Direct and simultaneous observation of ultrafast electron and hole dynamics in germanium

Understanding excited carrier dynamics in semiconductors is crucial for the development of photovoltaics and efficient photonic devices. However, overlapping spectral features in optical pump-probe spectroscopy often render assignments of separate electron and hole carrier dynamics ambiguous. Here, ultrafast electron and hole dynamics in germanium nanocrystalline thin films are directly and simultaneously observed by ultrafast transient absorption spectroscopy in the extreme ultraviolet at the germanium M4,5 edge. We decompose the spectra into contributions of electronic state blocking and photo-induced band shifts at a carrier density of 8 × 1020 cm−3. Separate electron and hole relaxation times are observed as a function of hot carrier energies. A first-order electron and hole decay of ∼1 ps suggests a Shockley–Read–Hall recombination mechanism. The simultaneous observation of electrons and holes with extreme ultraviolet transient absorption spectroscopy paves the way for investigating few- to sub-femtosecond dynamics of both holes and electrons in complex semiconductor materials and across junctions.

I nvestigation of the ultrafast photoexcited electronic response in semiconductors has provided invaluable insights into the carrier dynamics and dielectric properties of many materials. Germanium and its alloys with Si have promise for creating mid-infrared optoelectronics 1 and multijunction solar cells with higher efficiency 2 . However, the multiple energetically similar valleys complicate understanding of carrier thermalization and population inversion 3 following photoexcitation. A technique is clearly needed that can monitor carrier dynamics of both electrons and holes in the relevant valleys with fewfemtosecond time resolution. With the advent of femtosecond laser techniques, electron and hole scattering in semiconductors has been observed individually and characterized by optical and infrared pump-probe experiments 4,5 . However, the time-resolved observation and characterization of electron and hole kinetics simultaneously in the optical regime is challenging due to overlapping spectral signatures requiring narrowband excitation to separate pump from probe, which inherently limits the temporal resolution.
Ultrafast transient absorption spectroscopy in the extreme ultraviolet (XUV) is an important technique for studying electron dynamics at the sub-femtosecond and few-femtosecond timescale 6,7 . This technique has been successfully applied to investigate the dielectric response of insulators 8,9 and carrier dynamics in semiconductors 10,11 down to sub-femtosecond timescales. In ultrafast XUV transient absorption spectroscopy, a visible-to-near infrared (VIS-NIR) pump pulse excites carriers first, and after a given time delay t, a broadband pulse in the XUV consisting of an isolated attosecond pulse or a short attosecond pulse train generated by high harmonic generation 12 excites core-level electrons into the valence and conduction bands (CBs). The changes in electronic population in the valence and CBs lead to changes in XUV absorption from the core-level excitations, which we refer to as electronic state blocking (SB). In addition, the energy shift of valence and CBs induced by electron-hole excitations will also be recorded in XUV transient absorption. The core-level excitation with XUV photons is element specific, making the technique advantageous for investigating carrier dynamics in heteroatomic, ternary and quaternary systems. The photon energy of the broadband XUV pulse lies far above the interband excitation energies, and therefore this technique in principle allows simultaneous capture of electron and hole dynamics without further inducing interband carrier excitations.
Germanium is a group IV semiconductor with an indirect bandgap of 0.66 eV. Previous optical experiments observed electron scattering from one part of the band structure to another, that is, intervalley scattering, occurring on a few hundreds of femtosecond timescale [13][14][15][16] . In separate experiments, interband scattering of holes has been investigated 17,18 . Recently, studies of the intervalley scattering of carriers in germanium have attracted renewed interest due to the prospect of spin polarization induced in the scattering event, which is useful for the development of spintronics [18][19][20][21] .
Here, we address the need for measuring carrier dynamics more comprehensively by using ultrafast transient absorption in the XUV region of the spectrum, with specific application of the method to electron and hole dynamics in photoexcited germanium. A model is established that allows decomposing the measured raw transient absorption data into contributions of SB, shifts of the excited state spectrum and broadening. We show how the carrier dynamics can be extracted by separating spinorbit components in the XUV. Following this, XUV transient absorption at the germanium M 4,5 edge can spectrally resolve signatures of electrons in the CB and holes in the valence band (VB) simultaneously, which enables the time-and energyresolved tracking of electron and hole kinetics. Carrier-induced and heat-induced band shifts are observed in addition to state filling, synthesizing a complete picture of the photoinduced dynamics. The electron and hole relaxation and recombination times within a single experiment on nanocrystalline germanium thin films are successfully characterized.

Results
XUV transient absorption experiment on germanium. In Fig. 1 the XUV transient absorption experiment performed on a nanocrystalline germanium film is illustrated schematically. In the experimental apparatus (Fig. 1a), a time-delayed VIS-NIR pump pulse is collinearly superimposed with a broadband XUV pulse and focused onto the sample. The transient absorption signal DA meas (E, t) ¼ A p (E, t) À A(E) is defined by the difference between the absorbance of the excited (pumped) thin film A p and the static absorbance A. Depending on the time delay t between the VIS-NIR excitation (pump) pulse and XUV probe pulse, the transient absorption as a function of wavelength and time is recorded by spectrally resolving the XUV light after the sample using a spectrometer. A detailed explanation of the experimental apparatus and sample characterization is in the Methods section and Supplementary Note 1. The samples are produced by electron beam deposition and annealing, leading to nanocrystalline domains, most likely containing significant defects. The few-cycle VIS-NIR pulse spanning 1.2-2.5 eV photon energy promotes electrons across the bandgap from the VB into the CB (Fig. 1b). Given the spectrum of the exciting VIS-NIR pulse, the region of possible carrier excitation within the Brillouin zone of germanium is calculated (see Supplementary Note 2). Figure 1c summarizes the possible optical excitations from the heavy-hole, light-hole and split-off bands to the four low-lying CBs. All significant photon energies contained in the VIS-NIR pulse allow for a direct one-photon transition between the bands, hence indirect transitions assisted by phonons are expected to play a minor role for the pump pulse used 22   tracks the intervalley and intravalley scattering, that is, the thermalization of holes and electrons, and ultimately the recombination of the carriers. The broadband XUV pulse, whose spectrum spans the germanium M 4,5 absorption edge (Fig. 1d), allows to probe an electron originating from a germanium 3d core level to a respective VB or CB state. In general, the XUV pulse can probe the transient populations in both the CB and VB as both are of partial 4p orbital character (green colour code in Fig. 1b), while transitions to parts of the bands that are of 4s character are forbidden by dipole selection rules (red colour code in Fig. 1b) 23 . One expects to observe dynamical features of band shifts due to the excited electron-hole plasma 24 and phonon heating of the lattice 25,26 , carrier excitation-induced core-level shifts and spectral broadening of the M 4,5 edge ( Fig. 2b) 27 , and electronic SB from excited carriers. As a whole, the complexity of underlying effects that contribute to the observed transient absorption spectra require disentangling these from the raw data to study the contributions individually. We introduce an iterative procedure that accounts for possible spectral broadening, shifting and SB, and seeks to break up these individual contributions from the measured raw data. In this procedure, the individual contributions by shifting and broadening can be calculated from the measured static absorbance A(E) (Fig. 2b) of the sample using a linear energy shift DE shift and by convoluting the absorbance with a Gaussian of width s, respectively. The SB can be estimated in the first iteration by two Gaussians having a width corresponding to the bandwidth of the VIS-NIR pulse with opposite signs for electrons and holes spaced by the central energy of the VIS-NIR pulse and subsequently by considering the spin-orbit splitting of the core level. The assumption of a Gaussian shape of the SB is chosen as reasonable compromise between the expected initial distribution related to the NIR-VIS spectrum, which transitions into a Fermi-Dirac-type distribution after rapid thermalization, convoluted with the instrumental resolution of the instrument. The time dependence of the SB is first estimated as a single exponential decay with a characteristic time constant retrieved from the experimental data. By minimizing an error metric taking the sum of these components and the measured signal into account, in the first iteration the underlying time-dependent shift DE shift (t) and broadening s(t) are estimated. As a convention, here a positive value of DE shift (t) means a redshift of the excited state spectrum. In subsequent iterations, the SB component DA SB (E, t) ( Fig. 2c) is refined along with refinement of DE shift (t) and s(t), converging after five iterations. It is found that broadening DA broad (E,t) plays only a minor role in the experiments presented here and constitutes signals at the noise level (Fig. 2d). The major contributions arise from a time-dependent redshift DA shift (E,t) ( Fig. 2e) and the SB DA SB (E,t) ( Fig. 2c) following carrier excitation. A detailed description of the iterative procedure is given in Supplementary Note 3.
As noted, the SB transient obtained (Fig. 2c) contains contributions from both 3d 5/2 and 3d 3/2 core-level transitions. A Fourier reconstruction method (see Methods section and Supplementary Note 4) is applied to the transient absorption signal to retrieve the contribution from a single spin-orbit state transition (3d 5/2 ). Figure 2f shows the spin-orbit-separated SB transient DA SB;3d 5=2 E;t ð Þ that we will subsequently refer to as carrier dynamics.
In the time-dependent band shift DE shift (t) (Fig. 2g) at negative time delays, a constant redshift is measured that originates from a heat-induced band shift owing to the thin film being heated up in the multipulse exposure experiment. This transient feature allows retrieving the temperature of the thin film, which ranges from 325 to 475 K for the data presented here (see Methods section and Supplementary Note 5).
Feature assignment in experimental transient absorption data. Figure 3a shows the retrieved carrier dynamics DA SB;3d 5=2 E; t ð Þ (same as Fig. 2f) with assignments of the valleys in germanium (cf. Fig. 1b). In general, two features are apparent, a broad positive feature around 28.2 eV and a broad negative feature spanning 29.5 to 31.0 eV. The signs of the carrier dynamics features can be understood as follows. With VIS-NIR excitation of electrons from VB to CB, holes are created in the VB, resulting in an increase of XUV absorption below the M 4,5 edge. Oppositely, the VIS-NIRexcited electrons in the CB block XUV transitions from the core level, leading to a decrease in XUV absorption, and, thus, a negative DA. The carrier dynamics features as well as the band shift DE shift (t) rise within the instrumental response time (Fig. 3b) closely following the exciting VIS-NIR pulse. The slow rise of the signal before time zero can be associated with a small pedestal or prepulse, for which the allowed one-photon transition in this experiment is susceptible to observation, in contrast to previous semiconductor attosecond experiments that used nonlinear excitation schemes 10,11 .
To support the assignment of the features in the carrier dynamics signal, the profiles averaged over time delays between þ 8 to þ 12 fs (black rectangle in Fig. 3a) are compared to differential absorption profiles obtained by first-principles calculations (Fig. 3c). Here, real-time time-dependent density functional theory (TDDFT) was first employed to calculate the electronic excitation of crystalline germanium by a VIS-NIR pulse comparable to the one used in the experiments (see Methods section). From the retrieved populations in the CB and VB following excitation, an excited-state X-ray absorption spectrum (XAS) was then calculated to obtain a transient absorption spectrum (dashed lines in Fig. 3c) with respect to a calculated XAS ground state spectrum, where both spectra are calculated for a single spin-orbit state. The comparison of the calculated (dashes in Fig. 3c) and experimental transient absorption (blue solid area in Fig. 3c) strongly corroborates the assignment of electrons and holes to the negative and positive carrier dynamics feature, respectively. This assignment is further supported by the binding energy of the 3d 5/2 core level at 29.2 eV coinciding with the gap between the two main features relating to the bandgap. The deviations at higher electron and hole energies in Fig. 3c can be explained by the TDDFT calculation underestimating electronhole interaction effects and neglecting carrier-phonon scattering, carrier-carrier scattering and relaxation processes, thus overestimating state filling at higher energies. In the experiment, carriers can immediately scatter, assisted by phonons or other carriers, to regions in k space not accessible by direct optical transitions and hence on average leave more transitions open at a given energy, thus mitigating band-filling effects.
Electron kinetics. The carrier dynamics DA SB;3d 5=2 E;t ð Þ associated with the electrons (Fig. 4a) are analysed by singular value decomposition (SVD) to separate the underlying signals into different independent contributions of electrons and holes and to trace out their temporal evolution; see Methods section for details. The transient signal can be described by two major components whose spectrum is depicted in Fig. 4b along with the time dependence depicted in Fig. 4c. Calculating a carrier dynamics transient from these two SVD components (Fig. 4d) yields good agreement with the measured carrier dynamics transient (Fig. 4a). The component with larger amplitude and with the spectral distribution peaking near the band edge (red data points in Fig. 4b,c) can be assigned to the thermalized electrons. It exhibits a time dependence that can be fit with a single exponential decay with a time constant of t e,recomb ¼ (1,140±50) fs, which is characteristic of the recombination of the carriers, since this is the timescale when the transient carrier dynamics signal vanishes (Fig. 4a). The error bars are derived from the fitting to the experimental data. A single exponential describes the observed time dependence best and hints at a trap-assisted recombination process 28 , which is expected to be dominant in defect-rich samples 29 . The second largest component (blue data points in Fig. 4b,c) consists of higher-energy contributions and lower-energy components with reversed sign. This component (blue data points and fit with dashed black line in Fig. 4c) decays with t e,relax ¼ (110±30) fs, reversing its sign at the long time limit. This component of the SVD can be interpreted as high-energy electrons (hot carriers) ΔA calculated from TDDFT relaxing very rapidly from higher-energy states to fill available states at lower energies within this relaxation time.
In addition, single exponential functions were fitted to profiles along the time-delay axis at energies between 29.6 and 30.4 eV (Fig. 4e,f), with time-zero amplitudes and time constants as free parameters (Fig. 4g). The time-zero amplitudes relate to the initial distribution of electrons (green line with shaded error band in Fig. 4g) and show that initially a broad distribution of electrons is excited. The time constants (blue line with shaded error band in Fig. 4g) provide a global view of the energy-specific lifetimes. One observes generally an increasing lifetime of carriers closer to the CB edge. Further, the CB valleys such as X 1 , D 1 , G 2 0 and weakly L 1 stand out with significantly longer lifetimes, because the carriers relax to particular valleys and accumulate there.
Hole kinetics. Figure 5a shows the carrier dynamics signal that is associated with the holes, which is also analysed by SVD. The wide positive distribution at time zero spanning from 27.8 to 29.2 eV narrows down and decays as the time delay t increases, representing the hot carrier relaxation and recombination of holes. Decomposing the transient absorption signal by SVD reveals two major components. For the largest component, an assignment is made to a thermalized hole distribution (Fig. 5b, red line), whose dynamics show first a rise, t split-off,heavy hole ¼ (140±10) fs, after the initial excitation and a subsequent t h,recomb ¼ (1,080 ± 90) fs decay (Fig. 5c, red dots with solid black line) that are obtained by fitting a biexponential function. The assignment of the faster dynamics with the time constant t split-off,heavy hole to hole scattering pathways from the split-off band to heavy-hole band is based on the initial increase of the measured carrier concentrations. This is most clearly seen for time delays of 100-150 fs, where both the measured transients (Fig. 5a) and the thermalized component of the SVD (Fig. 5c) maximize. In Fig. 1c it is shown that excitation to the split-off band by the pump pulse, which is of partial 4s orbital character (Fig. 1b), is possible. Once carriers scatter from the split-off band to the heavy-hole band, which is predominantly of 4p orbital character, the carriers become observable to the XUV and increase the carrier dynamics signal. Since the VB is degenerate at the G point with predominantly 4p orbital character, another possible pathway would be hole relaxation within the split-off band towards the G point. However, this process has been previously found to be less efficient compared to the split-off band to heavy-hole band scattering process 17 . For other pathways such as carriers scattering from the heavy-or light-hole band into the split-off band, a reduction of amplitude, contrary to the observation, would be expected in the carrier dynamics signal. The second largest component features a positive distribution at larger (hot) hole energies (B28 eV) and a negative distribution near the band edge (Fig. 5b, blue line). The time evolution of this component starts at a positive value and becomes negative at the long time limit; it is fitted with a single exponential decay (Fig. 5c, blue dots with dashed black line) with a t relax ¼ (170 ± 10) fs time constant. This indicates that carrier dynamics, albeit a signal proportional to the population distribution, changes sign at the long time limit. Multiplying the time dynamics and the carrier dynamics component together suggests that the second largest component represents a depletion of hot holes and an increase of low-energy holes as the singular vector in time decays from positive to negative value (Fig. 5c, blue dots and dashed black line), representing the hot hole relaxation.
Time-dependent band shifts. The main contribution from the band shift observed here originates from the redshift of the CB, where the absorbance experiences a steep rise (inset Fig. 3c, upper  panel). The VB region is attributed to energies where the absorbance is relatively flat, indicating that band shifts in either direction would result in weak contributions on the VB side. It is a feature that XUV transient absorption measurements capture differential quantities, where underlying small shifts can cause measurably broad signals if these changes occur in steep sections of the underlying ground state absorbance (cf. Fig. 2e). These detectable shifts can be significantly below the actual spectral resolution of the instrument.
In Figure 6, the heat-induced band shift has been subtracted to focus on the band shift following photoexcitation using different intensities. The carrier-induced band shift scales with the third root of the number of excited carriers DE gap $ N 1 3 e (see Supplementary Note 6) and is thus time dependent as the carrier density changes 24 , for example, by recombination. In the experiment, the observed redshift increases with increasing intensity and thus increased initial carrier density N e in the CB. In Fig. 6 the time-dependent energy shift at one excitation intensity (black line in Fig. 6) is compared to DE gap ðtÞ $N e ðtÞ 1 3 (red dotted line in Fig. 6) using a carrier density decay that follows an exponential with a time constant of 1.1 ps with increasing delay t. The excellent agreement suggests that the observed band shift is predominantly caused by a carrier-induced dynamic redshift of the CB. The oscillations in the band shift signal may arise from phonon dynamics induced by carrier excitations. The phonon excitations lead to lattice strains that subsequently result in shifting of band energies. The impulsive and broadband excitation can cause excitation of overtones 30 and multiphonon processes 31 , which can lead to a large bandwidth of excited phonon modes giving rise to the fast oscillations observed in the bandshift signal.
The excited carrier densities range from 0.4 Â 10 21 to 1.2 Â 10 21 cm À 3 according to the TDDFT calculations using the pulse parameters of the experiment. In the inset of Fig. 6, the measured redshifts directly after excitation for the calculated carrier densities are compared to DE gap /2 (red dashed line, inset Fig. 6) to account for the fact that here only the redshift of the CB is observed. The overestimation of bandgap renormalization is due to the additional Burstein-Moss effect, which is a blueshifting of the absorption edges 32,33 at carrier densities exceeding B10 19 cm À 3 . In the X-ray absorption data this may also be counteracted by a core-level redshift due to the screening 10 , which is expected to be on the order of a few tens of meV. This, however, does not limit the extraction of the carrier dynamics contribution. An extended discussion can be found in Supplementary Note 6.

Discussion
In the present experiment, electron and hole dynamics are studied simultaneously and with high time and energy resolution.
The energetic separation of the electron and hole signals is of great value since band-related features and carrier-specific dynamics can be unambiguously assigned (Fig. 3a). The decoupling of the pump and probe spectral regions allows to excite samples with, for instance, ultrashort broadband laser pulses as in the present experiment to study ultrafast excitation dynamics and electronic relaxation processes of both carriers, or with longer monochromatic pulses for valley-specific excitation independent of the XUV probe pulse. The possibility for frequency-converting the ultrafast pump pulses will further extend the accessible materials. Converting the ultrashort pump pulse into the ultraviolet-to-blue spectral region 34 will allow the investigation of wide-bandgap materials. Furthermore, thermalization dynamics such as the B110 fs relaxation time observed for the CB electrons as well as capturing the fast oscillations in the band shift that are likely related to coherent phonons and their overtones require the time resolution that ultrafast XUV transient absorption spectroscopy offers, which can be expanded towards attosecond transient absorption spectroscopy by isolating a single attosecond pulse from the attosecond pulse train.
Discussing the electron dynamics, the assignment of the CB valleys in Fig. 4e, is essential. Using energy-resolved line profiles and fitting single exponential decays at each energy for near-edge CB states, energy-resolved carrier lifetimes are extracted. The initial amplitudes of the exponential fits (green line with shaded error bars in Fig. 4g) suggests that initially a broad distribution of electrons is excited. Following excitation, the carriers undergo intervalley and intravalley scattering with phonons, effectively populating all bands within the energy range and relaxing the electrons to respective valleys. The measured time constant versus energy (blue line with shaded error bar in Fig. 4g)   carrier states is the shortest, continually increasing towards the band edge. At the valleys, the measured time constants peak due to the carriers accumulating (see for example, energy assigned to X 1 and D 1 valleys in Fig. 4g). Hence, although the experiment averages over all k points in the Brillouin zone, the results suggest that characteristics of specific valleys can be extracted from the experiment due to the high DOS at the valleys, provided the involved valleys have sufficient energy spacing. At the germanium M 4,5 edge this is the case, for example, for the X 1 valley versus G 2 0 , but not for L 1 and G 2 0 . Here, the achievable resolution is mainly limited by the core-hole lifetime broadening, which constrains the energy resolution to dEE0.24 eV (ref. 23). For nanocrystalline samples, an additional broadening due to averaging over different crystal orientations is expected 35 . In the experiment, the slow time constants for decay in the G 2 0 and L 1 valleys are on the order of 1.1 ps, which is consistent with the carrier decay time in the SVD analysis. The SVD components (Fig. 4b) can be understood as an ensemble average and the measured B100 fs relaxation time in the second SVD component corresponds to an average relaxation time that also includes electrons at higher energies, where the signal strength prohibits meaningful line profile measurements. It is worth noting that the weak amplitude in the L 1 valley (Fig. 4g) is due to the high s orbital character of the states, which limits its visibility in the XUV spectrum. Further, G-L and X-L intervalley scattering occurs on a 200-300 fs timescale, which together with the observed lifetime of carriers at the CB edge gives rise to a low overall transient population observed in the L valley.
In the hole kinetics, the initial rise of the main component of the hole transient (Fig. 5c) has been assigned to holes scattering from the split-off band into the heavy-hole band based on comparison of orbital characters in the VB (Fig. 1b) and only one intervalley scattering pathway that allows for a signal increase. This assignment is supported by comparison with hole relaxation times measured in germanium by optical pump-probe techniques. Woerner et al. 17 reported intervalley scattering times of holes from the split-off band assisted by phonons to the heavyhole band in the G valley to be within their instrument response time of 250 fs and estimated a lifetime of B100 fs by calculations. This value corresponds well to the (140 ± 10) fs rise of the main component obtained in the present experiment. Although Woerner et al. 17 excited specifically to the split-off band from the heavy-hole band near the G point, their B0.5 eV photon energy results in comparable hole energies as in the present experiment, considering that a broadband pulse with B1.65 eV central photon energy excites across the bandgap (Z0.8 eV). Due to the higher effective mass of the heavy holes, the scattering process towards the heavy-hole band is expected to be dominant and the model calculations and experimental results in their paper suggest the interband scattering from the split-off band is more effective than interband relaxation. The long time decay of the hole signal on the order of B1.1 ps indicates the recombination of the holes with the electrons, which within the error bars have the same decay constant for the observed time window and accessible signal-to-noise ratio.
In the present experiment, the electrons and holes exhibit a similar lifetime, that is, the decay constants are B1.1 ps for both. A question that arises is whether the electrons and holes recombine or if they relax to states that are invisible for the experiment, for example, states of s orbital character. The in situ measurement of the band shift suggests an interpretation of carrier recombination within the observed timescale, because the band shift over time (cf. Fig. 6) also follows a behaviour that indicates a single exponential decay of the free carriers in the CB as is observed in the kinetic analysis of the carrier dynamics in the CB, with agreement of the time constants. The longest lifetime occurs in the lowest bands, compared to a shorter lifetime at higher energies, supporting that the carriers have first thermalized and subsequently recombine. A possible recombination mechanism that would support the observed fast decay rate suggests the involvement of intermediate trap states and a Shockley-Read-Hall recombination mechanism 29,36,37 . This can be understood based on the nanocrystalline samples. The nanocrystalline thin films have a grain size of B11 nm. Further, it is known that randomly oriented non-passivated nanocrystals exhibit a large number of trap states at the interfaces of the nanocrystallites. These trap states can act as recombination centres 38 . The recombination times are expected to be significantly accelerated compared to single-crystalline samples, since assisted by phonon scattering the carriers can quickly scatter into these trap states and subsequently recombine. Given the high carrier density initially excited, Auger recombination is expected to play an additional role, while time constants in the order of 5 ps would be expected from an Auger recombination process 39 .
The observed timescales are in good agreement with visible transient reflectivity measurements on germanium nanorods, where carrier lifetimes of B6 ps were measured for nanorods having a diameter of 18 nm (ref. 40), considering that grown nanorods in general are expected to have a lower number of trap states compared to annealed nanocrystalline thin films. Further, the smaller domain size in the present experiment and threedimensional versus two-dimensional trap state arrangement locations can explain the comparatively shorter measured lifetimes.
In summary, this paper presents the study of ultrafast carrier relaxation in germanium using NIR-XUV pump-probe spectroscopy with few-femtosecond time resolution. It is demonstrated that transient absorption in the XUV yields clear, spectrally resolved signatures for the energy distribution of electrons and holes simultaneously. This study on carrier dynamics of germanium using broadband ultrafast XUV transient absorption spectroscopy provides direct observation of the relaxation of both the electron and hole distributions in a semiconductor material after broadband (500-1,000 nm) excitation, which is unprecedented in conventional optical pump-probe studies. The transient absorption of XUV is further decomposed into contributions from electronic SB, band shifts and excited state broadening. To analyse the carrier dynamics in germanium, a Fourier transformbased technique is devised to retrieve the contribution from a single spin-orbit state at the germanium M 4,5 edge in the transient absorption data. Comparison with first-principles calculations supports the assignment of electron and holes in the retrieved carrier dynamics transient signal. Due to the high temporal and energy resolution, broadband ultrafast XUV transient absorption spectroscopy is an ideal tool for disentangling the contributions of relaxation and recombination processes. Clear signatures of hot carrier relaxation for both electrons and holes in germanium occurring below 200 fs is observed, which has been so far inaccessible through narrowband, pure optical/NIR pump-probe techniques. In addition, the presented experiment suggests that the retrieved time-dependent band shift can be decomposed into a heat-induced and, predominantly, a carrier-induced redshift of the CB. The heat-induced redshift allows for in situ characterization of the sample temperature. The temporal behaviour of the carrier-induced redshift supports the observed fast B1.1 ps carrier recombination of electrons and holes observed in the carrier dynamics. Moreover, scattering and relaxation rates can be measured in the time domain and by the additional high-energy resolution quantitative data can be retrieved for different parts of the band structure. This technique will be an invaluable tool for research on more complex semiconductor materials such as strained layers, heteroatomic, ternary, quaternary systems, junctions, two-dimensional material compounds and quantum confined systems, which are becoming increasingly important for applications such as solar energy production and highly efficient computer processors.

Methods
Experimental details and data analysis. VIS-NIR pump pulses with 5 fs pulse duration are used to optically excite germanium thin films. The nanocrystalline germanium films, fabricated by electron beam deposition and subsequent annealing, are 100 nm thick and supported by a 30-nm-thick silicon nitride membrane. The crystallites have an average size of 11 nm, see Supplementary Note 7 for extended characterization. A spectrally continuous XUV pulse spanning the germanium M 4,5 edge is generated by high harmonic generation in xenon assisted by polarization-assisted amplitude gating gating 41 . The XUV pulses are used to probe the transient population in the VB and CB in a transmission geometry and are subsequently resolved by a flat-field spectrometer. By changing the time delay t between the pulses the transient changes are measured. A shutter periodically blocks the VIS-NIR pump beam to obtain a differential absorption signal. The data set shown in the main text Figs 2 and 3 (panels a-c and black line in Fig. 6) and analysed in Figs 4 and 5 consists of five averages and was scanned with 0.6 fs time steps around time zero and with 3.3 fs time steps out to 1.5 ps. The additional intensities shown in Fig. 6 consist of 50 averages each with 26 fs time steps on an interval of t ¼ [ À 60, þ 550] fs. To compensate for time delay drifts in parallel to the germanium sample the differential absorption at the argon 3s3p 6 6p autoionizing state is measured 42 . The instrumental response time has an upper bound estimated to B6 fs by comparing these gas transient absorption measurements. At the same time, the spectrometer resolution can be estimated to dEE0.07 eV from known linewidths in gas absorption spectra. In the postexperimental data analysis, first the measured differential absorption data DA meas (E,t) data are decomposed into contributions from SB, band shifts and excited state broadening. The high-quality static absorbance used for decomposing these contributions was measured by averaging one hundred transmission spectra alternating between the sample and an empty silicon nitride membrane under otherwise identical conditions to those for the time-resolved measurements. The SB signal is subsequently separated into a single spin-orbit split state by dividing out a constant phase factor in the Fourier domain originating from the spacing (DE SO,Ge ¼ 0.58 eV, ref. 23) and degeneracy of the core levels (g 3d 5=2 ¼ 6 and g 3d 3=2 ¼ 4) exhibiting spin-orbit splitting and subsequent back transformation: The retrieval constitutes the signal probed from the 3d 5/2 state. Further details are given in Supplementary Note 4.
Excited carrier density. The number of excited electrons has been estimated in two ways. First, from the fluence F of the laser pulse with central frequency n and an absorption coefficient a of a sample with thickness d, one can estimate for a onephoton transition, where one absorbed photon excites one electron, that N e ffi F hnd 1 À exp À ad ð Þ ½ . For the presented experiment (I ¼ 2 Â 10 11 W cm À 2 , F ¼ 12.4 Â 10 À 3 J cm À 2 , d ¼ 100 nm, a ¼ 49 Â 10 3 cm À 1 , l 0 ¼ 760 nm) one gets N e D8 Â 10 20 cm À 3 . Second, the performed TDDFT calculations using k-dependent excitation and the spectrum of the experimental laser pulse an excitation fraction of 0.3% was calculated for a laser intensity of 2 Â 10 11 W cm À 2 . Using that a unit cell in germanium has a volume of V 0 ¼ 4.527 Â 10 À 23 cm 3 and that there are two atoms or eight valence electrons in this volume, one gets a valence electron density of d 0 ¼ 1.76 Â 10 23 cm À 3 . Hence, for the TDDFT-based calculation one gets N e D5 Â 10 20 cm À 3 .
Heat-induced band shift. In the time-dependent transient absorption experiments a transient signal at negative time delay, that is, NIR pump arrives after XUV probe, was consistently observed when measuring first the unexcited (cold) sample transmission followed by the excited (hot) sample spectrum to obtain a differential absorption spectrum. A comparison of the absorption change in a heated Ge film to this spectral feature suggests that it is related to an increased temperature of the germanium thin film after optical excitation causing a heat-induced band shift 43 . The absorption edge redshifts causing a positive differential signal. By resolving the underlying shift due to thermal expansion of the lattice DE ren,therm , a temperature can be assigned to the thin film. For example, for an average pump power of DP avg ¼ 0.39 mW the measured shift of the edge is DE ren,therm ¼ 72.8 meV, suggesting a temperature of T ¼ 475 K, which is confirmed by heat diffusion calculations using a finite element method (Supplementary Note 5).
First-principles TDDFT and XAS calculations for germanium. To support the assignment of spectroscopic features, X-ray absorption spectroscopy calculations based on the eXcited-electron Core-Hole (XCH) 44 approach were combined with first-principles TDDFT 45,46 . This first-principles treatment of the electric field dynamics described using real-time TDDFT 45,46 includes effects beyond the linear response of the material, and similar theoretical treatments reproduce electron tunnelling in silicon 10 and the dielectric breakdown in insulators 47 . The exciting VIS-NIR pump laser field models a few-femtosecond, 780 nm pulse with a peak intensity of 1 Â 10 11 W cm À 2 (see Supplementary Fig. 7a).
The excited-state electron and hole occupations generated by interaction with the pump laser pulse are obtained from the real-time TDDFT calculation in the form of a single-particle density matrix. This one-electron density matrix is subsequently projected onto an eigenbasis of the core-excited XCH Hamiltonian, along the lines described previously 10,48,49 , to estimate the M 4,5 edge XUV absorption in the presence of excited carriers. The difference between this excited state spectrum and the corresponding ground state spectrum in the absence of electron-hole pairs is compared to the 3d 5/2 core-level differential absorption retrieved from experiment at time delays directly after excitation, shown in Fig. 3c. Inclusion of the core-hole excitonic effects, at the level of the XCH approach, in the core-level absorption calculation was necessary to match the magnitude of the negative feature at 30 eV. In the first-principles calculations, the excited and ground state spectra are treated only for a single spin-orbit split state and shifts, such as core-level shifts or band shifts, are not included. We note that at the level of the adiabatic TDDFT and XCH theories employed here we expect state-blocking effects in the excited state M 4,5 edge absorption to be captured but many-body effects such as excited-state bandgap shifts and carrier lifetime modulation are not described. Therefore, the comparison with experiment is invoked primarily in the context of state blocking where good agreement is observed (Fig. 3c). Further details are given in Supplementary Note 10.
Singular value decomposition. The carrier dynamics can be decomposed into several carrier distributions with independent relaxation dynamics. Under such formulation, the transient absorption signal DA(E,t), with E as photon energy and t the delay time, can be decomposed into DA E; t ð Þ¼ X n s n u n E ð Þv n ðtÞ; ð2Þ where u n is the nth component of transient absorption, which corresponds to a distinct carrier distribution and n n its accompanying relaxation dynamics. s n is the singular value of the nth component, which signifies the importance of the component on the overall dynamics.
Germanium band structure calculation. The band structure of germanium ( Fig. 1b) is calculated by DFT within the Vienna ab initio simulation package 50,51 , using a generalized gradient approximation and Hubbard (GGA þ U) approach. The computational method and parameters are detailed in ref. 52.
Data availability. The experimental data can be obtained on reasonable request from the corresponding authors.