Growth Simulation and Actual MBE Growth of Triangular GaAs Nanowires on Patterned (111)B Substrates

Attempts were made to further elaborate our experimental growth method and the theoretical growth simulation method for formation of AlGaAs/GaAs QWRs on the (111)B substrates, paying attention to Al composition dependence of growth. A series of repeated growth experiments were carried out on simple one-sided mesa patterns, and from their analysis of the results led to determination of parameter values needed for computer simulation based on the continuum model. The experimental evolution of the cross-sectional structures was well reproduced by simulation, not only on one-side mesa, but also on mesa stripes actually used for wire growth. Finally, an optimum growth design was derived for growth of an array of GaAs triangular QWRs with 40 nm base width on GaAs (111)B substrate by the simulation, and the actual growth experiment confirmed its realization. [DOI: 10.1380/ejssnt.2006.19]


I. INTRODUCTION
Recently, intensive research efforts have been made on semiconductor quantum devices such as single electron transistors and quantum wire transistors. For the realization of large scale integrated circuits using such devices, it is necessary to form networks of high quality and highly uniform quantum structure in a size-and positioncontrolled fashion.
Selective molecular beam epitaxy (MBE) and metal organic vapor phase epitaxy (MOVPE) technique of III-V semiconductors on pre-patterned or masked substrates are two most promising technique for formation of positionand size-controlled arrays of quantum wires (QWRs) and quantum dots (QDs) [1][2][3][4]. Recently, we have reported that < 112 >-oriented QWRs and related network structures can be successfully formed for InP- [5] and GaAsbased heterostructure systems [6][7][8] on (111) patterned substrates by a selective MBE growth. However, growth on non-planar substrates usually involves various highindex facets simultaneously, and this complicates growth kinetics and growth control [9]. Thus, for precise control of wire cross-section and feature sizes, we need a quantitative modeling of the growth process based on understanding of the growth mechanism.
Regarding this point, a large number of efforts on numerical modeling of the crystal growth have been reported so far, not only on epitaxial growth on planar substrates [10][11][12][13], but also on growth on non-planar substrates [14][15][16][17][18]. For the latter, use of diffusion equations under the continuum approximation with phenomenological macroscopic parameters such as diffusion constants, migration length and incorporation rates has become a standard approach, because it can reproduce evolution of complex growth profiles of micron-meter sized structures qualita-tively or semi-quantitatively, as first indicated by a pioneering work by M. Ohtsuka and Miyazawa [14]. However, comparisons with experiments were poor in the previous works. Thus, it is not clear whether a suitable model can quantitatively reproduce the experimentally observed evolution of cross-sectional features of nanometer-sized quantum structures so that it can be utilized for design and control of growth of nanostructures. From such a view point, we have recently carried out a series of works on detailed comparison between growth experiments and the continuum modeling for selective MBE growth of GaAs QWRs on (001) and (111)B patterned substrates [19,20]. As a result, we established the basis of realistic simulation procedure by which actual growth can be predicted, designed and controlled. Our background motivation is to develop the growth method and the growth simulation method that can be applied for future growth of high density hexagonal nanowire networks which can be used for fabrication of quantum LSIs based on the hexagonal binary decision diagram (BDD) quantum circuit approach [21,22].
On the basis of our previous work, the purpose of this paper is to further elaborate the experimental growth method and the theoretical growth simulation method for realization of AlGaAs/GaAs narrow QWRs with controlled shape, size and position on the (111)B substrates. The reason why we pay a concentrated attention on growth on the (111)B orientation is that it is more suited than the (001) orientation in realizing uniform BDD node devices which have basic three-fold symmetry.
Specifically, we first carried out in this study a series of growth experiments to clarify the Al composition dependence of growth which we have not investigated previously. Then, attempts are made to reproduce experiments by computer simulation. Finally, we try to grow an array of GaAs triangular QWRs on GaAs (111)B substrate, using the conditions indicated by simulation. Such QWRs with triangular wire cross-section seem to be very much suited for tight gate control of QWRs by a nanometersized Schottky wrap gate (WPG).

II. EXPERIMENTAL
The sequence for selective MBE growth of GaAs QWRs used in this study is schematically shown in Figs. 1(a) and (b). As a template for selective MBE growth, array of < 112 >-oriented mesa stripe shown at the left of Fig. 1(a) was formed on semi-insulating (111)B GaAs substrates. To get basic information on growth by repeated growth, a one-sided mesa pattern shown at the right of Fig. 1(a) was also used. Patterns were prepared by the electron beam lithography and wet chemical etching.
Before loading into the MBE chamber, the patterned substrate was cleaned by acetone and ethanol with ultrasonic agitation, and then a light chemical etching was subsequently applied in the atmosphere, using a Semicoclean (Furuuchi Chemical Co., Ltd., Tokyo, Japan) solution. After loading into the MBE chamber, thermal cleaning was applied just prior to growth at a substrate temperature of 640 • C and under an arsenic pressure, monitoring the reflection high energy electron diffraction patterns.
The material supply sequences with growth time t g shown in Fig. 1(b) were used in this study for QWR growth and for the repeated growth of GaAs/AlGaAs layers. The latter was used to get basic growth information. As the standard growth conditions, growth temperature T sub was set to be 680 • C for (111)B substrates, and the V/III ratio was set to be 10 for all the experiments reported here. With these, a GaAs buffer layer was grown first on the patterned substrate in order to prepare a growth template for subsequent selective growth. This buffer growth on < 112 >-oriented mesa stripes on the (111)B substrates led to formation of a GaAs mesa structure defined by top (111)B and side (512) facets [23], as shown in Fig. 2(a). In this study, the buffer mesa width, W 0 , was changed from 200 nm to 1200 nm.
Then, growth of an AlGaAs/GaAs/AlGaAs sandwich layer on this buffer template led to formation of embedded trapezoidal GaAs QWRs on the top (111)B facets of an AlGaAs mesa structure with a reduced lateral wire width, W , as shown on Fig. 2(b). The Al composition, X, in Al X Ga 1−X As layer was changed from X = 0 to X = 0.5 in order to investigate the Al composition dependence on the various features of selective growth. Using the material supply sequence shown on the right of Fig. 1(b), repeated growth of wires on the same patterned substrate was also carried out in order to clarify evolution of complex cross-sectional features during growth and to extract characteristic parameter values necessary for simulation.

A. Selective MBE growth experiments
The growth experiments of single quantum wires confirmed the successful selective growth of quantum wire arrays for various values of the Al composition, and their cross-section was trapezoidal in agreement with in Fig.  2 Then, a series of repeated growth experiments were carried out on simple basic one-sided mesa step pattern. Resultant cross-sectional SEM images of the two samples grown on a (111)B substrate are shown in Figs. 3 (a) and (b). Here, Fig. 3(a) is for the sample obtained after repeated growth of GaAs(70 nm)/ AlAs(10 nm) layers where the AlAs layers were used as markers, and Fig.  3(b) is for the sample obtained after repeated growth of Al 0.3 Ga 0.7 As(100 nm)/ GaAs(10 nm) layers where GaAs layers were used as makers, respectively. Both samples are after a light stain etching of the cleaved cross-section. In both samples, the boundary separating the growth region between the top and the side facets can be clearly identified after repeated growth. We call this boundary the facet boundary (FB) as we did in our previous study [20]. Apparently, two FBs determine the lateral size of the bottom width of QWRs formed selectively on the mesa top. Thus, the facet boundary angle, θ b , with respect to the flat (111)B plane, is an important parameter in controlling the wire width by the present selective growth technique.
As seen in Figs. 3(a) and (b), the facet boundary angle keeps a constant value in the initial stage of selective growth. However, after the growth of the 3 or 4 layers, the boundary angles start to increase gradually and the facet boundaries become curved for both samples. It is e-Journal of Surface Science and Nanotechnology also seen that the value of θ b depends on the Al composition. Such complicated growth features are obviously correlated with the growth kinetics including the atom migration and incorporation on the grown surfaces. Figure 4 shows the plots of the facet boundary angle, θ b , measured from straight portions of FBs at the initial growth stage, as a function of the Al composition, X. The solid and dashed lines in Fig. 4 are simulated values explained later. As shown in Fig. 4, it was found that facet boundary angle changes significantly with the Al composition, increasing from 63.5 • up to 75 • for Al composition increase from 0 to 0.5.

Basic equations
It is obviously very difficult to describe the present complex behavior of the evolution of cross-sectional structures by a simple analytical equation. As a means of quantitative theoretical description, a computer simulation method which our group recently developed [19,20] was applied in an attempt to reproduce the present experimental results. This computer simulation was based on a phenomenological continuum growth model in which the growth process is described by a diffusion equation with macroscopic parameters such as diffusion constant and lifetime of adsorbed adatoms on growing surfaces.
In this modeling, the surface density of group III adatoms, n(x, t g ), at the lateral position, x, and the growth time, t g , is assumed to satisfy the following phenomenological equation: where n is the adatom density, G is the incoming molecular beam flux, and J is the surface diffusion flux of adatoms. τ (θ), D and U are the lifetime until incorporation, surface diffusion coefficient and chemical potential of adatoms on a facet, respectively. Here, θ?is the angle of the slope of the growing surface at x with respect to the (111)B plane, and we assume that the life time depend strongly on the surface slope, θ, indicated as τ (θ). In fact, the θ?dependence of τ (θ) gives appearance of new facets during growth as well as the growth selectivity between the neighboring facets. After the calculation of the adatom density, n(x, t g ) as a function of growth time, t g , the cross-sectional growth profile is obtained by plotting the vertical growth thickness, T (x, t g ), which is represented by the following equation, as a function of lateral position, x, and growth time, t g .
When there are more than one species of group III adatoms as in the case of growth of AlGaAs, Eq. (2) was solved separately for each species, and each contribution was added together in calculating Eq. (3).

Determination of parameter values for simulation
It is obvious that the surface lifetime, τ , and the diffusion coefficient, D, of group III adatoms in Eq. (1) and Eq. (2) are the two important parameters that determine the growth features. ?n our previous work [20], we have shown that the following theoretical formula for τ (θ) obtained by Ohtsuka, et al. [14,15] by solving a diffusion equation on a terrace with steps, gives satisfactory results in reproducing experiments. Therefore, this equation was used here again for simulation.
where τ (0) is the lifetime of adatoms at the singular surface appeared during the selective growth process. λ 0 is the diffusion length of adatoms defined as λ 0 = (τ (0) · D) 1/2 using the diffusion coefficient, D. The δ ± is the parameter related to the step densities, and it was used as the variable parameter for the fitting with the experiments. To use the above equations, we have to know which facets appear during growth and what are the values of τ (0) on the singular surfaces of these facets. For these, we know from our previous experience [20] that growth on (111)B substrates were characterized by the appearance of specific facets of (513) and (512) planes in addition to the initial (111)B plane. Based on the previous results, the relation of lifetime, τ 111B , τ 513 , and τ 5−12 , respectively on (111)B, (513) and (512) planes was set to be τ 111B : τ 512 : τ 5−13 = 1 : 0.9 : 2.5 with τ 111B = 1.5 sec.
As for the diffusion constant, it was assumed that D is strongly dependent on the temperature, T sub , as in the following equation.
where, E d is the activation energy of surface adatoms diffusion. In our previous papers [20], the value of D was estimated from the repeated growth experiments at various growth temperatures and the quantitative fitting of their growth profiles. Here, the values of D 0 = 1.0 × 10 −6 m 2 /s and E d = 1.2 eV were used for the growth of GaAs on the (111)B plane, namely the D(Ga) was set to be 4.45 × 10 −13 m 2 /s at T sub = 680 • . As for the Al composition dependence of D, the experimental results in the growth profile obtained in Figs. 3 (a) and (b) and the measured boundary angle shown in Fig.  4 suggest that the migration length of adatoms defined by λ = (Dτ ) 1/2 are considerably different between Ga and Al. In order to reproduce the growth profiles obtained in the present experiments, the calculation were carried out by changing the adatom migration length, λ(Ga) and λ(Al), systematically. The lines in Fig. 4 show results of simulation carried out to reproduce the experimentally observed dependence of the facet boundary angle, θ b on Al composition, X, for various values of γ defined by γ = λ(Ga)/λ(Al). From the repeated calculation, we found that the case of λ(Ga)= 10 2 λ(Al), i.e., the value of D for the Al atom being 10 4 times smaller that that of the Ga atom, gives the best result, in reproducing the experiments on the facet boundary angle shown in Fig. 4.

Reproduction of growth experiments by simulation
By using the above relation of λ(Ga)= 10 2 λ(Al) as well as the values of the lifetime mentioned previously, the growth profiles of AlGaAs layer on the GaAs mesa pattern were calculated. Figures 5(a) and (b) show the simulated growth profiles calculated for the growth on the < 112 >-oriented one-sided mesa on the (111)B substrate. It is seen that the experimental growth profiles well reproduced by the simulation. Namely, the facet boundaries are indeed formed with the same initial angles with experiments, and the boundary angles change with the growth time similarly with experiments.
It is seen both experimentally and theoretically that the top width of the growing AlGaAs layer first reduces with time, and some time later it starts to increase. This is a very interesting phenomenon caused by time-dependent increase of the face boundary angle, but it is very harmful for realization of narrow wires by selectively growth. In order to obtain narrow, it is important to know the evolution of the facet boundary angle with time, so that the growth of AlGaAs layer can be stopped before the start of the increase of the boundary angle. Figure 6 shows the measured top mesa width, W, of samples grown on three kind of buffer mesa stripes having the initial width, W 0 , of 350 and 740 and 1180 nm for a fixed Al composition of X = 0.3. For this, repeated growth experiments on mesa stripes were carried out.
The solid lines in Fig. 6 show the calculated results. The experimental results on mesa stripes, which are actually used for QWR growth, were well reproduced by the theoretical curves obtained in the present simulation. Thus, the present simulation is very powerful in predicting the evolution of growth behavior with time. Furthermore, it was found that the growth on a buffer mesa having a narrow initial width is advantageous for obtaining narrow QWRs, because the linear relation between the wire width and the growth time continues for a longer growth time, as seen in Fig. 6. In any case, the above results also indicate that the wire width formed on the top (111)B facet of the mesa can be kinetically controlled by the growth conditions and the growth time of the AlGaAs layer prior the start of the wire growth.

C. Growth design and actual growth of narrow triangular wire arrays
The present growth generally gives QWRs with a trapezoidal cross-section when the linearly width-reducing mode with time is used for the growth of both to the bottom AlGaAs layer and GaAs QWR layer. However, it should be possible to grow QWRs with a triangular cross-section by adjusting the growth conditions. Such triangular shaped QWRs seem to be ideally suitable for quantum devices with tight gate control by making the top AlGaAs layer thin and putting a Schottky wrap gate on the top of the barrier layer.
For this reason, the growth process to realize an array of triangular QWRs with a lateral base width of 40 nm, embedded in Al 0.3 Ga 0.7 As barrier layers, was designed for the growth on a GaAs buffer mesa with a top width W 0 = 200 nm. The growth temperature and time were varied to have the desired wire cross-section. The crosssectional structure of the wire obtained on computer by the optimal growth design is shown in Fig. 7(a). From the simulation result, the shape of QWRs expected to become a triangular one, having a side plane angle of 50 • ∼ 53 • corresponding to the angle of the (512) facet. Then, growth was actually carried out by using the optimum growth design on an initial buffer mesa with W 0 = 200 nm. Figure 7(b) shows the cross-sectional SEM images of the actually grown sample. The lateral wire width measured in the cross-sectional SEM image was about 40 nm which is very consistent with the theoretical growth profile shown in Fig. 7(a).
From these results, it can be concluded that the present simulation method with established parameter values is very powerful for the design of the shape, position and size of QWRs selectively grown on the patterned (111)B substrates. By further optimization of the growth conditions with the use of the present simulation, growth of QWRs with much narrower wire widths seems to be feasible.

IV. CONCLUSIONS
Continuing from our previous works, attempts were made in this paper to further elaborate the experimental growth method and the theoretical growth simulation method for formation of AlGaAs/GaAs QWRs on the (111)B substrates in view of future application to hexagonal BDD quantum LSIs. A series of repeated growth experiments were carried out on one-sided mesa patterns to clarify the Al composition dependence of growth which we have not investigated previously. Using values of lifetime and diffusion constant determined from comparison between theory and experiment on growth on the onesided mesa pattern, experimentally observed evolution of the cross-section of the grown structures was very well reproduced by computer simulation based on the continuum model. Finally, an optimum growth design was derived for growth of an array of GaAs triangular QWRs with 40 nm base width on GaAs (111)B substrate by the simulation, and the actual growth experiment confirmed its realization. Such QWRs with triangular wire cross-section seem to be very much suited for tight gate control of QWRs by a nanometer-sized Schottky wrap gate.