Direct Measurement of Nanoscale Lateral Carrier Diffusion: Toward

Article Cite This: ACS Photonics XXXX, XXX, XXX−XXX

Direct Measurement of Nanoscale Lateral Carrier Diffusion: Toward Scanning Diffusion Microscopy Mounir Mensi,† Ruslan Ivanov,† Tomas K. Uždavinys,† Kathryn M. Kelchner,‡ Shuji Nakamura,‡ Steven P. DenBaars,‡ James S. Speck,‡ and Saulius Marcinkevičius*,† †

Department of Applied Physics, KTH Royal Institute of Technology, Electrum 229, 16440 Kista, Sweden Materials Department, University of California, Santa Barbara, California 93106, United States

‡

S Supporting Information *

ABSTRACT: A multimode scanning near-field optical microscopy technique that allows the mapping of surface morphology, photoluminescence (PL) spectra in illumination and illumination-collection modes, and PL dynamics, all in one scan, has been developed along with a method to use it for evaluation of carrier diffusion. The method allows measuring diffusion lengths as small as ∼100 nm and their anisotropy and spatial distribution, parameters remaining inaccessible to conventional far-field techniques. The procedure has been applied to study ambipolar carrier diffusion in a nonpolar m-plane InGaN/GaN quantum well. The diffusion was found to be highly anisotropic with diffusion coefficients along and perpendicular to the wurtzite c axis equal to 0.4 and 1.9 cm2/s, respectively. The large diffusion anisotropy confirms band structure calculations that suggest that the topmost valence band in an m-plane InGaN quantum well is highly anisotropic. KEYWORDS: scanning near-field optical microcopy, photoluminescence, diffusion, recombination, InGaN, quantum well

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transport is often evaluated by measuring the increase of the luminescing area.3,4 However, this apparently simple method has several drawbacks. First, it is limited to materials with large diffusion lengths. For a far-field excitation, the size of the focused spot is defined by the diffraction limit and can hardly be below ∼1 μm. For a diffusion length on the order of 100 nm, the luminescing spot is barely larger than the excited area. Second, the spatial profile of the excitation power density is typically not uniform but rather has a Gaussian distribution. Different carrier concentration gradients in different regions of the excitation spot complicate evaluation of the diffusion parameters. Time of flight techniques5,6 measure the time of photoexcited carrier arrival at a particular marker (e.g., a region with a lower band gap). For these methods, the same spot-sizerelated issues remain. Moreover, for a fixed marker, the transport anisotropy cannot be assessed. Cathodoluminescence allows injecting electrons into a small, nanometer-size spot.7 Still, if high-energy electrons are injected, inelastic scattering of the primary electron beam increases the excitation volume. For an injection of low-energy electrons, only the electron transport in p-type materials can be studied; besides, transport measurements in specific layers of a heterostructure are hardly possible. Diffusion measurements using a transient grating require high power pulses and, therefore, may be affected by heating effects.8

ateral carrier diffusion in a semiconductor layer, e.g., a quantum well (QW), is a phenomenon that aids achieving spatial uniformity of injected carrier density in photonic devices, such as light-emitting diodes, laser diodes, and solar cells. Because of a larger hole effective mass and a smaller mobility, the limiting process for the spatial equilibration of the injected carrier concentration is the hole diffusion. The process is especially critical for ternary GaN-based AlGaN and InGaN compounds that are used in ultraviolet and visible light-emitting devices since the diffusion is strongly affected by nanometerscale band potential fluctuations and hole localization.1 Thus, to model devices based on the ternary nitrides or other materials and structures with band potentials varying on the nanoscale, experimental data on the local diffusion measured with a high spatial resolution are critical. Typically, diffusion parameters are evaluated from electrical mobility measurements via the Einstein relation. However, such measurements provide average values of the transport parameters. Besides, the hole transport is measured in p-type materials, which, for the case of III-nitrides, requires heavy doping that increases defect concentration and ionized impurity scattering. This results in lower mobility values as compared to intrinsic or lightly doped materials used in the active regions of the devices.2 In addition, complicated contact patterns are required in order to distinguish the transport anisotropy. Contactless methods rely on an optical detection of carrier motion. Their main categories are based on measurements of luminescence or light scattering at transient gratings. In photoluminescence (PL) experiments, the lateral carrier © XXXX American Chemical Society

Received: September 14, 2017 Published: November 21, 2017 A

DOI: 10.1021/acsphotonics.7b01061 ACS Photonics XXXX, XXX, XXX−XXX

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Figure 1. (a) Schematics of the carrier diffusion and recombination after photoexcitation via a near-field probe: 1, radiative recombination of carriers located under the probe and detected in the IC-mode; 2, radiative recombination detected in the I-mode for carriers that have diffused from under the probe; 3, radiative recombination escaping detection; 4, nonradiative carrier trapping or recombination. (b) Broadening of fine features in an ICmode PL intensity map as compared to an I-mode map.

demonstrated on a nonpolar m-plane InGaN/GaN quantum well known for its large band potential variations that should have a strong effect on the carrier transport via increased alloy scattering. In addition, m-plane QWs are anisotropically strained, which should also affect the hole transport. The study is important for several reasons. First, it presents a multimode SNOM technique. Second, it describes a way to use it for measurements of the local carrier transport and its anisotropy. Third, it provides information about the ambipolar carrier diffusion in a technologically important m-plane InGaN/ GaN QW. Fourth, the measured anisotropy of the hole transport experimentally confirms calculations of the valence band dispersion in strained InGaN QWs that show a large anisotropy of the hole effective mass.

In addition, this far-field method has a limited spatial resolution as well. Compared to the mentioned techniques, scanning near-field optical microscopy (SNOM) stands out in several aspects. First, it allows limiting the excitation spot to 50−100 nm, well below the diffraction limit. Second, the excitation is wavelengthtunable, which enables addressing a particular layer of a heterostructure. A two-probe experiment, in which one stationary fiber probe is used for an optical carrier injection and another one scans the sample collecting PL emitted by the diffused carriers, is the most straightforward approach to measure the lateral diffusion. However, for small, submicrometer diffusion lengths this technique becomes very complicated because of the mechanical difficulties in positioning the probes close to each other and the near-field interaction between them.9 An alternative approach is to compare single-probe SNOM PL intensity maps taken in the illumination-collection mode (IC-mode; PL excitation and collection in the near field through the probe) and in the illumination mode (I-mode; PL excitation through the probe, collection in the far field with a microscope objective) (Figure 1(a)). For PL intensity maps measured in the IC- and I-modes, one can notice feature broadening in the I-mode case (Figure 1(b)). Previously, this broadening has been directly related to diffusion with the half width at half-maximum (HWHM) of the PL intensity minimum assigned to the diffusion length.10 However, analysis of the SNOM operating principle shows that the situation is much more complicated and that broadening of a PL intensity profile cannot be related to diffusion parameters in a straightforward manner. In this paper, we present a multimode SNOM technique and a data analysis method that allow one not only to estimate small diffusion lengths but also to determine diffusion parameters over the whole scanned area, as opposed to a single point or a line. Scanning over small areas or applied on subareas of the scans, the technique allows measuring local diffusion parameters as well as their anisotropy. The method is

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PRINCIPLE OF THE DIFFUSION MEASUREMENT WITH A MULTIMODE SNOM First, let us recall the origin of the near-field PL registered in the different modes. In the IC-mode, the PL signal is generated by the radiative recombination of electrons and holes that, during their lifetime, remain under the probe, i.e., have not diffused out of the excitation area. In the I-mode, PL is generated by all radiatively recombining carriers, independently of their position with respect to the probe. Thus, PL intensity recorded at a measurement point of a scan (further denoted as a pixel) is affected by diffusion for the IC-mode, but not for the I-mode. While intuitively it is tempting to assign broadening of the PL intensity profile to the diffusion length, it is not a rigorous approach because SNOM is not an imaging, but a scanning, technique, and feature broadening in the I-mode map is not equivalent to the measurement of the spatial spread of the distribution of carriers excited with a stationary probe. In the scanning experiment, the probe moves step-by-step, acquiring data for the I- and IC-mode maps; thus, the feature broadening in the I-mode map is caused not by the diffusion of carriers excited at a specif ic pixel but by the transport and recombination properties of the surrounding pixels. B


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ACS Photonics Qualitatively, this is demonstrated in Figure 2. As a simplified example, let us consider a region with two discrete values of the

Thus, since the feature broadening in the I-mode maps is affected by the spatial lifetime variations, time-integrated PL intensity maps are not sufficient for the evaluation of the diffusion parameters. Data on the recombination rates are required. In addition, since the time-integrated PL intensity is determined by the internal quantum efficiency, spatial variations of the radiative and nonradiative recombination rates are not equivalent. Radiative and nonradiative lifetime maps, along with the time-integrated PL intensity maps, must be measured. To produce these maps, a multimode SNOM setup allowing one to map surface morphology, time-integrated PL spectra in I- and IC-modes, and time-resolved PL transients, all in the same scan, was constructed (see Methods). All the near-field data were recorded at room temperature. To calculate radiative and nonradiative lifetime maps from the maps of the PL transient amplitude and decay time, the procedure described in ref 12 was applied. The method relies on the premise that PL transient amplitudes are inversely proportional to the radiative lifetimes13,14 and an assumption that at low temperature PL decay at short times after the excitation is determined by the radiative recombination.14−16 With these assumptions, the spatially average room-temperature radiative lifetime was determined from far-field time-resolved PL measurements performed between 4 and 300 K with a streak camera. Details of the procedure are described in ref 15 (see data for sample A). This average value was used for normalization while transforming the inverse transient amplitude map into the radiative lifetime (τr) map. To evaluate the PL lifetime τPL, the PL decay was fit by a single-exponential function at each pixel of a scan, which is a good approximation for m-plane InGaN QWs (Figure 3(a)). The PL decay time and the radiative lifetime (τr) maps were used to calculate the nonradiative lifetime map using 1/τPL = 1/τr + 1/τnr. The calculated radiative and nonradiative lifetime maps are shown in Figure 3(b) and (c), respectively. The intensity maps are affected by the radiative and the nonradiative lifetimes and the diffusion. The key point of the method is to use the lifetime maps and an optimization procedure to simulate PL intensity maps measured in the I- and IC-modes with components of the anisotropic diffusion coefficient used as adjustable parameters. In the analysis, the diffusion is considered ambipolar because the majority electron concentration induced by the unintentional doping (∼1017 cm−3) is considerably smaller than the photoexcited carrier density, which, averaged over the carrier lifetime, is estimated as ∼4 × 1018 cm−3 (see Methods). Similar carrier densities are

Figure 2. Simulated envelope profiles (solid orange lines) of the timeintegrated PL intensity recorded in the I-mode in the vicinity of a region with a smaller carrier lifetime of different lateral extent Δx (a and b) and amplitude Δτ (b and c). Different diffusion lengths L for different carrier lifetime profiles show that the HWHM of the PL intensity dip may be affected not only by the diffusion but also by the recombination processes. Blue lines illustrate time-integrated spatial PL intensity profiles for carriers excited at different pixels of the scan (denoted as gray boxes).

radiative recombination rate. First, the HWHM of the PL intensity profile will depend on the width of the faster recombination region (cf. Figure 2(a) and (b)). Then, it will be affected by the difference of the recombination rates in the two regions (Figure 2(b) and (c)), because the high recombination rate area acts as a carrier sink, affecting the concentration gradient and the diffusion length.11

Figure 3. (a) Room-temperature near-field PL transients and spectra (inset) measured at points A and B indicated in part (c). (b and c) Maps of the radiative and nonradiative lifetimes, respectively. C


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Figure 4. Experimental (left column) and calculated PL intensity maps for IC- (top row) and I- (bottom row) modes, as well as PL intensity profiles along the lines shown in the maps.

and the internal quantum efficiency of the sample, η = τnr(x, y)/ [τr(x, y) + τnr(x, y)]. The time-integrated diffusion profile defined by eq 2 describes the spread of the carrier concentration and emitted PL intensity from an excitation spot and is used to construct PL intensity maps for the IC- and I- modes. For this purpose, each pixel of a map is assigned a carrier concentration profile with a unique shape defined by the local recombination times τr(x, y) and τnr(x, y) and a set of diffusion coefficient components (Dx , Dy , θ). The latter are considered to be the same for the whole map. Here the angle θ is an angle between the diffusion tensor eigenvector x and the horizontal scanning direction in the SNOM measurement. The calculated PL intensity maps are composed of the central pixels of these profiles for the IC-mode map and their spatial integrals for the I-mode map. By minimizing the difference between the calculated and the experimental PL intensity maps, the diffusion coefficient is extracted. Optimization is performed using a particle swarm algorithm embedded in the Matlab toolbox.

typical for LED QWs under operating conditions, making the determined diffusion parameters relevant for devices. Details of the procedure used to evaluate the diffusion coefficient are provided in the Supporting Information. In brief, the anisotropic diffusion−recombination process is described by the equation ∂N (x , y , t ) ∂ 2N (x , y , t ) ∂ 2N (x , y , t ) = Dx + D y ∂t ∂ 2x ∂ 2y −

N (x , y , t ) τPL(x , y , t )

(1)

where N(x, y, t) is a photoexcited electron−hole pair concentration, and Dx and Dy are diffusion coefficient components aligned along the eigenvectors of the diffusion tensor. Recombination is described by a single-exponential decay with a spatially varying PL decay time τPL(x, y). In the model, carrier drift in nanoscale electric fields caused by band potential fluctuations was neglected because for the ambipolar transport this effect is diminished by the Coulomb attraction between electrons and holes.4 Numerical solution of the nonlinear parabolic partial differential eq 1 requires large computational resources, and its solutions are not stable. Therefore, exact solvers for the diffusion profile are replaced by an approximation, in which the diffusion−recombination process is divided into an iterative sequence of independent diffusion and recombination events. The time-integrated PL intensity profile, generated during the excitation at a pixel (i, j) from the spatial spread of the carrier concentration, is calculated from Ii , j(x , y) = A

∫0

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RESULTS AND DISCUSSION PL intensity maps, obtained from the lifetime maps by solving the discussed optimization problem, are plotted alongside the experimental maps in Figure 4. As one can notice, the correspondence between the calculated and the measured maps, as well as the linear PL intensity profiles, is excellent. The optimization procedure was run eight times with only slightly varying results; the average values of room-temperature diffusion coefficients are Dx = 1.86 cm2/s, Dy = 0.41 cm2/s, and θ = −5.2° with standard deviations of 0.15 cm2/s, 0.05 cm2/s, and 1.4°, respectively. Diffusion tensor eigenvectors, obtained in the optimization procedure, coincide with the crystallographic axes; thus, the coefficients may be rewritten in terms of the c axis: D∥c= 0.4 cm2/s and D⊥c = 1.9 cm2/s. In semiconductors, hole mobilities are typically much smaller than

∞

N (x , y , t ) d t

(2)

where the coefficient A is a product of the measurement transfer function, which is considered to be spatially uniform, D


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and 0.05 cm2/s. Unexpectedly, no significant anisotropy in the mobility values was observed. Since in m-plane GaN the two topmost valence bands are close with an intervalence band energy splitting of ∼10 meV, holes in both bands participate in transport at room temperature. It has been suggested that, possibly, the anisotropy of those bands cancel each other. In the studied m-plane InGaN QW, however, the energy separation between the valence band levels is 30 meV (ref 21). If smoothing of the effective band potential by the carrier spread over several potential minima is taken into account, the effective interband energy is even larger, 90 meV. 22 Consequently, only holes in the first valence band level contribute to diffusion, revealing anisotropy of the topmost valence band. In conclusion, a SNOM-based method to measure the anisotropic carrier transport for diffusion lengths as small as 100 nm was developed. To that end, we have built a multimode SNOM setup allowing one to map surface morphology, PL spectra in illumination and illumination-collection modes, and PL dynamics, all in one scan. The method has been applied to study ambipolar carrier diffusion in an m-plane InGaN/GaN quantum well. The diffusion has been found to be highly anisotropic, with diffusion coefficients along and perpendicular to the wurtzite c axis equal to 0.4 and 1.9 cm2/s, respectively. The large diffusion anisotropy confirms the band structure calculations that have suggested a large anisotropy of the hole effective mass in the topmost valence band.

that of electrons, and the ambipolar diffusion is governed by the slower holes with Damb ≈ 2Dh. With this assumption, the hole diffusion coefficient components along and perpendicular to the c axis are ∼0.2 and ∼1.0 cm2/s, respectively. Corresponding hole mobilities, evaluated using the Einstein relation, are 8 and 40 cm2/(V s). The sensitivity of the optimization method to the values of the diffusion coefficient components is demonstrated in the Supporting Information. For values much smaller than the optimal, the I- and IC-mode maps barely differ because carriers, during their lifetime, hardly diffuse out of the aperture area. On the other hand, for much larger diffusion coefficients, the Imode map is much more blurred than the experimental one, indicating a wrong choice of the diffusion parameters. The calculated values correspond to the scanned area of 10 × 10 μm2; however, the demonstrated method can also be used on submaps of a larger scan or small (a few μm2) size regions in different places of a sample, revealing spatial variations of the diffusion parameters. Possibly, if performed on samples with a laterally nonuniform free carrier concentration, the measurements could reveal the concentration variations because of the varying ratio between the majority and photoinjected carriers and diffusion mechanisms switching from mono- to ambipolar. Our results show that the ambipolar carrier diffusion in the m-plane InGaN QW is anisotropic and almost 5 times faster along the [112̅0] a-direction than along the [0001] c-direction. In ternary nitrides, alloy scattering and optical phonon scattering are the prevailing scattering mechanisms, and mobility is inversely proportional to the effective mass. Consequently, the measured diffusion anisotropy directly indicates the anisotropy of the hole effective mass. Band structure k·p calculations for m-plane InGaN QWs show that two topmost valence bands experience a strong dispersion anisotropy near the center of the Brillouin zone.17,18 The anisotropy is caused by the biaxial strain experienced by InGaN grown on an m-plane GaN substrate. Calculated effective mass in the ground valence band level of a 3 nm In0.15Ga0.85N/GaN QW is about 5 times smaller in the a-direction (0.38m0) than along the c axis (1.91m0).17 Similar values are also obtained for bulk In0.15Ga0.85N. Diffusion anisotropy, obtained in our experiments, is in excellent agreement with the calculated anisotropy of the effective mass. It is instructive to compare the obtained values with other studies of hole transport in InGaN and GaN. Diffusion coefficient or mobility values in m-plane InGaN QWs are not available; however, polar c-plane QWs have been studied by various methods. The transport parameter values, however, differ by over 5 orders of magnitude. Forward-to-reverse bias step-recovery experiments provided a hole mobility of only 0.18 cm2/(V s), corresponding to a hole diffusion coefficient of ∼0.01 cm2/s.19 On the other hand, a differential transmission experiment produced an extremely large Damb = 3000 cm2/s.20 Other reported data have a smaller spread with diffusion coefficients in the range from 0.2 to 2 cm2/s. Confocal excitation and scanned PL collection experiments resulted in Damb = 0.25 cm2/s for a 3 nm In0.26Ga0.74N QW.5 Four-wave mixing performed on an In0.15Ga0.85N epilayer produced an ambipolar diffusion coefficient value of 2.1 cm2/s,7 which is close to our result for the direction perpendicular to the c axis. For m-plane materials, Hall transport of holes was studied in Mg-doped p-GaN.2 Mobility values were found to decrease with doping and were in the range from 25 to 1 cm2/(V s), corresponding to the hole diffusion coefficients between 1.2

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METHODS The experimental setup (Figure 5) is based on a commercial scanning near-field optical microscope (Multiview 4000 from Nanonics) operating at room temperature. In the experiments, the sample excitation was performed via a fiber probe. PL was

Figure 5. Experimental setup. E


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collected with the same probe (IC-mode) and with a 0.45 numerical aperture microscope objective located on the substrate side of the sample (I-mode). The IC-mode channel was split by a beam splitter and directed into a spectrometer and a time-correlated single photon counter (TCSPC) for spectrally- and time-resolved PL measurements, respectively. The fiber probe with an aperture diameter of ∼120 nm was etched from a multimode fiber and coated with a thin layer of Al. The probe was glued to a quartz tuning fork, and a constant probe−sample distance was maintained using a shear-force feedback, which also served for mapping of the surface morphology. PL was excited with second-harmonic pulses from a Ti:sapphire laser with a 150 fs pulse duration, a 380 nm central wavelength, and an 80 MHz pulse repetition rate. The average power coupled into the fiber was 0.8 mW, which, considering the probe aperture diameter and the probe throughput of 1 × 10−4, corresponds to the excitation energy density of 9 μJ/cm2 per pulse. However, the probe parameters, especially the throughput, are not well known; thus, the given value may vary within a factor of 2. PL spectra in the I- and ICmodes were measured with a spectral resolution of 0.5 nm using two spectrometers equipped with liquid N2 cooled CCD detectors (PYL-100BR and SPEC-10:100BR from Princeton Instruments). For the time-resolved measurements, QW PL was selected with a band-pass filter and directed into a photomultiplier (Hamamatsu R3809) that was connected to a single-photon counter (SIMPLE-TAU-150-DX from Becker & Hickl). The temporal response of the TCSPC system was 50 ps. The detectors were synchronized with a SNOM pixel clock; thus, the near-field PL spectra in the I- and IC-modes, the nearfield PL transients in the IC-mode, and the surface morphology were measured simultaneously at each pixel of a scan. One should note that recording near-field PL transients in the IC-mode provides the highest spatial resolution but has a drawback that PL decay may be affected not only by the recombination but also by the carrier diffusion from under the probe. In our sample, however, the carrier lifetimes are short; besides, the lateral diffusion effect is proportional to the carrier density gradient within the aperture area and is self-limiting, as described in the Supporting Information. The similarity of the PL decay time values measured in the near- and the far-field further justifies our approach to measure transients in the ICmode. The studied sample was a single In0.15Ga0.85N/GaN QW structure grown by metal−organic chemical vapor deposition on a low (∼5 × 106 cm−2) dislocation density bulk m-plane GaN substrate provided by Mitsubishi Chemical Corporation. The sample was grown on a substrate with a 1° miscut toward the [0001]̅ direction. The structure consisted of a 1.9 μm Sidoped GaN buffer, a 10 nm undoped InGaN QW, and a 3 nm undoped GaN cap layer. To enable SNOM measurements in the I-mode, the substrate side of the sample was polished by chemical−mechanical polishing. The top surface of the sample contained shallow diagonal striations, as discussed in ref 23. For a 10 × 10 μm2 area studied by SNOM, the surface roughness had a root-mean-square (rms) value of 2.1 nm. For the same area, the bottom surface had an rms of 3−5 nm, depending on the measurement position. The surface roughness had no influence on the measured near-field PL data, neither for the Inor for the IC-mode.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.7b01061. Detailed derivation of the equations used in the SNOM map simulation procedure, a discussion on the carrier lifetime measurements in I- and IC-modes, and PL intensity maps simulated with much larger and much lower diffusion coefficient values than the ones determined by the map fitting procedure (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail (S. Marcinkevičius): [email protected]. ORCID

Ruslan Ivanov: 0000-0002-5007-6893 Saulius Marcinkevičius: 0000-0002-4606-4865 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Leah Y. Kuritzky for valuable discussions. The research at KTH was performed within the frame of Linnaeus Excellence Center for Advanced Optics and Photonics (ADOPT) and was financially supported by the Swedish Research Council (contract no. 621-2013-4096). The work at UCSB was supported by the Solid State Lighting and Energy Electronics Center (SSLEEC).

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REFERENCES

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Direct Measurement of Nanoscale Lateral Carrier Diffusion: Toward

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