- Email: [email protected]
Desorption of InAs quantum dots
Journal of Crystal Growth 251 (2003) 218–222
Desorption of InAs quantum dots Ch. Heyn*, W. Hansen Institute for Applied Physics, University of Hamburg, Jungiusstr. 11, 20355 Hamburg, Germany
Abstract We study the desorption of self-assembled grown InAs quantum dots with in situ electron diffraction. The time up to quantum dot formation and the post-growth lifetime of the dots are recorded as a function of the temperature and of the arsenic flux. We find a reduction of the post-growth lifetime with temperature and a stabilization of the dots by a sufficient arsenic flux. The shorter lifetime at higher temperatures corresponds to a longer time needed for quantum dot formation. The indium sticking coefficient during growth as well as the post-growth desorption lifetime are calculated with a layer-based desorption model. The calculation results well describe our experimental data. r 2002 Elsevier Science B.V. All rights reserved. PACS: 61.14.Hg; 68.35.Rh; 81.05.Ys; 81.15.Hi Keywords: A1. Desorption; A1. Growth models; A1. Nanostructures; A1. Reflection high energy electron diffraction; A3. Molecular beam epitaxy; B2. Semiconducting III–V materials
1. Introduction The application of self-assembling mechanisms for the fabrication of nanostructures with controlled properties requires the detailed understanding of the dependencies of the processes on relevant growth parameters. In the case of selfassembled growth of InAs quantum dots on GaAs with the MBE technique, the most important parameters are the growth temperature T; the growth rate given by the indium flux FIn ; and the arsenic flux FAs : Clearly the surface kinetics of the adsorbants make the dot formation process strongly temperature dependent [1]. Furthermore, *Corresponding author. Tel.: +49-40-42838-2040; fax: +4940-42838-6332. E-mail address: [email protected] (C. Heyn).
it is well known that intermixing with Ga from the substrate layer beneath the InAs dots becomes increasingly important with increasing temperature [2]. Here we point out InAs desorption as an additional process that becomes crucial at high temperatures. This process turns out to be strongly dependent on the As pressure. The central experimental method is reflection high energy electron diffraction (RHEED) that allows for a quantitative characterization of the relevant time scales during the dot formation and in the post-growth regime. In particular, we measure the time tc up to quantum dot formation and the post-growth lifetime tD as a function of temperature at different values of the arsenic flux. The experimental data are interpreted in terms of desorption models that provide good reproduction of the measurements.
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-0248(02)02379-5
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
219
2. Experimental setup
3. Results and discussion Fig. 1 shows measured values of tc for a wide range of growth temperatures. We find a slightly rising tc with T up to approximately 5201C followed by an abrupt increase at higher temperatures. For temperatures higher than 5431C no clear 3D growth related RHEED pattern are found. The slight increase of tc at To5201C is attributed to a temperature dependent intermixing with Ga from the substrate, which reduces the strain energy and, thus, the driving force for dot formation [1]. The abrupt increase at T > 5201C cannot be explained with intermixing alone. We assume that desorption is the additional process responsible for the steep rise. In order to study the desorption process, we
70
tC (s)
The samples are grown in a solid source molecular beam epitaxy (MBE) system equipped with a valved cracker cell for arsenic. We use the valve for a precise and immediate control of the arsenic flux. The cracker zone temperature is chosen for As4 emission. To calibrate the arsenic flux we apply the method previously presented in Ref. [3]. Starting from (0 0 1)GaAs substrates, first a GaAs buffer layer is grown in order to smoothen the surface. During growth of the following InAs layer, the surface morphology evolution is studied in situ with RHEED at 12 kV. The InAs growth rate is FIn ¼ 0:04 ML/s for the measurements of tc and 0.1 ML/s for the desorption lifetime tD related experiments. The desorption experiments are started using an InAs coverage of 2.0 ML. We use the ½1% 1 0-azimuth for the RHEED measurements, where the 2D to 3D growth-mode transition becomes visible by two phenomena. First, the diffraction pattern change from 2D surface morphology related spots to transmission diffraction through 3D quantum dot islands [4], and second, additional chevrons arise, that are attached to the spots and are correlated with the island side-facets [5]. For an exact determination of the critical time tc up to the 2D to 3D transition and of the postgrowth evolution, we record the RHEED signal intensity as is described in detail in Refs. [1,4].
FIn = 0.04 ML/s FAs = 4.5 ML/s
60
50
40 440
460
480
500
520
540
Temperature (˚C) Fig. 1. Measured (symbols) and calculated (lines) values of the critical time tc up to quantum dot formation vs. temperature. The model used for the calculation considers desorption which is the dominant process at T > 5201C, while intermixing is neglected which is responsible for the slight rise of the experimental tc below To5201C. This explains the discrepancy at To5201C.
have performed further RHEED experiments in the post-growth regime. After growth is stopped by closing the In-shutter, the RHEED pattern are typical for 3D InAs quantum dot islands [4]. These 3D RHEED spots are stable for a certain time. Then they become weaker and finally disappear at a time tD : Now the RHEED pattern indicate a 2D surface morphology, with spots very similar to those in the 2D growth regime at the early stages of deposition [4]. We attribute this observation to the removal of the InAs quantum dots due to desorption. The dot removal is confirmed by our atomic force microscopy measurements. Here samples that are kept at growth temperature after growth for a time longer than tD show no dot-like structures, whereas those cooled down immediately show quantum dots. In Fig. 2 a typical time evolution of the RHEED reflex intensity is plotted. After growth is stopped we find an approximately constant intensity, followed by a nearly linear decrease up to the reappearance of the 2D related reflexes. From diffraction theory, the intensity of the 3D diffraction spots is proportional to the diffracting volume. The behaviour thus indicates an initially small and subsequently rapidly increasing desorption rate and, therefore, layer-by-layer
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
220
400 200 t D (s)
Layer-by-layer desorption 300
RHEED intensity
tD (s)
T = 518˚C
200
FAs= 4.8 ML/s 0 ML/s
0 0
2 4 6 FAs (ML/s)
100
0 480
500
520
540
560
Temperature (˚C) tD
In shutter opened
0
In shutter closed
Fig. 3. Measured (symbols) and calculated (line) temperature dependence of the post-growth quantum dot lifetime tD after deposition of 2.0 ML of InAs. Two values of the arsenic flux are chosen. The inset shows the dependence of the measured (symbols) and the calculated (line) time tD on the arsenic flux at a temperature T ¼ 5181C.
Switch from 3D to 2D reflex
100
50
150
Time (s) Fig. 2. Time evolution of the RHEED reflex intensity in ½1% 1 0azimuth (bulk line) together with the calculated quantum dot volume (dashed line). The quantum dot lifetime tD gives the time interval between growth stop and the re-appearance of a 2D related RHEED pattern. The inset illustrates the assumed layer-by-layer desorption.
desorption starting from the island top (see Fig. 2) instead of re-evaporation from the initially large side-facets. From our RHEED data, we precisely determine the post-growth lifetime tD up to the reappearance of a 2D surface morphology (Fig. 2). Fig. 3 shows such determined values of tD as function of the temperature in presence and absence of an arsenic flux. The experimental data obtained without arsenic flux are analysed in terms of a simple continuum model. We assume that after deposition of 2.0 ML InAs a 1 ML thick InAs wetting layer has been formed and that the remaining material is accumulated in the quantum dots. From atomic force microscopy, we find a typical dot density of 2.0 1010 cm2 corresponding to an average number of 51900 atoms per dot. Assuming pyramidshaped dots build of VP atoms and an angle a of 261 [6] between the substrate and the pyramid side-
facets, the initial pyramid height is hP ¼ 21 atoms. Later, due to the assumed layer-by-layer desorption, the pyramids become truncated. The size of the square area on top of a truncated pyramid build of V atoms is A ¼ ½6cot aðVP V Þ2=3 : An atom of species j may desorb from this area after its lifetime tj ¼ t0;j expðEj =kB TÞ; where t0;j is a prefactor, and Ej is the energy barrier for desorption. Since for desorption of both species, indium as well as arsenic, bonds between indium and arsenic layers must be broken, we take the respective lifetimes as approximately equal with t0;j ¼ tj;In ¼ tj;As ¼ t0 and Ej ¼ EIn ¼ EAs : Due to desorption, the volume of the pyramid is reduced according to dV ðtÞ=dt ¼ AðtÞ=tx: With V ðt ¼ 0Þ ¼ VP ; one 3 gets V ðtÞ ¼ VP 43 ðcot aÞ2 t=tx and for the pyramid height hðtÞ ¼ hP ðt=tx Þ: Therefore, the quantum dot lifetime becomes tD ¼ ðhP hc Þt0 expðE=kB TÞ and represents the reduction of the dot height from initially hP down to a critical height at which the 3D to 2D transition of the RHEED signals occurs. According to Ref. [1], we assume this critical height to be hc ¼ 3 ML. From the comparison with the experimental data for FAs ¼ 0 we obtain EIn ¼ EAs ¼ 3:53 eV and t0 ¼ 1:14 1023 s. We note the surprising small pre-factor, which is not well understood at present.
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
A similar finding is discussed in Ref. [7]. Fig. 2 shows the agreement between the such calculated quantum dot volume V ðtÞ and the 3D spot RHEED intensity. In addition, Fig. 3 presents the measured temperature dependence of tD for FAs ¼ 0: These experiments are quantitatively reproduced by the model as well. The experimental results obtained with As flux establish the stabilization of the quantum dots by an arsenic beam to the surface (Fig. 3). As an example, at T ¼ 5181C, tD increases from approximately 7 s without arsenic flux to more than 200 s by applying an arsenic flux FAs ¼ 6:0 ML/s. Desorption under As flux cannot be described with our simple continuum model. Therefore, a second layer-based model is developed that includes fluxes of In and As and allows us to calculate the stabilization of the dots by arsenic as well as the indium sticking coefficient during growth. In this model, the pyramids consist of alternating indium and arsenic layers with index i ¼ 1yhP ; starting from the bottom indium layer i ¼ 1: The maximum number of atoms in each layer Ai ¼ ½2cot aðhP iÞ2 reflects the pyramidshape. The quantity Xi describes the filling-level of the layer i and is unity directly after deposition. Desorption reduces Xi and, thus, the total island P volume V ¼ i Ai Xi : The filling-level evolution obeys for the indium layers: dXi =dt ¼ FIn ðXi Xiþ1 Þ=tIn and for the arsenic layers dXi =dt ¼ ðypr =ttr ÞðXi1 Xi Þ ðXi Xiþ1 Þ=tAs : This set of two coupled rate equations considers the capping of atoms in a certain layer by those in a layer over it. That means, atoms from layer i can only desorb once enough atoms from layer i þ 1 have already been removed. Furthermore, a precursor-state for the impinging As4 molecules is included. A model which accounts for direct incorporation of the As molecules without precursor-state yields a linear dependence of tD on the arsenic flux. This is not in agreement with the experimental finding of a nearly saturated tD at higher values of FAs (Fig. 3). Precursor states are well established in literature for description of
221
arsenic incorporation on GaAs surfaces [8]. As a simplification, we assume that the precursor state is populated with arsenic atoms and that every impinging As4 molecule produces two atoms according to the maximum sticking coefficient of 0.5. Arsenic atoms from the precursor state can either desorb after the lifetime tpr or transit into the chemisorbed state prior to incorporation after the lifetime ttr : In equilibrium, the coverage of the precursor layer is thus given by ypr ¼ FAs =ðt1 pr þ FAs Þ: The precursor layer related lifetimes are characterized by the energy barriers Epr ¼ 3:57 eV and Etr ¼ 3:24 eV. These values are determined by a fit to the flux dependent data of the inset in Fig. 3 assuming equal prefactors for all processes. With the layer-based model we calculate tD in the post-growth regime with a finite arsenic flux FAs > 0 but in the absence of an indium flux FIn : As described above, tD represents the time up to the reduction of the pyramid height down to hc ¼ 3:0 ML. Values of tD calculated for FAs ¼ 0 qualitatively reproduce the experiments and agree nearly exactly with the above continuum model (Fig. 3). As an additional important validation for the model, we calculate the temperature dependence of tD at a fixed arsenic flux FAs ¼ 4:8 ML/s using the same set of parameters. The very good agreement between the calculated and the measured lifetimes, visible in Fig. 3, demonstrates the ability of our model to describe the central desorption mechanisms during the post-growth regime. In the next step, we apply the layer-based model to calculate the desorption process during growth (FIn > 0). For that, the initial conditions are changed to Xi ðt ¼ 0Þ ¼ 0: We determine the indium sticking coefficient aIn ¼ yIn =ðFIn tÞ from the calculated indium coverage yIn for t ¼ 25 s deposition with FIn ¼ 0:04 ML/s. The wetting layer is neglected in this approach. Fig. 4 shows values of aIn calculated under variation of T for different values of FAs : In all cases, we find aIn E1 up to a certain temperature, followed by a strong decrease. A higher arsenic flux increases this transition temperature. A reduced sticking coefficient lengthens the time that is required for the deposition of a certain amount of material according to tc ¼ t0 =aIn ; where FIn t0 gives the film
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
222
1.0 0.8
α In
0.2
L/s 40 M
FAs =
0.4
4 ML/s L/s 0.4 M ML/s 0.04
0.6
FIn = 0.04 ML/s
0.0
[2], desorption of both In as well as of Ga might be of relevance. However, since desorption from pure GaAs surfaces is observed only at temperatures above 7501C [9], we assume that In desorption is dominant. This means that desorption results in two effects. First the volume of the quantum dots is reduced and second the dots become more Garich.
440 460 480 500 520 540 560 Temperature (˚C) Fig. 4. Calculated indium sticking coefficient aIn as function of the temperature at different values of the arsenic flux.
Acknowledgements
thickness at which dot formation occurs in the case of aIn ¼ 1: This fact explains the experimental temperature dependence of tc at growth temperatures higher than 5201C. Using t0 ¼ 45 s, we find good agreement as is demonstrated in Fig. 1.
The authors would like to thank S. Mendach for helpful discussions, S. Schnull . for operation of the MBE system, C. Weichsel for AFM measurements, and the ‘‘Deutsche Forschungsgemeinschaft’’ for financial support via HA 2042/3 and via SFB 508.
4. Conclusions In this work, we study the desorption of selfassembled InAs quantum dots. We find that, dependent on the arsenic flux, desorption takes place even at common growth temperatures. A continuum-based model is applied to describe the InAs dot desorption in absence of a beam flux. In the presence of an As flux, a layer-based model and a As precursor-state describe the data very well. The In sticking coefficient is calculated for dot growth in presence of In and As fluxes. Using these values, we can well reproduce the experimental temperature dependence of the critical time up to quantum dot formation. Considering a significant Ga content in the dots due to intermixing at higher growth temperatures
References [1] Ch. Heyn, Phys. Rev. B. 64 (2001) 165306. [2] P.B. Joyce, T.J. Krzyzewski, G.R. Bell, B.A. Joyce, T.S. Jones, Phys. Rev. B 58 (1998) R15981. [3] Ch. Heyn, M. Harsdorff, Appl. Surf. Sci. 100/101 (1996) 494. [4] Ch. Heyn, D. Endler, K. Zhang, W. Hansen, J. Crystal Growth 210 (2000) 421. [5] H. Lee, R. Lowe-Webb, W. Yang, P.C. Sercel, Appl. Phys. Lett. 72 (1998) 812. [6] J. Marquez, L. Geelhaar, K. Jacobi, Appl. Phys. Lett. 78 (2001) 2309. [7] C. Sasaoka, Y. Kato, A. Usui, Appl. Phys. Lett. 62 (1993) 2338. [8] C.T. Foxon, B.A. Joyce, Surf. Sci. 50 (1975) 434. [9] N. Sugiyama, T. Isu, J. Katayama, Jpn. J. Appl. Phys. 28 (1989) L287.
Desorption of InAs quantum dots Ch. Heyn*, W. Hansen Institute for Applied Physics, University of Hamburg, Jungiusstr. 11, 20355 Hamburg, Germany
Abstract We study the desorption of self-assembled grown InAs quantum dots with in situ electron diffraction. The time up to quantum dot formation and the post-growth lifetime of the dots are recorded as a function of the temperature and of the arsenic flux. We find a reduction of the post-growth lifetime with temperature and a stabilization of the dots by a sufficient arsenic flux. The shorter lifetime at higher temperatures corresponds to a longer time needed for quantum dot formation. The indium sticking coefficient during growth as well as the post-growth desorption lifetime are calculated with a layer-based desorption model. The calculation results well describe our experimental data. r 2002 Elsevier Science B.V. All rights reserved. PACS: 61.14.Hg; 68.35.Rh; 81.05.Ys; 81.15.Hi Keywords: A1. Desorption; A1. Growth models; A1. Nanostructures; A1. Reflection high energy electron diffraction; A3. Molecular beam epitaxy; B2. Semiconducting III–V materials
1. Introduction The application of self-assembling mechanisms for the fabrication of nanostructures with controlled properties requires the detailed understanding of the dependencies of the processes on relevant growth parameters. In the case of selfassembled growth of InAs quantum dots on GaAs with the MBE technique, the most important parameters are the growth temperature T; the growth rate given by the indium flux FIn ; and the arsenic flux FAs : Clearly the surface kinetics of the adsorbants make the dot formation process strongly temperature dependent [1]. Furthermore, *Corresponding author. Tel.: +49-40-42838-2040; fax: +4940-42838-6332. E-mail address: [email protected] (C. Heyn).
it is well known that intermixing with Ga from the substrate layer beneath the InAs dots becomes increasingly important with increasing temperature [2]. Here we point out InAs desorption as an additional process that becomes crucial at high temperatures. This process turns out to be strongly dependent on the As pressure. The central experimental method is reflection high energy electron diffraction (RHEED) that allows for a quantitative characterization of the relevant time scales during the dot formation and in the post-growth regime. In particular, we measure the time tc up to quantum dot formation and the post-growth lifetime tD as a function of temperature at different values of the arsenic flux. The experimental data are interpreted in terms of desorption models that provide good reproduction of the measurements.
0022-0248/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0022-0248(02)02379-5
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
219
2. Experimental setup
3. Results and discussion Fig. 1 shows measured values of tc for a wide range of growth temperatures. We find a slightly rising tc with T up to approximately 5201C followed by an abrupt increase at higher temperatures. For temperatures higher than 5431C no clear 3D growth related RHEED pattern are found. The slight increase of tc at To5201C is attributed to a temperature dependent intermixing with Ga from the substrate, which reduces the strain energy and, thus, the driving force for dot formation [1]. The abrupt increase at T > 5201C cannot be explained with intermixing alone. We assume that desorption is the additional process responsible for the steep rise. In order to study the desorption process, we
70
tC (s)
The samples are grown in a solid source molecular beam epitaxy (MBE) system equipped with a valved cracker cell for arsenic. We use the valve for a precise and immediate control of the arsenic flux. The cracker zone temperature is chosen for As4 emission. To calibrate the arsenic flux we apply the method previously presented in Ref. [3]. Starting from (0 0 1)GaAs substrates, first a GaAs buffer layer is grown in order to smoothen the surface. During growth of the following InAs layer, the surface morphology evolution is studied in situ with RHEED at 12 kV. The InAs growth rate is FIn ¼ 0:04 ML/s for the measurements of tc and 0.1 ML/s for the desorption lifetime tD related experiments. The desorption experiments are started using an InAs coverage of 2.0 ML. We use the ½1% 1 0-azimuth for the RHEED measurements, where the 2D to 3D growth-mode transition becomes visible by two phenomena. First, the diffraction pattern change from 2D surface morphology related spots to transmission diffraction through 3D quantum dot islands [4], and second, additional chevrons arise, that are attached to the spots and are correlated with the island side-facets [5]. For an exact determination of the critical time tc up to the 2D to 3D transition and of the postgrowth evolution, we record the RHEED signal intensity as is described in detail in Refs. [1,4].
FIn = 0.04 ML/s FAs = 4.5 ML/s
60
50
40 440
460
480
500
520
540
Temperature (˚C) Fig. 1. Measured (symbols) and calculated (lines) values of the critical time tc up to quantum dot formation vs. temperature. The model used for the calculation considers desorption which is the dominant process at T > 5201C, while intermixing is neglected which is responsible for the slight rise of the experimental tc below To5201C. This explains the discrepancy at To5201C.
have performed further RHEED experiments in the post-growth regime. After growth is stopped by closing the In-shutter, the RHEED pattern are typical for 3D InAs quantum dot islands [4]. These 3D RHEED spots are stable for a certain time. Then they become weaker and finally disappear at a time tD : Now the RHEED pattern indicate a 2D surface morphology, with spots very similar to those in the 2D growth regime at the early stages of deposition [4]. We attribute this observation to the removal of the InAs quantum dots due to desorption. The dot removal is confirmed by our atomic force microscopy measurements. Here samples that are kept at growth temperature after growth for a time longer than tD show no dot-like structures, whereas those cooled down immediately show quantum dots. In Fig. 2 a typical time evolution of the RHEED reflex intensity is plotted. After growth is stopped we find an approximately constant intensity, followed by a nearly linear decrease up to the reappearance of the 2D related reflexes. From diffraction theory, the intensity of the 3D diffraction spots is proportional to the diffracting volume. The behaviour thus indicates an initially small and subsequently rapidly increasing desorption rate and, therefore, layer-by-layer
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
220
400 200 t D (s)
Layer-by-layer desorption 300
RHEED intensity
tD (s)
T = 518˚C
200
FAs= 4.8 ML/s 0 ML/s
0 0
2 4 6 FAs (ML/s)
100
0 480
500
520
540
560
Temperature (˚C) tD
In shutter opened
0
In shutter closed
Fig. 3. Measured (symbols) and calculated (line) temperature dependence of the post-growth quantum dot lifetime tD after deposition of 2.0 ML of InAs. Two values of the arsenic flux are chosen. The inset shows the dependence of the measured (symbols) and the calculated (line) time tD on the arsenic flux at a temperature T ¼ 5181C.
Switch from 3D to 2D reflex
100
50
150
Time (s) Fig. 2. Time evolution of the RHEED reflex intensity in ½1% 1 0azimuth (bulk line) together with the calculated quantum dot volume (dashed line). The quantum dot lifetime tD gives the time interval between growth stop and the re-appearance of a 2D related RHEED pattern. The inset illustrates the assumed layer-by-layer desorption.
desorption starting from the island top (see Fig. 2) instead of re-evaporation from the initially large side-facets. From our RHEED data, we precisely determine the post-growth lifetime tD up to the reappearance of a 2D surface morphology (Fig. 2). Fig. 3 shows such determined values of tD as function of the temperature in presence and absence of an arsenic flux. The experimental data obtained without arsenic flux are analysed in terms of a simple continuum model. We assume that after deposition of 2.0 ML InAs a 1 ML thick InAs wetting layer has been formed and that the remaining material is accumulated in the quantum dots. From atomic force microscopy, we find a typical dot density of 2.0 1010 cm2 corresponding to an average number of 51900 atoms per dot. Assuming pyramidshaped dots build of VP atoms and an angle a of 261 [6] between the substrate and the pyramid side-
facets, the initial pyramid height is hP ¼ 21 atoms. Later, due to the assumed layer-by-layer desorption, the pyramids become truncated. The size of the square area on top of a truncated pyramid build of V atoms is A ¼ ½6cot aðVP V Þ2=3 : An atom of species j may desorb from this area after its lifetime tj ¼ t0;j expðEj =kB TÞ; where t0;j is a prefactor, and Ej is the energy barrier for desorption. Since for desorption of both species, indium as well as arsenic, bonds between indium and arsenic layers must be broken, we take the respective lifetimes as approximately equal with t0;j ¼ tj;In ¼ tj;As ¼ t0 and Ej ¼ EIn ¼ EAs : Due to desorption, the volume of the pyramid is reduced according to dV ðtÞ=dt ¼ AðtÞ=tx: With V ðt ¼ 0Þ ¼ VP ; one 3 gets V ðtÞ ¼ VP 43 ðcot aÞ2 t=tx and for the pyramid height hðtÞ ¼ hP ðt=tx Þ: Therefore, the quantum dot lifetime becomes tD ¼ ðhP hc Þt0 expðE=kB TÞ and represents the reduction of the dot height from initially hP down to a critical height at which the 3D to 2D transition of the RHEED signals occurs. According to Ref. [1], we assume this critical height to be hc ¼ 3 ML. From the comparison with the experimental data for FAs ¼ 0 we obtain EIn ¼ EAs ¼ 3:53 eV and t0 ¼ 1:14 1023 s. We note the surprising small pre-factor, which is not well understood at present.
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
A similar finding is discussed in Ref. [7]. Fig. 2 shows the agreement between the such calculated quantum dot volume V ðtÞ and the 3D spot RHEED intensity. In addition, Fig. 3 presents the measured temperature dependence of tD for FAs ¼ 0: These experiments are quantitatively reproduced by the model as well. The experimental results obtained with As flux establish the stabilization of the quantum dots by an arsenic beam to the surface (Fig. 3). As an example, at T ¼ 5181C, tD increases from approximately 7 s without arsenic flux to more than 200 s by applying an arsenic flux FAs ¼ 6:0 ML/s. Desorption under As flux cannot be described with our simple continuum model. Therefore, a second layer-based model is developed that includes fluxes of In and As and allows us to calculate the stabilization of the dots by arsenic as well as the indium sticking coefficient during growth. In this model, the pyramids consist of alternating indium and arsenic layers with index i ¼ 1yhP ; starting from the bottom indium layer i ¼ 1: The maximum number of atoms in each layer Ai ¼ ½2cot aðhP iÞ2 reflects the pyramidshape. The quantity Xi describes the filling-level of the layer i and is unity directly after deposition. Desorption reduces Xi and, thus, the total island P volume V ¼ i Ai Xi : The filling-level evolution obeys for the indium layers: dXi =dt ¼ FIn ðXi Xiþ1 Þ=tIn and for the arsenic layers dXi =dt ¼ ðypr =ttr ÞðXi1 Xi Þ ðXi Xiþ1 Þ=tAs : This set of two coupled rate equations considers the capping of atoms in a certain layer by those in a layer over it. That means, atoms from layer i can only desorb once enough atoms from layer i þ 1 have already been removed. Furthermore, a precursor-state for the impinging As4 molecules is included. A model which accounts for direct incorporation of the As molecules without precursor-state yields a linear dependence of tD on the arsenic flux. This is not in agreement with the experimental finding of a nearly saturated tD at higher values of FAs (Fig. 3). Precursor states are well established in literature for description of
221
arsenic incorporation on GaAs surfaces [8]. As a simplification, we assume that the precursor state is populated with arsenic atoms and that every impinging As4 molecule produces two atoms according to the maximum sticking coefficient of 0.5. Arsenic atoms from the precursor state can either desorb after the lifetime tpr or transit into the chemisorbed state prior to incorporation after the lifetime ttr : In equilibrium, the coverage of the precursor layer is thus given by ypr ¼ FAs =ðt1 pr þ FAs Þ: The precursor layer related lifetimes are characterized by the energy barriers Epr ¼ 3:57 eV and Etr ¼ 3:24 eV. These values are determined by a fit to the flux dependent data of the inset in Fig. 3 assuming equal prefactors for all processes. With the layer-based model we calculate tD in the post-growth regime with a finite arsenic flux FAs > 0 but in the absence of an indium flux FIn : As described above, tD represents the time up to the reduction of the pyramid height down to hc ¼ 3:0 ML. Values of tD calculated for FAs ¼ 0 qualitatively reproduce the experiments and agree nearly exactly with the above continuum model (Fig. 3). As an additional important validation for the model, we calculate the temperature dependence of tD at a fixed arsenic flux FAs ¼ 4:8 ML/s using the same set of parameters. The very good agreement between the calculated and the measured lifetimes, visible in Fig. 3, demonstrates the ability of our model to describe the central desorption mechanisms during the post-growth regime. In the next step, we apply the layer-based model to calculate the desorption process during growth (FIn > 0). For that, the initial conditions are changed to Xi ðt ¼ 0Þ ¼ 0: We determine the indium sticking coefficient aIn ¼ yIn =ðFIn tÞ from the calculated indium coverage yIn for t ¼ 25 s deposition with FIn ¼ 0:04 ML/s. The wetting layer is neglected in this approach. Fig. 4 shows values of aIn calculated under variation of T for different values of FAs : In all cases, we find aIn E1 up to a certain temperature, followed by a strong decrease. A higher arsenic flux increases this transition temperature. A reduced sticking coefficient lengthens the time that is required for the deposition of a certain amount of material according to tc ¼ t0 =aIn ; where FIn t0 gives the film
C. Heyn, W. Hansen / Journal of Crystal Growth 251 (2003) 218–222
222
1.0 0.8
α In
0.2
L/s 40 M
FAs =
0.4
4 ML/s L/s 0.4 M ML/s 0.04
0.6
FIn = 0.04 ML/s
0.0
[2], desorption of both In as well as of Ga might be of relevance. However, since desorption from pure GaAs surfaces is observed only at temperatures above 7501C [9], we assume that In desorption is dominant. This means that desorption results in two effects. First the volume of the quantum dots is reduced and second the dots become more Garich.
440 460 480 500 520 540 560 Temperature (˚C) Fig. 4. Calculated indium sticking coefficient aIn as function of the temperature at different values of the arsenic flux.
Acknowledgements
thickness at which dot formation occurs in the case of aIn ¼ 1: This fact explains the experimental temperature dependence of tc at growth temperatures higher than 5201C. Using t0 ¼ 45 s, we find good agreement as is demonstrated in Fig. 1.
The authors would like to thank S. Mendach for helpful discussions, S. Schnull . for operation of the MBE system, C. Weichsel for AFM measurements, and the ‘‘Deutsche Forschungsgemeinschaft’’ for financial support via HA 2042/3 and via SFB 508.
4. Conclusions In this work, we study the desorption of selfassembled InAs quantum dots. We find that, dependent on the arsenic flux, desorption takes place even at common growth temperatures. A continuum-based model is applied to describe the InAs dot desorption in absence of a beam flux. In the presence of an As flux, a layer-based model and a As precursor-state describe the data very well. The In sticking coefficient is calculated for dot growth in presence of In and As fluxes. Using these values, we can well reproduce the experimental temperature dependence of the critical time up to quantum dot formation. Considering a significant Ga content in the dots due to intermixing at higher growth temperatures
References [1] Ch. Heyn, Phys. Rev. B. 64 (2001) 165306. [2] P.B. Joyce, T.J. Krzyzewski, G.R. Bell, B.A. Joyce, T.S. Jones, Phys. Rev. B 58 (1998) R15981. [3] Ch. Heyn, M. Harsdorff, Appl. Surf. Sci. 100/101 (1996) 494. [4] Ch. Heyn, D. Endler, K. Zhang, W. Hansen, J. Crystal Growth 210 (2000) 421. [5] H. Lee, R. Lowe-Webb, W. Yang, P.C. Sercel, Appl. Phys. Lett. 72 (1998) 812. [6] J. Marquez, L. Geelhaar, K. Jacobi, Appl. Phys. Lett. 78 (2001) 2309. [7] C. Sasaoka, Y. Kato, A. Usui, Appl. Phys. Lett. 62 (1993) 2338. [8] C.T. Foxon, B.A. Joyce, Surf. Sci. 50 (1975) 434. [9] N. Sugiyama, T. Isu, J. Katayama, Jpn. J. Appl. Phys. 28 (1989) L287.