Ultrafast dynamics of hot charge carriers in an oxide … · 2019-09-20 · Figure 1: Hot charge...

Ultrafast dynamics of hot charge carriers in anoxide semiconductor probed by femtosecond

spectroscopic ellipsometry

Steffen Richterlowast12a Oliver Herrfurth2a Shirly Espinoza1 Mateusz Rebarz1Miroslav Kloz1 Joshua A Leveillee3 Andre Schleifedagger3 Stefan Zollner45

Marius Grundmann2 Jakob Andreasson16 Rudiger Schmidt-Grund2

1ELI BeamlinesFyzikalnı ustav AV CR vvi Za Radnicı 835 25241 Dolnı Brezany Czech Republic2Universitat Leipzig Felix-Bloch-Institut fur Festkorperphysik Linnestr 5 04103 Leipzig Germany

3University of Illinois Dep of Materials Science and Engineering 1304 W Green St Urbana IL 61801 USA4New Mexico State University Department of Physics PO Box 30001 Las Cruces NM 88003-8001 USA

5Fyzikalnı ustav AV CR vvi Sekce optiky Na Slovance 2 18221 Praha Czech Republic6Chalmers tekniska hogskola Institutionen for fysik Kemigarden 1 41296 Goteborg Swedenlowaststeffenrichtereli-beamseu daggerschleifeillinoisedu aThese authors contributed equally

Feb 2019

Many concurrent processes occur in highly photoexcited semiconductors such as interband and intrabandabsorption scattering of electrons and holes by the heated lattice Pauli blocking bandgap renormalizationand formation of Mahan excitons We disentangle their contributions and dynamics with broadband pump-probe ellipsometry performed on a ZnO thin film We directly obtain the real and imaginary parts of thetransient dielectric function and compare them with first-principles simulations The interband and excitonicabsorption are partially blocked and screened by the hot-electron occupation of the conduction band and holeoccupation of the valence band Simultaneously intra-valence-band transitions occur at sub-picosecond timescales after holes scatter to the edge of the Brillouin zone Our time-resolved ellipsometry experiments withintense UV excitation pave new ways to understanding non-equilibrium charge-carrier dynamics in materialsby distinguishing between changes in absorption coefficient and refractive index thereby separating competingprocesses This information will help to overcome the limitations of materials for high-power optical devicesthat operate in the ultrafast regime

Many-body systems under non-equilibrium condi-tions caused for instance by photo-excitation stillchallenge the limits of our understanding at micro-scopic length and ultrashort time scales [1 2 3] Ac-cessing and controlling emergent states experimentallyconstitutes one of the most exciting forefronts of con-temporary research in materials [4] In addition toadvancing fundamental understanding of exotic quan-tum states eg involving large densities of free charge-carriers [5] such many-body systems can provide tech-nological breakthroughs for novel applications includ-ing high-speed optical switching [6 7] and comput-ing [8 9] fast transparent electronics [10 11] lightharvesting [12] or even new means of propulsion ofspace crafts [13] The implementation of such next-generation devices requires development of techniquesto probe transient states of matter and control ultra-fast dynamics of excited electronic systems in solids

Despite tremendous promises understanding of thecoupling between fundamental electronic excitationsand the lattice mostly remains vague especially di-rectly after strong excitation Many experimental and

theoretical studies have aimed to separate fundamentalelectron-electron and electron-phonon effects as well asthe role of defect states in solids [14] As illustratedschematically in Fig 1 not only the population ofcharge-carrier states (as probed in luminescence exper-iments) matters in the highly excited regime but alsothe electronic structure Therefore the optically ac-cessible states change rapidly enabling or prohibitingcertain absorption channels but also changing othermaterial properties like the index of refraction Thisis especially significant for applications of transparentsemiconductors like wide-gap oxides [15]

Experimentally angular-resolved photo-electronspectroscopy is one of the most insightful probesfor the dispersion of populated electronic states[16 17] Beyond this optical spectroscopy accessesa convolution of joint density of states electron andhole populations and transition matrix elements viathe complex frequency-dependent dielectric function(DF) For excited systems conventional transientspectroscopy has been performed at different spectralranges [18 19 20 21] time-resolved sum-frequency

1

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15

Feb

2019

generation can be used to probe the dynamics ofelectronic transitions after excitation [22] A challengeis not only to achieve high time-resolution but todiscriminate different processes triggered by the exci-tation [14 23] One must understand the entire iespectral and complex-valued response of an excitedmaterial This requires obtaining both amplitude andphase information of a samplersquos DF as encoded in itscomplex reflection coefficient r

Conventional transient spectroscopy yields only am-plitudes and experimental data is often explainedby changes in the extinction coefficient κ neglectingchanges of the refractive index n a challenge for exci-tation spectroscopy that has already been discussed inthe 80rsquos [24] To compensate for the lack of phase infor-mation restrictive model assumptions ar required Analternative is to combine measurements from differentangles of incidence [25 26 27] or p- and s-polarization[28 29] Even these methods are only a work-aroundbut cannot directly obtain phase information On theother hand obtaining the full dielectric response fromthe time domain is only possible in the THz regime butnot for higher optical frequencies [30]

In ellipsometry the angles Ψ and ∆ offer rela-tive frequency-dependent amplitude and phase infor-mation on the physical response rprs = tan(Ψ)ei∆

(where indices refer to p- and s-polarizations) Thisprovides simultaneous access to the real and imag-inary part of the DF ε = ε1 + iε2 = (n + iκ)2Earlier ellipsometric pump-probe experiments sufferedfrom experimental challenges such as changing posi-tions of the probe spot on the sample or stability is-sues with the laser or were performed only at singlewavelengths [31 32 33 34 35 36] Time-resolvedsingle-wavelength ellipsometry has also been reportedin imaging mode [37 38 39]

In this article we present transient DF spectraof highly photo-excited ZnO with femtosecond time-resolution These spectra yield information on theultrafast dynamics of electron-electron and electron-phonon processes in this prototype oxide semiconduc-tor With its wide bandgap and excitons stable at roomtemperature ZnO is an ideal testbed Due to its strongpolarity strong electron-phonon coupling impacts theexciton dynamics Our improved ellipsometric ap-proach gives access the time-dependent DF of the ZnOfilm after excitation with about 100 fs time resolutionThese results are compared to first-principles simula-tions We separate Pauli blocking of absorption chan-nels from bandgap renormalization (BGR) induced byincreased screening of the electron-electron interactionby photo-excited electrons and holes From our anal-ysis we report direct observation of intra-valence-band(IVB) absorption in a spectral range that is normallytransparent in ZnO

Experimental Data

The experiments were performed on a ZnO thin filmpumped by 266 nm 35 fs laser pulses that created anelectron-hole pair density of 1020 cmminus3 Supercontin-

Figure 1 Hot charge carriers after strong excitation ofZnO with a UV pump pulse ab Within a few 100 fs af-ter excitation (violet arrows) scattering between charge carriersresults in the conduction band being occupied by excited elec-trons (filled circles) the valence band by holes (open circles) (a)The thermal distribution (Fermi-Dirac statistics) of the excitedelectrons (black) and holes (red) corresponds to effective tem-peratures Te Th of a few 1000 K (b) The quasi Fermi-energies(dashed lines) are shifted into the bandgap due to the high tem-peratures cd Within the first picoseconds scattering betweencharge carriers and phonons as well as recombination yield cool-ing and reduction of the density of excited electrons and holesStill charge-carrier temperatures are larger than the lattice tem-perature Tl Black arrows in a and c mark selected optical tran-sitions which are dynamically blocked (band-band transitions)or enabled (intra-valence-band transitions)

uum white-light pulses were used as a probe The tran-sient ellipsometric angles Ψ and ∆ obtained in the spec-tral range 20-36 eV are shown in Fig 2 Data wererecorded up to 2 ns with increasing delay steps but thestrongest response occurs on time-scales shorter than05 ps The pump-induced effect on the excitonic tran-sitions around 33 eV causes a sudden decrease in Ψand an increase in ∆

From the ellipsometric spectra we obtain the DFof the ZnO film for every pump-probe delay ∆t Fig-ure 3 illustrates the resulting DF ε = ε1 + iε2 at se-lected delays At negative ∆t the obtained DF coin-cides with the one obtained in standard ellipsometryThe peak around 335 eV comprises the excitonic tran-sitions (X) the peak around 342 eV is associated withexciton-phonon complexes (EPC) [40] There exist alsofurther complexes at slightly higher energy

For small positive ∆t the absorption at the bandedge and above is strongly damped as indicated by

2

Figure 2 Time-resolved ellipsometry data Ellipsometricangles Ψ (amplitude ratio) and ∆ (phase difference) of the ZnOthin film after non-resonant UV pump measured at 60 angle ofincidence Increases relative to the initial spectra before excita-tion (black) are shown in blue decreases in red The sketch atthe top illustrates the meaning of the ellipsometric parameters

the decreased ε2 (Fig 3) In particular the absorptionpeaks of exciton and EPC are both bleached within400 fs (Fig 4a) This is accompanied by a reducedrefractive index below the band edge as illustratedby ε1 Maximal absorption-suppression is reached at02 ps and lasts until approx 1 ps though we notethat the excitonic enhancement does not completelyvanish at any time as indicated by the peak struc-ture in ε2 The subsequent absorption recovery startsfrom higher energies approaching the fundamental ex-citonic absorption peak later (Fig 3) After 2 ps boththe exciton and EPC absorption peaks recover withtime constants of 3 ps slowed down after 10-20 ps witha non-exponential evolution

Rather simultaneously with the pump laser pulse(cf Fig 4a rise time 200 fs) a broad absorption bandopens up in the bandgap This low-energy absorptionreaches its maximum amplitude at ∆t = 02 ps andthen decreases with a time constant of 1 ps It vanishescompletely after 10 ps at which time the above-edgeabsorption has nearly completely recovered

As Fig 4 c indicates after an immediate redshift ofthe exciton by roughly 20 meV the energies increasewith a linear rate of approx 3 meVps during the first4 ps (red symbols in Fig 4 c) The EPC follows thetrend with even larger increase but without the initialredshift Another later redshift of both yields an en-ergy minimum at 100 ps At 2 ns the absorption edgeremains redshifted by approx 20 meV Furthermore itshould be noted that the energetic difference betweenthe exciton and EPC absorption peaks which had ini-tially increased by more than 30 meV approaches itsinitial value (50 meV) monotonically until complete re-laxation after several nanoseconds (Fig 4 c) Finallyour data shows that the spectral broadening of the ex-citon and EPC transitions is reduced as soon as thesample has been excited (ε2 in Fig 3) This reducedbroadening remains approximately constant for at least2 ns

Separating Physical Processes

Charge carrier excitation by 266 nm (467 eV) laserpulses in ZnO involves optical transitions from theheavy-hole light-hole and split-off valence-bands (VB)into the conduction band (CB) in the vicinity of theΓ point as indicated by the violet arrows in Fig 1aThe excited electrons carry excess energies of almost1 eV the excited holes almost 04 eV because of theirlarger effective mass The initial occupation of elec-tron and hole states due to the pump pulse is sharplypeaked and non-thermal It takes a few hundred fem-toseconds until a Fermi-Dirac distribution is estab-lished as sketched in Fig 1a and b Estimated effec-tive temperatures are reported in Table 1 The ini-tial thermalization is provided mainly through carrier-carrier scattering partially carrier-phonon scattering[14 23 41 42 43] The immediate effect on the opti-cal response spectra is three-fold First the occupationof the states leads to (partial) Pauli blocking (band fill-ing) and hence the observed absorption bleaching of theband-to-band and excitonic transitions The excitonicabsorption enhancement is also reduced by free-carrierscreening The reduced refractive index in the visiblespectral range results from the Kramers-Kronig rela-tions Second due to the flatness of the valence bandsexcited holes have enough excess energy to scatter to-wards the edge of the Brillouin zone (Fig 1a) and thuspromote IVB transitions which are observed as low-energy absorption Third the high density of photo-excited charge carriers yields BGR as seen by the red-shift of the exciton energy Additionally the excitedcarriers screen a static electric field in the film thatotherwise arises from Fermi-level pinning at the surfacecaused eg by donor-like oxygen vacancies [44] Whilethe steady-state broadening of the excitons is causedby the related band bending the charge carriers reduceit At large delay times the vacancies are still passi-vated by trapped electrons at the surface thus keepingthe excitonic peaks narrow Deeply trapped holes canremain for microseconds [45]

Analysis of the transients yields insights into individ-

3

Figure 3 Dielectric function at selected delay times Real (ε1 inset) and imaginary (ε2 parent figure) part of the DF ofthe ZnO thin film at pump-probe delays from -10 ps to 2000 ps

0 2 4 6 8 10 1001000-15

-10

-05

00

05

0 200 400 600 800-15

-10

-05

00

05

0 2 4 6 8 10 1001000

-20

0

20

40

60

80

100

21 23 25 27

00

03

d

1 ps

3 ps

cTime delay (ps)

a

Am

plitu

de d

iffer

ence

EPC

X

b

Am

plitu

de d

iffer

ence

Time delay (fs)

IVB

EPC-X

Time delay (ps)

Ener

gy (

meV

)

21 24 27 30 33

-30-25-20-15-10-050005

(2)

Photon energy (eV)

x10

(2)

Photon energy (eV)

Figure 4 Transient changes of absorption features Evo-lution of absorption amplitude (a b) and peak energy (c) ofthe exciton transition (red) and exciton-phonon complex (blue)as obtained from the maxima of ε2 Black symbols in (a b)depict the integrated value of ε2 in the spectral range 20 eVto 31 eV for different delay intervals The green symbols in (c)show the spectral difference between X and EPC which is relatedto an effective phonon energy Eph Its equilibrium value of about50 meV [40] is indicated by the dotted line Solid lines indicateexponential processes with their time constants d compari-son of computed (red) and experimental (black) ε2 at maximumchange

ual dynamics Charge-carrier thermalization is slightlyfaster for holes (200 fs) than for electrons (400 fs) be-cause of their lower excess energy This is observedin the experiment by a slightly faster rise of the

IVB absorption compared to the exciton bleaching(cf Fig 4a) The subsequent fast decay of the IVBabsorption is a consequence of the hole occupationfar from the Γ point Hence its 1e decay time of1 ps reflects mainly the hole cooling by scattering withphonons This process is also more effective for holesthan for electrons due to their higher effective mass[23]

A reduction of the number of excited charge car-riers (mostly Auger and defect recombination) is ex-pressed by the vanishing BGR within the first picosec-onds (cf exciton peak energy) The transient dy-namics of the absorption bleaching is however gov-erned by the decrease of electron and hole tempera-tures They approach each other due to cooling byscattering with optical phonons [23 46] resulting ina situation as sketched in Fig 1 cd The electron-LO-phonon (Froehlich) interaction is generally a fastprocess (asymp05 ps [23]) and very strong in the polarZnO However the excess energy of the charge car-riers yields an extraordinarily large population of LOphonon states and thus intermediately a non-thermalphonon distribution as sketched in Fig 5 A latticetemperature is not even well defined at this stateThese hot phonons slow down the electron relaxationthrough phonon re-absorption by the charge carriers[23 41 47] resulting in the plateau-like transient dur-ing the first 2 ps ( Fig 4b) It should be noted that thereturn of the EPC absorption (vanishing Pauli block-ing) starts earlier than for the excitons themselves be-cause the occupation of electronic states at energeti-cally higher levels decreases earlier than of those closerto the Γ point Finally the non-thermal phonon dis-tribution is also reflected by the increased energeticsplitting between exciton and EPC ( Fig 4 c) The ef-fective absorption peak of the EPC at 342 eV involves

4

Table 1 Statistics of the electron hole and lattice sub-systems Immediately after charge-carrier thermalization fol-lowing a pump laser pulse with 467 eV to excite 1020 cmminus3

electron-hole pairs in the ZnO thin film The increase of Tlafter complete equilibration is estimated to 50 K at most Seesupplementary information for details

temperature quasi Fermi-energy

electrons Te = 7000 K EeF minus ECB lt minus660 meV

holes Th = 2800 K EVB minus EhF lt minus260 meV

lattice Tl = 300 K

several optical phonons with an effective phonon en-ergy Eph on the order of 30 meV resulting in about50 meV splitting [40] The absorption and re-emissionof many optical phonons by the crystal increases the in-teraction probability of (high-energy) optical phononswith excitons while (low-energy) acoustic phonons areeffectively suppressed ie Ep increases

After more than 2 ps the charge carriers have cooleddown and the non-thermal phonons have disappeared(see Fig 5) In this picosecond regime the recoveryof the exciton and EPC absorption results from the re-duction of the excited carrier density mainly by nonra-diative Auger recombination [48] Its initial time con-stant is 3 ps At later times with lower carrier den-sities slower radiative electron-hole recombination isdominant The overshooting of the exciton amplitudeat later time is related to the reduced exciton broaden-ing as discussed above Equilibration with the latticecan be estimated to be accomplished approx 100 psafter excitation when the exciton energy reaches an-other minimum that indicates the highest achieved lat-tice temperature and thus bandgap shrinkage [49] As-suming a deposited energy density of 100 Jcm3 by thepump pulse a maximum temperature increase of 30-50 K can be expected If transferred entirely to the lat-tice this would correspond to a bandgap decrease ofapprox 25-30 meV at most This fits the experimentalobservation The following slow (approx 2 microeVps)heat dissipation lasts until at least 10 ns

Discussion

We use first-principles electronic-structure calculationsto explain the different effects near the band edgeTemperatures of electrons in the conduction and holesin the valence band are taken into account via Fermi-distributed occupation numbers in the absorption spec-trum for non-interacting electron-hole pairs Many-body perturbation theory including additional screen-ing and Pauli blocking due to the electrons and holesat 0 K is used to describe excitonic effects Com-parison with the experimental data in Fig 4d showsthat the observed reduction of the exciton absorptionis much less than what is expected from the calcula-tions An increased number of free charge carriers isknown to have two opposing effects on the band-edgeabsorption While the exciton is screened and shouldshift toward higher energies due to a reduced bind-ing energy the bandgap shrinks due to renormaliza-

0 20 40 60 8000

01

02

03

04

05

Non-thermal optical phonons

Acousticphonons

Energy (meV)

Occ

upat

ion

of p

hono

n st

ates

T = 300 K

Figure 5 Simplified distribution of hot phonons aftercharge-carrier relaxation The strong LO phonon interac-tion during cooling of the charge carriers yields a highly non-thermal occupation of optical phonons (gray) in contrast to theoccupation of mostly acoustic thermal phonons (blue) whichfollows a Bose-Einstein-distribution (red dashed line) before exci-tation and after lattice relaxation The phonon density-of-statesis taken from [61]

tion Both compensate each other in a good approxi-mation such that the absolute exciton energy remainsconstant [50 51 52 53] However when surpassingthe so called Mott transition excitons should ceaseto exist and BGR should take over That can ex-plain the initially observed redshift which has been ob-served earlier [27 53 54] Nevertheless we find thatthe excitonic absorption peak does not vanish entirelyat any time That reflects the difference between anequilibrated system and hot charge carriers In thecase of doping ZnO by 1020 cmminus3 excess electrons aBurstein-Moss blueshift of the absorption edge of morethan 200 meV would be expected [55] From density-functional-theory calculation approximately 370 meVcan be estimated It is clear that this does not applyto a hot electron-hole plasma where no strong blueshiftis observed [24 53 54] While BGR does generally notdepend on temperature [56] it should be slightly lessefficient for hot charge carriers [52] resulting in an ef-fectively higher Mott density Hence the Mott tran-sition might not be passed by the widely-distributedhot carriers although their density is well beyond theclassical threshold [57] According to [58] the frac-tion of carriers bound to excitons is rather small notexceeding 15 In this respect the non-vanishing exci-ton absorption peaks could indicate only partial Pauliblocking ie the ground-state occupation would neverexceed the Mott density Furthermore electron-holecoupling has indeed been observed to sustain the Motttransition albeit usually largely screened and broad-ened [52 59 60] Narrow exciton-like peaks have evenbeen observed well above the Mott transition in highlydoped GaN [5] The sustaining absorption peaks hereare likely to be Mahan excitons [59] but in the case ofexcited electrons and holes

The obvious explanation for photo-induced absorp-tion at lower photon energies would be due to the freecarrier response [62] However two Drude terms forelectrons and holes with the known densities and rea-sonable effective masses and mobilities cannot describe

5

the large absorption in ε2 Furthermore there are indi-cations for a maximum of ε2 around 19 eV and 21 eVhinting at IVB transition at the M point In a recentreport similar absorption features induced by lowerpump power and at much longer time scales were at-tributed to defect states [45] however defects cannotexplain the large absorption cross sections (ε2) we ob-serve Comparison of experiment and first-principlesdata for ∆(ε2) in Fig 4d (inset) shows good agree-ment in particular for energetic position and line shapeof spectral features The sub-gap energy-range be-tween 2 and 3 eV is dominated by contributions fromIVB transitions that become allowed in the presenceof free holes The computational results do not ac-count for phonon-assisted processes which likely ex-plains why the computational data underestimates theexperiment at these energies Conduction-conductionband transitions do not significantly contribute in thisenergy range The appearance of the low-energy ab-sorption indicate that the spectral weight of absorptionis transferred from the fundamental absorption edge tolower energies because the total number of charge car-riers remains constant which is known as sum rule [63]

Conclusion

The development of fs-time-resolved spectroscopic el-lipsometry allows to study the dynamics of the com-plex frequency-dependent dielectric function with sub-ps temporal resolution in a wide spectral range In-vestigating a UV-pumped ZnO thin film we wereable to discriminate different processes of the non-equilibrium charge-carrier dynamics of this highlyphoto-degenerate semiconductor We observe partialblocking and screening of near-band-edge and exci-ton absorption due to occupation of the electronicstates Non-vanishing excitonic absorption enhance-ment hints at the occurrence of Mahan excitons Intra-valence-band transitions become possible when holesscatter to the edges of the Brillouin zone Their fastresponse time renders them interesting for optoelec-tronic switching devices Finally there is evidence forhot-phonon effects by both a delayed relaxation andan increased exciton-phonon-complex energy The de-scribed dynamics are crucially dependent on the pumpenergy and hence excess energy of the carriers deter-mining their effective temperature From our data wecan also conclude that the high density of hot chargecarriers does not trigger the Mott transition The sur-vival of the excitonic absorption reflects directly thenon-equilibrium distribution of the excited charge car-riers These facts stimulate demand for new theo-ries regarding high-density exciton systems beyond thepresent state

MethodsWe used a c-plane oriented ZnO thin film grown by pulsed laserdeposition on a fused silica substrate The film thickness of 30 nmis sufficient to maintain bulk properties Only a very slight ex-citonic enhancement due to the confinement in the thin layer isexpected [64] At the same time 30 nm is thin enough to assumehomogeneous excitation by a 266 nm pump pulse (500 microJcm2

35 fs pulse duration) We therefore do not need to consider theambipolar diffusion of hot charge carriers We estimate the ex-cited electron-hole pair density to approx 1 times 1020 cmminus3 Theexperiment is performed at room temperature

Time-resolved spectroscopic ellipsometryWe employ time-resolved spectroscopic ellipsometry in a

pump-probe scheme An amplified TiSapphire laser (CoherentAstrella 35 fs 800 nm 1 kHz repetition rate) is used to generateits third harmonic as pump and continuum white-light in a CaF2

crystal as probe beam In a Polarizer-Sample-Compensator-Analyzer configuration we measure the transient reflectance-difference signal (∆RR)j at 60 angle of incidence for a seriesof different azimuth angles αj of the compensator The polarizerand analyzer are kept fixed at plusmn45 The probe spot had a 1e2

diameter of 200 microm the pump spot 400 microm (40 s-polarized)such that lateral carrier diffusion becomes negligible The cor-responding temporal and spectral bandwidths are estimated to100 fs and 5 nm in the UV respectively Spectra were capturedusing a prism spectrometer and a kHz-readout CCD camera(Ing-Buro Stresing) Most critical is the fluctuating probe spec-trum and amplitude due to the CaF2 crystal movement as well aswarm-up effects at the CCD camera Both occur mostly on timescales larger than a few milliseconds A two-chopper scheme inthe pump and probe paths is employed which allows us to obtaina wavelength-dependent live-correction for the pump-probe aswell as only-probe intensity spectra The obtained reflectance-difference spectra are applied to reference spectra in order toobtain the time-resolved ellipsometric parameters In order tominimize chirping of the probe pulse polarization optics involvea thin broadband wire grid-polarizer (Thorlabs) ahead of thesample The probe beam is focused by a spherical mirror Re-flected light is analyzed by an achromatic quarter-wave plateand Glan-type prism (both B Halle Nachfolger) We obtaintransient reflectance data by scanning of the delay line at var-ious compensator azimuth angles The transient ellipsometricparameters are computed from the reflectance-difference spec-tra The remaining chirp (few 100 fs difference between 20 eVand 36 eV - corresponding to roughly 3 mm dispersive material)induced by the CaF2 as well as the support of the wire grid po-larizer [28] is removed retroactively by shifting the zero-delay inthe data analysis using an even polynomial for its wavelengthdependence Further details can be found in the supplementaryinformation

Modeling of the ellipsometry data to obtain the materialrsquos DFis performed using a transfer matrix formalism [65] with the DFof ZnO parametrized by a Kramers-Kronig consistent B-splinefunction [66] In the model the film is assumed to be isotropicbecause the experimental configuration is mostly sensitive to theDF for ordinary polarization [67] The model is fitted to theMueller matrix elements N C S accounting also for spectralbandwidth The number of spline nodes was minimized in orderto capture all spectral features but avoid overfitting and artificialoscillations [68]

First-principles simulations of excited electron-holepairs at finite temperature

We use first-principles simulations based on many-body per-turbation theory to study the influence of electron-hole excita-tions on the optical properties of ZnO To this end we computeKohn-Sham states and energies within density functional theory(DFT) [69 70] and use these to solve the Bethe-Salpeter equation(BSE) for the optical polarization function [71] All DFT cal-culations are carried out using the Vienna Ab-Initio SimulationPackage [72 73 74] (VASP) and the computational parametersdescribed in Refs [75 76] All BSE calculations are performedusing the implementation described in Refs [77 78] In orderto describe excited electrons and holes we use and modify theframework described in Refs [76 79 80] and in detail in thesupplementary information

AcknowledgementsWe acknowledge Peter Schlupp for growing the thin film andMichael Lorenz (both Universitat Leipzig) for X-ray diffrac-tion measurements We gratefully acknowledge valuable discus-sions with Christoph Cobet Martin Feneberg Daniel Franta

6

Kurt Hingerl Michael Lorke Bernd Rheinlander Chris Sturmand Marcel Wille Parts of this work have been funded bythe Deutsche Forschungsgemeinschaft (DFG German ResearchFoundation) SFB 762 - Projektnr 31047526 (project B03)and FOR 1616 (SCHM27102) OH acknowledges the LeipzigSchool of Natural Sciences BuildMoNa Experimental develop-ment at ELI Beamlines was funded by the project rdquoAdvancedresearch using high intensity laser produced photons and parti-clesrdquo (ADONIS) Reg n CZ02101000016 0190000789from the European Regional Development Fund and the Na-tional Program of Sustainability II project ELI Beamlines- International Center of Excellence (ELISus) project codeLQ1606 SE was partially supported by the project Struc-tural dynamics of biomolecular systems (ELIBIO) reg noCZ02101000015 0030000447 from the European Re-gional Development Fund JAL and AS were supportedby the National Science Foundation under Grant Nos DMR-1555153 and CBET-1437230 and as part of the Blue Waterssustained-petascale computing project which is supported bythe National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois SZ was supported by the Na-tional Science Foundation Grant No DMR-1505172

Author contributionsSR OH SE MR and MK built the experimental setupand conducted the measurements MR wrote the computercode for data collection SR and OH wrote the computer codefor the data reduction and analysis AS and JAL performedand interpreted first-principles simulations JA RSG MGand SZ supervised the work and discussed approaches and re-sults SR and OH wrote the manuscript with inputs from allother authors

Additional informationSupplementary information is provided in the online version ofthis publication

Competing financial interestsThe authors declare no competing financial interests

Data availabilityMeasured and modeled data are available from the authors uponrequest

References[1] D S Chemla and J Shah Many-body and correla-

tion effects in semiconductors Nature 414549ndash557 2001doi10103835079000

[2] R Huber F Tauser A Brodschelm M Bichler G Abstre-iter and A Leitenstorfer How many-particle interactionsdevelop after ultrafast excitation of an electron-hole plasmaNature 414286ndash289 2001 doi10103835104522

[3] G R Fleming and M A Ratner Grand challenges inbasic energy sciences Phys Today 61(7)28ndash33 2008doi10106312963009

[4] E Baldini A Mann L Benfatto E Cappelluti A Aco-cella V M Silkin S V Eremeev A B Kuzmenko S Bor-roni T Tan X X Xi F Zerbetto R Merlin and F Car-bone Real-Time Observation of Phonon-Mediated σminusπ In-terband Scattering in MgB2 Phys Rev Lett 1190970022017 doi101103PhysRevLett119097002

[5] C Nenstiel G Callsen F Nippert T Kure S Schlicht-ing N Jankowski M P Hoffmann A Dadgar S FritzeA Krost M R Wagner A Hoffmann and F Bechst-edt Electronic excitations stabilized by a degenerate elec-tron gas in semiconductors Commun Phys 1(38) 2018doi101038s42005-018-0033-4

[6] P Colman P Lunnemann Y Yu and J Moslashrk Ul-trafast coherent dynamics of a photonic crystal all-optical switch Phys Rev Lett 117233901 2016doi101103PhysRevLett117233901

[7] Z Chai X Hu F Wang X Niu J Xie and Q GongUltrafast alloptical switching Adv Opt Mater 5(7) 2017doi101002adom201600665

[8] H Mashiko K Oguri T Yamaguchi A Sudaand H Gotoh Petahertz optical drive with wide-bandgap semiconductor Nat Phys 12741ndash745 2016doi101038nphys3711

[9] R Athale and D Psaltis Optical computing Pastand future Opt Photon News 27(6)32ndash39 2016doi101364OPN276000032

[10] H Ohta and H Hosono Transparent oxide optoelectron-ics Mater Today 7(6)42ndash51 2004 ISSN 1369-7021doi101016S1369-7021(04)00288-3

[11] H Frenzel A Lajn H von Wenckstern M LorenzF Schein Z Zhang and M Grundmann Recentprogress on ZnO-based metal-semiconductor field-effecttransistors and their application in transparent inte-grated circuits Adv Mater 22(47)5332ndash5349 2010doi101002adma201001375

[12] C S Ponseca Jr P Chabera J Uhlig P Persson andV Sundstrom Ultrafast electron dynamics in solar en-ergy conversion Chem Rev 117(16)10940ndash11024 2017doi101021acschemrev6b00807

[13] H A Atwater A R Davoyan O Ilic D Jariwala M CSherrott C M Went W S Whitney and J Wong Mate-rials challenges for the starshot lightsail Nat Mater 17861867 2018 doi101038s41563-018-0075-8

[14] S K Sundaram and E Mazur Inducing and prob-ing non-thermal transitions in semiconductors using fem-tosecond laser pulses Nat Mat 1217ndash224 2002doi101038nmat767

[15] M Lorenz M S Ramachandra Rao T Venkatesan E For-tunato P Barquinha R Branquinho D Salgueiro R Mar-tins E Carlos A Liu F K Shan M GrundmannH Boschker J Mukherjee M Priyadarshini N DasGuptaD J Rogers F H Teherani E V Sandana P Bove K Ri-etwyk A Zaban A Veziridis A Weidenkaff M Muralid-har M Murakami S Abel J Fompeyrine J Zuniga-PerezR Ramesh N A Spaldin S Ostanin V Borisov I Mer-tig V Lazenka G Srinivasan W Prellier M UchidaM Kawasaki R Pentcheva P Gegenwart F Miletto Gra-nozio J Fontcuberta and N Pryds The 2016 oxide elec-tronic materials and oxide interfaces roadmap J PhysD Appl Phys 49(43)433001 2016 doi1010880022-37274943433001

[16] S Mor M Herzog D Golez P Werner M EcksteinN Katayama M Nohara H Takagi T Mizokawa C Mon-ney and J Stahler Ultrafast electronic band gap control inan excitonic insulator Phys Rev Lett 119086401 2017doi101103PhysRevLett119086401

[17] A Zong A Kogar Y-Q Bie T Rohwer C Lee E Bal-dini E Ergecen M B Yilmaz B Freelon E J SieH Zhou J Straquadine P Walmsley P E Dolgirev A VRozhkov I R Fisher P Jarillo-Herrero B V Fine andN Gedik Evidence for topological defects in a photoin-duced phase transition 2018

[18] S A Donges A Sven O Khatib B T OrsquoCallahanJ M Atkin J H Park D Cobden and M B RaschkeUltrafast nanoimaging of the photoinduced phase transi-tion dynamics in VO2 Nano Lett 163029ndash3035 2016doi101021acsnanolett5b05313

7

[19] M Zurch H-T Chang L J Borja P M Kraus S KCushing A Gandman C J Kaplan M H Oh J S PrellD Prendergast C D Pemmaraju D M Neumark andS R Leone Direct and simultaneous observation of ul-trafast electron and hole dynamics in germanium NatureCommun 815734 2017 doidoi101038ncomms15734

[20] B Ziaja N Medvedev V Tkachenko T Maltezopou-los and W Wurth Time-resolved observation of band-gap shrinking and electron-lattice thermalization within x-ray excited gallium arsenide Sci Rep 518068 2015doi101038srep18068

[21] M Eisele T L Cocker M A Huber M PlanklL Viti D Ercolani L Sorba M S Vitiello and R Hu-ber Ultrafast multi-terahertz nano-spectroscopy with sub-cycle temporal resolution Nat Photon 8841 2014doi101038nphoton2014225

[22] L Foglia M Wolf and J Stahler Ultrafast dynamics insolids probed by femtosecond time-resolved broadband elec-tronic sum frequency generation Appl Phys Lett 109(20)202106 2016 doi10106314967838

[23] J Shah Ultrafast Spectroscopy of Semiconductors andSemiconductor Nanostructures Springer Series in Solid-State Sciences 115 Springer 2 edition 1999 ISBN 978-3-642-08391-4978-3-662-03770-6

[24] K Bohnert G Schmieder and C Klingshirn Gain andreflection spectroscopy and the present understanding of theelectron-hole plasma in II-VI compounds Phys Stat SolB 98(1)175ndash188 1980 doi101002pssb2220980117

[25] L Huang J P Callan E N Glezer and E MazurGaAs under intense ultrafast excitation Response ofthe dielectric function Phys Rev Lett 80185 1998doi101103PhysRevLett80185

[26] C A D Roeser A M-T Kim J P Callan L HuangE N Glezer Y Siegal and E Mazur Femtosecond time-resolved dielectric function measurements by dual-anglereflectometry Rev Sci Instrum 743413ndash3422 2003doi10106311582383

[27] T Shih M T Winkler T Voss and E Mazur Dielectricfunction dynamics during femtosecond laser excitation ofbulk ZnO Appl Phys A 96(2)363ndash367 2009 ISSN 0947-8396 doi101007s00339-009-5196-0

[28] F Boschini H Hedayat C Piovera C Dallera A Guptaand E Carpene A flexible experimental setup forfemtosecond time-resolved broad-band ellipsometry andmagneto-optics Rev Sci Instrum 86013909 2015doi10106314906756

[29] E Baldini A Mann S Borroni C Arrell F van Mourikand F Carbone A versatile setup for ultrafast broad-band optical spectroscopy of coherent collective modes instrongly correlated quantum systems Struct Dyn 3(6)064301 2016 doi10106314971182

[30] C Poellmann P Steinleitner U Leierseder P NaglerG Plechinger M Porer R Bratschitsch C SchullerT Korn and R Huber Resonant internal quantumtransitions and femtosecond radiative decay of excitonsin monolayer WSe2 Nat Mater 14889ndash893 2015doi101038nmat4356

[31] H R Choo X F Hu M C Downer and V P KesanFemtosecond ellipsometric study of nonequilibrium carrierdynamics in Ge and epitaxial Si1minusxGex Appl Phys Lett63(11)1507ndash1509 1993 doi1010631109671

[32] S Zollner KD Myers KG Jensen JM Dolan DWBailey and CJ Stanton Femtosecond interband hole scat-tering in Ge studied by pump-probe reflectivity SolidState Commun 104(1)51 ndash 55 1997 doi101016S0038-1098(97)00068-9

[33] H Yoneda H Morikami K-I Ueda and R M MoreUltrashort-pulse laser ellipsometric pump-probe experi-ments on gold targets Phys Rev Lett 91075004 2003doi101103PhysRevLett91075004

[34] V V Kruglyak R J Hicken M Ali B J HickeyA T G Pym and B K Tanner Measurement of hotelectron momentum relaxation times in metals by fem-tosecond ellipsometry Phys Rev B 71233104 2005doi101103PhysRevB71233104

[35] D Mounier E Morozov P Ruello J-M Breteau P Pi-cart and V Gusev Detection of shear picosecond acousticpulses by transient femtosecond polarimetry Eur Phys JST 153(1)243ndash246 2008 doirdquo101140epjste2008-00436-2

[36] C-K Min D G Cahill and S Granick Time-resolvedellipsometry for studies of heat transfer at liquidsolid andgassolid interfaces Rev Sci Instrum 81(7)074902 2010doi10106313465329

[37] S Rapp M Kaiser M Schmidt and H P Hu-ber Ultrafast pump-probe ellipsometry setup for themeasurement of transient optical properties during laserablation Opt Express 24(16)17572ndash17592 2016doi101364OE24017572

[38] J Csontos Z Toth Z Papa B Gabor M Fule B Giliczeand JBudai Ultrafast in-situ null-ellipsometry for studyingpulsed laser - silicon surface interactions Appl Surf Sci421325ndash330 2017 doi101016japsusc201703186

[39] T Pflug J Wang M Olbrich M Frank and A HornCase study on the dynamics of ultrafast laser heating andablation of gold thin films by ultrafast pump-probe reflec-tometry and ellipsometry Appl Phys A 124(2)116 2018doi101007s00339-018-1550-4

[40] S Shokhovets O Ambacher B K Meyer and G Gob-sch Anisotropy of the momentum matrix elementdichroism and conduction-band dispersion relation ofwurtzite semiconductors Phys Rev B 78035207 2008doi101103PhysRevB78035207

[41] A Othonos Probing ultrafast carrier and phonon dy-namics in semiconductors J Appl Phys 83(4) 1998doi1010631367411

[42] R R Gattass and E Mazur Femtosecond laser microma-chining in transparent materials Nat Photon 2219ndash2252008 doi101038nphoton200847

[43] S S Mao F Quere S Guizard X Mao R E RussoG Petite and P Martin Dynamics of femtosecond laserinteractions with dielectrics Appl Phys A 791695ndash17092004 doi101007s00339-004-2684-0

[44] M W Allen C H Swartz T H Myers T D Veal C FMcConville and S M Durbin Bulk transport measure-ments in ZnO The effect of surface electron layers PhysRev B 81075211 2010 doi101103PhysRevB81075211

[45] L Foglia S Vempati B T Bonkano M Wolf S Sadofevand J Stahler Revealing the competing contributions ofcharge carriers excitons and defects to the non-equilibriumoptical properties of ZnO 2018 URL arxivorgabs1811

04499

[46] J Shah Hot electrons and phonons under high intensityphotoexcitation of semiconductors Sol State Electron 2143ndash50 1978 doi1010160038-1101(78)90113-2

[47] W Potz and P Kocevar Electronic power transfer in pulsedlaser excitation of polar semiconductors Phys Rev B 287040ndash7047 1983 doi101103PhysRevB287040

8

[48] P C Ou J H Lin and W F Hsieh Spectral de-pendence of transient reflectance in a ZnO epitaxial filmat room temperature Appl Phys B 106399ndash404 2012doi101007s00340-011-4706-x

[49] R C Rai M Guminiak S Wilser and B Cai andML Nakarmi Elevated temperature dependence of energyband gap of ZnO thin films grown by e-beam deposition JAppl Phys 111(7)073511 2012 doi10106313699365

[50] J G Gay Screening of excitons in semiconductors PhysRev B 42567ndash2575 1971 doi101103PhysRevB42567

[51] C Klingshirn and H Haug Optical properties of highlyexcited direct gap semiconductors Phys Rep 70315ndash3981981 doi1010160370-1573(81)90190-3

[52] R Zimmermann Nonlinear optics and the Mott transitionin semiconductors Phys Stat Sol B 146371ndash384 1988doi101002pssb2221460140

[53] A Yamamoto T Kido T Goto Y Chen T Yao andA Kasuya Dynamics of photoexcited carriers in ZnO epi-taxial thin films Appl Phys Lett 75(4)469ndash471 1999doi1010631124411

[54] S Acharya S Chouthe H Graener T Bontgen C SturmR Schmidt-Grund M Grundmann and G Seifert Ultra-fast dynamics of the dielectric functions of ZnO and BaTiO3

thin films after intense femtosecond laser excitation JAppl Phys 115053508 2014 doi10106314864017

[55] H Fujiwara and M Kondo Effects of carrier concentrationon the dielectric function of ZnOGa and In2O3 Sn stud-ied by spectroscopic ellipsometry Analysis of free-carrierand band-edge absorption Phys Rev B 71075109 2005doi101103PhysRevB71075109

[56] C F Klingshirn Semiconductor Optics Springer 2012doi101007978-3-642-28362-8

[57] C Klingshirn R Hauschild J Fallert and H KaltRoom-temperature stimulated emission of ZnO Alterna-tives to excitonic lasing Phys Rev B 75115203 2007doi101103PhysRevB75115203

[58] M A M Versteegh T Kuis H T C Stoof and J TDijkhuis Ultrafast screening and carrier dynamics in ZnOtheory and experiment Phys Rev B 84035207 2011doi101103PhysRevB84035207

[59] G D Mahan Excitons in degenerate semiconductors PhysRev 153882ndash889 1967 doi101103PhysRev153882

[60] H Haug and T D B Tran Dynamical screening of excitonsby free carriers Phys Stat Sol B 85(2)561ndash568 1978doi101002pssb2220850219

[61] M Bachmann M Czerner S Edalati-Boostan andC Heiliger Ab initio calculations of phonon transportin ZnO and ZnS Eur Phys J B 85(5)146 2012doi101140epjbe2012-20503-y

[62] E Hendry M Koeberg and M Bonn Exciton andelectron-hole plasma formation dynamics in ZnO PhysRev B 76045214 2007 doi101103PhysRevB76045214

[63] D Franta D Necas and Lenka Zajıckova Application ofThomas-Reiche-Kuhn sum rule to construction of advanceddispersion models Thin Solid Films 534432ndash441 2013doi101016jtsf201301081

[64] D Pal J Singhal A Mathur A Singh S Dutta S Zoll-ner and S Chattopadhyay Effect of substrates andthickness on optical properties in atomic layer depositiongrown ZnO thin films Appl Surf Sci 421341 2017doi101016japsusc201610130

[65] M Schubert Polarization-dependent optical param-eters of arbitrarily anisotropic homogeneous lay-ered systems Phys Rev B 534265ndash4274 1996doi101103PhysRevB534265

[66] B Johs and J S Hale Dielectric function representa-tion by B-splines Phys Stat Sol A 205715ndash719 2008doi101002pssa200777754

[67] S Shokhovets L Spieszlig and G Gobsch Spectroscopicellipsometry of wurtzite ZnO and GaN examination ofa special case J Appl Phys 107(2)023509 2010doi10106313285485

[68] D V Likhachev Selecting the right number of knots forB-spline parameterization of the dielectric functions in spec-troscopic ellipsometry data analysis Thin Solid Films 636519ndash526 2017 doi101016jtsf201706056

[69] P Hohenberg and W Kohn Inhomogeneouselectron gas Phys Rev 136864ndash871 1964doi101103PhysRev136B864

[70] W Kohn and L J Sham Self-consistent equations in-cluding exchange and correlation effects Phys Rev 140A1133ndashA1138 1965 doi101103PhysRev140A1133

[71] G Onida L Reining and A Rubio Electronic ex-citations density-functional versus many-body Greens-function approaches Rev Mod Phys 74601 2002doi101103RevModPhys74601

[72] M Gajdos K Hummer G Kresse J Furthmuller andF Bechstedt Linear optical properties in the projector-augmented wave methodology Phys Rev B 730451122006 doi101103PhysRevB73045112

[73] G Kresse and D Joubert From ultrasoft pseudopotentialsto the projector augmented-wave method Phys Rev B591758ndash1775 1999 doi101103PhysRevB591758

[74] G Kresse and J Furthmuller Efficient iterative schemesfor ab initio total-energy calculations using a plane-wave basis set Phys Rev B 5411169ndash11186 1996doi101103PhysRevB5411169

[75] A Schleife C Rodl F Fuchs J Furthmuller and F Bech-stedt Optical and energy-loss spectra of MgO ZnO andCdO from ab initio many-body calculations Phys Rev B80035112 2009 doi101103PhysRevB80035112

[76] A Schleife C Rodl F Fuchs K Hannewaldand F Bechstedt Optical absorption in degener-ately doped semiconductors Mott transition or Ma-han excitons Phys Rev Lett 107236405 2011doi101103PhysRevLett107236405

[77] C Rodl F Fuchs J Furthmuller and F Bechst-edt Ab initio theory of excitons and optical prop-erties for spin-polarized systems Application to anti-ferromagnetic MnO Phys Rev B 77184408 2008doi101103PhysRevB77184408

[78] F Fuchs C Rodl A Schleife and F Bechstedt EfficientO(N2) approach to solve the Bethe-Salpeter equation forexcitonic bound states Phys Rev B 78085103 2008doi101103PhysRevB78085103

[79] A Schleife Electronic and optical properties of MgO ZnOand CdO Sudwestdeutscher Verlag fur Hochschulschriften2011 ISBN 3838127668

[80] K Kang A Kononov C-W Lee J A Leveillee E PShapera X Zhang and A Schleife Pushing the frontiers ofmodeling excited electronic states and dynamics to acceler-ate materials engineering and design Comput Mater Sci160207ndash216 2019 doi101016jcommatsci201901004

9

Supplementary informationUltrafast dynamics of hot charge carriers in an

oxide semiconductor probed by femtosecondspectroscopic ellipsometry

Steffen Richter12 Oliver Herrfurth2 Shirly Espinoza1 Mateusz Rebarz1Miroslav Kloz1 Joshua A Leveillee3 Andre Schleife3 Stefan Zollner45Marius Grundmann2 Jakob Andreasson16 Rudiger Schmidt-Grund2



5Fyzikalnı ustav AV CR vvi Sekce optiky Na Slovance 2 18221 Praha Czech Republic6Chalmers tekniska hogskola Institutionen for fysik Kemigarden 1 41296 Goteborg Sweden

Feb 2019

I Experimental setup

A schematic of the setup for femtosecond time-resolved spectroscopic ellipsometry is shown in figure S1 Thefundamental mode of the titanium sapphire laser TiSa is used for third harmonic generation THG (266 nm)employed as pump beam and guided through the chopper wheel C1 (f1 = 250 Hz) to the delay line DL andfocussed (lens L) onto the sample 1 of the laser power is used for supercontinuum white-light generationSCG in CaF2 employed as probe beam which passes through the chopper wheel C2 (f2 = 500 Hz) and isfocussed onto the sample S by a spherical mirror through the polarizer P The reflected light is collimated (lensL) and guided via compensator C and analyzer A to the prism spectrometer with CCD detector We refer alsoto reference [S1]

Figure S1 Schematic of the femtosecond time-resolved spectroscopic ellipsometry setup See also [S1]

i

II Measurement scheme and data reduction

In contrast to sapphire calcium fluoride-based white light generation offers more UV intensity up to 36 eV butthe crystal needs to be moved during creation of continuum white light in order to protect the crystal from heatdamage This movement and CCD warm-up yield fluctuating intensity spectra The situation is very differentfrom any other ellipsometer where the light source is stable at least over the time of a complete revolution ofthe rotating element We circumvent the problem by applying a two-chopper scheme as depicted in Fig S2Repeatedly four different intensity signals rdquopump amp proberdquo (S1) rdquopump onlyrdquo (S2) rdquoprobe onlyrdquo (S3) andrdquodarkrdquo (S4) are measured Hence at any time background- or even luminescence-corrected rdquopump amp proberdquo(Rp

j (E) equiv IS1 minus IS2) as well as rdquoprobe onlyrdquo (R0j (E) equiv IS3 minus IS4) spectra are obtained for each compensator

angle αj However they are still subject to intensity fluctuations as can be seen in Fig S3

a b

Figure S2 a Visualization of the two-chopper scheme b Example of a set of measured intensity spectra at ∆t = 400 fs andcompensator azimuth angle 100

Figure S3 Spectra of the ellipsometric parameters Ψ ∆ obtained from rdquoprobe onlyrdquo (S3) reflectance measurements The greenline indicates the average and the red line shows reference spectra obtained with a commercial ellipsometer Note that these spectraare only shown as a bechmark They are prone to offsets and modulations arising from long-term changes in the whitelight spectraor intensities The oscillations originate from the quarterwave plate For the time-resolved ellipsometry the reflectance differencesignal is evaluated instead These are robust against long-term changes

Evaluating only the reflectance-difference spectra (∆R(E)R(E))j = (Rpj (E) minus R0

j (E))R0j (E) equiv (IS1 minus

IS2)(IS3minus IS4)minus 1 is comparable to a multi-channel lock-in system and allows comparison of spectra measureda long time after each other Furthermore it minimizes systematic errors from polarization uncertainties

In order to compute the ellipsometric angles we utilize Moore-Penrose pseudo-inversion (ordinary least-squaresregression) in a Muller matrix formalism for each photon energy and delay time [S2] The Muller matrix of thesample in isotropic or pseudo-isotropic configuration is given as

ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11

1 minusN 0 0minusN 1 0 0

0 0 C S0 0 minusS C

= M11

1 minus cos(2Ψ) 0 0minus cos(2Ψ) 1 0 0

0 0 sin(2Ψ) cos(∆) sin(2Ψ) sin(∆)0 0 minus sin(2Ψ) sin(∆) sin(2Ψ) cos(∆)

For each compensator angle αj the Muller matrix Mdetj shall represent a respectively oriented compensatorfollowed by an polarizer (analyzer) as in the experiment Likewise Mprep shall represent the Muller matrixof a polarizer at the angle of the polarizer in the experiment Having measured N different configurations(compensator angles) j = 1 N we can introduce a 4timesN setup coefficient matrix Msetup Its jth column canbe written as

M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep

21 minusMdetj12 Mprep

11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)

With the row vector ~R containing the N intensity values Rj for each compensator angle αj it holds

M11 (1 NC S) = ~RMTsetup(MsetupM

Tsetup)minus1 (S3)

Instead of using the rdquopump amp proberdquo intensity spectra Rpj (E) the reflectance difference signal (∆R(E)R(E))j

is applied to ideal (theoretical) intensity spectra of the unexcited sample R00j (E) as computed from reference

spectra Rj = R00j (1 + (∆RR)j)

In a final step the Muller matrix elements can be transferred to ellipsometric angles and the degree ofpolarization (DOP )

Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)

DOP =radicN2 + C2 + S2 (S6)

requiring Ψ isin [0 90] and ∆ isin (90 270) if C lt 0 ∆ isin (0 90) cup (270 360) if C gt 0 It should be notedthat Ψ and ∆ are to first order unaffected by depolarization ie the above equations intrinsically involve onlythe non-depolarizing part of the Muller matrix Depolarization results in M22 6= M11 = 1 in contrast to Eq S1However as in the experimental configuration the input polarization was chosen to be linear at azimuth angleplusmn45 M22 is not probed and thus depolarization does not affect the data reduction The non-depolarizingMueller matrix is obtained by replacing (NC S) by (NC S)DOP

The ellipsometric parameters Ψ and ∆ are defined by the ellipsometric ratio

ρ =rp

rs= tan Ψei∆ (S7)

where rsp are the complex reflection coefficients for sp-polarized lightIn the experiments the compensator was rotated in 10 steps of 50 The polarizer was set at minus45 the

analyzer at +45 Each spectrum was averaged over 500 pulsesFinally the obtained data reveal an imprinted chirp of the white light ie propagation through the CaF2

window and the support of the wiregrid polarizers caused light of longer wavelength to arrive earlier at thesample than light of shorter wavelength This is illustrated in Fig S4 An even polynomial function is used todescribe this chirp and adjust the zero delay for each photon energy Data is interpolated accordingly

iii

Figure S4 Experimentally obtained Muller matrix elements N C S during the first picoseconds Top row Data as obtained fromthe experiment with clear indication of the chirped whitelight pulse Black curves show the polynomial function used to describethe true delay zero Bottom row Data after chirp correction by adjusting the zero positions for each photon energy

III General sample characterization

Time-resolved photoluminescence (PL) spectroscopy conducted with a streak camera reveals information on thetemporal evolution of the occupation of electronic states The sample was optically excited with 467 eV pulses ofa frequency-tripled TiSapphire laser (3 MHz150 fs1 nJ) Figure S5 a shows the transient photoluminescence atthe absorption edge of ZnO (328 eV) which is much less intense compared to the defect luminescence centeredat 24 eV This hints at the defect-rich crystal growth induced by the amorphous SiO2 substrate The ratio ofnear-band-edge to defect-related luminescence is not constant over the sample surface

We model the transient UV-PL (Fig S5 b c) with onset τo and decay time τd of roughly 4 ps which weexpect to be limited by the time resolution of our streak camera The preferred radiative recombination channelappears to be related to defect states having an order of magnitude higher onset τo = 60 ps as well as decay timesτd1 = 80 ps and τd2 = 415 ps These time constants match the late absorption recovery that is observed in thetime-resolved ellipsometry experiment The excited electron population seems to be not yet fully recombinedafter 2 ns corresponding to the time scale for vanished band bending observed in the time-resolved spectroscopicellipsometry data

The X-ray data (Fig S6) confirm c-plane orientation of the thin film and show the response of the amorphoussubstrate The FWHM of the ZnO (002) rocking curve is larger compared to other PLD-grown ZnO thin films[S3] The grain size is estimated to be on the order of the film thickness using the Scherrer formula

iv

Figure S5 a Time-resolved photoluminescence measured by a streak camera The dashed lines indicate the transients shown inpanel b c Blue (red) lines indicate an exponential model fit to obtain characteristic onset (decay) times

2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )

Figure S6 2θ-ω scan of the 30 nm thick ZnO film on an SiO2 substrate The inset shows the rocking curve of the ZnO (002) peak

IV Optical transitions in ZnO

With the symmetry assignments of the bands according to [S4] the dipole-allowed transitions for the electricfield oriented perpendicular to the optic axis (Eperpc) in wurtzite ZnO (space group 186) are listed in table IVcf also [S5 6] Only relevant bands at high-symmetry points of the Brillouin zone are considered and Kosternotation of the irreducible representations is used Transitions for E c are only allowed between states of thesame symmetry representation

direction point group dipole operator allowed transitions for Eperpcsymmetry representation

Γ ∆ A 6mm (C6v) Γ5 Γ1 harr Γ5 Γ2 harr Γ5

Γ3 harr Γ6 Γ5 harr Γ6

P K H 3mm (C3v) Γ3 Γ1 harr Γ3 Γ2 harr Γ3

Γ3 harr Γ3

U M L 2mm (C2v) Γ3 Γ1 harr Γ3 Γ2 harr Γ4

For the reciprocal-space directions corresponding to monoclinic CsC1h symmetry (R Σ as m and S T

v

as m) where the c-direction of the crystal is parallel to the respective mirror planes the assignment of bandsymmetries and transitions is generally more complex The dipole operator would transform generally like Γ1in some cases like Γ3

V Charge carrier density

Assuming linear absorption the density N of photo-excited electron-hole pairs in the film can be estimated as

N asymp Epulseλpump

hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with

quantity meaningEpulse = 1 microJ pump pulse energyλpump = 266 nm pump photon wavelengthdpump = 400 microm pump spot diameterθpump = 40 pump incidence angleθfilm = 19 pump angle in the film (with refr index n asymp 2)dfilm = 30 nm ZnO film thicknessαfilm = (50 nm)minus1 ZnO absorption coefficientR = 02 surface reflectance

This formula accounts for reflectance losses and an effectively enlarged pump spot as well as film thicknessat oblique incidence It does not account for reflectance from the film-substrate interface which increases theabsorption (in fact here it would increase the intensity available for absorption by about 1) With theexperimental parameters above the effective energy density of the pump was about 500 microJcm2 already takinginto account 20 reflection losses With a penetration depth of 50 nm in ZnO roughly 45 of the pump poweris absorbed in the film The substrate is transparent for light of 266 nm wavelength Furthermore only about87 of the entire pulse energy are contained within the 1e area which defines dpump However the latter iscompensated by the non-even beam profile as we probe only the central 200microm of the 400 microm diameter of theexcited area With the numbers above given one arrives at N asymp 975times 1019 cmminus3

It should be noted that we assume linear absorption In fact absorption bleaching of the material can also takeplace at the laser energy if the corresponding initial and final states are already empty or filled respectively Thiseffect can only matter if the excitation pulse is sufficiently short so that carrier scattering cannot compensate forthe bleaching during the time of the excitation pulse In other words there is a limit for the highest achievabledensity of excited electron-hole pairs for ultrashort laser pulses Even with higher pump power parts of thatlaser pulse would not be absorbed This could be an explanation why the excitonic absorption peaks do notcompletely vanish meaning the excitation density is overestimated However the estimated number of excitedelectron-hole pairs in the experiment here seems to be consistent with other works using different pulsed lasersources Finally there are preliminary indications that shorter laser pulses in the order of 20 fs instead of 35 fsinduce less IVB absorption This hints at absorption bleaching

VI Charge carrier statistics

Upon optical pumping with a 266 nm (Epump = 466 eV) laser pulse the excited electrons and holes obtaindifferent amounts of excess energy related to their effective masses (parabolic band approximation) [S7]

∆Ee =Epump minus Egap

1 +memh

∆Eh =Epump minus Egap

1 +mhme

With a bandgap energy of Egap asymp 34 eV electron effective mass me = 024m0 [S8] and hole effective massmh = 059m0 [S9] (m0 being the free electron mass) it follows ∆Ee asymp 090 eV and ∆Eh asymp 036 eV

Assuming the free-electronhole gas as an ideal gas an average kinetic energy corresponding to the excessenergy ∆Eeh is related to an effective temperature Teh by

∆Eeh =3

2kBTeh

with Boltzmann factor kB From this we can estimate initial effective temperatures for the charge carriers asTe asymp 7000 K and Th asymp 2800 K

vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b

00 02 04 06 08 10Fermi-Dirac distribution

00 02 04 06 08 10Charge carrier excess energy (eV)

Figure S7 Quasi Fermi-energies and distributions at high charge-carrier temperature a quasi Fermi-energies forelectrons (black) and holes (red) depending on the carrier temperature for a fixed carrier density of 1020 cmminus3 Lines representtheoretical results which are computed by evaluating the Fermi integral for electrons and holes using the ground-state density ofstates (DOS) computed within density functional theory (see b) Symbols represent the evaluation as discussed in the text herewith assumed non-parabolicity effect b First-principles numeric DOS (blue) and Fermi-Dirac distribution functions (redblacksolid lines) for the situation with Te = 7000 K and Th = 2800 K Dashed lines highlight the quasi Fermi-energies

While the effective charge-carrier temperatures express directly the average excess energy of excited electronsand holes their density Ne = Nh is given as [S10]

Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)

with the Fermi-Dirac integral F12 ECV are the energies of the conduction-band minimum and valence-bandmaximum respectively The effective densities of states (DOS) at the conduction band minimum and valenceband maximum are respectively

NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32

It can be estimated that NC(Te asymp 7000K) asymp 33 middot 1020 cmminus3 and NV(Th asymp 2800K) asymp 32 middot 1020 cmminus3 for theestimated carrier temperatures 1 However it should be noted that the temperature dependence of thoseeffective DOSrsquos results only from a substitution of the integrating variable from E to EkBTeh when expressing

Neh =int

DOS(E)(1 + e(EminusEehF )kBTeh) dE through F12 as above When estimating quasi Fermi-energies2

EehF for the hot charge carriers it is important to understand both their dependence on carrier density

and temperature Zero-temperature approximations do not hold At a given temperature a higher carrierdensity will clearly shift the quasi Fermi-energies towardsinto the respective bands ie Ee

F increases and EhF

decreases However the effect of high temperatures (at a given carrier density) is more sophisticated Evaluatingthe Fermi-Dirac integral with constant prefactors NCV shows that the quasi Fermi-energies would shift furthertowardsinto the bands if the effective temperatures are higher On the other hand the temperature dependenceof NCV yields exactly the opposite and is even more dominant Thus in total despite the high density of chargecarriers the quasi Fermi-energies are pushed into the bandgap due to the high carrier temperatures Fittingthe Fermi-Dirac integral to the initial density Neh asymp 1020 cmminus3 results in estimates on the order of Ee

FminusEC asymp-660 meV and EV minusEh

F asymp -260 meV for the above-obtained effective temperatures This means that both quasiFermi-energies are within the bandgap which is consistent with the numerical first-principles computations see

1At room temperature NC asymp 3 middot 1018 cmminus3 and NV asymp 1 middot 1019 cmminus3

2The term Fermi energy or Fermi level is used in consistency with most literature on semiconductors However precisely spokenwe refer actually to the chemical potential and note that Fermi energy is the limit of the chemical potential at zero temperature

vii

Fig S7 Compared with the intrinsic Fermi energy EF which is typically close the conduction-band minimumdue to intrinsic free electrons Ee

F is shifted even further into the bandgapIt should be noted that those estimates rely on parabolic approximations The non-parabolicity of the

bands yields another strong increase of the DOS through increasing effective masses for energies far from theminimum of the conduction and maximum of the valence band A doubled effective mass causes the distancesof the quasi Fermi-levels to the valenceconduction band maximumminimum to increase to roughly twice thecalculated values For the conduction band with the obtained carrier temperature Te we can estimate from anon-parabolicity parameter on the order of 04 eVminus1 [S11] that Ee

FminusEC should be in the order of 1 eV below theconduction band minimum [S12] Assuming a similar non-parabolicity for the valence band results consequentlyin EV minus Eh

F asymp minus300 meV

VII First-principles simulations of excited electron-hole pairs at finitetemperature

In order to describe excited electrons and holes we use the framework described in detail in Refs [S13 14 15]In the following we explain how this accounts for the effects of Pauli blocking bandgap renormalization (BGR)and additional free-carrier screening on exciton binding energies and the spectral shape of the dielectric function

We first focus on optically excited states at zero temperature In this case the lowest conduction-bandstates are occupied with free electrons of the density Ne and the highest valence states with holes of the samedensity Nh=Ne Hence transitions between these states are excluded This is described in our framework viaoccupation numbers of otherwise unchanged single-particle Kohn-Sham states To account for Pauli blockingwe adjust these occupation numbers according to Nh=Ne when computing the independent-particle dielectricfunction from the single-particle electronic structure and also when computing the BSE Hamiltonian

The effect of BGR due to free carriers in the optically excited state is a many-body effect and here we usethe model given by Berggren and Sernelius [S16 17] for doped systems to describe it as an effective bandgapshrinkage For a charge-carrier density of 1020 cmminus1 311 meV shrinkage is assumed [S18]

Finally our framework accounts for electronic interband screening of the electron-hole interaction in theBSE Hamiltonian using the static dielectric constant obtained in independent-particle approximation εeff=44In addition as discussed earlier for doped ZnO [S13] excited carriers modify the electron-hole interaction bycontributing intraband screening In our framework we approximate this contribution using the small-wave-vector limit of a static wave-vector (q) dependent Lindhard dielectric function which in the presence of freeelectrons and holes becomes [S13 14 15]

εintra(q) asymp 1 +q2TFe

q2+q2TFh

q2 (S8)

with the Thomas-Fermi (TF) wave vectors

qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)

The relative Fermi energies of electrons and holes EehF

EehF =

~2

2meh

(3π2Neh

)23 (S10)

refer to the conduction-band minimum and valence-band maximum EeF = Ee

F minus ECB and EhF = EVB minus Eh

Frespectively Eq (S8) then becomes

εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)

For equal excited electron and hole concentrations Nh=Ne equiv N this corresponds to

εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)

The term in parentheses in Eq (S12) resembles Eq (S10) and is thus a modified expression for the Fermi energythat accounts for excited electrons and holes in the screening expression Effective electron and hole masses areparametrized using parabolic fits to our first-principles band-structure data leading to me=03m0 For the hole

viii

effective mass in Eq (S12) we use the geometric average of the masses of the three degenerate uppermost valencebands ie mh=062m0 This approach is valid for zero temperature of the free carriers and its implementationin our BSE code [S13] allows us to compute the dielectric function including excitonic effects as a functionof free-carrier concentration N We refer to this quantity as εBSE(N)(E) in the following In contrast weuse εDFT(N)(E) to label the corresponding independent-particle dielectric-function that still accounts for Pauliblocking at zero temperature and BGR but neglects excitonic effects The difference between these two is∆εexc(NE)

∆εexc(NE) = εBSE(N)(E)minus εDFT(N)(E) (S13)

In order to account for the high carrier temperatures seen in the experiment we use Fermi-distributedoccupation numbers of electrons and holes This turns the eigenvalue problem for the excitonic Hamiltonianinto a generalized eigenvalue problem [S19] Here we avoid this increase in computational cost and insteadneglect the influence of temperature on excitonic effects we only use Fermi-distributed occupation numberswhen computing independent-particle spectra

We then compute independent-particle dielectric functions for valence-conduction-band transitions (εVBCBDFT(NT )(E))

intra-valence-band transitions (εIVBDFT(NT )(E)) and intra-conduction-band transitions (εICB

DFT(NT )(E)) The lat-ter two occur in the presence of holes in the valence and electrons in the conduction band respectively Whilethis describes Pauli blocking we use the same zero-temperature values for BGR to shift the bandgaps Finallywe compute the temperature-dependent dielectric function as the sum of these three temperature-dependentindependent-particle contributions and account for the influence of excitonic effects by approximating thosewith the zero temperature difference ∆εexc(NE)

ε(NTE) asympεVBCBDFT(NT )(E) + εIVB

DFT(NT )(E)

+ εICBDFT(NT )(E) + ∆εexc(NE) (S14)

Finally to compare with experimental pump-probe data we compute and visualize the difference

∆ε = ε(NTE)minus εBSE(N=0)(T = 0 K E) (S15)

Figure S8 shows such a visualization along with experimentally obtained data

-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)

calculation tSE no free carriers -10 ps hot carriers 02 ps

b

Figure S8 Obtained versus computed ε2 for high carrier excitation a Difference of the imaginary part ε2 of the DF forexcited and non-excited ZnO Symbols represent experimental data at 200 fs lines the computed DF assuming 1020 cmminus3 excitedcharge carriers with effective temperatures of 2800 K for holes and 7000 K for electrons The inset shows a zoom into the IVBabsorption range b Related ε2 spectra

ix

VIII Comparison of the dielectric function with existing models forhighly excited ZnO and conventional transient spectroscopy

0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)

Time delay (ps) Carrier density (cm-3) 00 2E17 02 1E19 10 5E19

c

Figure S9 Obtained DF vs DF model and conventional spectroscopy a DF of highly excited ZnO Symbols representthe spectra obtained in this work at three different delays after photo-excitation Lines show the expected spectra according to themodel of Wille et al for three different charge-carrier densities b Computed transient reflectance and c transmittance differencespectra at normal incidence for a 30 nm thin ZnO film on fused silica substrate according to the DFrsquos in a Note that althoughIVB absorption sets in transmittance at lower energies increases upon pumping while reflectance decreases This is caused by thelowered refractive index

In comparison to ellipsometry conventional reflectance and transmittance measurements lack any phaseinformation of the electromagnetic waves interacting with the sample This is usually compensated for bybefore-hand assumptions on the physical processes that however can lead to incorrect conclusions Reflectanceand transmittance spectra can be reconstructed from the knowledge of the DF We generate reflectance spectrabased on the DF obtained by time-resolved spectroscopic ellipsometry and compare them to theoretical valuesof Versteegh et al [S20] which were refined by Wille et al [S21] The underlying DF of Wille et al allowsto explain gain and lasing mechanisms in ZnO micro- and nanowires [S22] Both theoretical approaches arebased on a solution of the Bethe-Salpeter equation [S23] for a simplified ZnO-like bulk system The reflectancespectra are exemplary for various different pump-probe reflectance studies on ZnO [S24 25 26 27] Symbolsin Fig S9 show the DF as obtained in this work at selected pump-probe time delays lines represent theoreticalcurves according to Wille et al for various carrier densities Both studies find a decrease in the real and theimaginary part of the DF with increasing carrier density The model of Wille et al is about 100 meV blueshiftedand predicts ε2 lt 0 which can lead to optical gain and lasing This is not observed in our experiment due tothe reflection geometry Optical gain can only occur due stimulated emission which produces photons of equalwavevector (magnitude and direction) So-called gain spectroscopy was only reported in transmission geometryFurthermore it is seen that the theoretical curve of Wille et al is not able to explain the features related toexciton-phonon complexes at 34 eV since electron-phonon interaction is neglected in the model In the spectralrange far below the band gap which is not covered by Wille et al we find increased absorption which is relatedto the IVB absorption

The relative difference spectra of transmittance (panel b in Fig S9) and reflectance (panel c) are computed for

x

a structure consisting of 30 nm c-plane oriented ZnO on a fused SiO2 substrate which is equivalent to the samplestudied in this work Reflection from the substrate backside is ignored Changes around the absorption edgeof ZnO are on the same order of magnitude for both using the DF from theoretical model (lines) and applyingthe DF obtained in this work Surprisingly in the spectral range of the IVB aborption the transmittance isincreased although absorption appears It is clear that the increased transmittance is related to decreasedreflectance caused by the decrease in ε1 and hence refractive index This is in accordance with the Kramers-Kronig relations and is related to both the occurring IVB absorption as well as the absorption bleaching atthe absorption edge We would like to emphasize here that interpretation of the conventional reflectance ortransmittance changes can lead to erroneous conclusions about their physical origin because effects caused bychanges in the real and imaginary part of the DF cannot be separated Assuming a non-varying refractiveindex is insufficient and retrieval by exploiting the Kramers-Kronig relations is usually hampered by the limitedspectral range

References

[S1] M Rebarz M Kloz S J Espinoza Herrera and C D Brooks UV-VIS-NIR femtosekundovy elip-sometricky system uzitny vzor 30838 Ceska republika urad prumysloveho vlastnictvı 2017 URLhttpspisyupvczUtilityModelsFullDocumentsFDUM0030uv030838pdf

[S2] R A Chipman Polarimetry in Handbook of Optics chapter 22 McGraw-Hill 1995 ISBN9780070477407

[S3] M Lorenz Pulsed Laser Deposition of ZnO-Based Thin Films pages 303ndash357 Springer Berlin HeidelbergBerlin Heidelberg 2008 ISBN 978-3-540-73612-7 doi101007978-3-540-73612-7 7

[S4] U Rossler Energy bands of hexagonal II-VI semiconductors Phys Rev 184733ndash738 1969doi101103PhysRev184733

[S5] R C Casella Symmetry of wurtzite Phys Rev 1141514ndash1518 1959 doi101103PhysRev1141514

[S6] H W Streitwolf Selection rules for the space group c46v (wurtzite) Phys Stat Sol B 33225ndash233 1969doi101002pssb19690330120

[S7] J Shah Hot electrons and phonons under high intensity photoexcitation of semiconductors Sol StateElectron 2143ndash50 1978 doi1010160038-1101(78)90113-2

[S8] W S Baer Faraday rotation in ZnO Determination of the electron effective mass Phys Rev 154785ndash789 1967 doi101103PhysRev154785

[S9] K Hummer Interband magnetoreflection of ZnO Phys Stat Sol B 56249ndash260 1973doi101002pssb2220560124

[S10] M Grundmann The Physics of Semiconductors An Introduction Including Nanophysics and Applica-tions Springer third edition edition 2016

[S11] W A Hadi S K OrsquoLeary M S Shur and L F Eastman The sensitivity of the steady-state electrontransport within bulk wurtzite zinc oxide to variations in the non-parabolicity coefficient Solid StateCommun 151(12)874ndash878 2011 doi101016jssc201104004

[S12] R Beresford Statistical properties of an ideal nonparabolic Fermi gas J Appl Phys 70(11)6834ndash68411991 doi1010631349831

[S13] A Schleife C Rodl F Fuchs K Hannewald and F Bechstedt Optical absorption in degener-ately doped semiconductors Mott transition or Mahan excitons Phys Rev Lett 107236405 2011doi101103PhysRevLett107236405

[S14] A Schleife Electronic and optical properties of MgO ZnO and CdO Sudwestdeutscher Verlag furHochschulschriften 2011 ISBN 3838127668

[S15] K Kang A Kononov C-W Lee J A Leveillee E P Shapera X Zhang and A Schleife Pushingthe frontiers of modeling excited electronic states and dynamics to accelerate materials engineering anddesign Comput Mater Sci 160207ndash216 2019 doi101016jcommatsci201901004

[S16] K-F Berggren and B E Sernelius Band-gap narrowing in heavily doped many-valley semiconductorsPhys Rev B 24(4)1971ndash1986 1981 doi101103PhysRevB241971

xi

[S17] J Wu W Walukiewicz W Shan K M Yu J W Ager E E Haller H Lu and W J SchaffEffects of the narrow band gap on the properties of InN Phys Rev B 66(20)201403 2002doi101103PhysRevB66201403

[S18] A Kronenberger A Polity D M Hofmann B K Meyer A Schleife and F Bechstedt Struc-tural electrical and optical properties of hydrogen-doped ZnO films Phys Rev B 86115334 2012doi101103PhysRevB86115334

[S19] F Bechstedt Electron-Hole Problem pages 439ndash457 Springer Berlin Heidelberg 2015 ISBN 978-3-662-44593-8 doi101007978-3-662-44593-8 19

[S20] M A M Versteegh T Kuis H T C Stoof and J T Dijkhuis Ultrafast screening and carrier dynamicsin ZnO theory and experiment Phys Rev B 84035207 2011 doi101103PhysRevB84035207

[S21] M Wille C Sturm T Michalsky R Roder C Ronning R Schmidt-Grund and M Grundmann Carrierdensity driven lasing dynamics in ZnO nanowires Nanotechnology 27225702 2016 doi1010880957-44842722225702

[S22] M Wille T Michalsky E Kruger M Grundmann and R Schmidt-Grund Absorptive lasing mode sup-pression in ZnO nano- and microcavities Appl Phys Lett 109(6)061102 2016 doi10106314960660

[S23] H Haug and S W Koch Quantum Theory of the Optical and Electronic Properties of SemiconductorsWorld Scientific 1990 doi1011420936

[S24] T Shih E Mazur J-P Richters J Gutowski and T Voss Ultrafast exciton dynamics in ZnO excitonicversus electron-hole plasma lasing J Appl Phys 109(4)043504 2011 doi10106313549614

[S25] T Shih M T Winkler T Voss and E Mazur Dielectric function dynamics during femtosecond laserexcitation of bulk ZnO Appl Phys A 96(2)363ndash367 2009 ISSN 0947-8396 doi101007s00339-009-5196-0

[S26] P C Ou J H Lin and W F Hsieh Spectral dependence of transient reflectance in a ZnO epitaxialfilm at room temperature Appl Phys B 106399ndash404 2012 doi101007s00340-011-4706-x

[S27] C J Cook S Khan G D Sanders X Wang D H Reitze Y D Jho Y-W Heo J-M Erie D PNorton and C J Stanton Ultrafast carrier relaxation and diffusion dynamics in ZnO Proc SPIE 76037603ndash7603ndash14 2010 doi10111712845636

xii

generation can be used to probe the dynamics ofelectronic transitions after excitation [22] A challengeis not only to achieve high time-resolution but todiscriminate different processes triggered by the exci-tation [14 23] One must understand the entire iespectral and complex-valued response of an excitedmaterial This requires obtaining both amplitude andphase information of a samplersquos DF as encoded in itscomplex reflection coefficient r

Conventional transient spectroscopy yields only am-plitudes and experimental data is often explainedby changes in the extinction coefficient κ neglectingchanges of the refractive index n a challenge for exci-tation spectroscopy that has already been discussed inthe 80rsquos [24] To compensate for the lack of phase infor-mation restrictive model assumptions ar required Analternative is to combine measurements from differentangles of incidence [25 26 27] or p- and s-polarization[28 29] Even these methods are only a work-aroundbut cannot directly obtain phase information On theother hand obtaining the full dielectric response fromthe time domain is only possible in the THz regime butnot for higher optical frequencies [30]

In ellipsometry the angles Ψ and ∆ offer rela-tive frequency-dependent amplitude and phase infor-mation on the physical response rprs = tan(Ψ)ei∆

(where indices refer to p- and s-polarizations) Thisprovides simultaneous access to the real and imag-inary part of the DF ε = ε1 + iε2 = (n + iκ)2Earlier ellipsometric pump-probe experiments sufferedfrom experimental challenges such as changing posi-tions of the probe spot on the sample or stability is-sues with the laser or were performed only at singlewavelengths [31 32 33 34 35 36] Time-resolvedsingle-wavelength ellipsometry has also been reportedin imaging mode [37 38 39]

In this article we present transient DF spectraof highly photo-excited ZnO with femtosecond time-resolution These spectra yield information on theultrafast dynamics of electron-electron and electron-phonon processes in this prototype oxide semiconduc-tor With its wide bandgap and excitons stable at roomtemperature ZnO is an ideal testbed Due to its strongpolarity strong electron-phonon coupling impacts theexciton dynamics Our improved ellipsometric ap-proach gives access the time-dependent DF of the ZnOfilm after excitation with about 100 fs time resolutionThese results are compared to first-principles simula-tions We separate Pauli blocking of absorption chan-nels from bandgap renormalization (BGR) induced byincreased screening of the electron-electron interactionby photo-excited electrons and holes From our anal-ysis we report direct observation of intra-valence-band(IVB) absorption in a spectral range that is normallytransparent in ZnO

Experimental Data

The experiments were performed on a ZnO thin filmpumped by 266 nm 35 fs laser pulses that created anelectron-hole pair density of 1020 cmminus3 Supercontin-

Figure 1 Hot charge carriers after strong excitation ofZnO with a UV pump pulse ab Within a few 100 fs af-ter excitation (violet arrows) scattering between charge carriersresults in the conduction band being occupied by excited elec-trons (filled circles) the valence band by holes (open circles) (a)The thermal distribution (Fermi-Dirac statistics) of the excitedelectrons (black) and holes (red) corresponds to effective tem-peratures Te Th of a few 1000 K (b) The quasi Fermi-energies(dashed lines) are shifted into the bandgap due to the high tem-peratures cd Within the first picoseconds scattering betweencharge carriers and phonons as well as recombination yield cool-ing and reduction of the density of excited electrons and holesStill charge-carrier temperatures are larger than the lattice tem-perature Tl Black arrows in a and c mark selected optical tran-sitions which are dynamically blocked (band-band transitions)or enabled (intra-valence-band transitions)

uum white-light pulses were used as a probe The tran-sient ellipsometric angles Ψ and ∆ obtained in the spec-tral range 20-36 eV are shown in Fig 2 Data wererecorded up to 2 ns with increasing delay steps but thestrongest response occurs on time-scales shorter than05 ps The pump-induced effect on the excitonic tran-sitions around 33 eV causes a sudden decrease in Ψand an increase in ∆

From the ellipsometric spectra we obtain the DFof the ZnO film for every pump-probe delay ∆t Fig-ure 3 illustrates the resulting DF ε = ε1 + iε2 at se-lected delays At negative ∆t the obtained DF coin-cides with the one obtained in standard ellipsometryThe peak around 335 eV comprises the excitonic tran-sitions (X) the peak around 342 eV is associated withexciton-phonon complexes (EPC) [40] There exist alsofurther complexes at slightly higher energy

For small positive ∆t the absorption at the bandedge and above is strongly damped as indicated by

2








3


0 2 4 6 8 10 1001000-15

-10

-05

00

05

0 200 400 600 800-15

-10

-05

00

05

0 2 4 6 8 10 1001000

-20

0

20

40

60

80

100

21 23 25 27

00

03

d

1 ps

3 ps

cTime delay (ps)

a

Am

plitu

de d

iffer

ence

EPC

X

b

Am

plitu

de d

iffer

ence

Time delay (fs)

IVB

EPC-X

Time delay (ps)

Ener

gy (

meV

)

21 24 27 30 33

-30-25-20-15-10-050005

(2)

Photon energy (eV)

x10

(2)

Photon energy (eV)





4






lattice Tl = 300 K



Discussion


0 20 40 60 8000

01

02

03

04

05


Acousticphonons

Energy (meV)

Occ

upat

ion

of p

hono

n st

ates

T = 300 K




5


Conclusion












6

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii








3


0 2 4 6 8 10 1001000-15

-10

-05

00

05

0 200 400 600 800-15

-10

-05

00

05

0 2 4 6 8 10 1001000

-20

0

20

40

60

80

100

21 23 25 27

00

03

d

1 ps

3 ps

cTime delay (ps)

a

Am

plitu

de d

iffer

ence

EPC

X

b

Am

plitu

de d

iffer

ence

Time delay (fs)

IVB

EPC-X

Time delay (ps)

Ener

gy (

meV

)

21 24 27 30 33

-30-25-20-15-10-050005

(2)

Photon energy (eV)

x10

(2)

Photon energy (eV)





4






lattice Tl = 300 K



Discussion


0 20 40 60 8000

01

02

03

04

05


Acousticphonons

Energy (meV)

Occ

upat

ion

of p

hono

n st

ates

T = 300 K




5


Conclusion












6

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii


0 2 4 6 8 10 1001000-15

-10

-05

00

05

0 200 400 600 800-15

-10

-05

00

05

0 2 4 6 8 10 1001000

-20

0

20

40

60

80

100

21 23 25 27

00

03

d

1 ps

3 ps

cTime delay (ps)

a

Am

plitu

de d

iffer

ence

EPC

X

b

Am

plitu

de d

iffer

ence

Time delay (fs)

IVB

EPC-X

Time delay (ps)

Ener

gy (

meV

)

21 24 27 30 33

-30-25-20-15-10-050005

(2)

Photon energy (eV)

x10

(2)

Photon energy (eV)





4






lattice Tl = 300 K



Discussion


0 20 40 60 8000

01

02

03

04

05


Acousticphonons

Energy (meV)

Occ

upat

ion

of p

hono

n st

ates

T = 300 K




5


Conclusion












6

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii






lattice Tl = 300 K



Discussion


0 20 40 60 8000

01

02

03

04

05


Acousticphonons

Energy (meV)

Occ

upat

ion

of p

hono

n st

ates

T = 300 K




5


Conclusion












6

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii


Conclusion












6

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii

























7




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii




























04499



8



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii



































9







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii







Feb 2019




i





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii





a b







ii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii

Msample =

M11 M12 0 0M12 M11 0 0

0 0 M33 M34

0 0 minusM34 M33

(S1)

= M11


0 0 C S0 0 minusS C

= M11




M jsetup =

Mdetj11 Mprep

11 +Mdetj12 Mprep

21

minusMdetj11 Mprep


11

Mdetj13 Mprep

31 +Mdetj14 Mprep

41

Mdetj13 Mprep


31

(S2)



Tsetup)minus1 (S3)





Ψ =1

2tanminus1

(radicC2 + S2

N

) (S4)

∆ = tanminus1

(S

C

) (S5)




ρ =rp





iii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii






iv


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii


2 0 4 0 6 0 8 0 1 0 01 0

1 0 0

1 0 0 0

( 0 0 4 )

Inten

sity (c

ounts

)

2 θ ( deg )

( 0 0 2 )

1 4 1 6 1 8 2 0 2 2

5 0 x 1 0 2

1 0 x 1 0 3

1 5 x 1 0 3

2 0 x 1 0 3 Z n O ( 0 0 2 ) F W H M = 2 6 deg

Inten

sity

ω ( deg )








Γ3 harr Γ3



v





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii





hc0(1minusR)

[1minus exp

(minusαfilmdfilm

cos(θfilm)

)][cos(θpump)

(dpump2)2π

cos(θfilm)

dfilm

]

with







1 +memh


1 +mhme



∆Eeh =3

2kBTeh


vi

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii

0 1000 2000 3000 4000 5000 6000 7000-2

-1

0

1

2

3

4

5

6

0 2 4 6 8

electrons holes

Ener

gy (

eV)

Temperature (K)

a

Tmaxe h

numeric DOS

b





Ne = NC2

πF12

(Ee

F minus EC

kBTe

)

Nh = NV2

πF12

(EV minus Eh

F

kBTh

)


NC = 2

(mekBTe

2π~2

)32

NV = 2

(mhkBTh

2π~2

)32


Neh =int




F increases and EhF






vii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii












q2+q2TFh

q2 (S8)


qTFeh =

radic3Nehe2

2ε0εeffEehF

(S9)


EehF =

~2

2meh

(3π2Neh

)23 (S10)




εintra(q) = 1 +3e2

ε0εeff~2q2

(meNe

(3π2Ne)23

+mhNh

(3π2Nh)23

) (S11)


εintra(q) = 1 +1

q2

3Ne2

2ε0εeff

(2 (me +mh)

~2

1

(3π2N)23

) (S12)


viii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












xii








DFT(NT )(E)





-35

-30

-25

-20

-15

-10

-05

00

21 22 23 24 25 26 27 28

-01

00

01

02

03

04

(2)

a

600 550 500 450 400 350

Wavelength(nm)

(2)

Photon energy (eV)

20 22 24 26 28 30 32 3400

05

10

15

20

25

30

35

40

45

2 =

Im(

)

Photon energy (eV)


b


ix


0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












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0

1

2

3

4

5

6

7

-100

-80

-60

-40

-20

0

20

20 21 22 23 31 32 33 34 35-30

-20

-10

0

10

20

2

1

=

1+i

2

IVB absorption

a

normal incidence

RR

(

)

b

Photon energy (eV)

TT

(

)


c




x


References

















xi












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References

















xi












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Date post:	28-Jul-2020
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