3 Coherent Ballistic High-Field Transport in GaAs
3.4 Dynamic Quantum-Kinetic Polaron Model
We developed a new approach to get a more specific insight into the microscopic nonlinear dynamics of polarons on ultrafast time scales [120]. Calculations were performed within a nonlinear and a time-dependent extension of the linear model presented in Ref. [121]. We consider a single electron interacting with the local electric field in x-direction and with the phonon modes of the crystal via different types of electron-phonon interactions. The quantum mechanical Hamiltonian [122] reads:
H(t) =E(p) +exEloc(t) +
∑
b,q
Pb,q2 +ωb2(q)Q2b,q
2 (3.7)
+
∑
b,q
Mb(q)×
Pb,qcosqr+ωb(q)Qb,qsinqr .
r = (x,y,z) and p= (px,py,pz) denote the position and momentum operators of the elec-tron, respectively. The dispersive band structure of the lowest conduction band is described by E(p) which can be obtained from, e.g., pseudo-potential calculations [95, 123, 124]. For small excursions of the electron within the lowest minimum of the conduction band the effec-tive mass approximationE(p) =p2/2meff is sufficient to describe the polaron correctly. The local electric field Eloc(t) is the sum of the externally applied electric field and the field
re-3.4 Dynamic Quantum-Kinetic Polaron Model emitted by the coherent motion of all electrons. The latter contains the linear and nonlinear response of the system and accounts for the radiative damping of the electron motion [88].Qb,q andPb,q are the coordinate and the conjugate momentum of the phonon of branchbwith the wave vectorq= (qx,qy,qz)and angular frequencyωb(q). For simplicity, we limit our calcula-tions to the polar coupling to longitudinal optical phonons (b=LO) with a constant frequency ωLO(q) =ωLO= const.
and coupling to acoustic phonons (b=AC) via the deformation potential Ξwith an averaged sound velocitycS
εSis the static relative dielectric constant andε∞is the dielectric constant for frequencies well above the optical phonon frequency, but below electronic excitations. The differenceε∞−1−εS−1 is proportional to the polar electron-LO phonon coupling constantα[125].Vis the quantization volume which determines the discretization of thek- andq-space with the zone boundaryqzb. ρstands for the mass density of the crystal.
From Eq. (3.7) we derive the Heisenberg equations of motion for the expectation values of quantum mechanical operators likehxi,hpxi, etc. In this process new quantum mechanical op-erators containing combinations of canonical variables, e.g.,hPb,qsin(qr)i, appear on the right-hand side of the equations of motion. Since we are only interested in the expectation values of the relevant observables and since we would like to close the infinite hierarchy of equations at some level, we expand and subsequently approximate those expectation values. In lowest order one obtains exactly the equations of motion of the classical polaron [126,127]. The classical po-laron model predicts, however, an unrealistically high binding energy in the self-induced poten-tial as classical particles correspond to infinitely small wave packets. To overcome this problem one has to go one step further in the expansion of the expectation values of quantum mechanical operators and consider the finite size of the electron wave packet∆x2=hx2i − hxi2. As shown in detail in Refs. [128,129], the dynamics of∆x2is inherently connected with the dynamics of both the variance of its conjugate momentum∆p2x=hp2xi − hpxi2and∆xp=hxpx+pxxi/2− hxihpxi which is the covariance ofxandp. The main result of Refs. [128, 129] is that under certain cir-cumstances (which are fulfilled in our case) continuous position measurements of the electron caused by various fluctuating forces of the environment lead to decoherence phenomena in such a way that an initially Gaussian electron wave packet (in Wigner space) stays Gaussian in its further evolution and adjusts its size∆x2continuously to the respective momentum uncertainty according to∆x2=h¯2/4∆p2x (cf. minimum of Heisenberg’s uncertainty relation). Such contin-uous position measurements of the electron lead also to a small random walk in phase space, i.e., diffusion of both the position and momentum of the particle. According to the arguments of the authors of Refs. [128, 129] this diffusion is ineffectual in comparison to the wave packet
0 10 20 30 40 50 60
Figure 3.11: Quantum-kinetic decoupling: High electron velocities at 300 kV/cm cause a broad spectrum of the force (red line). The small overlap with narrow phonon modes (e.g., 3.5πa, blue area) explain the decreased friction force at high electron velocities.
localization and, thus, we completely neglect it in the following. The application of the approx-imations and arguments discussed above lead to the following system of equations of motion for the expectation values of the operators:
dhxi
The energy transfer of electrons to phonons is described in Eqs. (3.13) and (3.14). The driving term, e.g., in Eq. (3.13), consists of one oscillatory componentMb,qcosqxhxiand one amplitude factor exp −12qx∆x2
. The oscillatory component demonstrates, that an electron moving at a constant velocity vel=hxit exerts a force on the phonon oscillator with the frequencyqx·ve. A phonon mode is efficiently excited, only if this frequency matches the resonance condition ωph =qx·ve. The force created by electrons with the velocity of our experiment has a very broad spectrum with frequency components up to 60 THz (red line in Fig. 3.11). The very small spectral overlap with the phonon mode of 3.5πa (blue area) explains the inefficient generation of phonons and thus the weak friction force. This effect is called quantum-kinetic decoupling.
The friction force caused by LO phonons was already shown for different velocities in Fig. 3.3.
Beyond a critical velocity of 435 km/s, the friction force of the LO phonon mode decreases. The characteristic velocity is achieved for acoustic phonons already at several tens of km/s. This is due to the lower frequency and to the higherqvectors of the relevant acoustic phonon modes.
Furthermore, the amplitude factor demonstrates the importance of the polaron wave packet size
3.4 Dynamic Quantum-Kinetic Polaron Model (∼∆x2) for electron-phonon coupling. A large wave packet couples only to phonon modes with a small momentum|q|. The smaller the wave packet the higher the friction force, because more phonon modes with higher momenta contribute.
For simplicity we use in our model spherical Gaussian wave packets with the momentum hpxiinx-direction and isotropic momentum fluctuations∆p2x =∆p2y=∆p2z. Consequently, the expectation values of the kinetic energy hE(p)i=Ekin(hpxi,∆p2x) and the velocity operator hvxi=Vx(hpxi,∆p2x)are functions of bothhpxiand∆p2x. Both two-dimensional functions have been derived from pseudo-potential calculations [95, 123, 124]. The so far missing dynamical variable∆p2x(in turn determining∆x2=∆y2=∆z2=h¯2/4∆p2x) can be inferred from an equation of motion of the kinetic energy
dhE(p)i
using the following arguments: The temporal change of the total electron energydhE(p)i/dt splits naturally into a ballistic coherent (first term) and an incoherent contribution (second term) the latter of which is connected to the velocity fluctuations of the electron (see also discussion of equations (16) and (A4) of Ref. [130]). Since the acceleration of the electron in the external field does not change its momentum fluctuations, it exclusively contributes to the first term on the r.h.s. of Eq. (3.15). In general, the friction force due to phonon scattering (second term r.h.s.
of Eq. (3.12)) will contribute to both terms in Eq. (3.15). In the typical situation, however, the energy relaxation time is distinctly longer than the momentum relaxation time (cf. Fig. 13 of Ref. [108]). Thus, in good approximation we assume that the friction force exclusively con-tributes to the incoherent contribution of the electron’s energy change leading to the following implicit equation of motion for the expectation value of the momentum fluctuations∆p2x
∂Ekin(hpxi,∆p2x)
Emission and absorption of incoherent phonons is described by the energy relaxation rate Γloss(px,∆p2x,TL), which is generally a "slow" process occurring on a timescale of several hun-dreds of femtoseconds (cf. Fig. 13 of Ref. [108]). Thus, it can be well described by the Fermi’s golden rule approach like in the semi-classical Boltzmann transport equation. In absence of external electric fields this term relaxes the wave-packet size to its value at thermal equilibrium, i.e.,∆p2x =meffkBTL.
The incoherent energy relaxation heats up the polaron and minimizes the wave packet size.
This size determines the friction forces acting on the electron, since only phonons withq2<
1/∆x2 can couple efficiently [Eq. (3.12)]. Thus, for large wave packets the friction is weak, leading to ballistic transport, for small ones the friction is strong, leading to drift transport. In
10
Figure 3.12: (a) Calculated quasi-stationary high field transport of polarons (symbols) in GaAs.
For comparison, the blue solid lines show the result of the semi-classical Boltz-mann transport equation, i.e., Ensemble Monte Carlo simulations of M. V. Fis-chetti [108]. Stationary drift velocity of polarons (symbols) as a function of the applied electric field. The dashed lines show drift transport of polarons with vari-ous fixed values of the wave packet size∆x2=h¯2/4∆p2x. (b) Corresponding energy of the polarons in both theΓvalley and the X-valley of the crystal.
our experiment, we start with a large wave packet corresponding to a small∆p2x. To increase
∆p2x, incoherent energy has to be supplied by the friction force [Eq. (3.16)], which takes several hundreds of femtoseconds, leading on ultrafast time scales to negligible changes of ∆p2x and, thus, to ballistic transport.
Outside the quantum-kinetic regime our dynamic polaron theory and the semiclassical BTE [108] give identical results, e.g., for the stationary drift velocity in high fields [Fig. 1 (b)].
Therefore, we apply an electric field to Eqs. (3.11) – (3.14) and (3.16), which varies distinctly slower in time than the inverse of the incoherent energy loss rateΓloss(px,∆p2x,TL). The result of such a calculation is shown in Fig. 3.12.
The symbols in Fig. 3.12 (a) represent the drift velocity of polarons as a function of the applied electric field strength. If the electron-phonon matrix elements depend on the phonon wave vector|q|, one gets a common drift velocity-friction force characteristics for all electrons independent of the conduction band valley they drift in. A good agreement is found with the data calculated with the semiclassical BTE of Ref. [108] (blue solid line). The dashed lines mark drift
3.4 Dynamic Quantum-Kinetic Polaron Model
-10 0 10
1 2 3
-10 0 10
EmittedfieldE em (t)=E trans (t)-E in
(t)(kV/cm)
(a)
Quasi-ballistic Transport:
Polaron Model
Boltzmann Transport Equation:
Fermi's Golden Rule (b)
Tim e (ps)
Figure 3.13: (a,b) Red solid lines: emitted field transientEem(t) for an incident electric field Ein(t)with an amplitude of 300 kV/cm. (a) Black dashed line: result of the model calculation based on the polaron model. (b) Same experimental data as in (a), now compared to the results (blue dashed line) of a calculation assuming the intervalley scattering rates of Ref. [108]. At the time marked by the arrow the electron energies are high enough for scattering into the side valleys.
transport of polarons with variousfixedvalues of the wave packet size∆x2=h¯2/4∆p2x. Such a transport behavior is expected on a timescale shorter than the respective energy relaxation time, but long enough to assure transport in the drift limit, i.e., outside the quantum kinetic regime.
Fig. 3.12 (b) depicts the polaron energy as a function of the applied field strength. In contrast to the polaron velocity, it depends obviously on the electron’s valley. Here the energy of the polarons is shown in both theΓvalley (solid symbols) and theXvalley (open symbols).
Next, we compare our experimental results at short timescales below 200 fs with the cal-culated results of both models. The Monte-Carlo simulation of Ref. [108] predicts efficient scattering into the side valleys before reaching the negative mass regions. Since electrons in the side valleys have rather low velocities (<200 km/s) [108, 109, 131], it would result in a drastic reduction of the electron velocity and thus of the emitted field. Since the return of electrons into theΓvalley requires more than a picosecond [132], they would remain in the side valleys for the rest of the pulse. Accordingly, one expects a strong signal Eem(t)at the beginning of the pulse, but only very weak signals at later times, as shown by the dashed line in Fig. 3.13 (b).
This is in obvious disagreement with the experimental results.
The dynamic polaron theory instead is valid beyond Fermi’s golden rule and predicts correctly a quasi-ballistic high-field transport in the quantum kinetic regime. According to our model the size of the electron wave packet ∆x2=h¯2/4∆p2x remains almost constant in our experiments on ultrafast time scales. In particular, a wave packet with large ∆x2 couples only weakly to phonons with large wavevectors, responsible for scattering into the side valleys. One advantage of applying THz pulses instead of a static electric field to drive electrons, is that new phonon correlations are build up after each half-cycle. Therefore our THz pulse performs six times partial Bloch oscillations consecutively. Since intervalley scattering is effectively suppressed the first several hundreds of femtoseconds, we can observe ballistic transport even at the timet5in Fig. 3.6. Outside the quantum kinetic regime our dynamic polaron theory and the semiclassical BTE [108] give identical results, e.g., for the stationary drift velocity in high fields [Fig. 3.12 (b)]. For the long times inherent in stationary transport, enough energy can be supplied to the polaron to decrease its wave-packet size to very low values [130], leading to strong friction forces and thus to drift transport [131].