Strain relaxation of metastable SiGe/Si: Investigation with twocomplementary X-ray techniquesE. Kasper, N. Burle, S. Escoubas, J. Werner, M. Oehme et al. Citation: J. Appl. Phys. 111, 063507 (2012); doi: 10.1063/1.3694037 View online: http://dx.doi.org/10.1063/1.3694037 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i6 Published by the American Institute of Physics. Related ArticlesReflectance anisotropies of compressively strained Si grown on vicinal Si1−xCx(001) Appl. Phys. Lett. 102, 011902 (2013) Dislocation structure in AlN films induced by in situ transmission electron microscope nanoindentation J. Appl. Phys. 112, 093526 (2012) Free volume change of elongated polyethylene films studied using a positron probe microanalyzer Appl. Phys. Lett. 101, 203108 (2012) Strain and defects in Si-doped (Al)GaN epitaxial layers J. Appl. Phys. 112, 093102 (2012) On-chip stress relaxation testing method for freestanding thin film materials Rev. Sci. Instrum. 83, 105004 (2012) Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors

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Strain relaxation of metastable SiGe/Si: Investigation with twocomplementary X-ray techniques

E. Kasper,1,a) N. Burle,2 S. Escoubas,2 J. Werner,1 M. Oehme,1 and K. Lyutovich1

1Institut fur Halbleitertechnik, Universitat Stuttgart, Pfaffenwaldring 47, 70569 Stuttgart, Germany2IM2NP – CNRS 6242, Aix Marseille University, Campus St Jerome, 13397 Marseille cedex 20, France

(Received 9 December 2011; accepted 18 February 2012; published online 20 March 2012)

Metastable and strain relaxed SiGe layers with about 20% Ge content have been grown by

molecular beam epitaxy on Si substrates at 550 �C. The thickness regime of metastability and the

onset of strain relaxation were investigated on dust particle free surfaces obtained by careful

chemical cleaning and epitaxy loading under clean room conditions. Compared to earlier results

true metastable regime without misfit dislocations was obtained up to 140 nm thickness. The onset

of strain relaxation started with heterogeneous nucleation sites of misfit dislocations. X-ray

topography proved to be a unique monitoring tool to observe a low density of single dislocations.

From these results we suggested to define a critical thickness band with lower bound tcl from

dislocation nucleation to an upper bound tco (600 nm in our case) defined by the onset of

considerable strain relaxation. The strain relief was measured by X-ray diffraction (reciprocal

space mapping) and found to be very abrupt (76% strain relaxation at 800 nm thickness). VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.3694037]

I. INTRODUCTION

SiGe (silicon germanium) is lattice mismatched against

the Si substrate commonly used in microelectronics. The lat-

tice mismatch f is nearly linear with Ge content x (Vegard’s1

law: f ¼ 0.0417x). Exactly, a slight parabolic bow (Dismukes

law) is proven for both bulks SiGe (Ref. 2) and epitaxial

SiGe.3

f ¼ 0:036678xþ 0:005032x2: (1)

With x ¼ 0.2, the Vegard’s law gives aSiGe¼ 0.54765 nm and

f ¼ 0.835%, as Dismukes law leads to aSiGe¼ 0.54720 nm and

f ¼ 0.754%.

The lattice mismatch causes strained layer SiGe growth

(pseudomorphic SiGe) up to a critical thickness tc, Thicker

layers start to relax the strain by the introduction of misfit

dislocation networks at the interface. By kinetic reasons

(nucleation of dislocations at surface sites) the interface dis-

location network is connected to the surface by threading

dislocations. These threading dislocations that penetrate

devices in the SiGe layer prohibited a broad use of relaxed

SiGe layers in microelectronics whereas pseudomorphic

SiGe is now in commercial use in all high-end silicon cir-

cuits except memories. SiGe heterobipolar transistor (HBT)

holds speed records (transit frequency of 500 GHz (Refs. 4

and 5)), and it is used in microwave circuits.6 Since the 90

nm node logic complementary metal oxide silicon (CMOS)

circuits use local SiGe source/drain to boost p-channel per-

formance7 by strain.

We study SiGe layers with about 20% Ge content

because at higher mismatch values a third misfit accommo-

dation mechanism set in: Strong corrugation of the surface

or even islanding. The critical thickness tc under thermal

equilibrium is well understood both from an energetic point

of view (Van der Merwe approach8 and from force consider-

ations Blakeslee-Matthews approach9). Many differing equi-

librium curves have their origin in assumptions about

geometry and Burgers vector of the network, about core

energies and outer cutoff radii.

A typical tc value for 0.8% mismatch of SiGe layers is

about 10 nm. Rather high growth temperatures (900 �C) are

necessary to reach the equilibrium.10 At lower temperatures

(750 �C�550 �C) the onset of strain relaxation is delayed to

higher tc value that was impressively shown by a investiga-

tion of People/Bean.11 The pseudomorphic thickness range

between equilibrium curve and the 550 �C curve is often

named metastable regime12 although models of sluggish dis-

location movement favor an explanation as kinetically con-

trolled plastic deformation of the semiconductor on the

threshold from a deformable material at high temperatures to

a brittle material at room temperature. Even the exact value

of the critical thickness seems to depend on the accuracy

of the methods to measure strain and dislocation density.

The used methods include X-ray diffraction, Raman-

spectroscopy, defect etching, luminescence, and transmission

electron microscopy (TEM).

Characteristic for the relaxation process is a start with

heterogeneous nucleation centers on the surface. In this ini-

tial phase the strain relaxation is rather small and hard to

quantify even with X-ray diffraction (XRD). Needed are

methods to observe and analyze low density dislocation net-

works, later the networks are dense and the strain relaxation

is best followed with diffraction. In order to follow all stages

of the process from pseudomorphic to relaxed growth, we

used two complementary X-ray methods: X-ray topogra-

phy(XRT) for strained and low dislocated layers and X-ray

diffraction reciprocal space mapping (RSM) for middle and

high dislocated partially and fully relaxed layer.a)Electronic mail: kasper@iht.uni-stuttgart.de.

0021-8979/2012/111(6)/063507/10/$30.00 VC 2012 American Institute of Physics111, 063507-1

JOURNAL OF APPLIED PHYSICS 111, 063507 (2012)

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The classical work on strain relaxation was done with

laboratory experiments under not very well defined clean

room conditions. Submicron sized dust particles may stick to

the substrate surface and being overgrown by the epitaxial

layer they may act as nucleation centers for defects. We

processed the wafer in a clean room facility (Institute of

Semiconductor Engineering) using a molecular beam epitaxy

(MBE) growth equipment with surface down mounted sub-

strates. This procedure reduces the particle density. We

expect a strong influence on the initial phase of strain relaxa-

tion that we observe without limits given by a single charac-

terization method.

The onset of strain relaxation in the metastable regime

is retarded by dislocation nucleation barriers and by slow

motion of misfit dislocations. It is generally believed that ho-

mogeneous nucleation is energetically unfavorable13 for lat-

tice mismatch values below 2%. Heterogeneous nucleation

of dislocations on surface sites depends on occurrence of sur-

face defects.

II. EXPERIMENTAL

SiGe with lattice mismatch below 1% was grown epitax-

ially on dislocation-free Si substrates. We tried to suppress

the density of surface nucleation sites by careful pretreat-

ment before and during epitaxy. The first stages of misfit dis-

location generation were observed with X-ray topography,

which is the only technique allowing individual subsurface

dislocations to be detected. On the contrary, XRT is limited

to low densities (about 103 cm/cm2). Remarkable strain

relaxation sets in after generation of a high density of dislo-

cations; it was monitored by XRD. This technique allows

also layer thicknesses and Ge contents to be verified.

A. Molecular beam epitaxy (MBE) of SiGe layers on Si

SiGe layers with about 20% atomic percent (x) Ge con-

tent were grown in an MBE equipment14 the loading station

of which was connected to the clean room of the IHT, Stutt-

gart University. The nominal clean room class (class 100)

above the chemical bench and the loading station were

improved to be better than class 10 by a low throughput

cycle. The 100 mm diameter Si substrates (Czochalsky

growth, p-doped, 1–4 X�cm specific resistivity) were

removed immediately before the epitaxy process from the

sealed package, chemically precleaned (RCA etch) to get an

around 1 nm protective surface-oxide14 and loaded into the

wafer magazine of the MBE load lock, and after pump down

under ultrahigh vacuum (UHV) conditions transferred into

the MBE growth chamber. Here, as a second part of the Si

surface cleaning scheme an in situ anneal (900 �C, 5 min)

was performed in UHV vacuum that resulted in thermal dis-

sociation (SiO2Xþ Si ! 2SiO) and removal of the protec-

tive oxide. Epitaxy starts with a 100 nm Si buffer layer

(growth temperature: 600 �C) to cover the small residual

contaminations (<1/100 monolayer C, O, B). After the

buffer layer, the temperature was reduced to 550 �C and the

SiGe layer was grown from Si beam (source: e-gun evapora-

tor) and Ge beam (source: diffusion cell with graphite cruci-

ble). The beam intensities (nominal Ge/Si intensity of 0.2)

were controlled using a mass spectrometer with cross beam

ion source (Balzers). Growth rate was 0.1 nm/ s.

Thickness of the SiGe layers ranged from 50 nm to

800 nm. The thicknesses were monitored in situ with inter-

ferometric reflectometry15 and ex situ by surface stylus mea-

surement of the epitaxy edge. The sample identification

(A50 to A800) describes the process equipment (MBE A

with e-gun and effusion cell sources for Si and Ge, respec-

tively) and the thickness (in nm) of the SiGe layer. The

selected thickness range covers metastable strained layers

and partly relaxed layers.

B. X-ray topography

We used XRT transmission setting with Rigaku rotating

anode X-ray generator and Silver anode (kka¼ 0.057 nm).

Best resolution images are proceeded on silver photographic

emulsion on glass plates.

X-ray Topography (by transmission or reflection) is an

imaging technique based on Bragg’s law

2dhkl � sinh ¼ k (2)

in quite perfect monocrystals, which can be used in synchro-

tron apparatus or with laboratories X-ray sources. (In labora-

tories transmission settings conditions, sample are usually

about one square centimeter and some hundred micron

thick.) Crystal defects like dislocations or precipitates give a

contrast different from the perfect crystal area as far as they

introduce enough deformation (Fig. 1) of the lattice.16,17 On

the other hand, if the deformation is too high the Bragg posi-

tion varies a lot from one point to another so it is no longer

possible to image an entire sample. This technique has been

used for a long time in various materials like metals or

ceramics but it is perfectly suited also for semiconductors

like Si or Ge.18

In the case of epitaxial systems, the layer and the sub-

strate can be imaged either in the same time, or separately,

depending on conditions. Using the diffracting planes

perpendicular to the growth surface implies that layer and

FIG. 1. (Color online) Imaging XRT conditions from diffracting planes in

epitaxial layer (light-colored) and substrate (dark).

063507-2 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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substrate planes are in continuity with the same reticular dis-

tance, so they diffract with the same Bragg angle and are

imaged together. This is true as far as the layer relaxation is

low enough to resolve each dislocation as individual and

“far” from the others. (Fig. 1) The width resolution on the

imaging plate is about one micron, so the smallest defect that

can be observed must have a deformation field extended

over several microns and the nearest next defect-field should

be farther than some tens microns.

C. X-ray diffraction

High resolution X-ray diffraction is proceeded on a Pan-

Analytical X’Pert 4 circles goniometer. Sample positioning

is in reflection and the anode gets copper target source (kka¼1.541 A). In high resolution setup a monochromator is neces-

sary to select the Ka1 radiation and to get a parallel incident

beam, followed by an analyzer to achieve a better angular

resolution.19,20 For that purpose, we use a two bounces Ge

(220) monochromator associated with a parabolic variable-

step-multilayer mirror between the copper tube and the

monochromator. A three bounces Ge (220) analyzer is

mounted in front of the point detector. In order to extract the

relaxation rate and/or the Ge content in the SiGe layers, two

kinds of measurements can be performed: x�2h scans and

reciprocal space mappings (RSM).

The x�2h scans correspond to a rotation of the sample

with an x angle simultaneously with a rotation of the detec-

tor with a 2h angle. In a first step, x�2h scans are performed

on 004 planes, such symmetrical configuration corresponds

to a scan along qz vertical direction in the reciprocal space;

the position of the SiGe layer Bragg peak position with

respect to the Si substrate one allows to determine the per-

pendicular parameter a?SiGe with a precision of 0.002 A. If the

relaxation rate is known (for example, in the case of fully

strained samples as observed by XRT in Fig. 1) the germa-

nium content can be easily deduced as far as the Poisson

coefficient is determined. The precision on the composition

is limited by the acceptance of the Vegard’s law on Poisson

coefficient more than on the Bragg angle itself.

In order to extract directly axial strain values along the

two directions (in-plane and out-of-plane), asymmetric dif-

fraction21 or reciprocal space maps22 (RSM) are necessary

(see the 224 reflexion in Fig. 2). In the asymmetrical geome-

try, the diffusion vector ~q ¼ ~k � ~k0 is not perpendicular to

the sample surface, thus both vertical and horizontal compo-

nents of the scattering vector qz and qx are available:

qx ¼ qy ¼4pk

sinh sinðh� xÞ (3)

with h the Bragg angle and x the incident angle with respect

to the crystal surface.

We have measured two reflections for a better accuracy:

both 224 planes and 115 planes.

III. DISLOCATION NUCLEATION

The lattice mismatch f between SiGe and the substrate

Si causes a compressive strain e, which for a pseudomorphic

planar film is given by the same amount as the mismatch but

opposite sign: pseudomorphic film

e ¼ �f : (4)

The opposite sign stems from the usual definitions of lattice

mismatch (positive for the larger film constant) and strain

(negative for compressive strain).

The biaxial film strain is correlated with an in plane

stress r whose magnitude depends on the elastic properties.

r ¼ E

1� ve; (5)

where E is the elastic modulus (defined in uniaxial deforma-

tion), � is the Poisson coefficient. For Si � value is 0.278; E

value is 131GPa in h100i direction; for Ge E[100] is equal to

107GPa and � to 0.27.

In biaxial deformation, the shear modulus G is generally

used, with the relation

2Gð1þ vÞ ¼ E: (6)

The shear modulus G is equal to 51GPa for Si and 40GPa in

Ge. The elastic moduli of SiGe are slightly smaller if one

assumes a linear interpolation between Si and Ge. For SiGe

(x ¼ 0.2) we expect a reduction in stiffness by about 5%.

The relevant stress for misfit dislocation glide is the resolved

excess stress rex. This resolved stress is depending on the

geometric characteristics of the glide plane and the Burgers

vector by respect with the main strain direction; it has been

given by Dodson and Tsao23 as

rex ¼ 2Gjej 1þ v

1� vcos k cos u� Gb� cosulnðat=bÞ

4ptð1� vÞ : (7)

The first term is caused by the misfit strain e, the second

term is related to the self energy of dislocations.

FIG. 2. (Color online) Geometry of a 224 reciprocal space map. In the

asymmetrical configuration the scattering vector q (~q ¼ ~k � ~k0 ) has two

components qz and qx.

063507-3 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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In the diamond cubic lattice structures of Si, SiGe, and

Ge the dislocations propagate primarily by glide on (111)

planes inclined to the (001) interface. The geometry of the

interfacial dislocation array is therefore defined by the inter-

section of the inclined glide planes with the interface. The

misfit dislocation lines run along orthogonal [110] directions.

The dislocation properties are defined by b, a, k, u, the

film thickness is t. The Burger’s vector (amount b) of a disloca-

tion in the diamond lattice points in a [1/2]h110i direction

(length b is 0.384 nm in Si), a is a factor that describes the dislo-

cation core energy. Usually selected values of a ranges from 1

to 3. The exact choice is not very critical as long as the thickness

t is much larger than the Burger’s vector length b. Then the

strain energy around the dislocation dominates the total disloca-

tion energy. The angles k and u are defined by Burger’s vector

to dislocation normal in the interface and by glide plane normal

to interface normal, respectively. The values of cos k and cos uare 0.5 and 0.58, respectively, for the 60� dislocation (60� angle

between line and Burger’s vector) on (001) interfaces.

In an unrelaxed layer, the strain and therefore the stress

are constant with the layer regardless of thickness. The elas-

tic energy stored in the layer increases with thickness. The

relaxation begins when the stored energy is high enough to

overlay the formation of dislocation line. The extension of

misfit dislocation is only allowed above the equilibrium criti-

cal thickness tc where the excess stress is above zero. The

relative contribution of the self energy (second term in Eq.

(7)) shrinks rapidly with thickness t, then we can assume a

resolved excess shear stress rex ¼ 430 MPa for pseudomor-

phic or weakly relaxed layers.

The pseudomorphic growth and the early stages of dislo-

cation nucleation under the influence of the excess stress

were observed by XRT. The layers with thicknesses (A50,

A100) up to 100 nm proved to be dislocation free. Some-

times, the substrate showed reflections from precipitates

deep in the substrate. These precipitates are probably SiO2

complexes created in oxygen rich regions of the Czochralski

grown silicon substrate during the preepitaxial heat treatment

(900 �C). The XRT image (Fig. 3) is shown here to demon-

strate that substrate precipitates are not connected with heter-

ogeneous dislocation nucleation.

First misfit dislocations are observed with A200 (Fig. 4)

These misfit dislocations belong to heterogeneous nucleation

centers (density below 100/cm2), which generate one or sev-

eral dislocation half loops with a misfit dislocation segment

in the interface and two threading dislocation arms connect-

ing the interface with the surface.

The threading arms in the thin layer (200 nm) cannot be

detected directly with XRT but according to the dislocation

theory, requests that single dislocation cannot end inside the

volume. Stronger contrast from some nucleation centers indi-

cates multiple generations of dislocations from the same source.

In the 400 nm thick sample (A400) multiple generation

is seen frequently (Fig. 5) and the length of the dislocation

segments (several mm) is now high enough to obtain cross-

ings between dislocations of different sources. Such cross-

ings can act as second generation nucleation sites (Frank-

Read source12 as a general multiplication mechanism and

Hagen-Strunk source24 as specific source for strained layers).

We will see later that even the misfit dislocation network in

A400 does not relax the strain considerably. A mean dislocation

spacing is hardly to define in this heterogeneous nucleation

phase but from large area observation we obtain 600 lm and

about 150 lm for 200 nm and 400 nm thickness t, respectively.

FIG. 3. XRT image of a section of sample A100 showing precipitate (prob-

ably SiO2) contrast within the substrate. No dislocations from the 100 nm

thick SiGe layer are visible.

FIG. 4. XRT image of a section of sample A200. Typical straight misfit dis-

location contrast in both h110i directions is visible. The h110i directions are

the intersections of the (001) interface with the (111) glide planes.

FIG. 5. XRT image of a section of sample A400. Multiple dislocation gen-

eration and overlap of dislocations from different sources happens

frequently.

063507-4 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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Larger relaxation is evidenced in the thickest sample

A800. Here, the density of misfit dislocations (Fig. 6) is too

high to be measured by XRT. The “wavy” or “grainy” struc-

ture must not be correlated with the dislocation shape: The

stresses induced by interactions between dislocations are

overlapping all other contrasts and no information can be

extracted about individual dislocations.

The lengths of dislocations from individual sources

(Figs. 4 and 5) let us assess the dislocation velocity under

the given temperature (550 �C) and the given excess

stress (400 MPa), assuming that the dislocations have

been gliding above a thickness of 150 nm. Misfit disloca-

tion length is between 1 and 5 mm for 500 s to 2500 s

gliding time.

A mean dislocation velocity of 2 lm/s is extracted. That

is higher than data from Hull25 and Houghton26 but fits with

an extrapolation of Yamashita data27 extracted from lower

shear stresses. The reason for our higher velocity data could

be the low nucleation density that allows the dislocation arm

to move without interaction with other dislocations.

Strain measurements are performed with XRD x�2hscans and reciprocal space maps (RSM) around Si Bragg

peaks. In Fig. 7 are presented the symmetrical (004) x�2hscans for the fully strained SiGe layers (A100, A200 and

A400). The SiGe peak angular position allows determining

the perpendicular lattice parameter and thus the Ge content,

assuming no relaxation. Values are slightly larger (22.2%,

see Table I) than the nominal 20% one. If the layers are per-

fect with planar interfaces, interference fringes can be

observed, which spacing (in reciprocal space units) is inver-

sely proportional to the SiGe layer thickness t ¼ 2p/Dqz.

This is the case for samples A100, A200, and A400 (Fig. 7);

the calculated thickness values confirm the nominal thick-

nesses within a few percent (Table I).

Figure 8 represents the (224) RSM of sample A200,

with the axis in hkl units. The relation between reciprocal

space vector coordinates and hkl units are the following:

qx ¼2ph

aSi; qy ¼

2pk

aSi; qz ¼

2pl

aSi; (8)

with aSi the silicon lattice parameter (in nm).

On the asymmetrical RSM, the Si and SiGe Bragg peaks

are aligned along the vertical direction that confirms that the

relaxation is quite zero on these samples, as previously

observed by XRT. The vertical position of the SiGe peak,

with respect to the Si one allows to determine the Ge content

in the layer and is in agreement with the one deduced from

the (004) x�2h scan (see Table I).

IV. METASTABLE CRITICAL THICKNESS AND STRAINRELAXATION

A. Brief report on metastability and relaxation rate

Above the critical thickness tc the strain e in the partially

relaxed layer is reduced by the introduction of a misfit dislo-

cation network. This network is best characterized by a mean

distance p between the misfit dislocations lying in the

FIG. 6. (Color online) XRT image of a section of sample A800. The high

density dislocation network cannot be resolved with XRT. But the grainy

structure indicates variable strain levels probably caused by the inhomoge-

neous dislocation nucleation.

FIG. 7. (Color online) (004) x� 2hscans of samples A100, A200, A400,

reported in qz units.

063507-5 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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interface. (Note: The usual dislocation density notation based

on dislocation lengths divided by volume is not appropriate

for an interface network.) On (001) Si the misfit dislocation

lines create an orthogonal network along h110i directions

that are the intersections of the f111g glide planes with the

(001) substrate surface (Fig. 9). The Burger’s vector of the

dislocation is [1/2]h110i, the one lying 60� to the dislocation

direction is preferred. (Note: The Burger’s vector parallel to

the dislocation line gives a screw dislocation that is ineffec-

tive in misfit strain relaxation, it would cause a twist.) For

strain relaxation the edge component of the Burger’s vector

projected to the normal of the dislocation is the active

component

b0 ¼ b� cosk: (9)

Each dislocation displaces the film lattice planes by b’. The

(1/p) dislocations per unit length reduce the compressive

strain (-e) to

ð�eÞ ¼ f � b0

p: (10)

Total strain relaxation (e ¼ 0) is obtained by a relaxation

spacing prel

prel ¼b0

f: (11)

For our experimental set of f ¼ 0.008 we obtain a relaxation

spacing prel ¼ 24 nm when we assume b0 ¼ b/2 ¼ 0.192 nm.

At higher strain values dislocation reactions28 may occur

that deliver sessile Lomer dislocations for which b0 ¼ b.

John Bean’s group discovered that at moderate growth

temperatures (550 �C) the measured critical thickness for

onset of misfit dislocation generation is much higher than the

equilibrium value. Most cited is a fit through the experimen-

tal points given by People/Bean.11 Let us assign the metasta-

ble critical thickness tcp to this fitting curve.

The value of tcp29 for the given lattice mismatch is tcp ¼

150 nm, compared to the equilibrium value of tc ¼ 10 nm.

tcp

b� f 2 ¼ 1

200ln

tcp

b

� �: (12)

It is an ongoing debate (for a review, see Ref. 12) if the met-

astable critical thickness is due to a retarded relaxation pro-

cess where the given value tcp only marks a detectable

dislocation spacing or if it is a real metastable process

defined by a nucleation barrier for dislocation generation.

The degree of strain relaxation r is now defined as

r ¼ f þ ef¼ prel

p: (13)

The degree of relaxation can be extracted from measurement

of strain e or measurement of dislocation spacing p following

Eq. (10).

B. Experimental measurements of relaxation

In the case of 20% Ge layers on Si, the misfit would be

fully relaxed (assuming 60� misfit dislocations) by a squared

dislocation network with an average distance p about 25 nm

between two neighboring dislocations, which is 4�105

TABLE I. Ge contents x (from (004) XRD, (224) and (115) RSM), thick-

nesses tSiGe from XRD fringes (except A800, which gives the nominal value

as fringes are only visible in unrelaxed layers) and relaxation r (from XRT

and RSM).

Sample x Ge % tSiGe (nm) r %

A100 23.0 9362 0 (XRT)

A200 22.9 19562 0.004 (XRT)

A400 21.2 405612 0.016 (XRT)

A800 21.6 800(nom) 7464 (RSM)

FIG. 8. (Color online) 224 RSM of sample A200. The axis are in hkl units.

Si Bragg peak is located at h ¼ k ¼ 2 and l ¼ 4, while the SiGe one appears

at lower l value (strain).

FIG. 9. Geometry of a parallel set of misfit dislocations along [ �110 ]. The

orthogonal dislocation set is not shown to avoid overload of the drawing.

The dislocation spacing is given by the distance p. The Burger’s vector b,

and its projection length b0 on the interface normal to the dislocations are

shown for the (111) glide plane

063507-6 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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dislocation lines/cm in cross section or 8�105 dislocation line

cm/cm2 in plane view. So the highest density that can be

clearly imaged by XRT corresponds to a relaxation rate

below 1%.

The relaxation rate cannot be measured by RSM when it

is lower than a few percent. It can be only estimated from

the observed dislocation densities in XRT images; we found

0% in A100 (Fig. 3), 0.004% (30 cm/cm2) in A200 (Fig. 4),

and 0.016% (130 cm/cm2) in A400 (Fig. 5) assuming b0¼0.192 nm.

On the contrary, the thicker sample A800 exhibits dislo-

cation density higher than some 104 cm/cm2 so we can only

say from XRT that r is at least 10% (Fig. 6). In that case, the

relaxation rate can be precisely measured by RSM (Fig. 10).

The RSM were recorded around the 224 reflection. As

seen previously, an identical in-plane reciprocal vector for

the Si substrate and SiGe film indicates a pseudomorphic

structure (Fig. 8, A200). As expected from the XRT results,

we cannot detect any relaxation up to A400 because disloca-

tion spacings above 25 lm result in less than 0.1% relaxa-

tion. In a relaxed structure the Bragg peak would shift to

smaller reciprocal space value for a tensile strain and to

higher value for a compressive strain. For the sample A800

(Fig. 10), both qx and qz SiGe positions are lower than the Si

Bragg peak, which means that the layer started to relax. We

measure a relaxation rate equal to 76% for this sample (see

Table I).

The whole results extracted from HR-XRD (scans and

RSM) are presented in Table I.

V. DISCUSSION

Let us discuss the results keeping in mind the open ques-

tion if the metastable critical thickness (as given, for

instance, by the People-Bean interpolation, Eq.(12)) marks

the easy visibility of misfit dislocations. We remind the

reader that experiments described in this paper were per-

formed under state of the art clean room conditions for

substrate cleaning and epitaxy loading to avoid particle

contamination.

One sees immediately that the first dislocations (A200)

appear above the People-Bean curve. This observation con-

firms the thickness area below the People-Bean curve (and

above the equilibrium curve) to be a true metastable region pro-

tected (at 550 �C) from heterogeneous nucleation by an energy

barrier if strong disturbances as surface particles are absent.

Speculations seem valid that moderate misfit dislocation den-

sities observed earlier at the People-Bean curve are connected

to particle contamination under routine laboratory conditions.

The second observation is the slow onset of relaxation

that gives an apparent higher critical thickness from relaxa-

tion measurements (X-ray diffraction, Raman shift belong to

this type of measurement, but also TEM because of the high

magnification method). For a broad range of measurement

methods it seems therefore more appropriate to define a criti-

cal thickness band instead a curve. The lower critical thick-

ness bound is defined by misfit dislocation counting

(extrapolation to zero density), the upper critical thickness

bound is defined by relaxation measurement (extrapolation

to zero relaxation).

On the search for an extrapolation scheme, let us con-

sider first the equilibrium case. Here, the strain e and the mis-

fit dislocation distance p are given by a minimum of the total

energy (per area) Etot, which is composed of the contribu-

tions of the film strain energy Ef and the dislocation net-

works energy En.

Etot ¼ Ef þ En; (14)

where isotropic elastic theory delivers for a biaxial stressed

layer

Ef ¼ C1e2t;

C1 ¼ 2G½ð1þ vÞ=ð1� vÞ� (15)

and dislocation theory,30 we use here for simplicity the edge

dislocation energy, delivers for a single dislocation

Ed ¼ C2½1þ ln ðRa=riÞ�;

C2 ¼ Gb2=½4pð1� vÞ�: (16)

The network energy En is then given by (the factor 2 refers

to the orthogonal network on (001) orientations)

En ¼2

pEd; (17)

The quantities ri and Ra in Eq.(16) are the inner and outer

cutoff radii, respectively.

The inner cutoff radius ri is a fraction of the Burgers

vector length b, its exact value depends on the dislocation

core energy, and this value is related to the value a in Eq. (7).

a ¼ 2eðb=riÞ; (18)

where e is the base of the natural logarithm (2.7). The outer

cutoff radius is about twice the film thickness t at low

FIG. 10. (Color online) 224 RSM of sample A800. The axis are in hkl units.

Si Bragg peak is located at h ¼ k ¼ 2 and l ¼ 4, while the SiGe one appears

at lower l and h values (larger lattice parameters).

063507-7 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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dislocation densities. This value is assumed for the critical

thickness calculation. At higher dislocation densities

(p/2� t) the mutual dislocation interaction drops the outer

cutoff radius to p/2.

Ra ¼ 2t ðregime I; t < p=2Þ;

Ra ¼ p=2 ðregime II; t � p=2Þ: (19)

For more refined calculations a smooth transition between

the two regimes is assumed,31 which is neglected for our

purposes.

The minimum energy condition is found considering the

relation between e and p (Eq. (10))

@Etot

@e¼ 2C1etþ

2

p

@Ed

@eþ Ed

2

b0: (20)

For regime I (Eq.(19)) the solution is straight forward

because Ra does not depend on e, that means @Ed=@e ¼ 0.

This leads for regime I to

et ¼ �ðEd=C1Þb0 ¼ C3½1þ ln ð2t=riÞ�;

C3 ¼ �C2

C1b0¼ �b½8pð1þ vÞ cosk��1; (21)

For regime II the term @Ed=@e reads C21p

b0

ðeþ f Þ2 which to-

gether with Eqs. (10), (16), (20) gives

et ¼ C3 2þ lnb0

2r1ðeþ f Þ

� �� �: (22)

The equilibrium critical thickness tc immediately follows

from Eq. (21) when one sets –e ¼ f and t ¼ tc

f � tc ¼ �C3½1þ ln ð2t=riÞ�; (23)

which is equivalent to the force-considerations (Matthews-

Blackeslee approach) when replacing the inner cutoff radius

ri by Eq. (18) (ri ¼ 2eb/a).

For comparing an experimental group of samples with

some scattering in the Ge content and therefore mismatch f,the consideration of the relaxation degree is more general.

We obtain by combining Eqs. (13), (20), (22) and (23) an

equilibrium relation for the relaxation branch above the criti-

cal thickness (t larger tc).

� etf¼ ð1� rÞt ¼ tc

1þ ln ð2t=riÞ1þ ln ð2tc=riÞ

: (24)

The left side of Eq. (24) is proportional to the force with

which a strained layer bends the substrate (note: Stress

times thickness describes the force from a unit length). In

Fig. 11 the experimental term (1�r)t is given as a function

of the thickness t. We see the linear increase of force with

thickness for pseudomorphic and weakly relaxed (r< 0.02)

layers and with higher thickness (below 800 nm) a sudden

decrease caused by strong relaxation. As an aid for the

eye, a solid line with strong relaxation onset at 600 nm is

drawn.

As the right term in Eq. (24) is only slowly varying with

lnt we alternatively can suggest a double logarithmic presen-

tation with the variables ln(1�r) versus lnt (Eq. (24))

ln ð1� rÞ ¼ �lntþ lntc þ ln1þ ln ð2t=riÞ1þ ln ð2tc=riÞ

: (25)

Figure 12 exhibits log(1�r) versus logt according to the

form of Eq. (25). Again the straight line is an aid for the eye

demonstrating the sudden decrease in strain by the strong

relaxation.

Both diagrams (Figs. 11 and 12) deliver from the extrap-

olation of experimental points in their relaxed branch a value

of the upper bound tco of the metastable critical thickness

band.

The strong onset of relaxation is linked to dislocation

multiplication sources that can work after crossings of dislo-

cations from different first generation heterogeneous sources

occur. Within the metastable critical thickness band, a pre-

sentation following Eq. (24) is useless because of relaxation

degrees much smaller than 1, which lets the factor (1�r) vir-

tually unchanged. For this band regime between lower bound

tcl and upper bound tco of the critical thickness, we use a

simplified version of a general relaxation scheme given by

Dodson-Tsao25 as extrapolation relation for tcl

FIG. 11. The relative force of a strained layer (1�r) � t vs thickness (logt).Increase of the force up to the upper bound tco then sudden decrease at the

relaxation branch. The force causes a curvature of the wafer.

FIG. 12. Experimental (crosses) relative strain values (1�r) vs thickness

(logt). The strain is nearly constant up to an upper bound tco where strong

relaxation sets in.

063507-8 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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dedt¼ Constðe� eequÞ2 ðf þ eþ esÞ: (26)

For the critical thickness band the conditions jej jeequjand jf þ ej < es are valid, with eequ the equilibrium strain

and es a strain value characteristic for the heterogeneous

nucleation site density. The simplification under this condi-

tion reads

dedt¼ Const ðese

2Þ (27)

with the solution

1

fþ 1

e¼ �Const ðt� tclÞes (28)

if relaxation starts at the lower bound tcl. Relaxation r is then

a linear function of the thickness

r ¼ �Const ðt� tclÞ ees: (29)

Figure 13 gives the presentation of r versus t below the upper

bound thickness. A lower bound of about 140 nm is found.

The samples with thicknesses below the lower bound A100,

A50 had zero misfit dislocation densities as demonstrated by

X-ray topography.

In the absence of particle contamination the heterogene-

ous nucleation starts below a thickness of 200 nm which is

similar to the critical thickness predicted by the People-Bean

interpolation formula. Serious strain relaxation (more than

2%) may be found at higher thickness values. We define a

critical thickness band with lower bound given by the onset

of dislocation nucleation and an upper bound defined from

strain relaxation measurements.

For the given experimental MBE conditions:

Ge content about 20% (lattice mismatch f ¼ 0.8%,

growth temperature 550 �C, growth rate 0.1 nm/s, p-substrate

doping ffi 1016=cm3, wafer surface finish from supplier

A) the metastable critical thickness band is given by

tcl¼ 140 nm to tco ¼ 600 nm.

VI. CONCLUSIONS

We investigated the strain relaxation mechanisms of epi-

taxial SiGe layers at 550 �C growth temperature. A rather

low Ge content of 20% was chosen to avoid competition in

compressive strain relaxation mechanism between misfit

dislocation generation and surface roughening via Stranski-

Krastanov growth mode.32 At the selected growth tempera-

ture (550 �C) the strain relaxation in the heterostructure layer

with about 0.8% lattice mismatch is given by misfit disloca-

tion network generation at the substrate-film interface. At

higher growth temperatures the competition from surface

roughening occurs at lower mismatch values as already

shown in the first work on SiGe/Si heterostructure stability.33

The distinct growth temperature of 550 �C was chosen to

allow a comparison with the classical People/Bean11 publica-

tion on non equilibrium critical thickness. In their original

paper11 People/Bean claimed to have found an energy bal-

ance calculation different from v. d. Merwe8 but now the

research community (see Refs. 25 and 31) agrees that the

People/Bean critical thickness curve separates metastable

strained layer growth from partially strain relaxed films. Ear-

lier work (Refs. 25, 28, and 12) considered the curve as a

detection limit of dislocations in thin films. Our approach

differed twofold compared with earlier work:

(i) The epitaxy under modern clean room conditions

avoided dust particle contamination of the wafer surface.

Even very small submicron dust particles act as strong heter-

ogeneous dislocation sources when overgrown by a film.

(ii) Observation of dislocation networks by two comple-

mentary X-ray methods (topography, diffraction) allowed a

rigorous dislocation and strain monitoring on the same sam-

ple series. Especially, X-ray topography is a unique method

to monitor low dislocation densities from heterogeneous

nucleation sites because it combines sensitivity to single dis-

locations with large area observations. Competing methods

suffer from low sensitivity to single dislocations (e.g.,

haze34) or from small area observations (e.g., atomic force

microscopy of slip steps). In principle selective defect etch-

ing can be applied to low density defects but with thin films

the etching depth is restricted, which in practice demands

high magnification observation of small areas.

X-ray diffraction methods are established for strain

relaxation measurements with more than 2% relaxation

degree,35 which covers the high density network side. Simi-

lar restrictions are valid for other strain measurement meth-

ods such as Raman spectroscopy.36 Under these

experimental conditions we found the following:

(i) The Si0.8Ge0.2 films are growing (at 550 �C) misfit

dislocation free up to a critical thickness of 140 nm

(much thicker than the equilibrium thickness of

10 nm, roughly about the People/Bean curve). The

dislocation freedom proves the region between 10 nm

and 140 nm thickness to be a true metastable regime

where a nucleation barrier avoids dislocation genera-

tion if dust particle contamination is deleted.

(ii) Measurable strain relaxation (>2%) is shifted to

thickness values (about 600 nm) considerably higher

than the People/Bean curve. Similar observations are

FIG. 13. Relaxation r vs thickness t. The early stage of heterogeneous

nucleation is represented by a straight line the extrapolation of which to

r¼ 0 gives the lower bound tcl of the critical thickness band.

063507-9 Kasper et al. J. Appl. Phys. 111, 063507 (2012)

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also made with chemical vapor deposition epitaxy35

but at different temperature (650 �C). We assume

therefore that this observation is general for epitaxy

under particle free condition.

(iii) We propose a metastable critical thickness band with

a lower and an upper bound, respectively. The lower

bound tcl (140 nm) marks the onset of heterogeneous

nucleation whereas the upper bound tco (600 nm)

marks the onset of measurable strain relaxation.

We could not identify the nature of the dislocation

nucleation sites in particle free epitaxy. However, X-ray to-

pography demonstrated clearly that oxide precipitations

inside in annealed Czochralski grown wafers are not respon-

sible for the nucleation. We speculate that surface step accu-

mulations (ripples), self assembled Ge content variations or

mechanical surface polishing defects could deliver that

remaining nucleation sites.

The metastable growth regime with fully strained SiGe

layers (pseudomorphic growth) could possibly extended even

more if these nucleation sites may be reduced or the barrier

height may be increased. These future investigations need close

cooperation with wafer manufacturers because on commercial

wafers the surface finishing is not well known to the customer.

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