using standard prb s - Purdue Physics - Purdue University

PHYSICAL REVIEW B

VOLUME 60, NUMBER 15

15 OCTOBER 1999-I

Transient enhanced intermixing of arsenic-rich nonstoichiometric AlAs/GaAs quantum wells R. Geursen, I. Lahiri, and M. Dinu Department of Physics, Purdue University, West Lafayette, Indiana 47907-1396

M. R. Melloch School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907-1285

D. D. Nolte Department of Physics, Purdue University, West Lafayette, Indiana 47907-1396 共Received 10 March 1999兲 The superlattice intermixing of arsenic-rich nonstoichiometric AlAs/GaAs quantum wells grown at lowsubstrate temperatures around 300 °C is enhanced by several orders of magnitude relative to diffusion in stoichiometric structures grown at ordinary substrate temperatures. The transient enhanced intermixing is attributed to a supersaturated concentration of group-III vacancies grown into the crystal by low-temperature growth conditions. The enhanced diffusion decays during moderate-temperature annealing between 600 °C and 900 °C for annealing times between 30 and 1000 s. First-order and second-order decay kinetics were both found to agree equally well with diffusion data obtained from isochronal and isothermal annealing. However, both of these kinetics require a thermally activated annihilation enthalpy to explain temperature-insensitive behavior observed in the time-dependent diffusion coefficient. The activation enthalpy H a for the decay is between 1.4 and 1.6 eV, which is compared with the migration enthalpy H m ⫽1.8 eV of the gallium vacancy in GaAs. For the strongest annealing, the diffusion length approaches the self-diffusion values observed in isotopic superlattices of stoichiometric GaAs. 关S0163-1829共99兲03439-6兴

I. TRANSIENT-ENHANCED DIFFUSION

Transient-enhanced diffusion in semiconductor crystals occurs when the mechanisms responsible for selfdiffusion are augmented by nonequilibrium conditions. Intrinsic defects such as vacancies and interstitials are the most likely candidates for the mechanisms for selfdiffusion. Enriched concentrations of these defects increase self-diffusion in the crystal. Transient-enhanced diffusion is commonly generated by ion implantation of dopant atoms.1,2 The implantation causes damage that consists of significant concentrations of intrinsic defects as well as impurity complexes. Enhanced diffusion in Si after implantation can be a detrimental effect, especially in view of decreasing silicon device dimensions, which has attracted significant attention to transientenhanced diffusion in this material system.3 Nonstoichiometric growth of semiconductors also leads to perturbations from equilibrium defect concentrations. Nonstoichiometric growth of GaAs is possible using molecular beam epitaxy by holding the substrate temperature at lowerthan-normal temperatures during growth.4 The nonstoichiometry is taken up primarily by excess arsenic, but also to a lesser degree by gallium vacancies.5 Self-diffusion in GaAs is considered to be controlled primarily by the diffusion of gallium vacancies,6 so an enhanced vacancy concentration is expected to produce enhanced diffusion. One of the chief areas of interest concerning arsenic-rich nonstoichiometric GaAs is the study and explanation of the relaxation processes that take place in nonstoichiometric GaAs during annealing. For instance, the evolution of the excess arsenic into arsenic precipitates through the process of Ostwald ripening has been well documented.7 In addition, 0163-1829/99/60共15兲/10926共9兲/$15.00

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the changes in arsenic anti-site concentrations have been traced as a function of increasing annealing for weak to moderate annealing up to 600 °C using magnetic circular dichroism and infrared absorption.8 Vacancy concentrations have been inferred from the ratio of ionized arsenic antisite defects and have been traced as a function of growth temperature using positron annihilation spectroscopy.5,8 However, these direct techniques are not generally useful for strongly annealed materials in which the defect densities decay below the sensitivity limits. An alternative spectroscopic approach uses the effect of excess vacancy concentrations on the intermixing of quantum wells. Quantum-confined excitons are sensitive to small changes in the potential profile caused by interface intermixing. We previously used modulation spectroscopy techniques to track the shift of the excitonic transition in nonstoichiometric quantum wells as a function of increasing annealing.9 The two dramatic features in this early study were the unprecedented enhancement in the diffusion and interface intermixing in these materials, and the observation of an apparently athermal diffusion coefficient for moderate annealing temperatures between 700 °C and 900 °C. The effective activation enthalpy for the diffusion coefficient in these materials was measured to be anomalously small, between 0.3 and 1 eV.9,10 This abnormally small diffusion enthalpy was also verified in an independent experiment using electron diffraction analysis.11 While the small enthalpy is consistent with the migration enthalpy of interstitials,12 the high-temperature limit would lead to unphysically large diffusion lengths, which were not observed in our data. If an explicit time-dependent decay of the enhanced diffusion is assumed,9 then the small diffusion enthalpy in the 10 926

©1999 The American Physical Society

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TRANSIENT ENHANCED INTERMIXING OF ARSENIC- . . .

nonstoichiometric material could be explained as an effective enthalpy that was the difference between the vacancy migration enthalpy H m and the vacancy annihilation enthalpy, H eff⫽Hm⫺Ha . The small value we observed for H eff strongly suggests that H a ⬇H m , which is plausible on the grounds that a vacancy must first migrate in order to annihilate. However, this relation was too simple to extract accurate values for H a . A more complete and accurate fitting of the decay kinetics to the measured data is necessary to find a quantitative value for H a . In this paper, we describe a comprehensive study of the transient enhanced diffusion in low-temperature-grown 共LTG兲 AlAs/GaAs quantum wells. By performing a matrix of isochronal and isothermal annealing, and tracking the quantum-confined exciton transition energies, we have compiled the most complete picture yet of the decay of the nonstoichiometric material towards equilibrium. The background on nonstoichiometric quantum wells is presented in Sec. II, which describes the optoelectronic properties of LTG quantum wells and the growth and processing of the materials. In Sec. III we describe our use of phase-shifted modulation spectroscopy that is uniquely suited to the light-induced screening of built-in fields in p-i-n structures. The photomodulation spectra are then compared in Sec. IV against a calibration of the excitonic energy shift vs diffusion length. The energy shifts are translated into diffusion lengths, which are compared against first-order and second-order decay kinetics in Sec. V. Remaining questions and issues are discussed in Sec. VI, such as the role of nonlinear diffusion, and the relationship between interface intermixing and interface roughening.

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transition. The advantages of the as-grown LTG quantum wells quickly degrade with increasing annealing temperature as the excitonic transition broadens because of the interface roughening.10 B. Growth and preparation of LTG quantum wells

The samples used in this study were p-i-n multiple quantum well diodes grown by molecular-beam epitaxy on n ⫹ GaAs substrates. Contact and stop-etch layers of n-type material were grown on the substrate at 600 °C, which were then followed by a 150-period superlattice of 100-Å GaAs wells and 35-Å AlAs barriers grown at 310 °C. Because of the low-growth temperature, approximately 0.2% excess arsenic was incorporated in the multiple quantum well 共MQW兲.21 A 2000-Å p-Al0.3Ga0.7As (1⫻1018 cm⫺3) buffer layer was formed on top of this low-temperature-growth layer, followed by a 2000-Å cap of p-GaAs (1 ⫻1019 cm⫺3), both grown at 450 °C. Because the 450 °C growth temperature acts as a weak in situ annealing for the LTG GaAs, arsenic precipitates form in the MQW. The growth was divided into samples that underwent rapid thermal annealing 共RTA兲 for several different of temperatures of 650, 700, 750, 800, 850, and 900 °C and different annealing times of 30, 100, 300, and 1000 s. Physical limitations in the RTA equipment at high temperature and long times, and the uncontrolled out diffusion of arsenic at these extremes, excluded the 1000 s at 900 °C annealing sample from our results. The isothermally and isochronally annealed samples were epoxied to glass and their substrates were removed to conduct infrared transmission measurements.

II. NONSTOICHIOMETRIC QUANTUM WELLS A. Semi-insulating optoelectronic quantum wells

Many optoelectronic quantum well applications require the dual properties of ultrafast lifetimes and sharp quantum confined excitons. Ultrafast lifetimes require high-defect densities, but these usually broaden the excitonic transition. Recent developments have achieved a compromise between broadening and speed in AlAs/GaAs quantum wells grown at low-substrate temperatures.13 The low-substrate growth temperature 共310 °C兲 incorporates excess arsenic and other intrinsic defects such as group-III vacancies that act as trapping and recombination centers. Electron lifetimes of several picoseconds were achieved in as-grown and weakly annealed materials, while retaining excellent quantum-confined Stark effects. The ability to achieve ultrafast lifetimes and sharp excitons in this class of materials has opened the door on a host of applications such as photorefractive nanostructures,14,15 and has potential for low-dark current photodetectors,16 electroabsorption modulators,17 ultrafast striplines,18 and electroabsorption sampling.19 For a complete review, see Ref. 20. When a vacancy crosses the interface between AlAs and GaAs, an Al and Ga atom swap places. Diffusion therefore rounds the conduction- and valence-band potential profiles, producing smaller quantum confinement, but increasing the exciton transition energy because of the increase of aluminum in the wells. The enhanced diffusion also causes enhanced interface roughening, which broaden the excitonic

III. PHASE-SHIFTED PHOTOMODULATION SPECTROSCOPY A. Phase-shifted detection

Our previous measurements13 on LTG AlAs/GaAs MQW’s utilized electromodulation spectroscopy with applied electric fields to identify the excitonic transition energies as functions of annealing time and temperature. Modulation spectroscopy is routinely used to identify critical points in optical absorption spectra, because the optical properties at the critical points are sensitive to small perturbations.22 In electromodulation spectroscopy, an external voltage applied to the device produces a Stark shift in the quantum-confined exciton that is detected as a change in transmission. The transition energy of the lowest-energy excitonic transition is obtained by identifying the photon energy for which the electroabsorption has a maximum derivative with respect to energy. This standard electromodulation technique led us to the original discovery of transient enhanced intermixing and the small effective migration enthalpies in the LTG quantum wells.9 However, the applied electric field could heat the device, producing uncertainties in the subsequent determination of the heavy- and light-hole exciton positions in the spectra. In this paper, we employ the method of phase-shifted photomodulation spectroscopy. The modulation of the heterostructure optical properties in this technique is provided by a low-power laser diode 共15 mW兲 at a wavelength of 690 nm.

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GEURSEN, LAHIRI, DINU, MELLOCH, AND NOLTE

No electric field is necessary because our quantum well structures were grown as p-i-n diodes that have a built-in electric field. Photomodulation spectroscopy relies on the generation of free carriers by illumination with above-gap light. The charges redistribute to screen the built-in electric field, producing a flat-band condition in the quantum wells and causing a blueshift of the Stark-shifted excitons. The experiments are performed in transmission to minimize sensitivity to surface band-bending effects. The infrared radiation from a quartz halogen lamp is monochromated using a 1-m double-grating spectrometer, and the transmission through the sample is detected with a silicon photodiode using a long-pass filter for visible light rejection. The use of the built-in field of the p-i-n diodes ensures the conditions of weak perturbation and consistent electric field conditions for all samples, therefore, making it possible to accurately quantify transition energies. In our use of phase-shifted photomodulation spectroscopy the transmitted probe light signal is detected 90° out of phase from the modulation stimulus. This is performed by synchronizing a lock-in amplifier to the modulating signal, then detecting the modulation of the probe light in the opposite quadrature. This approach guarantees that any contamination from the modulated pump is removed from the transmission signal. The technique of phase-shifted photomodulation spectroscopy requires the special condition that the pump beam must be modulated at a frequency that is comparable to the dielectric relaxation time of the quantum wells. The screening of the built-in field by the space-charge occurs at the dielectric relaxation rate, and the screening process will lag in time behind the pump light by as much as one quarter of the modulation period, representing a 90° shift in the lock-in detection. By detecting the modulated transmission in quadrature at 90° to the direct modulation, we obtain maximum signal with the best rejection of scattered pump light. The optimum modulation frequency in our experiment was determined by increasing the chopping frequency and detecting the differential transmission in quadrature. The normalized difference between the transmission with and without above-gap illumination gives the characteristic spectra of the Stark-shifted quantum-confined excitons. This is a normalized quantity, which reduces the effect of Fabry-Perot fringes that are present in the spectra because of the thinsemiconductor film. We performed differential transmission measurements for all the samples annealed under all conditions. The maximum differential transmission occurs for an optimum chopping frequency of ⬇144 Hz. The low value of the optimum frequency indicates a slow response time, and can be explained by the potential barriers that the electrons must penetrate to rearrange themselves under the influence of the built-in electric field and free-carrier diffusion. The electrons move along the 具100典 crystallographic direction, perpendicular to the interfaces, and must tunnel through or be thermally emitted over the AlAs barriers. The low probability of vertical transport strongly slows the response time of the device, delaying the quenching of the built-in field by the desired 90° phase shift.

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FIG. 1. Transmission 共a兲, and differential transmission 共b兲, spectra for isochronal annealing of 30 s for increasing annealing temperatures. The spectra are offset by a constant value to help visualization.

annealing temperatures are shown in Figs. 1 and 2. The transmission spectra show an excitonic blueshift as a function of annealing temperature, accompanied by broadening of the excitonic transitions. In the two figures, the effect is most pronounced for the 300 s annealing, with an extremely large shift observed in the 900 °C spectrum. Relatively sharp quantum-confined excitonic features are observed for the low temperature 共⬍700 °C兲 annealing. As the annealing temperature is increased, broadening of the exciton transition begins to wash out the distinct Stark shift present at lower temperatures. The excitonic transition is not apparent in the transmission spectra for temperatures greater than ⬃750 °C, but remain well defined in the differential transmission spectra. There is also a dependence of the sharpness of the excitonic peaks with annealing time, accompanied by a reduction in the modulated signal. The devices were not antireflection coated, therefore Fabry-Perot fringes are evident in the spectra below the band gap, but they are suppressed in the differential transmission spectra and do not affect the position or sharpness of the excitonic features of interest. For the modulation experiments, a marginally higher frequency 共277 Hz兲 than the optimum value was employed in the photomodulation measurements to obtain the best signal-to-noise ratio. IV. DIFFUSION LENGTH ANALYSIS

B. Photomodulation spectra

A. Calibration

Representative transmission and differential transmission spectra for the samples annealed at 30 and 300 s for all

The principal signature in the modulation experiments is the shift of the heavy-hole exciton to higher energies with


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FIG. 2. Transmission 共a兲, and differential transmission 共b兲, spectra for isochronal annealing of 300 s for increasing annealing temperatures. The spectra are offset by a constant value to help visualization.

increasing annealing. This increase in transition energy arises primarily from the increased aluminum concentration in the wells as the aluminum diffuses out of the AlAs barriers and into the GaAs wells. In addition, the confinement energy is reduced by rounding of the quantum well potentials. To translate an excitonic energy shift into a diffusion length, it is necessary to calculate a calibration curve of the energy shift as a function of diffusion length. If the diffusion is strictly linear, with no concentration dependence, then the diffusion profile of the quantum wells is a simple complimentary error function. The diffusion length is the only adjustable parameter in the function, and all other parameters are fixed by the growth. While concentration-dependent diffusion is certainly expected to occur in the AlAs/GaAs quantum wells, deviations from linear diffusion are expected to be weak. The limits of validity for the assumption of linear diffusion are discussed at the end of this paper. The concentration profile C(x) of aluminum for a barrier of width b centered at the origin is given by

C共 x 兲⫽

冋冉冊冉冊册

1 erfc 2

x⫹

b 2

2L D

x⫺

⫺erfc

b 2

2L D

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FIG. 3. 共a兲 Concentration profile for aluminum assuming a diffusion length of 1 nm, and 共b兲 the corresponding confinement potential for electrons, and the electron effective mass.

ⵜ 2␺共 x 兲⫽

2m * 共 x 兲关 V 共 x 兲 ⫺E 兴 ␺ 共 x 兲 ប2

共2兲

is solved for the potential-well profile for both the electrons and the holes, using a fourth-order Runge-Kutta algorithm. The wells were assumed to be uncoupled, the excitonic binding energy was neglected, and no band nonparabolicity was included. These conditions collectively limit the accuracy of the calibration to about a 5% uncertainty. The uncertainty on the experimental values of the energy shifts are 10% to 20% relative uncertainty. Therefore, the omissions in the numerical calibration do not significantly affect our results. Calculated differences in energies, relative to the transition energy of an as-grown potential well, are given in Fig. 4 for the

共1兲

for a diffusion length (L D ). The concentration profile also defines an effective mass m * 关 C(x) 兴 for the electrons and heavy holes. The concentration, carrier confinement potentials and effective masses are shown in Fig. 3. In the onecarrier two-band model, the Schro¨dinger equation

FIG. 4. Calibration curve to translate experimentally measured energy shifts to diffusion lengths. The calibration of the energy shifts of the heavy-hole exciton assumes linear diffusion.

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GEURSEN, LAHIRI, DINU, MELLOCH, AND NOLTE TABLE I. Peak heavy-hole exciton energy 共and full width at half maximum values兲, in eV 共meV兲, respectively, as a function of annealing time and temperature from the Gaussian fitting function. Annealing temperature As-grown 650 °C 700 °C 750 °C 800 °C 850 °C 900 °C

Annealing Time 30 s

100 s

1.4750 共4.88兲 1.4784 共4.69兲 1.4812 共4.80兲 1.4854 共6.85兲 1.4869 共7.23兲 1.4970 共8.98兲

1.4774 共4.70兲 1.4796 共4.57兲 1.4843 共11.11兲 1.4870 共7.20兲 1.4959 共7.48兲 1.5230 共9.45兲

300 s

1000 s

1.4796 共4.90兲 1.4863 共14.04兲 1.4880 共17.04兲 1.4883 共10.58兲 1.5011 共7.58兲 1.5295 共8.68兲

1.4819 共4.70兲 1.4904 共10.47兲 1.4888 共21.47兲 1.4968 共10.52兲 1.5320 共8.23兲

1.4738

heavy-hole exciton transition as a function of the diffusion length. This curve is used to convert the experimental values for the energy shifts into diffusion lengths, within the assumptions and approximations discussed above.

B. Intermixing-induced energy shifts and diffusion lengths

Higher annealing temperatures for longer times produces pronounced intermixing and roughening of the well-barrier interface, causing a broadening, and ultimately a ‘‘wash out,’’ of the excitonic transition. Due to this broadening of the excitonic transition, determination of the heavy-hole exciton energy in the photomodulation spectra becomes increasingly difficult for higher annealing temperatures and times. Therefore, a curve fitting process was employed to extract heavy-hole exciton energies, which isolated the heavy-hole peak and fit a Gaussian function, with an added linear term 共to account for the continuum兲, to the curve. The fitting parameters, for all isochronal and isothermal anneals, are shown in Table I. The measured change in heavy-hole transition energies for the isochronal and isothermal annealing conditions are shown as the data in Fig. 5 for the four annealing times of 30, 100, 300, and 1000 s. Also shown in the figure is a theoretical fit described in the next section that assumes firstorder kinetics for the decay of the enhanced diffusion. The striking feature of the data are the relatively flat regions for temperatures between 700 and 800 °C. The energy shift becomes insensitive to annealing temperature in this regime, exhibiting apparent athermal behavior, which had been observed previously.9 Using the calibration of Fig. 4, the energy-shift data in Fig. 5 are converted to diffusion lengths and are shown in Fig. 6, again with the theoretical fits that assume first-order decay kinetics and that H a ⫽H m . The curves are primarily to demonstrate the striking temperature independence caused by the H a ⫽H m constraint. More accurate fits of first-order and second-order decay kinetics to the data are presented in the next section.

V. TRANSIENT-ENHANCED DIFFUSION AND DECAY KINETICS

The diffusion lengths in Fig. 6 for quantum well intermixing in nonstoichiometric quantum wells are enhanced over the steady-state values of intermixing in GaAs by as much as three orders of magnitude for the intermediate annealing temperatures used in this study. The enhanced diffusion is a nonequilibrium situation that must decay in time towards the steady-state values. The nature and origins of this decay are among the outstanding problems in the physics of nonstoichiometric semiconductors. The following discussion assumes that the primary mechanism for quantum well intermixing in both stoichiometric and nonstoichiometric quantum wells is the migration of group-III vacancies. This assumption is highly plausible, based on the considerable theoretical and experimental literature discussed in the introduction. Under this assumption, the decay of the enhanced diffusion will reflect a decay in the vacancy concentration. Therefore, the time and temperature

FIG. 5. Experimentally measured energy shifts for the array of isochronal and isothermal annealing are shown as data points compared against the theory for transient enhanced intermixing that assumes linear diffusion and first-order decay kinetics with H a ⫽H m .


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FIG. 6. Diffusion lengths derived from the experimental energy shifts of Fig. 6 compared against the theory for first-order decay kinetics.

dependence of the intermixing diffusion length L D provides a means for analyzing the properties of vacancy diffusion in nonstoichiometric heterostructures. The time- and temperature-dependent diffusion length for vacancy selfdiffusion satisfies the differential rate equation6 2 dL D,V

dt

1 ⫽ d 2 ⌫ m n exp共 ⫺G F /k B T 兲 , 6

共3兲

where d⫽a/& is the second-nearest-neighbor distance, n ⫽12 is the number of second-nearest-neighbor sites, and a is the lattice constant. The free energy of formation for the vacancy is G F . The jump probability for vacancy migration is given by ⌫ m ⫽ ␯ m exp共 ⫺G m /k B T 兲 ,

共4兲

where ␯ m is the attempt frequency and G m is the free energy of migration. The diffusion equation for quantum well intermixing is related to the self-diffusion equation by 2 dL D,QW

dt

1 关 N␯兴 ⫽ d 2 n⌫ m 6 关 N III 兴

共5兲

assuming that group-III vacancy migration is the dominant diffusion mechanism for intermixing, where the ratio of the group-III vacancy concentration to the concentration of group-III sites is given by 关 N␯兴 ⫽N 0 exp共 ⫺G F /k B T 兲 ⫹N 1 共 t 兲 , 关 N III 兴

共6兲

in which the first term is the equilibrium term, and the second term is the excess fraction of vacancies that decays to zero during annealing after nonequilibrium growth. Equation 共6兲 allows Eq. 共5兲 to be written as 2 dL D,QW

dt

1 ⫽ d 2 n⌫ m 关 N 0 exp共 ⫺G F /k B T 兲 ⫹N 1 共 t 兲兴 , 6

共7兲

which defines the equation for the intermixing diffusion length L D,QW for transient-enhanced diffusion. Solving Eq. 共7兲 requires knowledge of the time dependence of N 1 (t), the fraction of excess group-III vacancies. This function must decay to zero at long times. Vacancies

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can decay through many channels associated with crystal defects. These include the trapping of vacancies at other point or extended defects, the formation of divacancies, and annihilation with interstitials. The vacancies can also annihilate at surfaces. In nonstoichiometric materials, in addition to the obvious surface of the sample, there are also internal surfaces between the semiconductor and the arsenic precipitates that form upon annealing. Almost all of these decay channels should lead to nonsingle-exponential decay. For instance, the removal of mobile vacancies by the formation of divacancies would produce strictly second-order kinetics. Most of the other decay channels also approximate second-order kinetics, such as annihilation at the surfaces of precipitates 共which become less dense with increasing annealing兲, or the formation of point defect complexes. In these decay channels, the trapping site densities also decay with annealing. This would produce decays with stretched exponentials because the vacancy trapping rate would be decreasing with annealing time. The many channels for vacancy decay can be visualized through a generalized ‘‘decay diagram’’ in which the vacancy interacts with a trap species N i to produce a product species P i . The diagram can also run backwards in time in which the product P i decays into a free vacancy. The (2N ⫹1) coupled differential equations describing N decay channels are N

N

d关V兴 B i关 N i 兴 ⫹ A i关 P i 兴 ⫽⫺ 关 V 兴 dt n⫽1 n⫽1

兺

兺

d关 Pi兴 ⫽⫺A i 关 P i 兴 ⫹ 关 V 兴 B i 关 N i 兴 dt d关 Ni兴 ⫽⫺ 关 V 兴 B i 关 N i 兴 ⫹A i 关 P i 兴 , dt

共8兲

where the N i species can be group-III vacancies, arsenic antisites, gallium antisites, arsenic precipitates, dislocations, etc. These coupled equations cannot be solved without a detailed knowledge of the all the decay products of vacancy annihilation. On the other hand, there are several simple decay channels that may be expected to dominate. For instance, if P i ⫽N i , then the decay obeys strictly first-order kinetics because the trap concentration remains constant. This situation may not be plausible, because most trapping densities will decay with increasing annealing. But this system is mathematically the simplest to solve. Another simple example is when N i is the vacancy concentration. This leads to strictly second-order decay kinetics in which vacancies form divacancy pairs. Although even this situation may not be the exclusive decay channel, it provides a model for the vacancy decay in which the decay rate decreases with increasing annealing as the density of trapping sites decreases. The solution of Eq. 共8兲 in terms of second-order kinetics therefore, may be more plausible as a means of describing transient enhanced diffusion. For these two simple cases, the general first- and secondorder kinetic time dependence of the diffusion length and the decay of the transient enhance diffusion are described by

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GEURSEN, LAHIRI, DINU, MELLOCH, AND NOLTE 2 dL D,QW

dt

the growth process, so only require the enthalpy H m to participate in diffusion. The enhancement term decays with the decay time ␶ a . The time-dependent diffusion coefficient produces a time-dependent diffusion length, which is then given by

⫽D 0 exp关 ⫺ 共 H f ⫹H m 兲 /k B T 兴 ⫹D 1 exp共 ⫺H m /k B T 兲 y 共 t 兲

dy 1 ⫽⫺ exp共 ⫺H a /k B T 兲 y N , dt ␶a

共9兲

where H m and H f are the migration and formation enthalpies of the group-III vacancy, respectively, y is the concentration of group-III vacancies, and the decay of the enhanced diffusion is assumed to be thermally activated with an annihilation enthalpy H a . The first term in the first equation describes the equilibrium diffusion and the second term describes the transient contribution to the time evolution of the squared diffusion length. The second equation describes the time dependence of the enhanced diffusion. The decay order is given by N. First-order decay kinetics have N⫽1 and describe single exponential decay. Second-order decay kinetics have N⫽2 and describe stretched exponential decay. Single-exponential decay requires that the excess vacancies trap at a fixed concentration of trapping sites whose density is unaffected by annealing. Stretched exponential decay is caused by a decay of the trapping site concentration. A. First-order decay kinetics

The opportunities for first-order decay kinetics with single exponential decays are limited, requiring a concentration of vacancy trapping or annihilation sites that remains independent of annealing. While known point defects, such as arsenic antisites, decay during annealing and therefore cannot provide this decay mechanism, the analysis of first-order decay kinetics helps to qualitatively explain the flat plateau regions in the measured diffusion length. Assuming first-order kinetics with N⫽1 in Eq. 共9兲, the supersaturated vacancy concentration decays exponentially and gives rise to a time-dependent diffusion coefficient. The vacancy annihilation time ␶ a is thermally activated according to 1 ⫽ ␯ a exp共 ⫺H a /k B T 兲 , ␶a

共10兲

where H a is the annihilation enthalpy, and ␯ a is the attempt frequency for annihilation. The excess vacancy concentration introduces a time-dependent term in the diffusion coefficient given by D 1 共 t 兲 ⫽D 1 exp共 ⫺H m /k B T 兲 exp共 ⫺t/ ␶ a 兲 .

共11兲

The general time-dependent diffusion coefficient is D 共 t 兲 ⫽D 0 exp关 ⫺ 共 H f ⫹H m 兲 /k B T 兴 ⫹D 1 exp共 ⫺H m /k B T 兲 ⫻exp共 ⫺t/ ␶ a 兲 .

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共12兲

This expression contains the steady-state term that is caused by an equilibrium concentration of vacancies at a given temperature. To participate in diffusion, the equilibrium vacancies must first be thermally generated with a formation enthalpy H f , and then must migrate with a migration enthalpy H m . On the other hand, in the case of nonstoichiometric growth the excess vacancies have already been formed by

L D,QW 共 t 兲 2 ⫽D 0 t exp关 ⫺ 共 H f ⫹H m 兲 /k B T 兴 ⫹D 1 ␶ a exp共 ⫺H m /k B T 兲关 1⫺exp共 ⫺t/ ␶ a 兲兴 . 共13兲 There are several interesting asymptotic limits contained in Eq. 共13兲. For very long annealing times, the enhancement term becomes negligible, and equilibrium diffusion with a net enthalpy of H f ⫹H m is recovered. For intermediate annealing times, when the enhancement term dominates over the steady-state term, the diffusion length becomes L D,QW 共 t 兲 2 ⫽D 1 ␶ a exp共 ⫺H m /k B T 兲 ⫽

D1 exp关 ⫺ 共 H m ⫺H a 兲 /k B T 兴 , ␯a

共14兲

which describes diffusion with an effective migration eneff ⫽Hm⫺Ha . If H m and H a are comparable thalpy given by H m in magnitude, then this intermediate-time regime would produce apparent athermal diffusion, which would exhibit plateau in the diffusion length as a function of temperature. For very short-annealing times, Eq. 共13兲 describes a quasisteady-state diffusion length given by L D,QW 共 t 兲 2 ⫽D 1 t exp共 ⫺H m /k B T 兲 ,

共15兲

which is characterized only by the migration enthalpy H m without the need to form vacancies with the enthalpy H f . To make the comparison with theoretical models and experimental data, it is helpful to plot the data of Fig. 6 in terms of the square of the diffusion length, divided by the annealing time. In the case of steady-state diffusion, this procedure gives a single curve that is the steady-state diffusion coefficient as a function of temperature. In the case of transient-enhanced diffusion, this procedure produces a family of curves indexed by the annealing time that all asymptote to the steady-state values for high temperatures and long annealing times. The square of the diffusion length data from Fig. 6 divided by the annealing times are plotted in Fig. 7共a兲 as a function of inverse temperature. The plateau regions noticed in Figs. 5 and 6 remain prominent features in the intermediate temperature range. For high temperatures, the data approach the steady-state diffusion coefficient measured by Wang et al. for isotopic superlattices of GaAs.23 The solid curves in the figure are given by a chi-squared fit to the diffusion data. The parameters D 0 ⫽43 cm2/s and H f ⫹H m ⫽4.24 eV have been taken by the values from Wang et al.23,24 When the annihilation enthalpy H a was allowed to vary, the chi square was minimized for H a ⫽1.4 eV, D 1 ⫽4.8⫻10⫺8 cm2/s and 1/␯ a ⫽4.1 ␮ s. The best fit is shown in Fig. 7共a兲. The enthalpy value is close to H m ⫽1.8 eV, which would be required if the vacancies must first migrate in order to annihilate.


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ics. When the annihilation enthalpy is allowed to vary, we obtain a minimum chi-squared for H a ⫽1.6 eV with D 1 ⫽6.0⫻10⫺8 cm2/s and 1/v a ⫽100 ns. When we constrain H a ⫽H m ⫽1.8 eV, the data are best fit with the parameters D 1 ⫽6.0⫻10⫺8 cm2/s and 1/v a ⫽10 ns. In comparing the fits in Figs. 7共a兲 and 7共b兲, the fits are equally good. Therefore, we cannot distinguish between first-order and second-order decay kinetics in this experiment, leaving the order of the decay kinetics an important open question. VI. DISCUSSION

FIG. 7. Effective diffusion coefficient plotted as a function of inverse temperature. The high-temperature data asymptote to the equilibrium curve of Haller et al. 共Ref. 23兲. The fit to first-order kinetics is shown in 共a兲 and the fit to second-order kinetics is shown in 共b兲. B. Second-order decay kinetics

The most likely decay kinetics for the decay of the supersaturated vacancy concentration are second-order decay kinetics, in which the decay rate also decays in time because the sites that bind up the vacancies also anneal away. The time dependence of the diffusion length and the decay of the transient enhance diffusion are given by 2 dL D,QW

dt

⫽D 0 exp关 ⫺ 共 H f ⫹H m 兲 /k B T 兴 ⫹D 1 exp共 ⫺H m /k B T 兲 y 共 t 兲

dy ⫽⫺ ␯ a exp共 ⫺H a /k B T 兲 y 2 , dt

共16兲

where most of the parameters are the same as in the case of first-order kinetics, and y is the relative excess in vacancy concentration. The square of the diffusion length divided by the annealing time is shown in Fig. 7共b兲, fit to the second-order decay kinetics. Second-order decay kinetics lead to the same qualitative trends as the first-order decay. The plateaus are still present, although they are not as flat as for first-order kinet-

The principal finding of this work is the identification of the annihilation enthalpy H a and the assignment of a value between 1.4 and 1.6 eV that is approximately equal to the suggested migration enthalpy H m ⫽1.8 eV for the gallium vacancy. The somewhat smaller values than H m may be attributed to experimental uncertainty, or may reflect a small difference in the migration enthalpy of vacancies in LTG materials. Evidence for suppressed migration enthalpy has been observed previously for LTG materials.21,25,26 The similarity between H a and H m also lends support for the dominant role played by group-III vacancies in the diffusion enhancement, and weighs against interstitial diffusion mechanisms.12 It is important to compare the magnitude of the transient enhancement we observe in the LTG quantum wells with the steady-state enhancements obtained by other techniques, such as doping with Si during growth, or annealing under an SiO2 capping layer. Direct comparison of diffusion coefficients is not possible between steady-state and transientdiffusion conditions, because the diffusion coefficient is time dependent in the transient case. However, to establish the order of magnitude of the effect, we can compare our value of L D 2 /t with the values for D obtained by Si doping of different concentrations by Mei et al.27 For our shortest duration annealing of 30 s, the value for L D 2 /t is comparable to the steady-state value of the diffusion coefficient for a Si concentration of approximately 2⫻1018 cm⫺3. Enhanced disordering of GaAs quantum wells through encapsulation with SiO2 typically yields diffusion coefficients in the range of 10⫺15 cm2/s. 28,29 Important issues and questions remain unanswered concerning the mechanisms for the relaxation of nonstoichiometric GaAs to its equilibrium condition. The question of the order of the decay kinetics remains an important open question. Second-order kinetics would be most expected in this material, because all the likely traps for vacancies also decay in time during annealing. However, the uncertainties in our data cannot rule out first-order decay kinetics. First-order decay is more difficult to explain, because it requires a constant density of vacancy annihilation sites that are unaffected by annealing. Furthermore, the constant trap densities that would be needed to trap the vacancies are anomalously high, requiring a vacancy to be trapped or annihilated within several hundred Å. This length is much smaller than the distance to external surfaces or even to the surfaces of the coarsening precipitates. Despite this circumstantial evidence against first-order decay kinetics, further work will be required to put this important question to rest. The possible role of nonlinear diffusion has not been ad-

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GEURSEN, LAHIRI, DINU, MELLOCH, AND NOLTE

dressed in this paper. Nonlinear diffusion in the nonstoichiometric quantum wells can arise from two sources: 共1兲 the concentration of grown-in vacancies can vary between the AlAs and GaAs layers; and 共2兲 the migration enthalpy can be a function of Al fraction. Preliminary Monte Carlo simulations show that the first nonlinear diffusion mechanism is negligible because the initial segregation of vacancies is rapidly erased during even short-time annealing. In addition, the simulations of the second condition show that any difference in the migration enthalpy between AlAs and GaAs works to suppress intermixing. Therefore, the enhancements we infer by assuming linear diffusion would be lower bounds. The

1

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important question that remains is the quantitative effect nonlinear diffusion would have on the value of H a obtained through our linear analysis. This questions awaits further investigation.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of Grant No. NSF-9400415-DMR through the MRSEC, and Grant No. NSF-9708230-ECS. Helpful discussions with Sunder Balasubramanian are acknowledged.

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