Basic Concepts

Eighty years ago, Felix Bloch showed that electron wave functions in the Coulomb potential of the nuclei in a crystal are periodically modulated plane waves [94]. The spatially periodic modulation of these Bloch functions restricts the allowed energies of the electrons, leading to a dispersive band structureE(¯hk)containing both allowed (bands) and forbidden energy regions (gaps) [95].

An external electric field accelerates conduction band electrons (charge −e) according to Newton’s law with:

F=ma, (3.1)

⇔ −eE=h dk/dt.¯ (3.2)

Starting at theΓ point with k=0, Fig. 3.1 (a) illustrates the increasing quasi-momentum k during acceleration by an external electric field. Various incoherent scattering processes change

(a) (b)

Figure 3.1: Free conduction band electrons are driven by an externally applied electric field.

(a) Scattering randomizes the momentum distribution. Only a small average mo-mentumkof the entire electron ensemble remains, causing a drift current. (b) With-out scattering the electron traverses again and again the Brillouin zone, resulting in Bloch oscillations in direct space (inset).

the momentum of individual electrons and randomize the momentum distribution. Only a small average momentum of the entire electron ensemble remains. This drift-like electron transport is well-known as the ohmic current [l.h.s. of inset in Fig. 3.1 (a)]. Without any scattering processes, ballistic transport occurs as shown in Fig. 3.1 (b). The electrons are expected to follow the dispersion of their band at a constant rate in momentum space [96]. The Brillouin zone in (100) direction of a zincblende crystal with a lattice constant ofaextends from−^2π_a <

k<^2π_a . For each electron leaving the Brillouin zone atk= ^2π_a, one indistinguishable electron will enter at k=−^2π_a. Accordingly, the electron in Fig. 3.1 (b) traverses again and again the Brillouin zone. The effective mass of a band electron is given by the curvature of its band,m_eff=

¯

h²[d²E(¯hk)/dk²]⁻¹. In the conduction band of GaAs, the effective mass is positive around theΓ and theX points and negative around the band maxima. In this regions the acceleration changes its sign, if the electric field points constantly into one direction. The electron velocityvin real space is given by the derivative of Eq. (3.2) with respect to the space coordinates:

v=h¯⁻¹∇_kε(¯hk). (3.3)

The integration of Eq. (3.3) along t yields the actual electron motion in real space. It is depicted in the inset of Fig. 3.1, that the electron undergoes coherent periodic Bloch oscillations.

The observation of Bloch oscillations [94] in bulk crystals was prevented so far by effective ultrafast scattering processes. However, several experimental tricks allowed to observe Bloch oscillations in artificial structures. In the first experiments the Brillouin zone was minimized to such an extent, that it could be traversed by electrons even within the momentum relaxation.

Since the extension of the Brillouin zone is inversely proportional to the lattice constant a, a periodic structure with a large lattice constant was required. Since the 80’s of the last century

3.1 Basic Concepts semiconductor superlattices with a lattice constant of few nanometers were grown by molecular beam epitaxy (MBE). These heterostructures facilitated the first observation of Bloch oscilla-tions in 1992 by Feldmannet al. using the four-wave mixing method [97]. Several years later Waschkeet al. detected the THz radiation, which was emitted by Bloch oscillations, under an applied voltage [98]. The inverse effect, i.e., the manipulation of the an electrical current by incident THz photons was demonstrated by Unterraineret al. [99]. Next people investigated systems with low scattering rates to induce Bloch oscillations. Bloch et al. observed Bloch oscillations of cold atoms in optical lattices [100], Delahaye et al. found them in Josephson junctions [101] and Christodoulideset al. induced Bloch oscillations in optical waveguide ar-rays [102]. In our experiment we concentrated on the last remaining parameter, i.e., we reduced the oscillation time of Bloch oscillations. The Brillouin zone is traversed before scattering pro-cesses suppress the coherent electron motion. To achieve the required electron velocity, strong electric fields of several hundred kV/cm have to be applied on a short timescale. This is accom-plished using nonlinear THz spectroscopy as described in Section 2.3.

The prerequisite of Bloch oscillations in bulk GaAs, ballistic transport, was investigated using different approaches. The first experiments in the 80’s of the last century used Raman spectra or cross correlations of two transmitted near-infrared pulses to estimate a scattering time of elec-trons in the femtosecond time range [29, 103]. Pump-probe experiments [104] and in particular luminescence experiments performed by Kashet al.[30, 92, 105] found scattering times from theΓinto theX valley of 180 fs and into theLvalley of 540 fs. The intensity of the lumines-cence as a function of frequency exhibits characteristic kinks at energies, where scattering into other valleys sets in. While different values in the range of several hundred femtoseconds have been obtained for the scattering rates out of theΓvalley [104, 106], it was generally accepted, that scattering back into theΓvalley lasts about 100 ps. The intraband motion of electrons was probed by Leitenstorferet al. in Refs. [31, 107]. Electric fields of 130 kV/cm were employed inside a p-i-ndiode made of GaAs. The carriers were photo-injected using a broadband 10 fs pulse and accelerated by the static built-in electric field. The emitted electric field was focused on a ZnTe crystal and detected with electrooptic sampling. The applied optical setup causes a detected signal, that is proportional to theaccelerationof the carriers. The authors of Ref. [31]

find that electrons are scattered from theΓvalley into theLvalley only 20 fs after the electrons achieved sufficient kinetic energy. The effective electron mass in theLvalley is substantially larger, which decelerates the electron and emits a field of opposite sign. Surprised by such a short scattering time, the authors state: “This regime is especially interesting since the accel-eration period in theΓvalley becomes shorter than the oscillation cycle of the phonons. [...]

To our pleasant surprise, we find that the experimental data are in quantitative agreement with theoretical predictions of nonequilibrium spatial transport based on Monte Carlo simulations (of Ref. [108]).”

All previous experiments excited simultaneously electrons and holes in GaAs. The influence of concomitantly excited holes in these experiments were investigated by Abeet al.[109]. After comparison of the experimental results with a Monte Carlo simulation, the authors conclude a significant influence from the hole population for electric fields above 20 kV/cm. Su et al.

generated electron-hole pairs in GaAs and investigated the carrier transport in subsequent THz pump-probe experiments [22]. The analysis of the experimental data is complicated because of the thickness of the sample. However, after comparison with two simulations, they deduct for electric fields up to 170 kV/cm intervalley scattering within the pulse duration. Quasi-ballistic transport and side valley transfer after several hundred femtoseconds was found by Schwanhäußeret al.[107].

Figure 3.2: Scattering rates obtained with Boltzmann’s transport equation and Fermi’s golden rule according to Ref. [108]. Scattering rates of (3 fs)⁻¹are predicted for electrons near the negative mass region.

0 1000 2000 3000 4000 5000

0 1

positive and negative

dif f erential mobility

Electron velocity (km/s)

Frictionforce(arb.u.)

Figure 3.3: Quantum-kinetic decoupling occurs between electrons and surrounding LO phonon for electron velocities above 435 km/s (taken from Ref. [4]). The negative differen-tial mobility (blue solid line) changes sign and the electron experiences a reduced friction force (solid orange line).