Ultrafast Dynamics of Gallium Vacancy Charge States in β-Ga_2O_3
Arjan Singh, Okan Koksal, Nicholas Tanen, Jonathan McCandless, Debdeep Jena, Huili, Xing, Hartwin Peelaers, Farhan Rana
UUltrafast Dynamics of Gallium Vacancy Charge States in β -Ga O Arjan Singh, ∗ Okan Koksal, Nicholas Tanen, Jonathan McCandless, Debdeep Jena,
1, 2
Huili (Grace) Xing,
1, 2
Hartwin Peelaers, and Farhan Rana School of Electrical and Computer Engineering,Cornell University, Ithaca, New York 14853, USA. Department of Materials Science and Engineering,Cornell University, Ithaca, New York 14853, USA. Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA. (Dated: February 10, 2021)Point defects in crystalline materials often occur in multiple charge states. Although many ex-perimental methods to study and explore point defects are available, techniques to explore the non-equilibrium dynamics of the charge states of these defects at ultrafast (sub-nanosecond) time scaleshave not been discussed before. We present results from ultrafast optical-pump supercontinuum-probe spectroscopy measurements on β -Ga O . The study of point defects in β -Ga O is essentialfor its establishment as a material platform for high-power electronics and deep-UV optoelectronics.Use of a supercontinuum probe allows us to obtain the time-resolved absorption spectra of materialdefects under non-equilibrium conditions with picosecond time resolution. The probe absorptionspectra shows defect absorption peaks at two energies, ∼ ∼ I. INTRODUCTION β -Ga O , an ultra-wide bandgap material, is a verypromising candidate for high power electronic devices,solar-blind UV photodetectors, and sensors [1–5]. Theavailability of good quality large-area Ga O substrates,the high breakdown electric field of the material, theability to n -dope the material over a wide concentrationrange, the decent mobility of electrons, and the relativelylong recombination times of photoexcited carriers in thematerial have all contributed to this promise. Most ofthese properties can be significantly impacted by mate-rial defects [6]. β -Ga O can have many intrinsic andextrinsic point defects, including vacancies, interstitials,and impurities [7–9]. The behavior of many of these pointdefects is not well understood. Developing a better un-derstanding of the properties of these defects is criticalin realizing the material’s promise.First-principles calculations have been instrumental indetermining the formation energies, charge states, opti-cal and thermodynamic transition energies, and the cor-responding optical cross-sections of point defects in β -Ga O [10–20]. Among the intrinsic point defects, Gaand O vacancies and interstitials have been theorized tohave small formation energies. Ga vacancies, in particu- ∗ [email protected] lar, have the smallest formation energies in n -doped β -Ga O grown under oxygen-rich conditions [13–17, 19].In n -doped β -Ga O , Ga vacancy is a deep compen-sating acceptor and, depending on the Fermi level, itcan be present in various charge states. Many differ-ent experimental techniques, including scanning probeand transmission electron microscopy [21], cathode- andphoto-luminescence spectroscopy [22], electron spin res-onance spectroscopy [23, 24], and deep level transientspectroscopy [25] have been used to explore point defectsin β -Ga O . However, none of these techniques have al-lowed simultaneous probing of different charge states ofdefects at ultrafast time scales. Since carrier capture bydefects in β -Ga O occurs on picosecond to nanosecondtime scales [26], it is important to be able to probe defectdynamics on these ultrafast time scales.In this paper, we present results from ultrafast optical-pump supercontinuum-probe spectroscopy of defects in β -Ga O , combined with first-principles calculations.Pump-probe spectroscopy is especially useful in exploringdefects because, first, it allows for synchronized lock-indetection, which enables detection of fractional changesin light intensity as small as 10 − [27]. Such a de-gree of sensitivity is useful given the relatively small op-tical absorption due to defects. Second, pump-probespectroscopy allows us to probe materials under non-equilibrium conditions. In n -doped materials, defectstates are typically filled with electrons, disallowing opti-cal transitions from the valence band (VB) to these defect a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b states. In this work, we excite electrons out of the defectstates using an optical pump pulse and probe the subse-quent relaxation of the defect states, as they transitionthrough different charge states toward their equilibriumstate. We probe this defect relaxation using a broadbandsupercontinuum optical pulse, which is frequency filtered,to obtain the transient absorption by the defects at differ-ent energies. This yields the time-dependent absorptionspectra of the defects with picosecond time resolution. (a) (b) E V E C 𝐕 𝐆𝐚(cid:2879)𝟑
Pump E V E C 𝐕 𝐆𝐚(cid:2879)𝟏 𝐕 𝐆𝐚(cid:2878)𝟏 𝐕 𝐆𝐚𝟎 𝑫 𝒑(cid:2879)𝟏 Pump Probe Probe E f Energy 𝒌 𝒌𝚪 𝚪 𝑫 𝒑𝟎 𝑫 𝒏𝟎 𝑫 𝒏(cid:2878)𝟏 FIG. 1. (a) The conduction and valence bands of n -doped β -Ga O are depicted along with the optical excitation scheme.Also shown are the Ga vacancies in their equilibrium -3 chargestate. (b) A snapshot of the non-equilibrium state some timeafter the optical excitation. The Ga vacancies are present indifferent charge states which allow optical transitions from thevalence band and these transitions contribute to the transientabsorption experienced by the probe pulses. Our results show that transient optical absorption bydefects in β -Ga O depends sensitively on the polariza-tion of the probe pulse. Absorption is maximum for theprobe polarized along the crystal c-axis. The probe ab-sorption spectra shows peaks at two energies, ∼ ∼ II. EXPERIMENTS AND RESULTS
The samples studied in this work were obtained fromthe Tamura Corporation and consisted of bulk Sn-doped(010) β -Ga O substrates, with an electron density, n ≈ × cm − and a thickness of ∼ µm . The sampleswere grown by the EFG method [28, 29] in oxygen-richconditions. A 405 nm ( ∼ ∼ ∼ ∼ and ∼ µ J/cm , respectively. λ probe = 800 nm x -4 T = 300K c‐axisa*‐axis
FIG. 2. Differential probe transmission ∆
T /T for n-doped(010) β -Ga O is plotted as a function of probe delay. Theprobe wavelength is 800 nm. When the probe is polarizedalong the a*-axis (perpendicular to the b- and c-axis), thechange in transmission is due to optical absorption from pho-toexcited free electrons (intra-conduction band absorption[30]). As the polarization of the probe is changed away fromthe a*-axis toward the c-axis, additional absorption due todefects is observed that makes ∆ T /T more negative in thefirst few hundred picoseconds.
Fig. 2 shows the normalized differential transmission∆
T /T , as a function of the pump-probe delay, of an 800nm probe pulse polarized along different crystal axes for(010) β -Ga O . The observed negative values of ∆ T /T signify an increase in the optical absorption induced bythe photoexcitation of electrons by the pump pulse. Asseen in the figure, the measured ∆
T /T is highly polariza-tion dependent. Much larger peak values of | ∆ T | /T areobserved when the probe is polarized along the c-axis,and the peak values steadily decrease as the probe po-larization is changed to be along the orthogonal a*-axis.Very notably, the shape of the ∆ T /T transient is also po-larization dependent suggesting that different loss mecha-nisms are contributing to the probe absorption when theprobe is polarized along the c-axis and a*-axis. In re-cently reported work [30], we have examined the ∆
T /T transient for probe polarization along the a*-axis in detailand shown that, for this polarization, the probe experi-ences optical loss only due to intra-conduction band tran-sitions (a form of free-carrier absorption) characterizedby a 1 /ω frequency dependence, where ω is the centerfrequency of the probe pulse. Optical absorption relatedto defects is not observed for probe polarized along thea*-axis. When the probe is polarized away from the a*-axis, we observe additional absorption (i.e. in additionto free-carrier absorption) that keeps increasing with thepump-probe delay for the first few hundred picoseconds.We attribute this additional absorption to optically ac-tive defect states. As can be seen in Fig. 2, defect absorp-tion is maximum for probe polarized along the c-axis.In order to better quantify the defect absorption, wemeasure ∆ T /T for different probe wavelengths. Sincefree-carrier (intra-conduction band) absorption is ex-pected to be polarization independent [31], we sub-tract the measured ∆
T /T along the a*-axis from thatalong the c-axis to obtain the defect absorption con-tribution to ∆
T /T . We refer to this modified normal-ized differential transmission as δ (∆ T /T ). δ (∆ T /T ) asa function of pump-probe delay, for various probe wave-lengths is shown in Fig. 3(a). Interestingly, the shapes ofthe δ (∆ T /T ) transients are wavelength dependent (i.e. δ (∆ T /T ) transients for different wavelength are not justscaled versions of each other). The corresponding time-dependent defect absorption spectra for different probedelays are shown in Fig.3(b). As can be seen in this Fig-ure, the defect absorption spectra (proportional to themagnitude of δ (∆ T /T )) for all probe delays) can be fitusing two Gaussian absorption coefficients, one centeredat 1.63 eV ( ∼
761 nm) and the other at 2.2 eV ( ∼
940 nm 800 nm 660 nm600 nm 515 nm (a)(b) 𝜹 FIG. 3. (a) The difference δ (∆ T /T ) between the ∆
T /T tran-sients measured for probe polarized along the c-axis and a*-axis is plotted as a function of the probe delay, for differentprobe wavelengths. (b) The defect absorption spectra (pro-portional to the magnitude of δ (∆ T /T )) are shown for differ-ent probe delays. The defect absorption spectra for all probedelays can be fitted using two Gaussian absorption coefficientscentered at 1.63 eV ( ∼
761 nm) and 2.2 eV ( ∼
564 nm). Therelative weights of these two Gaussian absorption coefficientschange with time (but their widths stay constant).
III. DISCUSSION
The experimental observations allow us to conclude thefollowing. First, the polarization dependent defect ab-sorption in Fig. 2 has been observed previously in bothheavily and mildly n -doped β -Ga O samples of differ-ent crystal orientations [26], and is therefore unrelatedto doping. Second, the relatively large strength of theabsorption (proportional to the maximum magnitude of δ (∆ T /T ), which is of the order of 10 − ) signifies a fairlylarge concentration of the defects. Third, the increase inabsorption in the first few hundred picoseconds after thepump pulse points to optical transitions from the valenceband to the defect states being responsible for the defect-related optical absorption. Therefore, in the discussionthat follows, optical transition will refer to the process inwhich an electron transitions from the valence band toa defect state after absorbing light. Fourth, the probewavelength-dependent transients point to either multipledefects with different absorption spectra but the samepolarization selection rule or to a single defect with mul-tiple charge states. A. The nature of defect states: first-principlescalculations
The experimental data was analyzd with the help offirst-principles calculations. We used density functionaltheory as implemented in the
VASP code [32], usingprojector augmented wave potentials [33] with an energycutoff of 400 eV and a 2 × × k -point grid in a 120-atom 1 × × V OI , V OII , V
OIII ), as these be-have as deep donors, with the +2 to 0 thermodynamictransition occurring at Fermi energies larger than 2.5 eV(measured from the valence-band maximum) [14, 17, 19].Note that thermodynamic transition energies and opti-cal absorption energies are not the same because the for-mer include the effect of full lattice relaxation, whichdecreases the thermodynamic transition energy with re-spect to the optical absorption energy. Although the +1charge state of an oxygen vacancy is not stable, opticaltransitions from +2 to +1 or from +1 to 0 charge statesare possible but require larger photon energies than theones observed in our experiments [14, 38]. Ga vacancies,on the other hand, have characteristics that match allour experimental observations. A large concentration ofGa vacancies is expected to be present in our samplesbecause of their very low formation energies in n -doped β -Ga O [13–15, 19]. In particular, the Ga(I) divacancy-interstitial complex ( V ic Ga in the notation of Ingebrigtsen et al. [19]) has the lowest formation energy of all intrinsicdefects.Ga vacancies can readily form complexes with otheratoms and defects, such as hydrogen [14]. We find thatthe absorption features seen in Fig. 2 and Fig. 3 do notchange upon annealing at 1100 K in 80% O ambient. Wetherefore exclude the possibility of hydrogenated Ga va-cancies being responsible for these absorption features.In our Sn-doped samples, Ga vacancies can also formcomplexes with Sn dopants [21]. The calculated forma-tion energy diagram (Fig. 4(a)) indicates that these Sn- V Ga complexes can readily form and that they have ther-modynamic transition levels at energies similar to thoseof Ga vacancies. Our first-principles calculations showthat optical transitions from +1 to 0 and from 0 to -1charge states of the V icGa complex take place at 1.8 eVand 2.5 eV, respectively, both of which are very closeto the experimentally observed absorption energies. Thecorresponding transitions for the complex with Sn are 1.7eV (+1 to 0) and 2.7 eV (0 to -1). To further distinguishbetween these two defects, we calculated the light polar-ization dependence expected for optical transitions fromthe valence band to the defect states. For these chargestate transitions, the V icGa complex shows a strong polar- FIG. 4. (a) Formation energies as function of Fermi level for V ic Ga and Sn- V Ga complexes. The kinks in the curves indicatethe thermodynamic transition levels. Both Ga-rich and O-rich conditions are shown. (b-c) Wavefunction isosurface at10% of the maximum value corresponding to the unoccupieddefect state of the +1 charge state of the (b) V ic Ga complexand (c) the Sn- V Ga complex. ization dependency, with light polarized along the c-axisleading to the strongest absorption in agreement withour experiments. In contrast, the absorption for the Sn- V Ga complex does not depend on the polarization withrespect to the a , b , and c crystal axes. The differencecan be understood by looking at the wavefunction of theunoccupied defect state (+1 charge state): for the V ic Ga complex (Fig. 4(b)) the wavefunction is mainly orientedalong the c-axis, while for the Sn- V Ga complex (Fig. 4(c))the wavefunction is not oriented along any of the crystalaxes. This polarization dependence allows us to excludethe Sn- V Ga complexes, and strengthens our conclusionthat the measured absorption is due to the charge statesof the Ga(I) vacancy.Our calculations show that optical transitions from -1to -2 and from -2 to -3 charge states of the V icGa complexare nearly polarization independent and do not show apreference for light polarization along the c -axis. We cantherefore exclude these transitions playing a dominantrole in causing probe absorption in our experiments. Fi-nally, our calculations show that optical transitions from+2 to +1 charge states of the V icGa complex do show astrong preference for light polarization along the c -axisbut the calculated absorption energy of 1.4 eV for thistransition is smaller than the measured energies. Wetherefore conclude that the probability that the pumppulse leaves the defect in charge state +2 is low. A pos-sible explanation for this is as follows.Our earlier work [26] showed that optical transitionscaused by a ∼ V icGa to conduction band transitions, which giveenergies of 4.2, 4.6, and 5.2 eV for -1 to 0, 0 to +1, and +1to +2 transitions, respectively. In these transitions, anelectron is excited from the defect to the conduction bandvia a single-photon absorption process. Since these calcu-lated energies are much larger than the ∼ E o , then the strength of the correspondingtwo-photon transition can be approximately described bya universal function that peaks when the photon energyequals ∼ . E o and rapidly approaches zero when thephoton energy equals ∼ . E o [39]. It follows that thepump used in our experiments is much more likely tocause -1 to 0 and 0 to +1 charge state transitions thancause the +1 to +2 transition. Furthermore, as discussedbelow, a minimum model based on probe-induced opticaltransitions from +1 to 0 and from 0 to -1 charge states ofthe V icGa complex can explain our data well. It is there-fore safe to conclude that our pump pulse is not likely toput the V icGa complex into the +2 charge state. B. A rate equation model for the defect statedynamics
Next, we present coupled rate equations for model-ing the non-equilibrium dynamics of the charge states ofGa vacancies and show that the computed wavelength-dependent and time-dependent ∆
T /T transients, assum-ing that the defect optical absorption is due to the chargestates of Ga vacancies, agree very well with our measure-ments. The probe frequency dependent ∆
T /T can bewritten as,(∆
T /T ) = e − [ n ( τ ) − n o ] σ fc ( ω ) L i − n d σ d ( ω ) fL i − ≈ − [ n ( τ ) − n o ] σ fc ( ω ) L i − n d σ d ( ω ) f L i (1)where, L i is the pump-probe interaction length deter-mined by the spatial overlap of the pump and probebeams in the sample ( ≈ µ m in our measurementscheme), n o is the equilibrium electron density ( ∼ × cm − ), n ( τ ) is the total electron density in the con-duction band at time τ , σ fc ( ω ) is the absorption cross-section associated with free-carrier intra-band absorption[30]. As shown recently by the authors [30, 31], σ fc ( ω ) is proportional to ω − , where ω is the frequency of theprobe. n d is the defect density. σ d ( ω ) is the defect ab-sorption cross-section. f equals 1 (or 0) for probe polar-ization along the c-axis (or a*-axis). σ d ( ω ) can be writtenas, σ d ( ω ) = (cid:80) σ j ( ω ) P j ( τ ). P j ( τ ) is the time-dependentprobability of a Gallium vacancy being in the j chargestate. σ +1 ( ω ) and σ ( ω ) are the defect absorption cross-sections when the defect is in the +1 and 0 charge states,and are assumed to be Gaussians centered at 1.63 eV and2.2 eV, respectively. The widths of the Gaussians arechosen to fit the measured defect absorption spectra (seeFig. 3), and the peak values of the Gaussians are used asfitting parameters. Note that, δ (∆ T /T ) ≈ − n d σ d ( ω ) L i . P j ( τ ) are calculated using a defect-assisted carrier recom-bination rate equation model, dndτ = − (cid:88) j D jn n n d P j n d dP j dτ = − (cid:0) D jn n n d + D jp p n d (cid:1) P j + D j +1 n n n d P j +1 + D j − p p n d P j − dpdτ = − (cid:88) j D jp p n d P j (2)Here, n (or p ) is the density of electrons in the con-duction band (or of holes in the valence band), D jn (or D jp ) is the electron (or hole) capture rate constant forthe defect in the j charge state. Given that we don’t seethe experimentally measured absorption peak (centeredat 2.2 eV) corresponding to the 0 charge state decreasewith the pump-probe delay, we assume that charge states − − ∼ P − such that P − remains zero in the first ∼ n d , the ini-tial values P j ( τ = 0) ( j = +1 , , − σ j ( ω ) ( j = +1 , T /T transients forboth probe polarizations, for all probe wavelengths, andfor all probe delays. The free-carrier absorption cross-section σ fc ( ω ) is determined as discussed in our earlierwork [30].Fig. 5(a) shows the measured and the computed ∆ T /T transients for probe polarized along the a*-axis, thecase in which only free-carrier absorption contributes to∆
T /T , for different probe wavelengths. Fig. 5(b) showsthe measured and the computed δ (∆ T /T ) transients, towhich only the defect absorption contributes. As men-tioned earlier, δ (∆ T /T ) is obtained by subtracting themeasured ∆
T /T along the a*-axis from that along thec-axis. The parameter values used to fit the data aregiven in Table II. Fig. 5 shows that the model fits thedata very well for all probe wavelengths and polariza-tions, and at all probe delays. From the fits, the defect (a)(b)
T = 300K940 nm 800 nm600 nm 515 nm 660 nmData T = 300K940 nm 800 nm600 nm 515 nm 660 nmDataProbe pol. || a* axisProbe pol. || c axis
FIG. 5. (a) The measured (dashed) and the computed(solid) ∆
T /T transients for probe polarized along the a*-axis,the case in which only free-carrier absorption contributes to∆
T /T , are plotted for different probe wavelengths. (b) Themeasured (dashed) and the computed (solid) δ (∆ T /T ) tran-sients, to which only the defect absorption contributes, areplotted for probe polarized along the c-axis for different probewavelengths. density was found to be ≈ . × cm − . The holecapture rate constants are found to be larger than theelectron capture rate constants. This is why the maxi-mum magnitude of δ (∆ T /T ) occurs at τ >
0, long afterthe pump has passed. The parameter values in TableII are similar in magnitude to the ones determined byKoksal et al. for defect states in Sn-doped ¯201 β -Ga O using a much simpler model [26].Fig. 6(a) shows the computed values of P +1 , P ,and P − plotted as a function of the probe delay, τ . The values of the products n d σ +1 | peak P +1 ( τ ) and n d σ | peak P ( τ ), which are the peak absorption coeffi-cients for charge states +1 and 0, respectively, can alsobe extracted directly from the data plots shown in Fig. 3.Fig. 6(b) plots these extracted values (solid squares)along with those computed using the rate equations (solidlines). The agreement between the data and the modelis again very good. Parameter Value Unit n d (1 . ± × cm − D +1 n (2 . ± . × − cm /sD n (1 . ± . × − cm /sD p (2 . ± . × − cm /sD − p (2 . ± . × − cm /sσ +1 | peak (7 . ± × − cm σ | peak (5 . ± × − cm TABLE I. Extracted values of model parameters 𝑽 𝑮𝒂(cid:2878)𝟏 𝑽 𝑮𝒂𝟎 (a)(b) 𝑷 (cid:2878)𝟏 𝑷 𝑷 (cid:2879)𝟏 FIG. 6. (a) The computed values of the probabilities P +1 , P , and P − for the defect (Gallium vacancy) to be in chargestates +1, 0, and −
1, respectively, are plotted as a function ofthe probe delay, τ . (b) The computed values of the products n d σ | peak P ( τ ) and n d σ | peak P ( τ ), which are the peak opti-cal absorption coefficients for the defect charge states +1 and0, respectively, are plotted as a function of the probe delay(solid lines). Also plotted are the values of these absorptioncoefficients extracted directly from the data plots shown inFig. 3 (solid squares). IV. CONCLUSION
In conclusion, we reported experimental resultsfrom ultrafast optical-pump supercontinuum-probe spec-troscopy of non-equilibrium defect absorption in β -Ga O . Our experimental and theoretical results showthat the measured absorption features are due to opti-cal transitions from the valence band to different chargestates of Ga(I) vacancies. Good agreement betweenour first principles calculations and the experimentaldata, and the ability of our rate equations to model themeasured transients for different probe wavelengths andpolarizations at all probe delays show that our modelcaptures the underlying physics well. Our results alsodemonstrate that broadband ultrafast supercontinuumspectroscopy can be a valuable tool to explore defectsstates and defect dynamics in semiconductors. ACKNOWLEDGMENTS
This work made use of the Cornell Center for MaterialsResearch Shared Facilities which are supported through the NSF MRSEC program (DMR-1719875). This workwas also supported by AFOSR under Grant No. FA9550-18-1-0529. Computing resources were provided by theExtreme Science and Engineering Discovery Environ-ment (XSEDE), which is supported by NSF grant num-ber ACI-1548562. The authors acknowledge helpful dis-cussions with Dr. Shin Mou (AFRL).
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