First-principles Approach to Ultrafast Pump-probe Spectroscopy in Semiconductors

Abstract

Pump-probe spectroscopy is a powerful technique for investigating ultrafast exciton dynamics. However, developing a theoretical framework for modeling the transient response in photoexcited materials has remained a challenge. Here, we present a first-principles approach based on a non-equilibrium extension to the Bethe-Salpeter equation to simulate pump-probe spectroscopy and disentangle electronic and thermal contributions to the transient response. Applied to three prototypical semiconductors, i.e., the transition-metal dichalcogenide WSe₂, the metal halide perovskite CsPbBr₃, and the transition-metal oxide TiO₂, the method obtains the transient spectra in excellent agreement with experiment. Our analysis reveals that distinct renormalization mechanisms shape the spectral shifts: Photoinduced Coulomb screening, as the dominant electronic effect, drives excitonic blueshifts, while Pauli blocking plays a minor role. Thermal effects induce redshifts on the picosecond time scale. We further demonstrate how key parameters, such as carrier population distribution, pump wavelength, and pump polarization, impact the transient absorption spectra, which offer direct control over the exciton resonance energy. Our approach establishes a framework for interpreting and tailoring pump-probe spectra, providing guidelines for exciton engineering and thus contributing to the design of energy-selective optoelectronic devices.

Introduction

Pump-probe spectroscopy is a state-of-the-art time-resolved technique for investigating light-matter interaction, particularly non-equilibrium processes like exciton dynamics¹. In a typical pump-probe absorption experiment, the material is optically excited by a pump pulse, and the resulting transient changes in transmission are measured with a time-delayed probe pulse, uncovering information about exciton dynamics on ultrafast timescales². Advances in laser technology enable this technique to access a broad range of timescales and energy ranges, from attosecond to picosecond regimes^1,3,4, and from the infrared and visible⁵ to the extreme ultraviolet (XUV)⁶ and X-ray domains⁷.

Pump-probe experiments often face limitations in resolving distinct dynamical processes in periodic systems, due to complex many-body interactions. This motivates the development of theoretical methods for simulating transient phenomena. Early theoretical descriptions of transient absorption (TA) spectra were formulated within density-matrix theory^8,9,10,11, relying on model Hamiltonians and thus lacking first-principles accuracy. Beyond these approaches, real-time time-dependent density-functional theory (RT-TDDFT) is widely used as a first-principles method to simulate TA spectra^12,13,14. While efficient and thus applicable to large systems, RT-TDDFT within a mean-field framework lacks an explicit treatment of many-body interactions. In contrast, many-body wavefunction approaches, such as the time-dependent configuration interaction singles approach¹⁵ and coupled cluster theory¹⁶, offer a high-accuracy description of electron correlation. However, they are computationally quite involved and thus limited to small atomic or molecular systems. The Bethe-Salpeter equation (BSE) of many-body perturbation theory offers an accurate treatment of electron-hole interaction. It is regarded as the state-of-the-art approach for computing absorption spectra in equilibrium¹⁷. However, the effect of photoexcited carriers is not included in standard implementations. Later, this formalism was generalized to the non-equilibrium regime, validated on a simple four-band model system¹⁸, and subsequently used to simulate the transient reflectivity spectrum of silicon¹⁹. More recently, we took the first step towards treating such pump-probe spectra from first principles, which allowed us to quantify how photoexcited carriers impact the TA spectra in the wide band-gap material ZnO²⁰.

In this work, we present an approach integrating RT-TDDFT, constrained density-functional theory (cDFT), and a non-equilibrium extension to BSE that considers both electronic and thermal contributions fully ab initio. We apply our method to three prototypical materials: tungsten diselenide (WSe₂), cesium lead bromide (CsPbBr₃), and anatase titanium dioxide (TiO₂). These compounds are representative of transition-metal dichalcogenides (TMDCs), metal halide perovskites, and transition-metal oxides, respectively, and are widely used in photodetection²¹, photocatalysis^22,23, and solar energy conversion^24,25. We disentangle the spectral features into electronic and thermal contributions, revealing how each modulates the exciton resonance. Our analysis not only enables us to qualitatively interpret and understand available experimental TA spectra, but also predicts how the transient response can be controlled via the carrier population and pump parameters. Our work thus provides a powerful tool for interpreting pump-probe experiments and gaining insight into transient response, offering guidelines for the design of energy-selective optoelectronic devices.

Results and Discussion

First-principles formalism to simulate TA spectra

Figure 1 presents the computational workflow used to simulate pump-probe spectra and to disentangle the electronic and thermal contributions. This approach consists of five main steps: computing the ground state, determining photoexcited carrier distributions and expanded lattice structure, solving the equilibrium and non-equilibrium BSE, evaluating the core-level response, and separating the transient response into electronic and thermal contributions. The left panel of Fig. 1 illustrates the optical pump–X-ray probe scheme, while the right panel outlines the computational procedure adopted in this work. Within this framework, the pump-induced non-equilibrium electronic effects are described using either RT-TDDFT or cDFT, depending on the timescale. The pump-induced thermal effect is modeled within density functional theory (DFT) by considering a thermally expanded lattice structure. The probe response is simulated by the BSE formalism. The computational workflow is summarized as follows:

Fig. 1: Workflow for simulating transient absorption (TA) spectra.

Left: Scheme of the pump-probe process. A pump pulse photoexcites the semiconductor, initiating transitions from valence-band (VB) to conduction-band (CB) states (purple arrow). A time-delayed pulse subsequently probes the transient response of the material by promoting core electrons to CB states (pink arrow). Right: Computational workflow for disentangling electronic and thermal contributions to the TA spectra. fs and ps denote femtosecond and picosecond timescales, respectively. cDFT and NE BSE denote the constrained density functional theory and non-equilibrium Bethe-Salpeter equation methods, respectively. Pauli and Screening represent the photoinduced Pauli blocking and Coulomb screening effects. The blue (red) workflow indicates the procedure to isolate the electronic (thermal) contribution.

• Step 1: DFT ground-state. We first perform a self-consistent DFT calculation to obtain the Kohn-Sham (KS) eigenvalues and orbitals $\{{\varepsilon }_{n{\bf{k}}}^{0},{\psi }_{n{\bf{k}}}^{0}\}$, which serve as the input for the equilibrium and non-equilibrium BSE calculations.

• Step 2: Photoexcited carrier distribution via two routes.

RT-TDDFT for the femtosecond timescale

The time evolution of photoexcited carrier distributions at femtosecond time delays following optical pumping is simulated using RT-TDDFT. The pump field is modeled through a time-dependent vector potential A(t) (velocity gauge), yielding the time-dependent KS Hamiltonian²⁶:

$$\widehat{H}({\bf{r}},t)=\frac{1}{2}{\left(-{\rm{i}}\nabla +\frac{1}{\rm{c}}{\bf{A}}(t)\right)}^{2}+{V}_{{\rm{K}}{\rm{S}}}({\bf{r}},t),$$

(1)

where V_KS(r, t) is the time-dependent KS potential, and c is the speed of light. Starting from the initial condition ${\psi }_{n{\bf{k}}}({\bf{r}},t=0)={\psi }_{n{\bf{k}}}^{0}({\bf{r}})$, the KS orbitals evolve according to the time-dependent KS equations:

$$i\,\frac{\partial }{\partial t}\,{\psi }_{n{\bf{k}}}({\bf{r}},t)=\widehat{H}({\bf{r}},t)\,{\psi }_{n{\bf{k}}}({\bf{r}},t).$$

(2)

To obtain time-dependent photoexcited carrier distributions, the propagated orbitals are projected onto the ground-state KS eigenstates {${\psi }_{c{\bf{k}}}^{0}$, ${\psi }_{v{\bf{k}}}^{0}$}. The conduction-band (c) occupations are computed as:

$${f}_{c{\bf{k}}}(t)=\mathop{\sum }\limits_{n}{f}_{n{\bf{k}}}^{0}{| \langle {\psi }_{c{\bf{k}}}^{0}| {\psi }_{n{\bf{k}}}(t)\rangle | }^{2},$$

(3)

and the valence-band (v) occupations are obtained accordingly,

$${f}_{v{\bf{k}}}(t)={f}_{v{\bf{k}}}^{0}-\mathop{\sum }\limits_{n}{f}_{n{\bf{k}}}^{0}{| \langle {\psi }_{v{\bf{k}}}^{0}| {\psi }_{n{\bf{k}}}(t)\rangle | }^{2},$$

(4)

where ${f}_{n{\bf{k}}}^{0}$ denotes the ground-state KS occupations.

cDFT for the picosecond timescale

At picosecond time delays following the optical pumping, the photoexcited carrier distributions are modeled using cDFT. In the semiconductors investigated in this work (WSe₂, CsPbBr₃, TiO₂), carrier-carrier and carrier-phonon scattering lead to rapid thermalization on sub-picosecond timescales^27,28,29. Therefore, on the picosecond timescale, electrons and holes have already relaxed to the vicinity of the conduction-band minimum (CBM) and valence-band maximum (VBM), respectively, such that the photoexcited carrier distribution can be described by Fermi-Dirac distributions characterized by an electronic temperature T:

$${f}_{c{\bf{k}}}={f}_{c{\bf{k}}}^{{\rm{e}}}=\frac{1}{1+\exp \left[\frac{{\varepsilon }_{c{\bf{k}}}-\mu }{{k}_{{\rm{B}}}T}\right]},$$

(5)

$${f}_{v{\bf{k}}}={f}_{v{\bf{k}}}^{0}-{f}_{v{\bf{k}}}^{{\rm{h}}}={f}_{v{\bf{k}}}^{0}-\frac{1}{1+\exp \left[\frac{\mu -{\varepsilon }_{v{\bf{k}}}}{{k}_{{\rm{B}}}T}\right]}.$$

(6)

Here, k_B is the Boltzmann constant, μ is the Fermi-level energy in equilibrium, and ${f}_{c{\bf{k}}}^{e}$ (${f}_{v{\bf{k}}}^{h}$) denotes the electron (hole) occupation numbers in the conduction (valence) states. The excited electron and hole populations are constrained to satisfy charge neutrality and to match the total number of photoexcited carriers per unit cell, N_exc:

$$\mathop{\sum }\limits_{{\bf{k}}}{w}_{{\bf{k}}}\,{f}_{c{\bf{k}}}^{{\rm{e}}}=\mathop{\sum }\limits_{{\bf{k}}}{w}_{{\bf{k}}}\,{f}_{v{\bf{k}}}^{{\rm{h}}}={N}_{{\text{exc}}},$$

(7)

where w_k are the weights of the k-points in the Brillouin zone.

• Step 3: Non-equilibrium BSE. The electron-hole wavefunctions are formulated in the basis of ground-state KS orbitals, i.e., the exciton wavefunctions are expanded as

$${\phi }^{\lambda }({{\bf{r}}}_{{\text{e}}},{{\bf{r}}}_{{\text{h}}})=\mathop{\sum }\limits_{vc{\bf{k}}}{A}_{vc{\bf{k}}}^{\lambda }\,{\psi }_{v{\bf{k}}}^{0* }({{\bf{r}}}_{{\text{h}}})\,{\psi }_{c{\bf{k}}}^{0}({{\bf{r}}}_{{\text{e}}}),$$

(8)

with the expansion coefficients ${A}_{vc{\bf{k}}}^{\lambda }$. The BSE is written as an eigenvalue problem for an effective two-particle Hamiltonian, according to the standard equilibrium formulation³⁰:

$$\mathop{\sum }\limits_{{v}^{{\prime} }{c}^{{\prime} }{{\bf{k}}}^{{\prime} }}{H}_{vc{\bf{k}},{v}^{{\prime} }{c}^{{\prime} }{{\bf{k}}}^{{\prime} }}^{BSE}\,{A}_{{v}^{{\prime} }{c}^{{\prime} }{{\bf{k}}}^{{\prime} }}^{\lambda }={E}_{\lambda }\,{A}_{vc{\bf{k}}}^{\lambda },$$

(9)

with excitation energies E_λ. The BSE Hamiltonian is decomposed as:

$${H}^{{\text{BSE}}}={H}^{{\text{diag}}}+2{H}^{{\text{x}}}+{H}^{{\text{dir}}},$$

(10)

where H^diag describes independent-particle transitions, H^x contains the repulsive exchange interaction mediated by the bare Coulomb potential $v({\bf{r}},{\bf{r}}{\prime} )$, and H^dir accounts for the attractive electron-hole interaction, which contains the statically screened Coulomb potential ${W}_{{\bf{G}}{{\bf{G}}}^{{\prime} }}({\bf{q}})$. It incorporates the inverse dielectric function, ϵ⁻¹, at ω = 0,

$${W}_{{\bf{G}}{{\bf{G}}}^{{\prime} }}({\bf{q}})={v}_{{\bf{G}}}({\bf{q}})\,{\epsilon }_{{\bf{G}}{{\bf{G}}}^{{\prime} }}^{-1}({\bf{q}},\omega =0),$$

(11)

where the microscopic dielectric function, ϵ⁻¹, is calculated within the random-phase approximation (RPA) from the independent-particle polarizability χ⁰. We evaluate χ⁰ by replacing the equilibrium carrier distribution with the photoexcited carrier distribution obtained in Step 2:

$${\chi }_{{\bf{G}}{{\bf{G}}}^{{\prime} }}^{0}({\bf{q}},\omega )=\mathop{\sum }\limits_{vc{\bf{k}}}\frac{{f}_{v{\bf{k}}}-{f}_{c,{\bf{k+q}}}}{\omega -({\varepsilon }_{c,{\bf{k+q}}}^{0}-{\varepsilon }_{v{\bf{k}}}^{0})+i\eta }\,{M}_{vc}^{{\bf{G}}}({\bf{k}},{\bf{q}})\,{[{M}_{vc}^{{{\bf{G}}}^{{\prime} }}({\bf{k}},{\bf{q}})]}^{* },$$

(12)

where η is a positive infinitesimal and ${M}_{vc}^{{\bf{G}}}({\bf{k}},{\bf{q}})=\langle {\psi }_{v{\bf{k}}}^{0}| {e}^{i({\bf{q}}+{\bf{G}})\cdot {\bf{r}}}| {\psi }_{c,{\bf{k+q}}}^{0}\rangle$ are the plane-wave matrix elements.

• Step 4: Core-level response. In the optical limit (q → 0), the macroscopic dielectric function ε_M(ω) is obtained as³¹:

$${\varepsilon }_{M}(\omega )=1+\frac{8\pi }{\Omega }\mathop{\sum }\limits_{\lambda }\frac{| {t}_{\lambda }({\bf{q}}\to 0){| }^{2}}{{E}_{\lambda }-\omega -i\eta },$$

(13)

where Ω is the unit-cell volume, and t_λ(q → 0) is the transition coefficient:

$${t}_{\lambda }({\bf{q}})=\mathop{\sum }\limits_{vc{\bf{k}}}{A}_{vc{\bf{k}}}^{\lambda }\,\sqrt{{f}_{v{\bf{k}}}-{f}_{c,{\bf{k}}+{\bf{q}}}}\,\frac{\langle v{\bf{k}}| \widehat{p}| c,{\bf{k}}+{\bf{q}}\rangle }{{\varepsilon }_{c,{\bf{k}}+{\bf{q}}}^{0}-{\varepsilon }_{v{\bf{k}}}^{0}},$$

(14)

with $\widehat{p}$ being the momentum operator. From the complex dielectric function, ε_M(ω), both the absorption α(ω) and the reflectivity coefficients R(ω), can be evaluated³².

• Step 5: Electronic and thermal contribution to the TA spectra.

Electronic contribution

To quantify the electronic contribution to TA spectra, we solve the non-equilibrium BSE using the photoexcited carrier distribution obtained in Step 2, yielding a non-equilibrium absorption spectrum α^neq(ω, t). The electronic TA spectrum at a pump-probe delay t is defined as the difference between the non-equilibrium and equilibrium spectra α^eq(ω) as

$$\Delta \alpha (\omega ,t)={\alpha }^{{\text{neq}}}(\omega ,t)-{\alpha }^{{\text{eq}}}(\omega ).$$

(15)

It contains contributions from two many-body effects: (i) Pauli blocking, arising from occupation-induced suppression of optical transitions, and (ii) photoinduced Coulomb screening, i.e., the modification of the electron-hole interaction due to the screening by photoexcited carriers. We disentangle the two contributions by selectively introducing the photoexcited carrier distribution either in the transition coefficients through the factor $\sqrt{{f}_{v{\bf{k}}}-{f}_{c,{\bf{k+q}}}}$ (Step 4, Eq. (14)) to isolate Pauli blocking, or in the RPA polarizability used to construct the statically screened Coulomb interaction (Step 3, Eq. (11)-(12)) to isolate photoinduced Coulomb screening.

Thermal contribution

At picoseconds time delays after photoexcitation, energy relaxation mediated by exciton-phonon scattering has brought the electronic and lattice subsystems into thermal equilibrium, resulting in an approximately homogeneous lattice temperature. Therefore, the lattice response can be approximated by a spatially uniform thermal expansion³³. We model pump-induced heating by constructing a structure with an expanded lattice at an elevated temperature T, and compute the spectra using equilibrium BSE on top of it. Note that exciton-phonon coupling can also modify the spectra, which is not explicitly included here. Accordingly, the thermal contribution to the TA spectrum is approximated as

$$\Delta \alpha (\omega ,T)=\alpha (\omega ,T)-{\alpha }^{{\text{eq}}}(\omega ).$$

(16)

Finally, we note that the approach presented here is not a formal realization of non-equilibrium BSE derived on the Keldysh contour. A fully non-equilibrium BSE treatment would require explicit two-time propagation³⁴, whose computational cost scales unfavorably with the number of bands, k-points, and time steps, making it impractical for realistic materials³⁵. To enable pump-probe simulations at an affordable cost, we have adopted three approximations. First, the adiabatic approximation assumes that the pump-driven system is frozen during the probe process, and the optical response at each pump-probe delay is evaluated from an instantaneous non-equilibrium carrier distribution¹⁸. Second, electronic and thermal effects are separated based on different timescales. Third, thermal effects are approximated by lattice expansion. This approach thus has its limitations: the lack of real-time propagation in the BSE prevents the description of the real-time formation, evolution, and relaxation of excitons. The absence of explicit electron-phonon interaction prevents the description of exciton-phonon scattering. Treating the screening at the static RPA level neglects the dynamical screening, memory effects, and vertex corrections. Despite these limitations, the present approach provides a computationally tractable way to simulate and predict the pump-probe spectra, offering an interpretation of the physical origin of the transient spectral features.

Transient absorption at the Se M edge in WSe₂

TMDCs are layered materials whose two-dimensional monolayers are weakly bound by van der Waals forces^36,37,38. They exhibit strong photoluminescence, spin-control capabilities, and efficient light absorption. Most of the remarkable properties of TMDCs originate from tightly bound excitons. Therefore, understanding exciton dynamics is essential for advancing TMDC-based optoelectronic and photonic devices^36,39,40. Pump-probe TA spectroscopy has revealed that exciton dynamics in TMDCs are highly sensitive to factors such as exciton density and dielectric environment^41,42. However, the microscopic mechanisms underlying these observations remain elusive, highlighting the need for first-principles investigations. Here, we focus on WSe₂ as a representative TMDC, due to its oxidation stability, high carrier mobility, and ambipolar characteristics^43,44, and perform first-principles calculations of TA spectra. To consider the electronic effect, RT-TDDFT and cDFT calculations adopt the pump parameters and excitation densities reported in experiment²⁷ to simulate the photoexcited carrier distribution. To model thermal effects, we choose 700 K as a conservative upper bound that amplifies thermal effects for a clean separation from the electronic counterpart, while remaining below the air-oxidation threshold of 773 K⁴⁵.

In a first step, we discuss the XUV absorption spectra at the Se M_4,5 edge in equilibrium. As observed in Fig. 2a, b, the calculated spectrum (black curve) agrees well with the measured counterpart (shaded gray area), reproducing the two-peak structure and the relative peak positions. The additional measured feature at around 54 eV originates from transitions involving the W O₂ edge, which is not included in the calculations. Importantly, the excitonic effects are essential to achieve this level of agreement, since the independent-particle approximation (IPA) cannot even qualitatively describe the peak structure observed in the experiment (Fig. S1). This finding contradicts earlier claims that IPA can yield good qualitative agreement^27,46.

Fig. 2: Electronic and thermal contributions to transient absorption (TA) spectra at the Se M_4,5-edges of WSe₂.

a Equilibrium (black curve) and non-equilibrium absorption (yellow/green curve) at femtoseconds, reflecting electronic contributions for different excitation densities n_e. b Non-equilibrium absorption on the picosecond scale, showing the electronic effect (blue curve) at an excitation density of 1.0 × 10²⁰ cm⁻³ and thermal contribution (red curve) at a lattice temperature T_l of 700 K. c, d TA spectra corresponding to a and b, respectively. Experimental spectra (gray area) from ref. ²⁷ are shown for comparison. e, f Decomposition of the electronic TA spectra into photoinduced Coulomb screening (blue area) and Pauli blocking (red area) at femtosecond and picosecond delays. Insets: Zoom into the Pauli blocking effect. g Electron (red circles) and hole (blue circles) distributions in k-space, shown for localized, intermediate, and delocalized carrier populations at a fixed excitation density of 1.0 × 10²⁰ cm⁻³. h Corresponding TA spectra computed using the carrier distributions shown in g. The spectra are shown in terms of the absorption coefficient, α in a and b, and differential absorption, Δα in c–f and h.

Figure 2a, b show the electronic (yellow, green, and blue curves) and thermal (red curve) contributions to the non-equilibrium absorption spectra on the femtosecond and picosecond timescales. The electronic effect arises from photoexcited carriers at different excitation densities (n_e), while the thermal effect reflects the impact of lattice expansion at 700 K. Both contributions renormalize exciton resonance energies: photoexcited carriers induce a blueshift that increases with excitation density, while lattice expansion results in a redshift. The changes relative to the equilibrium spectra are shown in Fig. 2c, d. At a delay of 4 fs, photoexcited carriers dominate the TA spectra. The calculated spectrum at n_e = 2.0 × 10¹⁹ cm⁻³ (yellow curve in Fig. 2a) reproduces the experimental features between 55 ~ 56.5 eV (shaded area) very well. In contrast, the peaks observed below 55 eV and above 56.5 eV are likely to be attributed to thermal effects, similarly to what we show for the picosecond timescale (red curve in Fig. 2d). The TA amplitude grows non-linearly with excitation density, since a 15-fold increase in n_e does not yield a proportional increase in spectral intensity. At a delay of 5 ps, lattice heating becomes important, as evidenced by the good agreement between the calculated signal near 55 eV and experiment (Fig. 2d).

Furthermore, we disentangle the electronic TA spectra into contributions from Pauli blocking and Coulomb screening by photoexcited carriers, as illustrated in Fig. 2e, f. Pauli blocking (red area) plays a minor role on both the femtosecond and picosecond timescales. Conversely, photoinduced Coulomb screening (blue area) accounts for nearly all features in the TA spectra. By weakening the effective Coulomb interaction between core holes and valence electrons, the exciton binding energies are reduced, giving rise to a spectral blueshift.

Finally, we examine how the TA spectra depend on the carrier distribution in k-space. For a fixed excitation density of 1.0 × 10²⁰ cm⁻³, Fig. 2g displays the localized, intermediate, and delocalized carrier distribution in k-space. The corresponding TA spectra are depicted in Fig. 2h, indicating that a more delocalized distribution of excited electrons and holes across k-space leads to enhanced screening. This, in turn, amplifies the TA response, with positive and negative features becoming more pronounced. The effect tends to saturate at the intermediate carrier distribution in k-space. This result shows that carrier delocalization in k-space directly determines the strength of screening and the amplitude of the TA signal.

Transient absorption at the Br K-edge in CsPbBr₃

Metal halide perovskites have garnered significant interest over the past decade, owing to their unique properties, such as high carrier mobility, tunable band gaps, and high power-conversion efficiencies^47,48. These excellent properties extend their broad applications in next-generation optoelectronic devices, including photovoltaics, light-emitting diodes, and photodetectors⁴⁹. While the performance of such devices has advanced rapidly, it is now approaching theoretical limits^47,50. Therefore, understanding the microscopic mechanisms of exciton dynamics remains in great demand for device optimization.

X-ray transient absorption (XTA) with element specificity offers simultaneous sensitivity to both electronic and lattice degrees of freedom, providing valuable insights into excitonic behavior⁵¹. Despite its great potential, achieving ultrafast XTA measurement remains technically challenging. This highlights the importance of first-principles calculations, which can interpret and predict XTA spectra, providing practical guidelines for experimental design. Here, we focus on the all-inorganic perovskite CsPbBr₃, owing to its high photoluminescence quantum yield and exceptional stability⁵². We disentangle the electronic and thermal contributions to its XTA spectra using our first-principles approach. Pump parameters for the RT-TDDFT simulations and excitation densities for the cDFT calculations are obtained from experimental works^53,54,55. The thermal expansion is evaluated at 350 K, i.e., below the phase transition that occurs around 360 K⁵⁶.

We first calculate the Br K-edge absorption spectrum in equilibrium, depicted in Fig. 3a, b (black curve), which shows good agreement with experiment (shaded gray area, ref. ⁵³). Upon incorporating the effect of photoexcited carriers, the spectrum experiences a blueshift that increases with excitation density (yellow and green curves). The XTA amplitude shown in Fig. 3c increases nonlinearly with the excitation density. At a delay of 4 fs, it is approximately an order of magnitude larger than that on the picosecond timescale (Fig. 3d), owing to the higher excitation density shortly after the pump pulse. The electronic XTA signal (yellow, green, and blue curves in Fig. 3c, d) reproduces most experimental features, except for the initial positive peak around ~ 13.466 keV, which we assign to thermal effects, as discussed below.

Fig. 3: Electronic and thermal contributions to transient absorption (TA) spectra at the Br K-edge of CsPbBr₃.

a Equilibrium (black curve) and non-equilibrium absorption (yellow/green curve) at femtoseconds, reflecting electronic contributions for different excitation densities n_e. b Non-equilibrium absorption on the picosecond scale, showing the electronic effect (blue curve) at an excitation density of 6.0 × 10¹⁸ cm^-3 and thermal contribution (red curve) at a lattice temperature T_l of 350 K. c, d TA spectra corresponding to a, b, respectively. Experimental spectra (gray area) from ref. 53 are shown for comparison. e, f Decomposition of the electronic TA spectra into photoinduced Coulomb screening (blue area) and Pauli blocking (red area) at femtosecond and picosecond delays. Insets: Zoom into the Pauli blocking effect. g Electron (red circles) and hole (blue circles) distributions in k-space, shown for localized, intermediate, and delocalized carrier populations at a fixed excitation density of 6.0 × 10¹⁸ cm^-3. h Corresponding TA spectra computed using the carrier distributions shown in (g).

Thermal expansion at 350 K induces a renormalization of the exciton resonance at a time delay of 1 ps, manifested as a redshift in the absorption spectra (red curve in Fig. 3b). This shift originates from the reduced energy separation between the Br 1s core level and the conduction bands. To date, femtosecond XTA experiments on CsPbBr₃ have not been reported. Our first-principles investigation can predict the XTA response on a short timescale for guiding future femtosecond XTA experiments.

Figure 3e, f show different effects contributing to the electronic components of the XTA spectra, for both the femtosecond and picosecond timescales. The dominant effect is Coulomb screening by excited carriers (blue area). The impact of Pauli blocking (red area) plays a minor role; it nevertheless leads to a positive signal in the pre-edge region around ~ 13.468 keV, corresponding to core transitions to depopulated orbitals, as well as a negative peak around ~ 13.473 keV, resulting from blocked transitions to occupied states. Photoinduced Coulomb screening reduces electron-hole Coulomb interactions, decreases exciton binding energies, and is responsible for the blueshifted non-equilibrium absorption spectra in Fig. 3a, b. These results demonstrate how photoexcited carriers manipulate Coulomb screening. We conclude that precise control over excitonic resonance energies can be achieved by adjusting the excitation density.

We further investigate the effect of photoexcited carrier distributions on the Coulomb screening by employing Fermi-Dirac statistics at different electronic temperatures in Fig. 3g, while keeping the excitation density fixed at 6.0 × 10¹⁸ cm⁻³. The corresponding XTA spectra in Fig. 3h reveal that a more delocalized distribution in k-space lead to a larger XTA amplitude, indicating enhanced screening. This effect saturates at the intermediate carrier distribution in k-space, consistent with our observations in WSe₂. These results demonstrate that in this material, not only excitation density but also carrier distribution can modulate photoinduced Coulomb screening and exciton resonance energies.

Transient absorption at the Ti K-edge in anatase TiO₂

Anatase TiO₂ is one of the most promising metal-oxide semiconductors for photocatalysis⁵⁷. Upon photoexcitation, electron-hole pairs are created: electrons at Ti sites drive reduction reactions, while holes at O sites mediate oxidation. XTA spectroscopy can track changes in the oxidation state and the local atomic structure, which can be used to identify the active sites and guide the design of effective catalysts⁵⁸. However, only a few experiments have targeted XTA spectra at the Ti K-edge, due to the considerable technical demands of high-energy X-ray measurements^58,59. Also here, we can bridge the gap between experiment and theory, providing predictive insight into the XTA spectra at the Ti K-edge of anatase TiO₂. Pump parameters for RT-TDDFT and excitation densities for cDFT calculations are taken from experimental works^59,60. Thermal effects are evaluated at 800 K, well below the phase transition of TiO₂ occurring around 870 K⁶¹.

The equilibrium absorption spectrum at the Ti K-edge is depicted in Fig. 4a, b, showing good agreement with the experimental counterpart (shaded gray area, ref. ⁵⁸). We first examine the electronic contributions to the absorption spectra (yellow, green, and blue curves). The non-equilibrium spectra at both femtosecond and picosecond time delays exhibit a blueshift, which is more pronounced in the femtosecond case, due to the higher excitation density. This carrier-induced blueshift produces a negative signal near 4.984 keV in the XTA spectra (Fig. 4c), consistent with experiment⁵⁸. We next analyze the thermal contributions arising from lattice expansion at 800 K and a time delay of 1 ps (red curve in Fig. 4b). In this case, a redshift of the exciton resonance occurs, originating from the reduced energy separation between the Ti 1s core level and the conduction band. This thermal effect produces a positive peak around 4.982 keV in the XTA spectrum (Fig. 4d), a feature also observed in experiment, but not reproduced when considering the electronic effect alone. This thermal contribution is scaled by a factor of 0.2 for comparison with the electronic TA spectra.

Fig. 4: Electronic and thermal contributions to transient absorption (TA) spectra at the Ti K-edge of anatase TiO₂.

a Equilibrium (black curve) and non-equilibrium absorption (yellow/green curve) at femtoseconds, reflecting electronic contributions for different excitation densities n_e. b Non-equilibrium absorption on the picosecond scale, showing the electronic effect (blue curve) at an excitation density of 1.8 × 10¹⁹ cm^-3 and thermal contribution (red curve) at a lattice temperature T_l of 800 K. c, d TA spectra corresponding to a, b, respectively. Experimental spectra (gray area) from ref. 58 are shown for comparison. e, f Decomposition of the electronic TA spectra into photoinduced Coulomb screening (blue area) and Pauli blocking (red area) at femtosecond and picosecond delays. Insets: Zoom into the Pauli blocking effect. g Electron (red circles) and hole (blue circles) distributions in k-space, shown for localized, intermediate, and delocalized carrier populations at a fixed excitation density of 1.8 × 10¹⁹ cm^-3. h Corresponding TA spectra computed using the carrier distributions shown in (g).

Like the other materials, we disentangle the effects that shape the transient response in Fig. 4e, f. Photoinduced Coulomb screening (blue area) dominates the XTA spectra across both timescales. In contrast, Pauli blocking (red area) plays a comparatively minor role. Its influence is more noticeable at early femtosecond delays, consistent with our findings at the Br K-edge in CsPbBr₃. Likewise, with increasing excitation density, the enhanced photoinduced Coulomb screening further decreases the exciton binding energy (Fig. S2), leading to a blueshift of the exciton resonance in the absorption spectra (Fig. 4a, b). These results highlight Coulomb screening as the key mechanism modulating exciton resonance in TiO₂ and demonstrate its universality across different material classes.

We further investigate the role of carrier distribution using Fermi-Dirac statistics at different electronic temperatures for a fixed excitation density of 1.8 × 10¹⁹ cm⁻³ (Fig. 4g). The corresponding XTA spectra are shown in Fig. 4h. The delocalization of carrier distribution in k-space enhances dielectric screening, thereby weakening the Coulomb interaction between electrons and holes. A progressively more delocalized carrier distribution gives rise to an increasing XTA amplitude. This increase saturates at the intermediate carrier distribution in k-space. The overall behavior is consistent with our observations in WSe₂ and CsPbBr₃.

Beyond TA spectra, we further investigate the electronic contribution to the transient reflectivity (TR) spectra on both femtosecond and picosecond timescales. As shown in Fig. 5a, b, electronic excitations induce a blue shift in the spectrum, which becomes increasingly prominent at higher excitation densities, all concomitant to the behavior of the TA spectra. Obviously, this shift arises primarily from photoinduced Coulomb screening (Fig. 5e, f). In contrast, the thermal contribution to the spectra (red curve in Fig. 5b) leads to a red shift relative to the equilibrium response. The presence of delocalized carrier distributions further amplifies the response in the TR spectra (Fig. 5g, h), as observed also for the TA spectra. The TR spectra for WSe₂ (Se M_4,5-edge) and CsPbBr₃ (Br K-edge) are provided in Fig. S3-S4 in the Supplementary Note 4.1.

Fig. 5: Electronic and thermal contributions to the transient reflectivity (TR) spectra at the Ti K-edge of anatase TiO₂.

a Equilibrium (black curve) and non-equilibrium reflectivity (yellow/green curve) at femtoseconds, reflecting electronic contributions for different excitation densities n_e. b Non-equilibrium reflectivity on the picosecond scale, showing the electronic effect (blue curve) at an excitation density of 1.8 × 10¹⁹ cm^-3 and thermal contribution (red curve) at a lattice temperature T_l of 800 K. c, d TR spectra corresponding to a, b, respectively. e, f Decomposition of the electronic TR spectra into photoinduced Coulomb screening (blue area) and Pauli blocking (red area) at femtosecond and picosecond delays. Insets: Zoom into the Pauli blocking effect. g Electron (red circles) and hole (blue circles) distributions in k-space, shown for localized, intermediate, and delocalized carrier populations at a fixed excitation density of 1.8 × 10¹⁹ cm^-3. h Corresponding TR spectra computed using the carrier distributions shown in g. The spectra are shown in terms of the reflectivity coefficient, R in (a, b), and differential reflectivity, ΔR in (c-f) and (h).

Control of photoinduced Coulomb screening and exciton resonance

For the three materials studied so far, we have observed that higher excitation densities and more delocalized carrier distributions in k-space enhance photoinduced Coulomb screening. Beyond these two parameters, we consider now the polarization direction of the pump beam. We take anatase TiO₂ as an example, as it exhibits a well-known anisotropic dielectric response, with stronger absorption when the electric field aligns with the z-direction⁶². At a delay of 20 fs, our simulations show that a pump polarization oriented along the x-direction yields a smaller excitation density compared to the polarization along z (Fig. S5a). In both cases, the degree of delocalization in k-space is similar (Fig. 6a). The higher excitation density induced by a z-polarized pump gives rise to a stronger XTA signal (Fig. 6b), indicating stronger Coulomb screening. This finding opens the possibility for tuning exciton resonances through the polarization of the pump beam.

Fig. 6: Femtosecond control of core-exciton resonances by Coulomb screening at the Ti K-edge in anatase TiO₂.

Time evolution of photoexcited electron (red circles) and hole (blue circles) distributions following femtosecond optical excitation with varying a pump polarizations (x and z), and c pump wavelengths (355 nm and 266 nm). b, d Corresponding transient absorption (TA) spectra.

The pump wavelength is another degree of freedom to explore, since it may also affect exciton resonances. Exciting anatase TiO₂ with different wavelengths at the same pump fluence results in nearly the same excitation densities (Fig. S5b), yet with a more pronounced XTA spectrum for the shorter wavelength (Fig. 6d). The excitation with shorter wavelength populates the electronic states over a broader energy range in k-space, also further away from the band edges (Fig. 6c), which increases their polarizability and consequently enhances the dielectric screening. These findings identify the pump wavelength as an additional control parameter for tuning exciton resonance energies.

Controlling exciton screening in view of modulating exciton resonances offers broad opportunities for optoelectronic applications. Such control enables the tuning of the detection windows of X-ray detectors without altering the device structure. It also allows control over the energy range contributing to second-harmonic generation (SHG), thereby enabling the modulation of the generated second-harmonic wavelength. Overall, controlling exciton screening enables tunable light-matter interaction across a broad spectral range and is promising for the design of energy-selective photonic devices, such as photodetectors, nonlinear optical devices, and optical filters.

In summary, we have developed an approach that combines RT-TDDFT and cDFT with the non-equilibrium extension to the BSE to simulate pump-probe spectra and disentangle electronic and thermal contributions to the TA spectra. The present implementation relies on several controlled approximations, including an adiabatic treatment of photoexcited states, a timescale-based separation of electronic and thermal effects, and an approximate treatment of thermal effects by lattice expansion. These approximations enable the simulation of the pump-probe spectra in realistic materials at an affordable computational cost. We have applied this method to three prototypical materials, i.e., WSe₂, CsPbBr₃, and anatase TiO₂, achieving excellent agreement with experiments.

Our results reveal that thermal effects caused by lattice heating induce redshifts of exciton resonances. Photoinduced Coulomb screening dominates the electronic effects that reduce exciton binding energies, thus resulting in blueshifts of the spectra. Pauli blocking is a minor electronic effect. Moreover, we have demonstrated that exciton resonance energies can be tuned through excitation density, carrier distribution, pump polarization, and wavelength. These insights provide practical guidelines for tailoring optical parameters to design energy-selective optoelectronic devices.

The ab initio approach presented here becomes a reference scheme for simulating pump-probe spectra across a wide range of materials. It can capture the dominant mechanisms governing transient response across timescales, ranging from attoseconds to picoseconds and energy ranges, spanning the visible and near-infrared to extreme ultraviolet and X-ray regimes. This capability is crucial for identifying future targets for attosecond core-level transient spectroscopy, which remains challenging to achieve experimentally. This approach is also well-suited for modeling transient reflectivity spectra. Looking ahead, it can be extended to simulate other time-resolved probes, such as time-resolved terahertz spectroscopy and time-resolved photoluminescence.

Methods

Ground-state calculations

All calculations were performed using the all-electron, full-potential package exciting. Brillouin zone integrations were performed using a Monkhorst-Pack k-point grid of 18 × 18 × 5 for WSe₂, 6 × 6 × 4 for CsPbBr₃, and 8 × 8 × 6 for TiO₂. The Perdew-Burke-Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) was employed to describe exchange-correlation effects. Scalar relativistic effects, including the mass-velocity and Darwin terms, were treated using the atomic ZORA (zero-order regular approximation) scheme, while spin-orbit coupling was neglected.

Excited-state calculations

Table 1 shows the computational parameters employed in the cDFT, RT-TDDFT, and BSE calculations for WSe₂, CsPbBr₃, and TiO₂. The k-point meshes define the Brillouin-zone sampling, whereas ${R}_{MT}| {\bf{G}}+{\bf{k}}{| }_{\max }$ determines the LAPW basis-set cutoff. In RT-TDDFT, the pump photon energy (E_hv), polarization, and pulse duration specify the frequency, orientation, and temporal profile of the external field. In the BSE calculations, ${| {\bf{G}}+{\bf{q}}| }_{\max }$ defines the cutoff for the KS response function, dielectric screening, and the Coulomb-potential expansion. The broadening defines the Lorentzian linewidth applied to the spectra. N_empty determines the unoccupied orbitals included in the calculation of screening and electron-hole interaction.

Table 1 Computational parameters employed in the cDFT, RT-TDDFT, and BSE calculations for WSe₂, CsPbBr₃, and TiO₂

All spectra were computed within the BSE formalism using Kohn-Sham eigenvalues and eigenvectors from ground-state DFT calculations. Since the G₀W₀ approach is insufficient to yield reliable quasiparticle corrections for core states⁶³, we adopted a scissor correction by aligning the calculated equilibrium spectra with the experimental counterparts. The scissor correction has been widely employed in previous studies of core excitations, and is generally sufficient to describe the spectral lineshape and relative peak positions^64,65. Given that the main objective of our work is to resolve pump-induced transient spectra, i.e., the difference between equilibrium and non-equilibrium spectra, this approach is justified. The screened Coulomb interaction W entering the BSE kernel was evaluated without scissor correction, in line with standard BSE practice.