Excitation dynamics and ultrafast band-gap renormalization

For the chosen excitation conditions the optically induced interband polarizations pre-dominantly lead to the generation of an uncorrelated electron-hole plasma. Based on the description of sec. 2.3.3, the generated plasma effectively screens the Coulomb interaction between electrons and holes leading to the replacement of unscreened HF renormalizations by their screened counterparts. Details in the dephasing of the in-terband polarizations play only a minor role for the carrier-relaxation dynamics [IV].

In this context it was sufficient to treat the polarization equation (2.45) on the level of the screened HF approximation within the pump simulations.

Based on the conclusions of secs. 2.3.3 and 2.3.4, the quantum-Boltzmann-like scat-tering rates to describe the intraband carrier-relaxation dynamics for the electron-distribution functions were implemented as

d dtf_ik^e

_el.

= 2π

~

X

q6=0

W_q^X

j,p

(V_q −V_k−pδ_i,j)δ(˜Σ^eik −˜Σ^eik−q− ˜Σ^ejp+ ˜Σ^ejp−q)

×ⁿf_ik^ef¯_ik−q^e f_jp−q^e f¯_jp^e −f¯_ik^ef_ik−q^e f¯_jp−q^e f_jp^{e o} + 2π

~

X

q6=0

W_q^X

j,p

V_qδ(˜Σ^eik+q − ˜Σ^eik− ˜Σ^hjp+ ˜Σ^hjp−q)

×ⁿf_ik^ef¯_ik+q^e f_jp^hf¯_jp−q^h −f¯_ik^ef_ik+q^e f¯_jp^hf_jp−q^{h o}, (4.9) d

dtf_ik^e

ph.

= 2π

~

X

q6=0

gqg⁰_qδ˜Σ^eik+q − ˜Σ^eik−~ωq

h(1 +nq)f_ik+q^e f¯_ik^e −nqf_ik^ef¯_ik+q^{e i} + 2π

~

X

q6=0

g_qg⁰_qδ˜Σ^eik−q − ˜Σ^eik+~ω_q ^hn_qf_ik−q^e f¯_ik^e −(1 +n_q)f_ik^ef¯_ik−q^{e i}, (4.10) where contributions from the microscopic polarizations were considered small within the remaining scattering terms. The equations for the scattering dynamics of the hole-distribution functions are analog. In eqs. (4.9) and (4.10),W_q and g_q denote the

time-dependent screened Coulomb and Fr¨ohlich interactions that were computed from their respective bare counterpartsV_q andg_q⁰ with the aid of the Lindhard polarization function (2.59) in the static limit. Furthermore, n_q = _eβ~ωq¹−1 denotes the phonon occupation number corresponding to the LO-phonon energy of ~ωq = 27.72 meV in monolayer MoTe2 [45]. Energy conservation of the scattering processes was assured using the numerical energy-conservation function πδ_η(x) = _x2+(^~η~η)², where a small broadening ~η in the range of 1-5 meV was employed.

The scattering dynamics of the electron and hole distribution functions f_ik^λ couple among others to distribution functions f_ik−q^λ . However, f_ik^λ is only defined on a rigid, equidistant|k|-grid. Therefore, distribution functions of typef_ik−q^λ were approximated using a quadratic spline

s^λ_k =f_k^λ

l+1 + (k−k_l+1) ∂f_k^λ

∂k

_k=¯k

12

+(k−k_l+1)(k−k_l+2) 2

∂²f_k^λ

∂k²

_k=¯k

12

(4.11) for crystal momentak_l+1 ≤k < k_l+2between the two grid points, where the shorthand notations

∂f_k^λ

∂k

_k=¯k

12

= f_k^λ

l+2−f_k^λ

l+1

k_l+2−k_l+1 , ∂²f_k^λ

∂k²

_k=¯k

12

=

∂f_k^λ

∂k

_k=k

l+2

− ^∂f_∂k^kλ

_k=k

l+1

k_l+2−k_l+1 , (4.12) with ¯k₁₂= (k_l+1+k_l+2)/2 andk_l+1 =k_l+dk, are introduced. It is easily verified that s^λ_kl+1 =f_k^λ_l+1 and s^λ_kl+2 =f_k^λ_l+2 hold for the boundaries of the spline. Since a quadratic polynomial requires three coefficients, the final coefficient results from the condition that the curvature of the spline is given by the second derivative of the distribution function in the center of the interval. Employing the symmetric difference quotient for the first derivative at grid point k_l, ^∂f_∂k^kλ|_k=kl = ^fkl+1_k^λ_l+1^−f_−k^kl−1_l−1^λ , yields the numerically implemented form

f_k^λ ≈f_k^λ_l+1+xk

f_k^λ_l+2 −f_k^λ_l+1−x_k(1−x_k) 4

f_k^λ_l+3−f_k^λ_l+2−f_k^λ_l+1+f_k^λ_l, (4.13) where x_k = (k −k_l+1)/dk. Note that the interpolation is only possible for crystal momenta k < k_Nk−2 where N_k denotes the number of |k|-grid points. Therefore, the |k|-grid had to be wide enough to ensure that the distribution functions decay sufficiently towards the end of the grid.

For the calculation of the excitation and subsequent carrier-relaxation dynamics in monolayer MoTe2 the DBEs were solved in the time domain employing the widely used 4th-order Runge-Kutta algorithm [72]. The results are shown representatively for the A-band electron distribution function f_A,k^e (t) in fig. 4.4. We predicted extremely efficient intraband carrier-carrier scattering driving the distribution functions into hot quasiequilibrium states near the band edges and away from the excitation energy within a few femtoseconds. In particular, carrier temperatures above 2000 K were observed at the time where the pump maximum hits the sample. Consequently, an

accumulation of carriers at the excitation energy was almost completely avoided. This led to a highly efficient generation of about 5.20×10¹³ electron-hole pairs per cm² for the given excitation conditions. Carrier-phonon scattering then drove the hot carriers towards a quasiequilibrium at the temperature of the phonon bath within 2.5 ps.

Figure 4.4.: Dynamics of the A-band electron-distribution function in the vicinity of the K/K⁰ points after excitation with a 300 fs pump pulse with peak amplitude of 1.25 MV/cm at t=t₀. (Adapted from ref. [IV].)

The calculation of the carrier-carrier scattering rates (4.9) constitute the major numerical challenge because of the fivefold integration, whereas the numerical com-plexity of carrier-phonon scattering rates (4.10) is comparable to the calculation of screened HF renormalizations. To reduce the numerical effort, scattering rates were not recalculated in every time step of the integration, but only when the electron and hole-distribution functions changed significantly since the previous calculation.

The change in the distribution functions was quantified by computing the scattering deviation

Dscatt. = ^X

λ,i,k

dk k f_ik^λ,new−f_ik^λ,old f_ik^λ,new+f_ik^λ,old

!2

in every time step prior to the prompt of recalculating the next carrier-carrier scatter-ing rates. The recalculation was triggered wheneverD_scatt. hit the designated threshold value. In this case

• the grids Πq(t) and Wq(t) for the Lindhard polarization function and screened Coulomb interaction,

• s-type angular-averaged Coulomb matrix elements W_i,kk^λµµ0^0λ0(t),

• carrier-carrier scattering rates _dt^df_ik^e

el.,

• the grid g_q(t) for the screened Fr¨ohlich interaction,

• and carrier-phonon scattering rates _dt^df_ik^e

ph.

were recalculated in the stated order. The threshold value for D_scatt. to provide nu-merically stable simulations is typically found by trial and error.

Numerically verified, the excitation-induced band-gap renormalization of the GEs on the level of screened HF approximation is contained within the renormalized electron and hole energies

˜Σ^eik(t) =^e_ik−^X

k⁰

dk⁰k^{0 h}W_i,kk^cccc0(t)−W_i,kk^cvcv0(t)ⁱf_ik^e0(t), (4.14)

˜Σ^hik(t) =^h_ik+^X

k⁰

dk⁰k^{0 h}W_i,kk^vvvv0(t)−W_i,kk^vcvc0(t)ⁱ 1−f_ik^h0(t). (4.15) Eqs. (4.14) and (4.15) are stated in the numerically implemented form using the angular-averaged Coulomb matrix elements W_i,kk^λµµ0^0λ0(t). Evidently, time-dependent in-traband energy differences, e.g. ˜Σ^e/hik (t)−˜Σ^e/hik−q(t), enter the Lindhard polarization and energy-conserving functions. Knowing that the screened HF renormalizations predom-inantly introduce a rigid shift to the unrenormalized single-particle energies, intraband energy differences, ˜Σ^e/hik (t)−˜Σ^e/hik−q(t)≈^e/h_ik −^e/h_ik−q, remain almost unchanged through the course of the excitation. Therefore, Lindhard polarization and energy-conserving functions were computed based on the unrenormalized single-particle energies im-mensely reducing the numerical effort without noticeable impact on the results. Here, energies of type ^e/h_ik−q were interpolated based on the previous description.

The excitation-induced band gap for given spin and valley indices was estimated from the renormalized electron and hole energies via

E_gⁱ(t) = ˜Σ^hik=0(t) + ˜Σ^eik=0(t). (4.16) In fig. 4.5 the time evolution of the excitation-induced band-gap renormalization E_gⁱ(t)−E_gⁱ(0) (black) and the corresponding total carrier densityn_i(t) =n^h_i(t) +n^e_i(t) (red) are shown. A- and B-band properties are plotted in solid and dashed lines, re-spectively, and the gray shaded area indicates the envelope of the optical pump pulse.

For the given excitation conditions slightly higher carrier densities were induced within the B bands than in the A bands. Due to opposite spin, intravalley relaxation processes between B and A bands were considered negligible.

The generation of excited charge carriers within the respective bands was found to be accompanied by a huge and almost instantaneous reduction of the band gap. As pointed out in sec. 4.1, the band-gap reduction results from combined screening and phase-space-filling effects. Supported by the ultrafast Coulomb scattering the major contribution to screening develops during the build-up of the excited charge carriers and therefore directly correlates to the width and strength of the pump pulse. For the MoTe2 monolayer we particularly observed a large band-gap shrinkage of about 390 meV already 0.4 ps after the pump-pulse maximum hit the sample. At this point,

the amount of excited charge carriers had just saturated such that further reduction of less than 20 meV on the time scale of the thermalization could be assigned to phase-space filling.

Figure 4.5.: Time evolution of the excitation-induced band-gap renormalization (black) and the density of excited charge carriers (red). The solid lines correspond to the A-band properties, whereas the dashed lines show the B-band properties. The gray shaded area indicates the envelope of the optical pump pulse. (Adapted from ref. [IV].)

Im Dokument Microscopic theory of the linear and nonlinear optical properties of TMDCs (Seite 65-69)