x-ray scattering on semiconductor nanostructures: gamnas€¦ · x-ray scattering on semiconductor...

X-ray Scattering on Semiconductor Nanostructures:

GaMnAs

J. Matejova, V. Holy, L. Horak

Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic

Abstract. GaMnAs is a diluted magnetic semiconductor material with possible application

in spinotronics. Developing such a material, one encounters problems due to undesired

defects in crystal lattice of GaMnAs: interstitial Mn ions. While searching for the way

to remove them, we treat nanostructured samples (substrate, layer and nanowires on top)

that are strained (including lateral strain). The strain is caused by lattice misfit between

substrate, layer and nanowires and by the stress due to the local increase of GaMnAs

lattice parameter in presence of interstitials. X-ray diffraction is a powerful method for

investigating such strained samples. We carried out a theoretical computation of reciprocal

space maps for these samples and compared with experimental data. By fitting, we yielded

an estimate of the physical constants important for description of the processes occuring in

GaMnAs while removing the interstitials.

Introduction

Nowadays, design of new electronic devices is of increasing interest, which is interconnected with

new requirements for materials. Due to the trend of miniaturization, nanostructured samples are being

treated. One of the most powerful methods to investigate such samples is X-ray diffraction, which is

at once undestructive and very sensitive for strain in crystal lattice [e.g. Pietsch et al., 2004]. GaMnAs

belongs to the group of diluted magnetic semiconductors (DMS), which are expected to be applied in

spintronics [e.g. Jungwirth et al., 2006]. The challenge is to reach magnetic ordering at room temperature

[e.g. Ohno, 1998].

There are two types of Mn atoms in the GaMnAs lattice: 1. substitutional Mn ions on Ga sites

(MnGa), which act as acceptors and provide local magnetic moments, and 2. interstitial Mn ions (Mni),

which act as double donors and destroy ferromagnetism.

Because the solubility of Mn is low, MnAs precipitates appear at usual growth temperatures. To

avoid this, GaMnAs layers are grown at low temperatures (around 200

C). However, due to the low

growth temperature, AsGa antisites and Mni appear. The latter suppress magnetic ordering, so reduction

of Mni rate is needed [e.g. Edmonds et al., 2004].

There is an evidence, that the post-growth annealing reduces Mni rate [Holy et al., 2006]. Supposed

mechanism for removing the interstitials during the annealing is their diffusion towards the surface and

surface recombination. Nevertheless, the annealing reduces Mni rate only in air (not in vacuum), so the

detailed explanation of the processes on the GaMnAs layer surface is a matter of interest. Moreover, the

knowledge of Mni concentration profile in subsequent phases of the post-growth annealing is important

for a variety of physical problems concerning GaMnAs.

Samples

To investigate the process of Mni outdiffusion, a layer of GaMnAs was grown on a GaAs substrate.

To approve, that the outdiffusion occurs only at the free surface, stripes of GaAs were periodically placed

on the top of the layer. There appears lateral gradient of concentration in such a sample and the crystal

lattice is therefore laterally strained, which can be suitably investigated by reciprocal space mapping

(RMS). Exact parameters of the layer are listed in Tab. 2, whereas the shape of the sample is sketched

in Fig. 1. The measurement has been carried out at two subsequent phases of the annealing with the

same sample – see Tab. 1.

Table 1.: Description of the samples.

D065#7 annealed at 200

C for 20 minutes

D065#7a2 annealed at 200

C for another 3 hours

78

WDS'10 Proceedings of Contributed Papers, Part III, 78–85, 2010. ISBN 978-80-7378-141-5 © MATFYZPRESS

MATEJOVA ET AL.: X-RAY SCATTERING ON SEMICONDUCTOR NANOSTRUCTURES

Table 2.: Parameters of sample D065#7.

thickness of GaMnAs layer 50 nm

lateral period 3.2 µm

lateral width of GaAs stripes 1.6 µm

thickness of GaAs stripes 10 nm

total rate of Mn ions 6.5%

Figure 1.: A sketch of sample D065#7.

Diffusion and surface recombination

The diffusion process is described by drift-diffusion equations [Olejnik et al., 2007] for concentrations

n and p of Mn2+

ions and holes (eq. (1) and eq. (2) respectively).

∂n

∂t

= ∇ · (Dn∇n + 2µnn∇φ), (1)

∂p

∂t

= ∇ · (Dp∇p + µpp∇φ), (2)

where dn, Dp are the diffusion constants and µn, µp are the mobilities of Mn2+

ions and holes respectively.

The diffusion constants are related to the mobilities by the Einstein relation (eq. (3).)

Dx =kBT

e

µx, (3)

where kB is the Boltzmann constant, T is a temperature, e is the elementary charge and x = n, p. The

electric potential φ is given by the Poison equation (eq. (4)) [Olejnik et al., 2007].

∇2φ = (cs − p − 2n)

e

ε

, (4)

where cs is a concentration of MnGa and ε is the permitivity of GaMnAs.

The surface recombination is approached by a container, that is gradually filled by Mn interstitials

[Olejnik et al., 2007]. The formula for the flux jnS of Mni at the free surface is given by eq. (5.)

jnS = S0

(

1 −NS(t)

NSmax

)

nS(t), (5)

where S0 is the surface recombination rate, NSmax is the maximal number of Mni, that can be trapped

in the container, and NS(t) is the actual number of Mni in the container.

The set of equations eq. (1) – eq. (5) was solved by FEM method (Comsol Multiphysics). The results

for parameters given by Tab. 3 are depicted in Fig.2. Vertical cross-section of the concentration profile

at different times is shown in Fig. 3. The concentration of Mn interstitials in both figures is normalized

to the concentration of substitutional Mn and the time unit δt is so far unknown because of unknown

Dn.

Strain field computation

The strain field was computed by FEM method (Comsol Multiphysics again).The tensor of elastic

constants for the Hook’s law was set to the values according to [NSM] for GaAs. The initial strain for

79


x [µ m]

z [µ

m]

Concentration profile n(x,z) at time step t=100

−3 −2 −1 0 1 2 30.04

0.05

0.06

0.2

0.4

(a) t = 100 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 30.04

0.05

0.06

0

0.2

0.4

(b) t = 400 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 30.04

0.05

0.06

0.2

0.4

(c) t = 800 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 30.04

0.05

0.06

0

0.2

0.4

(d) t = 2400 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 30.04

0.05

0.06

0

0.2

0.4

(e) t = 4400 · δt

Figure 2.: Theoretical concentration profile of Mn interstitials at different phases of the annealing for

the simulation parameters given by Tab. 3.

Figure 3.: Vertical cross section of the concentration profile for simulation parameters given by Tab. 3.

the computation was set according to [Masek et al., 2003], whence the lattice constant of GaMnAs is

given by formula eq. (6).

a(cs, ci, ca) = a0 + 0.02cs + 1.05ci + 0.69ca [A], (6)

where cs, ci and ca are relative concentrations of substitutional Mn ions, interstitial Mn ions and AsGa

antisites, respectively.

Table 3.: Set of the simulation parameters, for which are the concentration profiles and the displacement

fields depicted in the present study. Space unit δx = 5 A and time unit δt is going to be obtained by

fitting.

n0 0.4

S0 5 × 103 · δx

δt

NSmax 104

µp

µn10

8

Dn 1 · δx2

δt

80


x [µ m]

z [µ

m]

Displacement ux(x,z) [Angstroem] at time step t=100

−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

−0.1

−0.05

0

0.05

0.1

(a) t = 100 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(b) t = 800 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

(c) t = 4400 · δt

Figure 4.: Theoretical displacement ux(x, z) at different phases of the annealing for the simulation

parameters given by Tab. 3.

x [µ m]

z [µ

m]

Displacement uz(x,z) [Angstroem] at time step t=100

−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(a) t = 100 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(b) t = 800 · δt

x [µ m]

z [µ

m]


−3 −2 −1 0 1 2 3−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(c) t = 4400 · δt

Figure 5.: Theoretical displacement uz(x, z) at different phases of the annealing for simulation param-

eters given by Tab. 3.

The computed equilibrium displacement ux(x, z) in lateral direction and uz(x, z) in vertical direction

for the parameters given by Tab. 3 are shown in Figures 4 and 5, respectively. Again, different times of

the simulation are depicted.

Reciprocal space mapping (RSM)

Theory

According to [Pietsch et al., 2004] we derived a formula for diffraction intensity map (see eq. (7)

and eq. (8).)

I~h(Qx, Qz) ∼

∑

Hx

δ (Qx − hx − Hx)

∣

∣

∣

∣

∫

dzξ~hHx(z)e

−i(Qz−hz)z

∣

∣

∣

∣

2

, (7)

where ~h = (hx, hy, hz) =

2πa

(h, k, l) are indices of a particular diffraction, Hx =2πD

is a reciprocal grating

vector and D is a period of the grating.

ξ~hHx(z) ∼

∫ D/2

−D/2

dxχ~h(x, z)e

−i(hxux+hzuz)e−iHxx

, (8)

where ux, uz are components of a displacement field ~u(x, z) and χ~h(x, z) = χ~h

Ω(x, z), where χ~his a

susceptibility of a material, corresponding with a diffraction ~h, and Ω(x, z) is the shape function of the

sample.

81


Experiment

The measurement on the samples D065#7 and D065#7a2 was executed at ID10B beamline in

ESRF, Grenoble. C(111) double crystal monochromator was set, so as the primary beam energy was

E = 7.59 keV. The primary beam cross section was determined by a pair of slits in front of the sample

stage to be 0.5 × 0.3 mm2. The sample was placed into evacuated Be dome to avoid oxidation of the

sample. 2D RSMs were measured in the vicinity of the symmetric 004 and asymmetric 224 reciprocal

lattice points. The linear positional sensitive detector (PSD) Vantec was employed for the experiment.

Results

For our computation, the diffusion constants Dn and Dp had been unknown as well as the surface

recombination rate S0. That’s why the time t in the simulation is given as a multiple of δt unit. If we fit

the difraction map of the sample at a known phase of the annealing to the computed map at a specific

time t, we can find out, what is the real value of the time unit δt. The diffusion constants Dn and Dp

could be determined from this value.

Every unknown quantities have to be fitted. Their list is in Tab. 4.

Table 4.: A list of unknown quantities to be fit.

time units δt

surface recombination rate S0

diffusion constant of Mni Dn

ratio of diffusion constants kp = Dp/Dn

capacity of the surface container NSmax

initial rate of Mn interstitials n0

Fitting D065#7a2 data as first. We started with fitting difraction map 224 for D065#7a2. The

best fit with group of parameters B at time t = 500 · δt (see Tab. 5) is shown in Fig. 6a.

Table 5.: Parametres of the best fits. Space units: δx = 5 A, time units δt, δt∗

are going to be

determined in following text. n0 is normalized to the concentration cs of MnGa.

A: S0 = 5 × 103 · δx

δtNSmax = 10

4n0 = 0.45 µp/µn = 10

8

B: S0 = 5 × 103 · δx

δtNSmax = 10

4n0 = 0.41 µp/µn = 10

8

We know, that D065#7a2 had been annealed for 3 hours and 20 minutes, so the real time of the

annealing is treal = 12×103

s, whereas the time of the annealing determined from the fit is tfit ≈ 500 ·δt.

Now we can find out the value of the time unit δt (because treal should be equal to tfit): δt = 24 s.

Consequenly, the estimate of the diffusion constants for GaMnAs with 6.5% Mn is Dn ≈ 10−20

m2s−1

,

Dp ≈ 10−12

m2s−1

. The surface recombination rate is then S0 ≈ 103

As−1

.

For these values of Dn, Dp and S0, the initial rate of the Mn interstitials is n0 = 0.41.

From the known time unit δt, we can count the time t′

of the simulation, corresponding to the time

after the first annealing (that is to the diffraction map for D065#7). Then t′

= 55 · δt. For parameters

B (Tab. 5) and time t′

= 55 · δt we computed a diffraction map 224, which can be seen in Fig. 6b. The

theoretical map evidently doesn’t agree with the experimental map.

Fitting D065#7 data as first. Let’s try to start from the other end of the problem. If we fit

the D065#7 data (before the 2nd annealing) as first, we get the best agreement with the theory for

parameters A and time t = 400 · δt∗ (see Tab. 5). This fit is shown in Fig. 7a.

Analogically, we can derive the time unit δt∗

: t∗

real = 1200 s, t∗

fit = 400 · δt∗

, soδt∗ ≈ 2.2 s and

therefore the diffusion constants and the surface recombination rate are D∗

n ≈ 10−19

, D∗

p ≈ 10−11

,

S∗

0 ≈ 104

As−1

.

Using the time unit δt∗

, we can compute the simulation time t∗′

corresponding to the time after the

2nd annealing: t∗′

= 4000 · δt∗. For parametres A (Tab. 5) and time t∗′

= 4000 · δt∗, we can compute

the diffraction map 224 and compare with the experimental data for D065#7a2, which is depicted in

Fig. 7b. Again, the experiment doesn’t agree with the theory.

82


(a) (b)

Figure 6.: Diffraction map 224 for parametres B (see Tab. 5) and time t = 500 · δt (Fig. 6a) and

t′

= 55 · δt (Fig. 6b) in comparison with experimental data D065#7.

Discussion

Let’s analyse the disagreement between the experimental data and the computed results. According

to eq. (6), the change ∆a of the lattice constant due to the Mn ions, normalized to the MnGa concentration

cs, is given by eq. (9.)

∆a(x, z, t)

cs

= 0.02 + 1.05n0 − 1.05∆ci(x, z, t), (9)

where ∆ci(x, z, t) = ci(x, z, t) − n0 and n0 is initial homogeneous concentration of Mni in the layer and

ci(x, z, t) is interstitial concentration profile for coordinates x, z as a function of time t. If we choose a

particular point (x, z), then we can plot∆acs

as a function of ∆ci(t), while ∆ci(t) time dependence is

unknown (that is what we get from the simulation of the diffusion).

In Fig. 8, there is shown an illustration of the time dependence of∆acs

(∆ci(t)) for ∆ci as an arbitrary

function ∆ci(t) of the time. The important fact is, that for varying n0, we get graphs of the function∆acs

(t), which are only shifted in vertical direction relatively to each other (see eq. (9).) So for different

n0 we get some kind of iso-lines (see Fig. 8, the dashed lines).

At the dashed lines, there are three triangular markers. The first one is the supposed initial state

(t = 0, homogeneous concentration and therefore ∆ci = 0), the second one is for the measurement after

the first annealing (D065#7) at real time t∗

= 1200 s with the initial rate of Mni fitted to be n0 = 0.45

and the time fitted to be t∗

= 400 · δt (see parameters A, Tab. 5).

The third marker is for the measurement after the second annealing (D065#7a2) at real time t∗ =

12000 s with the initial rate of Mni fitted to be n0 = 0.41 and the time fitted to be t = 500 · δt (see

parameters B, Tab. 5).

According to the eq. (6) [Masek et al., 2003], the quantity∆acs

should trace any of the dashed iso-lines

in Fig. 8 during the annealing.

However, according to our experiment, the quantity∆acs

should trace the solid line in Fig. 8 during

the annealing. This implies, that to explain our experiment properly, eq. (6) from Masek et al. [2003]

must be generalized.

83


(a) (b)

Figure 7.: Diffraction map 224 for parametres A (see Tab. 5) and time t = 400 · δt∗

(Fig. 7a) and

t∗′

= 4000 · δt∗ (Fig. 7b) in comparison with experimental data D065#7a2.

∆ci(t)

∆ci(t=0)

∆ci(t=500 δt)

n0=0.45

n0=0.41

∆ci(t=400 δt)

∆ a/ cs

Figure 8.: Illustration for analysis of the results.

Conclusion

As an approach to outdiffusion of Mni in GaMnAs, we used model defined by eq. (1) – eq. (5). For

the simulation of the strain field, we used formula Masek et al. [2003]. By comparison of the computed

and the measured RSMs, we got ambivalent results. Anyway, we yielded rough estimates of the diffusion

constants Dn and Dp and of the surface recombination rate S0 (see previous text). The estimated values

are in following ranges: Dn = 〈10−20

; 10−19〉m

2s−1

, Dp = 〈10−12

; 10−11〉m

2s−1

, S0 = 〈103; 10

4〉 As−1

.

In section Discussion, we sketched an analysis of using formula from Masek et al. [2003]. This analysis

implies, that the formula eq. (6) should be generalized, so as the lattice constant of GaMnAs be a

nonlinear function of ci. A more detailed experiment must be carried out for such a generalization.

84


Acknowledgments. This work is a part of the research program MSM0021620834 that is financed

by the Ministry of Education of the Czech Republic, the work has been also supported by the EU FP7

Project NAMASTE (contract No. 214499). We acknowledge the support of the staff of the beamline

ID10B at ESRF Grenoble.

References

Edmonds, K., Bogus lawski, P., Wang, K., Campion, R., Novikov, S., Farley, N., Gallagher, B., Foxon,

C., Sawicki, M., Dietl, T., et al., Mn Interstitial Diffusion in (Ga, Mn) As, Physical review letters, 92 ,

37 201, 2004.

Holy, V., Matej, Z., Pacherova, O., Novak, V., Cukr, M., Olejnik, K., and Jungwirth, T., Mn incorpo-

ration in as-grown and annealed (Ga, Mn) As layers studied by x-ray diffraction and standing-wave

fluorescence, Physical Review B , 74 , 245 205, 2006.

Jungwirth, T., Sinova, J., Masek, J., Kucera, J., and MacDonald, A., Theory of ferromagnetic (III, Mn)

V semiconductors, Reviews of Modern Physics, 78 , 809–864, 2006.

Masek, J., Kudrnovsky, J., and Maca, F., Lattice constant in diluted magnetic semiconductors (Ga, Mn)

As, Physical Review B , 67 , 153 203, 2003.

NSM, NSM archive, http://www.ioffe.rssi.ru/SVA/NSM/Semicond/.

Ohno, H., Making nonmagnetic semiconductors ferromagnetic, Science, 281 , 951, 1998.

Olejnik, K., Novak, V., Cukr, M., Pacherova, O., Matej, Z., Holy, V., and Marysko, M., GaMnAs

annealing under various conditions: air vs. As cap, Physics of Semiconductors; Part B , 893 , 1219–

1220, 2007.

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nanostructures, Springer Verlag, 2004.

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