the density of states at the Fermi energy is a square root function of the Fermi energy (6.71). In the presence of a magnetic field the density of states diverges every time that a new cylinder (with a one-dimensional density of states, (6.79)) touches the Fermi surface at $E_{\mathrm {F}}$. In real systems, the divergence will be smoothed, however, a pronounced peak or the periodic nature of the density of states is often preserved.

Fig. 15.11

Based on [1123]

3D electron gas in an external magnetic field. a Allowed states in $\mathbf{k}$-space for magnetic field along the z direction. b Density of states (DOS) $\rho$ versus energy (in units of $\hbar \omega _{\mathrm {c}}$). Dashed line is three-dimensional DOS without magnetic field.

The period is given by the number $n_{\mathrm {m}}$ of cyclotron orbits (Landau levels) within the Fermi surface.

$$\begin{aligned} \left( n_{\mathrm {m}}+\frac{1}{2} \right) \hbar \omega _{\mathrm {c}}= E_{\mathrm {F}} \;. \end{aligned}$$

(15.32)

If the number of carriers is constant, the density of states at the Fermi energy at varying magnetic field varies periodically with 1/B. From the conditions $(n_{\mathrm {m}}+\frac{1}{2})\hbar e B_1/m =E_{\mathrm {F}}$ and $(n_{\mathrm {m}}+1+\frac{1}{2})\hbar eB_2/m=E_{\mathrm {F}}$ with $1/B_2=1/B_1+1/\Delta B$ we find,

$$\begin{aligned} \frac{1}{\Delta B}=\frac{e \, \hbar }{m^* \, E_{\mathrm {F}}} \;. \end{aligned}$$

(15.33)

This periodicity is used to determine experimentally, e.g., the properties of the Fermi surface in metals using the Shubnikov-de Haas oscillations (of the magnetoresistance) or the de Haas-van Alphén effect (oscillation of the magnetic susceptibility).

Equation (15.31) needs to be extended to account for the splitting (15.7) of the Landau level due to the electron spin. According to [1452], the electron Landau level energy can be written as

$$\begin{aligned} E_n =\left( n+\frac{1}{2} \right) \, \frac{\hbar \, e \, B}{m^*(E)} \pm g^*_{\mathrm {e}}(E) \, \mu _{\mathrm {B}} \, B \;, \end{aligned}$$

(15.34)

with energy dependent effective mass and g-factor

$$\begin{aligned} \frac{1}{m^*(E)}= & {} \frac{1}{m^*(0)}\,\frac{E_{\mathrm {g}}\,(E_{\mathrm {g}}+\Delta _0)}{3E_{\mathrm {g}}+2\Delta _0}\,\left( \frac{2}{E+E_{\mathrm {g}}} + \frac{1}{E+E_{\mathrm {g}}+\Delta _0} \right) \end{aligned}$$

(15.35a)

$$\begin{aligned} g^*_{\mathrm {e}}(E)= & {} g^*_{\mathrm {e}}(0) \,\frac{E_{\mathrm {g}}(E_{\mathrm {g}}+\Delta _0)}{\Delta _0} \left( \frac{1}{E+E_{\mathrm {g}}} - \frac{1}{E+E_{\mathrm {g}}+\Delta _0}\right) \end{aligned}$$

(15.35b)

The band edge value $m^*(0)$ of the effective mass is given by (6.43) and that of the g-factor by

$$\begin{aligned} g^*_{\mathrm {e}}(0) = 2 \, \left[ 1-\frac{2 \,{\Delta }_0}{3 \, E_{\mathrm {g}}\,(E_{\mathrm {g}}+{\Delta }_0)}\,E_{\mathrm {P}}\right] \;. \end{aligned}$$

(15.36)

For large spin-orbit splitting, the value of the g-factor deviates strongly from 2 and becomes negative.

2.4 Magnetic Field Effect on Impurities

 

The Zeeman effect is the magnetic field dependence (and the lifting of degeneracies) of spectral lines. For semiconductor physics this is a tool to investigate properties of defect related states with sufficiently sharp spectral features. As example we refer to the investigation of the Si:P donor system (cmp. Fig. 9.33) and its linear and quadratic Zeeman effect [1453] as visualized in Fig. 15.12. The difference in transition energies for lines 4 and 1 depends linearly on the magnetic field and is equal to the electron spin splitting. The quadratic Zeeman effect is also termed ‘diamagnetic shift’ and allows a determination of the wavefunction size. In the present case, the electron Bohr radius at the neutral phosphorus donor in Si was found to be $a_{\mathrm {D}}=1.33(5)$ nm (cmp. (7.22)) [1453].

Fig. 15.12

Adapted from [1453]

a Schematic level scheme and transitions (and their polarization) of donor bound excitons at Si:P. b Experimental magnetic field dependent shift of photoluminescence transition energies (from about $E=1.150$ eV at $B=0$ T) (symbols) together with fits for heavy (dashed lines) and light (solid lines) hole transitions.

2.5 Magnetic Field Effect on Excitons

 

The effect of an magnetic field on excitons is similar to that on the wave function at an impurity, except that the electron and hole are effected simultaneously. The effect is two-fold: The Zeeman-like effect on the spin states leads to a splitting of the exciton lines which become circularly polarized; for ‘weak’ fields the splitting is linear with the field. Additionally, the magnetic field leads to a reduction of exciton size leading to a quadratic shift of the line center to larger energies, the so-called diamagnetic shift. Calculations for various confinement geometries have been given in [1454]. The change of exciton size leads also to modification of localization effects in disordered quantum wells [1455]. As example the magnetic field dependence of quasi two-dimensional excitons in a WS$_2$ monolayer is shown in Fig. 15.13 (cf. Sect. 13.2.4).

Fig. 15.13

Adapted from [1313], reprinted under Creative Commons Attribute (CC BY 4.0) license

Magnetic field effect on excitons in WS$_2$ monolayer. a Normalized transmission spectra. b Energy position of $E_{\sigma ^+}$ and $E_{\sigma ^-}$ of (circularly polarized) spectral lines of the 1–5 s exciton lines. c Calculated energy $(E_{\sigma ^+}+E_{\sigma ^-})/2$, showing the diamagnetic shift. Solid lines are model calculation with reduced exciton mass $m_{\mathrm {r}}^*=0.175$ (and further parameters), d Zeeman splitting $E_{\sigma ^+}-E_{\sigma ^-}$ of the 1–3 s states together with linear fits (dashed line), g-factors are around 4, as labelled.

2.6 Energy Levels in a 2DEG

 

In a 2D electron gas (2DEG), e.g., in a quantum well or a potential well at a modulation-doped heterointerface , a free motion in z is not possible and $k_z$ is quantized. The energy levels (for each 2D subband) are only given by the cyclotron energy (Fig. 15.14a). The density of states is a sequence of $\delta$-like peaks (Fig. 15.14b). Each peak contributes (degeneracy $\hat{g}$ of a Landau level) a total number of

$$\begin{aligned} \hat{g}=\frac{e \, B}{h} \end{aligned}$$

(15.37)

states (per unit area without spin degeneracy and without the degeneracy of the band extremum). In reality, disorder effects lead to an inhomogeneous broadening of these peaks. The states in the tails of the peaks correspond to states that are localized in real space.

Fig. 15.14

Based on [1123]

2D electron gas in an external magnetic field. a Allowed states in $\mathbf{k}$-space. b Density of states (DOS) $\rho$ versus energy. Dashed line is two-dimensional DOS without magnetic field. Thick vertical lines: $\delta$-like DOS without broadening, curves: broadened DOS.

Also, in a 2D system several physical properties exhibit an oscillatory behavior as a function of Fermi level, i.e. with varying electron number, and as a function of the magnetic field at fixed Fermi energy, i.e. at fixed electron number (Fig. 15.15).

Fig. 15.15

Adapted from [1123, 1456]

Oscillatory (theory, $T=6$ K) behavior of a 2DEG (GaAs/(Al,Ga)As) in a magnetic field: a Fermi level, b magnetization, c specific heat, d thermoelectric power. A Gaussian broadening of 0.5 meV was assumed.

2.7 Shubnikov-De Haas Oscillations

 

From the 2D density of states (per unit area including spin degeneracy) $D^{\mathrm {2D}}(E)=m^*/ \pi \hbar ^2$ the sheet density of electrons $n_{\mathrm {s}}$ can be expressed as a function of the Fermi level (at $T=0$ K without spin degeneracy)

$$\begin{aligned} n_{\mathrm {s}}=\frac{m^*}{2\pi \, \hbar ^2} \, E_{\mathrm {F}} \;. \end{aligned}$$

(15.38)

Using (15.33) we thus find (without spin degeneracy, without valley degeneracy), that the period of 1/B is $\propto n_{\mathrm {s}}$:

$$\begin{aligned} \frac{1}{\Delta B}=\frac{e}{h} \, \frac{1}{n_{\mathrm {s}}} \;. \end{aligned}$$

(15.39)

The carrier density of a 2DEG can therefore be determined from the oscillations of magnetoresistance, and is proportional to the density of states at the Fermi level (Shubnikov-de Haas effect). A corresponding measurement with varying field and fixed electron density is shown in Fig. 15.16. The periodicity with 1/B is obvious. Since only the component of the magnetic field perpendicular to the layer affects the (x, y) motion of the carriers, no effect is observed for the magnetic field parallel to the layer.

Fig. 15.16

Data from [1457]

Shubnikov-de Haas oscillations at a modulation-doped (Al,Ga)As/GaAs heterostructure with a 2DEG, $n=1.7 \times 10^{17}$ cm$^{-2}$ and $\mu =11\,400$ cm$^2$/Vs.

In another experiment the carrier density was varied at constant field (Fig. 15.17). The electron density in an inversion layer in p-type silicon is (linearly) varied with the gate voltage of a MOS (metal-oxide-semiconductor) structure (inset in Fig. 15.17, for MOS diodes cf. Sect. 24.5). In this experiment, the Fermi level was shifted through the Landau levels. The equidistant peaks show that indeed each Landau level contributes the same number of states.

Fig. 15.17

Data from [1458]

Shubnikov-de Haas oscillations of a 2DEG at the (100) surface of p-type silicon (100 $\Omega$ cm) at a magnetic field of $B=3.3$ T and $T=1.34$ K. The inset shows schematically the contact geometry.

2.8 Quantum Hall Effect

 

In high magnetic fields, at low temperatures and for high-mobility, 2D electron gases exhibit a deviation from the classical behavior. We recall that the classical Hall effect (i.e. considering the Lorentz force, classical Drude theory), the generation of a field $E_y$ perpendicular to a current flow $j_x$ (cf. Sect. 15.2.1), was described with the conductivity tensor $\mathbf{\sigma }$ (here, for the (x, y)-plane only)

$$\begin{aligned} \mathbf{\sigma }= & {} \frac{\sigma _0}{1+\omega _{\mathrm {c}}^2 \, \tau ^2} \left( \begin{array}{cc} 1 &{} \omega _{\mathrm {c}} \, \tau \\ -\omega _{\mathrm {c}} \, \tau &{} 1 \end{array} \right) \end{aligned}$$

(15.40a)

$$\begin{aligned} \sigma _{xx}= & {} \sigma _0 \, \frac{1}{1+\omega _{\mathrm {c}}^2 \, \tau ^2} \rightarrow 0 \end{aligned}$$

(15.40b)

$$\begin{aligned} \sigma _{xy}= & {} \sigma _0 \, \frac{\omega _{\mathrm {c}} \, \tau }{1+\omega _{\mathrm {c}}^2 \, \tau ^2} \rightarrow \frac{n \, e}{B} \;, \end{aligned}$$

(15.40c)

where $\sigma _0$ is the zero-field conductivity $\sigma _0=n e^2 \tau /m^*$ (8.5). The arrows denote the limit for $\omega _{\mathrm {c}} \tau \rightarrow \infty$, i.e. large fields. The resistivity tensor $\mathbf{\rho }=\mathbf{\sigma }^{-1}$ is given by

$$\begin{aligned} \mathbf{\rho }= & {} \left( \begin{array}{cc} \rho _{xx} &{} \rho _{xy} \\ -\rho _{xy} &{} \rho _{xx} \end{array} \right) \end{aligned}$$

(15.41a)

$$\begin{aligned} \rho _{xx}= & {} \frac{\sigma _{xx}}{\sigma _{xx}^2+\sigma _{xy}^2} \rightarrow 0\end{aligned}$$

(15.41b)

$$\begin{aligned} \rho _{xy}= & {} \frac{-\sigma _{xy}}{\sigma _{xx}^2+\sigma _{xy}^2} \rightarrow -\frac{B}{n \, e} \;. \end{aligned}$$

(15.41c)

2.8.1 Integer QHE

Experiments yield strong deviations from the linear behavior of the transverse resistivity $\rho _{xy} \propto B$ (15.41c) for large magnetic fields at low temperatures for samples with high carrier mobility, i.e. $\omega _{\mathrm {c}}\tau \gg 1$ (Fig. 15.18). In Fig. 15.19a, b, Hall bars are shown for 2DEGs in silicon metal-oxide-semiconductor field-effect transistor (Si-MOSFET) electron inversion layers and at GaAs/(Al,Ga)As heterostructures, respectively.

Fig. 15.18

Reprinted with permission from [1459], ©1982 APS

Hall resistivity $\rho _{xy}$ and longitudinal resistivity $\rho _{xx}$ for a modulation-doped GaAs/(Al,Ga)As heterostructure ($n=4\times 10^{11}$ cm$^{-2}$, $\mu =8.6\times 10^4$ cm$^2$/Vs) at 50 mK as a function of magnetic field (10 kG=1 T). The numbers refer to the quantum number and spin polarization of the Landau level involved. The inset shows schematically the Hall bar geometry, $V_{\mathrm {L}}$ ($V_{\mathrm {H}}$) denotes the longitudinal (Hall) voltage drop.

Fig. 15.19

Reprinted with permission from [1460]

a Silicon MOS (metal-oxide-semiconductor) structure of K. v. Klitzing’s et al. original experiments. b GaAs/(Al,Ga)As heterostructure sample grown with molecular beam epitaxy for QHE measurements, chip carrier and bond wires.

The Hall resistivity exhibits extended Hall plateaus with resistivity values that are given by

$$\begin{aligned} \frac{1}{\rho _{xy}}=\nu \, \frac{e^2}{h} \;, \end{aligned}$$

(15.42)

with $\nu \in \mathbb {N}$, i.e. integer fractions of the quantized resistance $\rho _0=h/e^2=25812.807$... $\Omega$, which is also called the von-Klitzing constant. In Fig. 15.18, a spin splitting is seen for the $n=1$ Landau level (and a small one for the $n=2$). We note that the topmost Hall plateau is due to the completely filled $n=0$ Landau level; the resistance is $\rho _0/2$ due to the spin degeneracy of 2.

The integer quantum Hall effect, first reported in [1461, 1462], and the value for $\rho _0$ are found for a wide variety of samples and conditions regarding sample temperature, electron density or mobility of the 2DEG and the materials of the heterostructure. Besides the Hall effect in Si-based structures and record mobility (Al,Ga)As/GaAs heterostructures (cf. Fig. 12.37), also (Mg,Zn)O/ZnO heterostructures exhibit the effect [1463, 1464] in the same fashion.

Within the plateau, the resistivity is well defined within 10$^{-7}$ or better up to the $10^{-9}$ regime. A precise determination allows for a new normal for the unit Ohm [1389, 1465], being two orders of magnitude more precise than the realization in the SI system, and an independent value for the fine-structure constant $\alpha =(e^2/h)/(2\,c\,\epsilon _0)$. At the same time, the longitudinal resistivity, starting from the classical value for small magnetic fields, exhibits oscillations and eventually it is close to zero for the plateaus in $\rho _{xy}$. For $\rho _{xx}$ values of 10$^{-10}$ $\Omega /\Box$ have been measured, which corresponds to 10$^{-16}$ $\Omega$/cm for bulk material, a value three orders of magnitude smaller than for any non-superconductor.

The interpretation of the quantum Hall effect(s) is discussed in [1466] among many other treatises. The simplest explanation is that the conductivity is zero when a Landau level is completely filled and the next is completely empty, i.e. the Fermi level lies between them. The temperature is small, i.e. $kT \ll \hbar \omega _{\mathrm {c}}$, such that no scattering between Landau levels can occur. Thus no current, similar to a completely filled valence band, can flow. The sheet carrier density $n_{\mathrm {s}}$ is given by counting the i filled Landau levels (degeneracy according to (15.37)) as

$$\begin{aligned} n_{\mathrm {s}}=i \, \frac{e \, B}{h} \;. \end{aligned}$$

(15.43)

In the transverse direction energy dissipation takes place and the Hall resistivity $\rho _{xy}=B/(n_{\mathrm {s}} e)$ takes the (scattering-free) values given in (15.42).

However, this argument is too simple as it will not explain the extension of the plateaus. As soon as the system has one electron more or less, the Fermi energy will (for a system with $\delta$-like density of states) be located in the upper or lower Landau level, respectively. Then, the longitudinal conductivity should no longer be zero and the Hall resistivity deviates from the integer fraction of $\rho _0$. In general, a single particle picture is insufficient to model the IQHE.

The accepted model for the explanation of the QHE, supported now with plenty of experimental evidence, is the edge state model where quantized one-dimensional conductivity of edge channels, i.e. the presence of conductive channels along the sample boundaries, is evoked [1475].

The most fundamental arguments for the origin of the IQHE come from gauge invariance and the presence of a macroscopic quantum state of electrons and magnetic flux quanta [1468]. This model holds as long as there are any extended states at all in the inhomogeneously broadened density of states (Landau levels) (Fig. 15.14). These edge states arise from the fact that the quantum Hall state has another topology than the (topologically trivial) surrounding [463, 1469, 1470]. Each plateau corresponds to a distinct topological phase characterized by a Chern number (cmp. (6.29)) related to the ‘Hofstadter butterfly’ [1471, 1472] (cf. Sect. G.3.1).

It is remarkable that the classical Hall effect is based on homogeneous conduction for a proper evaluation (cf. Fig. 15.7e), while the quantum Hall effect involves electrical transport only along the edges. A detailed microscopic picture of the edge channels is also of interest. Due to depletion at the boundary of the sample, the density of the 2DEG varies at the edge of the sample and ‘incompressible’ stripes develop for which $\partial \mu /\partial n_{\mathrm {s}}\rightarrow \infty$. When the filling factor is far from an integer, the Hall voltage is found to vary linearly across the conductive channel and the current is thus homogeneous over the sample (Fig. 15.20d). In the Hall plateau, the Hall voltage is flat in the center of the channel and exhibits drops at the edges, indicating that the current flows along the boundary of the sample (edge current) [1473] in agreement with predictions from [1476]. Although the current pattern changes with varying magnetic field, the Hall resistivity remains at its quantized value.

In graphene, the IQHE plateaus take on the resistance values [1249, 1477],

$$\begin{aligned} \frac{1}{\rho _{xy}}=\pm 4\, \left( n+\frac{1}{2} \right) \, \frac{e^2}{h} \;, \end{aligned}$$

(15.44)

with $n \in \mathbb {N}_0$. This new ‘half integer’ quantization condition can be translated into the quantized filling factor $\nu =\pm 4\,(n+1/2)$ in the usual QHE formalism of (15.42). The condition (15.44) is due to the different nature of Landau levels for a linear dispersion (Dirac spectrum) [1478]. The $+/-$-sign denotes the QHE for electrons and holes, respectively. The factor $g=4$ stems from the degeneracy of the Landau levels, accounting for spin degeneracy (when the Landau level separation is much larger than the Zeeman spin splitting) and for the sublattice degeneracy. The two $n=0$ Hall plateaus have even been observed (for $B=45$ T) at room temperature [1479].

Fig. 15.20

Adapted from [1473]

Normalized Hall potential profile for different magnetic fields around filling factor $\nu =2$. The overall voltage drop corresponds to 20 mV. The insets show the sample geometry and transport data. The 2DEG is from a GaAs/Al$_{0.33}$Ga$_{0.67}$As modulation-doped heterostructure, $n_{\mathrm {s}}=4.3\times 10^{11}$ cm$^{-2}$, $\mu =5\times 10^5$ cm$^2$/Vs, $T=1.4$ K.

Fig. 15.21

Adapted from [1480], reprinted with permission, ©1990 AAAS

Hall resistance $R_{xy}$ and magnetoresistance $R_{xx}$ of a two-dimensional electron system (GaAs/(Al,Ga)As heterostructure) of density $n=2.33 \times 10^{11}$ cm$^{-2}$ at a temperature of 85 mK, versus magnetic field B. Numbers identify the filling factor $\nu$, which indicates the degree to which the sequence of Landau levels is filled with electrons. Plateaus are due to the integral ($\nu =i$) quantum Hall effect (IQHE) and fractional ($\nu =p/q$) quantum Hall effect (FQHE).

2.8.2 Fractional QHE

 

For very low temperatures and in the extreme quantum limit, novel effects are observed when the kinetic energy of the electrons is smaller than their Coulomb interaction. New quantum Hall plateaus are observed at various fractional filling factors $\nu =p/q$. We note that the effects of the fractional quantum Hall effect (FQHE) in Fig. 15.21 mostly arise for magnetic fields beyond the $n=1$ IQHE plateau. The filling factor $\nu =n/(eB/h)$ is now interpreted as the number of electrons per magnetic flux quantum $\phi _0=h/e$.

The effects of the FQHE cannot be explained by single-electron physics. The plateaus at fractional fillings $\nu$ occur when the Fermi energy lies within a highly degenerate Landau (or spin) level and imply the presence of energy gaps due to many-particle interaction and the result of correlated 2D electron motion in the magnetic field.

A decisive role is played by the magnetic flux quanta. The presence of the magnetic field requires the many-electron wavefunction to assume as many zeros per unit area as there are flux quanta penetrating it. The decay of the wavefunction has a length scale of the magnetic length $l_0=\sqrt{\hbar /(eB)}$. Since the magnetic field implies a $2 \pi$ phase shift around the zero, such an object is also termed a vortex, being the embodiment of the magnetic flux quanta in the electron system. Such a vortex represents a charge deficit (compared to a homogeneous charge distribution) and thus electrons and vortices attract each other. If a vortex and an electron are placed onto each other, considerable Coulomb energy is gained. At $\nu =1/3$, there are three times more vortices than there are electrons, each vortex representing a charge deficit of $1/3\,e$. Such a system is described with many-particle wavefunctions, such as the Laughlin theory for $\nu =1/q$ [1468] and novel quasi-particles called composite fermions [1481, 1482] for other fractional fillings. For further reading we refer readers to [1483] and references therein.

Fig. 15.22

Reprinted with permission from [1484], ©1991 APS

Weiss oscillations: a magnetoresistance and b Hall resistance of an antidot lattice (inset in (a)) with pattern (solid lines) and without pattern (dashed lines) at $T=1.5$ K. b Schematic of the different orbits: (‘p’: pinned, ‘d’: drifting, ‘s’: scattered).

2.8.3 Weiss Oscillations

 

In Fig. 15.22, measurements are shown for a Hall bar in which an array of antidots (in which no conduction is possible) has been introduced by dry etc.hing. The antidot size is 50 nm (plus depletion layer) and the period is 300 nm. These obstructions for the cyclotron motion lead to a modification of the magnetotransport properties.

Before etc.hing of the antidot array the 2DEG has a mean free path length of 5–10 $\upmu$m at 4 K for the mobility of $\approx 10^6$ cm$^2$/Vs. At low magnetic fields there is a strong deviation of the Hall resistivity from the straight line to which the QHE levels converge. Similarly, $\rho _{xx}$ shows a strong effect as well.

These effects are related to commensurability effects between the antidot lattice and the cyclotron resonance path. When the cyclotron orbit is equal to the lattice period, electrons can fulfill a circular motion around one antidot (pinned orbit, Fig. 15.22b) that leads to a reduction of conductivity. At high fields, drifting orbits for which the cyclotron orbit is much smaller than the lattice period occur. At small fields, scattering orbits also contribute for which the cyclotron radius is large and the electron has antidots from time to time. Resonances in the Hall resistivity have been found due to pinned orbits enclosing 1, 2, 4, 9 or 21 antidots.