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Buried dislocation networks for the controlled growth of III–V semiconductor nanostructures
ARTICLE IN PRESS
Journal of Crystal Growth 275 (2005) e1647–e1653 www.elsevier.com/locate/jcrysgro
Buried dislocation networks for the controlled growth of III–V semiconductor nanostructures F. Glas, J. Coelho, G. Patriarche, G. Saint-Girons CNRS/LPN, Laboratoire de Photonique et de Nanostructures, route de Nozay, 91460 Marcoussis, France Available online 24 December 2004
Abstract Shallowly buried dislocation networks (DNs) can be used to organize the formation of epitaxial semiconductor quantum dots, via the periodic strain field induced at the growth front. We fabricate such DNs by bonding epitaxially two GaAs crystals twisted and tilted with respect to each other and studied them by transmission electron microscopy. The DNs differ greatly from the mere superposition of a square screw DN and of a one-dimensional mixed DN accommodating, respectively twist and tilt. We present a detailed quantitative analysis of these DNs and explain how their properties relate to the crystal disorientations. Dislocation interaction generates a surface pattern of dilatational and compressive strain favouring the ordered growth of (Ga)InAs nanostructures. r 2004 Elsevier B.V. All rights reserved. PACS: 61.72.Lk; 61.72.Mm; 68.37.Lp; 68.35.Gy; 68.65.k Keywords: A1. Interfaces; A1. Line defects; A1. Nanostructures; A1. Transmission electron microscopy; A3. Epitaxial wafer bonding; B2. Semiconducting III–V materials
1. Introduction Quantum dots (QDs) may be formed spontaneously when a semiconductor is deposited epitaxially on a mismatched substrate [1], but the sizes, shapes, compositions and spatial distribution of the QDs usually remain fairly uncontrolled. However, by homogenizing these quantities, better properties could be achieved for existing devices Corresponding
author. Tel.: +33 1 69 63 60 79; +33 1 69 63 60 06. E-mail address: [email protected] (F. Glas).
fax:
and, by tailoring then, we could fabricate new devices. A promising way to order QDs is to use the strain field induced at the growth front by a shallowly buried periodic dislocation network (DN), which affects the surface potentials of the deposited species and the nucleation sites of the QDs. By choosing appropriate DNs, it might become possible to order laterally QDs having identical shapes an sizes. Such DNs can be obtained by epitaxial wafer bonding [2,3]. Elasticity calculations show that edge dislocations should be most efficient because they induce
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.11.219
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dilatational surface strains [4,5]. However, their periods, governed by the misfit between the two bonded crystals, are not easily adjusted. On the other hand, the period of a screw DN can be selected by simply choosing the twist angle between the two crystals, but such DNs induce pure shear strains at the surface, which affect only weakly the deposited species [4]. So far, lateral organization of QDs mediated by DNs has been reported for metals [6] and for Ge on Si [7] but not for III–V materials. In this paper, we present a detailed analysis of DNs fabricated for growing (Ga)InAs QDs on GaAs. We obtain shallowly buried DNs by bonding epitaxially a GaAs host substrate to a GaAs substrate on which several lattice-matched layers have been grown. Details are given elsewhere [8]. The final assembly is composed of a 20 nm thick GaAs layer tilted and twisted with respect to the host substrate to which it is bonded. The DNs lie in the interface, which is a grain boundary (GB). Tilt is achieved by using vicinal substrates with normals slightly and equally disoriented from [0 0 1] around a /1 0 0] direction. We bond them with their one-dimensional (1D) sets of surface steps nearly orthogonal, so that the resultant tilt axis [3] is close to a /1 1 0S direction, hereafter noted [1 1 0]. A small additional twist is set up by rotating the crystals by up to a few degrees around the normal to the interface. We study the DNs by transmission electron microscopy (TEM).
2. Structure and first analysis of the DNs In these materials, the Burgers vectors (BVs) are of type a=2h1 1 0i: Taken independently, twist may be accommodated by a square two-dimensional (2D) network of screw dislocations with ‘in-plane’ a=2h1 1 0 BVs, whereas tilt accommodation requires a BV edge component normal to the GB [9]. From the previous studies of Si/Si [10] and GaAs/ InP bonding [3], we expect the latter to be provided by a 1D network of mixed dislocations oriented along the tilt axis (601 dislocations if this axis is /1 1 0]) (Table 1). For these two ideal DNs, the dislocation period is D ¼ 12 b0 =sinðy=2Þ; with y
Table 1 Moduli of the BV components of ideal screw and 601 mixed dislocations Dislocations
Screw 601 mixed
Burgers vector component Screw
Edge, in GB
Edge, normal to GB
pffiffiffi a 2=2 pffiffiffi a 2=4
0 pffiffiffi a 2=4
0 a/2
Fig. 1. TEM weak beam plan-view images of the GB: (a) g ¼ 2 2 0; showing mainly screw subnetwork #2; (b) g ¼ 2¯ 2 0; showing screw subnetwork #1 (full arrow) and mixed DN (dashed arrow).
Table 2 Characteristics of the DNs of the sample studied
DN
Index j
Nearest /1 1 0S direction oj ð1Þ
Mixed Screw #1 Screw #2 Ideal screw
m 1 2
[1 1 0] [1 1 0] ½1¯ 1 0
2.870.8 2.171.7 0
Dj (nm) 27.972.8 21.070.6 15.070.3 13.972.1
All values are measured, except those on the last line (period calculated from ytwist ).
the crystal disorientation and b0 the BV component accommodating it (Table 1). In this section, we give a general analysis of the DNs and illustrate it for a typical sample (Fig. 1 and Table 2). We might expect the actual DNs to be a mere superposition of the two abovementioned ideal DNs. Indeed, TEM analysis of the GB (Fig. 1) shows a 2D DN with in-plane a=2h1 1 0i BVs (reminiscent of the ideal screw DN) and a 1D set of mixed dislocations oriented close
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to the twist axis, having both in-plane and normal BV components. However, the major differences are: (i) the periods of the two subnetworks of the 2D DN are different; (ii) these subnetworks are slightly disoriented with respect to the /1 1 0] directions. We note o the disorientation of a DN with respect to its neighbouring /1 1 0] direction and D its period; indices m, 1 and 2 specify, respectively, the mixed DN and the two subnetworks of the 2D DN oriented close to [1 1 0] and ½1¯ 1 0 (Fig. 2, drawn for om o0). We measure o1 ; o2 and all periods by TEM (Table 2), and the actual twist angle ytwist (here equal to 1.6570.251), but not the tilt ytilt ; by electron diffraction. Moreover (iii) each mixed dislocation interacts with the nearly orthogonal dislocations of subnetwork #2 and becomes a sequence of segments disoriented from the average line direction. Each interaction shifts the dislocation of subnetwork #2 by half a period (Figs. 1(a) and 3). Point (iii) manifests energy-minimizing shifts and rotations already observed when mixed dislocations interact with screw or edge DNs [3,9–11]. Our mixed dislocations have eight possible BVs, each having two components of modulus a=2; respectively along an in-plane /1 0 0] direction and [0 0 1] (Table 1). We assume that all
screw #1
010
110
110
1 2
m b2
b1
screw #2
mixed
100
100
010 Fig. 2. Orientations of the interfacial DNs. All directions are relative to a given crystal. Dashed lines: crystallographic directions. Full and dotted lines: average dislocation directions. Note the rotation of the figure with respect to Fig. 1.
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screw #2
110
screw #2 110
110 110 m D1
011 101
101 011 011
101
011 101
Dm D2 (a)
110
110
(b) screw #2
screw #2
Fig. 3. (a) Schematics of the DNs of Fig. 1, oriented accordingly. Full lines: screw dislocations; dotted lines: mixed dislocations. (b) Interaction between mixed dislocations (m) and screw subnetwork #2. Arrows give line orientation. BVs are indicated for each segment with factor a/2 omitted. BV sequences ‘a’ and ‘b’ appear, respectively above and below the mixed dislocation line. Dashed line: average mixed line.
normal components have the same sign (arbitrarily taken as positive along [0 0 1]); this amounts to assuming that there are no more mixed dislocations than necessary for tilt accommodation. This leaves four possible BVs. Nodal conservation rules [9] imply that each segment of mixed dislocation has its own BV and that only two ‘a’ and ‘b’ sequences are possible (Fig. 3(b)). Any sequence is fully determined once one segment is known, but sequences corresponding to two dislocations are a priori independent. If the segments of the mixed dislocations are of equal lengths, the in-plane component of the average BV is along [1 1 0] and ½1¯ 1¯ 0 for ‘a’ and ‘b’ dislocations, respectively. Point (i) can be understood if ‘a’ sequences outnumber ‘b’ sequences. Then, the in-plane component of the average BV of the whole mixed DN is along [1 1 0], i.e. parallel to the BV of subnetwork #1. Since the mixed dislocations are also oriented close to [1 1 0], twist accommodation is thus shared between the mixed DN and subnetwork #1. No such sharing occurs with subnetwork #2, which has orthogonal BVs. Hence, less #1 dislocations are needed than expected, D1 is larger than expected and D2 as expected. This is exactly what we observe (Table 2). Finally, point (ii) implies that the dislocations of the 2D DN have small edge components. Since the two bonded substrates have identical lattice parameters, these must compensate edge components associated with the
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mixed DN. For simplicity, we shall call ‘screw’ the dislocations of the 2D DN, although this is not strictly appropriate.
3. Quantitative analysis of the DNs These conclusions are supported by a quantitative analysis of the GB. In the sample studied, o1 and o2 have the same sign (Table 2). Since twist is accommodated chiefly by the screw dislocations, the projections of the BVs of the latter along their lines must be of the same sign; pffiffiffi we arbitrarily choose these BVs to be b ¼ ða= 2Þu1 1 0 and b2 ¼ 1 pffiffiffi ða= 2Þu1¯ 1 0 ; respectively, where uhkl is a unit vector along [hkl] (Fig. 2). We note p1 0 ; p0 1 ; p1¯ 0 and p0 1¯ pthe four ffiffiffi probabilities pffiffiffi of the p ffiffiffi possible BVspffiffiffiða= 2Þu1 0 1 ; ða= 2Þu0 1 1 ; ða= 2Þu1¯ 0 1 and ða= 2Þu0 1¯ 1 of the mixed dislocations, which verify 0ppij p1 and p1 0 þ p0 1 þ p1¯ 0 þ p0 1¯ ¼ 1: The geometry of all combinations of periodic linear interfacial DNs compatible with given disorientations between two crystals is prescribed by Frank’s formula [9]. If vector V of one crystal transforms into V0 (having the same coordinates) in the other, this formula gives the ‘closure defect’ BðVÞ ¼ V V0 in terms of the geometry and BVs of the dislocations. Taking n as having a (large) positive component along u0 0 1 and applying Frank’s formula to vectors u1 1 0 and u1¯ 1 0 and to our three DNs (#1, #2 and mixed), we get
where a ¼ p1 0 p1¯ 0 þ p0 1 p0 1¯ and b ¼ p1¯ 0 p1 0 þ p0 1 p0 1¯ characterize the projection pffiffiffi in the GB of the average BV b¯ m ¼ 14 a 2ðau1 1 0 þ bu1¯ 1 0 Þ þ 12 u0 0 1 of the mixed dislocations. We retain only the terms of first order in angles o1, o2, om, ytwist and ytilt ; which are all small. If ytwist 40 corresponds to a rotation from u1 1 0 towards u1¯ 1 0 ; in terms of crystal disorientations, the closure defects are then simply Bðu1 1 0 Þ ¼ ytwist u1¯ 1 0 þ om ytilt u0 0 1 and Bðu1¯ 1 0 Þ¼ ytwist u1 1 0 ytilt u0 0 1 : Using these formulas in Eqs. (1) and (2), projecting on u0 0 1 ; u1 1 0 and u1¯ 1 0 and setting d 0 ¼ pffiffiffi a 2 yields: ytilt ¼ a=2Dm ;
(3)
aom ¼ 2o1 Dm =D1 ;
(4)
bom ¼ 2Dm ð1=D2 ytwist =d 0 Þ;
(5)
a ¼ 2Dm ð1=D1 ytwist =d 0 Þ;
(6)
b ¼ 2o2 Dm =D2 :
(7)
Eq. (3) simply expresses tilt accommodation by the mixed DN and yields ytilt ¼ 0:58 0:06 (compatible with substrate vicinalities of 0:5 0:1 ). From (4)–(7) and the experimentally determined quantities and their uncertainties (Table 2), we want to get om, a and b. Since the DN periods are measured more precisely than the disorientations, we first calculate a refined value of ytwist rather than using our measurement. From (4)–(7), we get om ¼
Bðu1 1 0 Þ sin o1 a cos o2 a pffiffiffi u1 1 0 pffiffiffi u1¯ 1 0 D1 D2 2 2 " pffiffiffi # sin om a 2 a ðau1 1 0 þ bu1¯ 1 0 Þ þ u0 0 1 ; 4 2 Dm
¼
ð1Þ Bðu1¯ 1 0 Þ cos o1 a sin o2 a pffiffiffi u1 1 0 pffiffiffi u1¯ 1 0 ¼ D1 D2 2 2 " pffiffiffi # cos om a 2 a ðau1 1 0 þ bu1¯ 1 0 Þ þ u0 0 1 ; þ 4 2 Dm ð2Þ
o1 1 ytwist D2 =d 0 ¼ : 1 ytwist D1 =d 0 o2
(8)
Hence, ytwist verifies a second order equation, of which only one solution is compatible with the experiments, namely ytwist ¼ d 0 =D2 þ o1 o2 d 0 = ðD1 D2 Þ: This value depends only weakly on o1 and o2 and can be used to obtain accurate values of the other parameters. Inserting the value so calculated (ytwist ¼ 1:52 0:04 ; compatible with but more precise than our electron diffraction measurement) and the measurements of D1 and Dm in (6) leads to 0:78pap1:34: Since jajp1; we get: 0:78pap1 and, from (7) and the values of D2, Dm and o2 ; 0:023pbp0:277: Since p1 0 þ p0 1 ¼ 1 1 2 ð1 þ aÞ and p1¯ 0 þ p0 1 ¼ 2 ð1 þ bÞ; the final constraints on the pij are 0:89pp1 0 þ p0 1 p1;
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p0 1 X0:4; p1 0 X0:6 and 0pp1¯ 0 þ p0 1¯ p0:11: This confirms that b¯ m is oriented close to [1 1 0]. Moreover, from (8), om ¼ 7:9 3:5 ; so that om ; difficult to measure, is non-zero and has a sign of opposite to that of o1 and o2 : The results confirm and develop the conclusions of Section 2. To summarize, a 2D network of ‘screw’ dislocations accommodates mainly the twist; this DN is neither square nor exactly along the /1 1 0S directions. Tilt is accommodated by a 1D network of mixed dislocations. Their average BV has a large component a along [1 1 0], so that collectively the mixed dislocations have a large screw component and accommodate twist significantly, which increases the period of the screw subnetwork oriented close to [1 1 0]. Such BV selection must be driven by the reduced interface energy accompanying this increased periodicity. The disorientation of the mixed dislocations with respect to [1 1 0] and the component b of their BV along ½1¯ 1 0 produce an edge component cancelled by the disorientations of the screw dislocations with respect to /1 1 0S. More details are given elsewhere [12].
4. Designing DNs for the ordered growth of QDs We now explain how this analysis allowed us to design DNs adapted to the ordered growth of QDs. Fig. 4 details a unit cell of the hexagonal DN formed by the interaction of the mixed dislocations with those of subnetwork #2. The only arbitrary choice is that of the sign of the BV common to all #2 screw lines, here called 12 ½1 1¯ 0a (i.e. the opposite of Fig. 3(b)). Recall that two sequences ‘a’ and ‘b’ are possible for each mixed dislocation. From Fig. 4, which shows the screw and edge component of each BV, two conclusions can be drawn. First, if the segments of each mixed dislocation are of equal length, their edge components cancel. Second, a ‘b’ sequence produces large screw BV components; the sum of two consecutive components is along ½1¯ 1¯ 0 and induces a twist in the same rotational direction as subnetwork #2; conversely, an ‘a’ sequence produces a smaller twist in the opposite direction. Hence, subnetwork #1 can be totally eliminated, provided ‘b’ se-
_
a2
m2
011 N2
+
screw
110 100 a1 m1
b1 screw
_
b2
m2
a1
m1
011
101
010
a2
101
+
b2 1 10
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110
_ 011
101 _
N1 011
+ +
101
b1 screw
Fig. 4. Unit cell of the DNs used for growth. Inset: crystalline directions. m1, m2: mixed dislocations. Large arrows as in Fig. 2(b). Near each segment are given the possible BVs (with factor a/2 omitted) and the screw and edge components of their inplane projection (small arrows). Full (resp. dashed) ellipses indicate segments strongly contrasted in TEM images formed with diffraction vectors along 200 (resp. 020). ‘+’ and ‘’ give the type of strain in the region above the interface (see text).
quences dominate and the balance between ‘b’ and ‘a’ sequences produces exactly the same twist as subnetwork #2. This is possible when the density of mixed dislocations is larger than twice the nominal screw density (Table 2, bottom line) by a factor between 1 and 2 (depending on the angle between segments). This is the ultimate stage of the mechanism explained in Section 3, the partial ‘replacement’ of subnetwork #1 by the screw components of the mixed DN now becoming total. Then, the DN consists solely of hexagonal cells. Such DNs were fabricated by selecting smaller crystal disorientations. The twist was set as close to zero as possible but we kept enough tilt to have D2 bDm : TEM confirms the absence of subnetwork #1. As mentioned above, this requires a given proportion of dominant ‘b’ and minority ‘a’ sequences. This balance can be visualized by TEM by selecting a reflection g of 200 type (Fig. 5). The rules governing dislocation contrast [13] imply that every second segment of each mixed dislocation should be strongly contrasted for g ¼ 200; the other half of the segments being weakly contrasted. The pattern should reverse for g ¼ 020: All screw segments should be strongly contrasted for both g (Figs. 4 and 5(b)). Thus, in Fig. 5(a), the segments of mixed dislocations in contrast for
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Fig. 5. (a) TEM plan-view 200 dark field image of the GB. Some hexagonal cells are highlighted and steps s, s’ and nonstandard quadruplet of hexagons (P) indicated. (b) Segments of screw and mixed dislocations contrasted strongly (full lines) and weakly (dashed lines) for diffraction vector along 200. Directions and sequence types as in Fig. 4. Dotted line: average orientation of the dark lines of (a).
g ¼ 200 have BV 12 ½1¯ 0 1a (resp. 12 ½1 0 1a) if they belong to sequences ‘b’ (resp. ‘a’). Moreover, upon going from a mixed dislocation to the next one (m1 to m2), the pattern of contrasted and faint segments should shift laterally by 12 D2 if and only if the sequences of the two dislocations are of the same type. In Fig. 5(a), we observe parallel and equidistant dark lines. Each line has steps (e.g., s and s0 ) shifting the lines in the same direction. This can be interpreted as follows (Fig. 5(b)). The portion of line between two steps is constituted by ‘b’ segments linked by barely visible short screw segments. A step denotes the absence of lateral shift between the segments in contrast for two consecutive mixed dislocations and corresponds to the insertion of a single ‘a’-type segment between two ‘b’-type series. Since the remaining mixed
segments appear faintly, each hexagon can be reconstituted. Between steps, the strongly contrasted lines should thus be parallel and equidistant. This is always verified (Fig. 5(a)). Moreover, their steps should be correlated, with two nonstandardly contrasted hexagons (h1 and h2 in Fig. 5(b)) between steps belonging to two neighbouring lines. This is frequently the case, but we also find sets of four non-standardly contrasted hexagons (e.g., around P in Fig. 5(a)); such phase shifts could be due to disruptions of the sequences by cavities or to missing screw segments. Constituting such patterns is essential for the subsequent growth of nanostructures. Indeed, the edge component of two consecutive segments of each mixed dislocation have opposite directions (Fig. 4). Hence, the strains induced on a given side of the interface by such a pair also alternate between dilatation and contraction (normal to the line), as indicated by ‘+’ and ‘’ in Fig. 4. These signs remain arbitrary as long as we do not specify relatively to which crystal Fig. 4 is drawn. Let us assume that ‘+’ corresponds to an expansion in the crystal on which growth will proceed. Fig. 6 shows several hexagons of the DN in a ‘step-free’ area and the dilatational surface strain induced by each mixed segment. Screw segments remain neutral, since they only induce shear and, for symmetry, dilatation cancels at the centre of the hexagons. Thus, dislocation interaction and elimination of subnetwork #1 generates a periodic 2D pattern of alternating compressed and dilated areas at the surface. Such a pattern should affect the deposition of both lattice-matched and lattice-mismatched material. We expect GaAs to grow preferentially in unstrained areas (dashed ovals in Fig. 6) and InAs
-
+
-
+
-
+
+
-
+
-
+
-
-
+
-
+
-
+
+
-
+
-
+
-
Fig. 6. Hexagonal cells of the DN. ‘+’ and ‘’ indicate dilatation and contraction above the interface. Full (resp. dashed) ellipses indicate regions where (Ga)InAs (resp. GaAs) grows preferentially.
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to grow preferentially in dilated areas (‘+’ in Fig. 6). Our first experiments confirm that this is the case. Indeed, we observe that during a standard growth sequence, the thicknesses of both the GaAs buffer layer and the InAs layer vary. Moreover, the periods, orientations and positions of these variations are correlated with those of the underlying DN. More details are given elsewhere [12,14]. To summarize, we studied in detail the dislocation networks generated at the interface obtained by bonding two slightly twisted and tilted GaAs crystals. Dislocation interaction leads to the formation of a periodic pattern of alternating dilatational and compressive strain above the interface. Since the geometrical parameters of this pattern can easily be adjusted by selecting appropriate twist and tilts, the method offers a large flexibility for controlling the geometrical parameters of the surface strain field, and hence, hopefully, of the subsequently grown nanostructures. Acknowledgements This work was partly supported by Re´gion Ile de France, SESAME project No 1377 and Conseil Ge´ne´ral de l’Essonne.
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References [1] L. Goldstein, F. Glas, J.Y. Marzin, M.N. Charasse, G. Le Roux, Appl. Phys. Lett. 47 (1985) 1099. [2] Y.H. Lo, R. Bhat, D.M. Hwang, M.A. Koza, T.P. Lee, Appl. Phys. Lett. 58 (1991) 1961. [3] G. Patriarche, F. Jeanne`s, J.L. Oudar, F. Glas, J. Appl. Phys. 82 (1997) 4892. [4] A. Bourret, Surf. Sci. 432 (1999) 37. [5] A.E. Romanov, P.M. Petroff, J.S. Speck, Appl. Phys. Lett. 74 (1999) 2280. [6] H. Brune, M. Giovannini, K. Bromann, K. Kern, Nature 394 (1998) 451. [7] F. Leroy, J. Eymery, P. Gentile, Appl. Phys. Lett. 80 (2002) 3078. [8] J. Coelho, G. Patriarche, F. Glas, G. Saint-Girons, I. Sagnes, J. Phys. C. 16 (2004) 7941. [9] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley, New York, 1982, pp. 26, 703–8, 723. [10] M. Benamara, A. Rocher, A. Laporte, G. Sarrabayrousse, L. Lescouze`res, A. PeyreLavigne, M. Fnaeich, A. Claverie, Mater. Res. Soc. Symp. Proc. 378 (1995) 863. [11] J.G. Zhu, C.B. Carter, Philos. Mag. 62 (1990) 319. [12] J. Coelho, G. Patriarche, F. Glas, G. Saint-Girons, I. Sagnes, L. Largeau, Phys. Rev. B 70 (2004) 155329. [13] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy of Thin Crystals, Krieger, Malabar, 1977, p. 252. [14] J. Coelho, G. Patriarche, F. Glas, G. Saint-Girons, I. Sagnes, J. Crystal Growth, this issue.
Journal of Crystal Growth 275 (2005) e1647–e1653 www.elsevier.com/locate/jcrysgro
Buried dislocation networks for the controlled growth of III–V semiconductor nanostructures F. Glas, J. Coelho, G. Patriarche, G. Saint-Girons CNRS/LPN, Laboratoire de Photonique et de Nanostructures, route de Nozay, 91460 Marcoussis, France Available online 24 December 2004
Abstract Shallowly buried dislocation networks (DNs) can be used to organize the formation of epitaxial semiconductor quantum dots, via the periodic strain field induced at the growth front. We fabricate such DNs by bonding epitaxially two GaAs crystals twisted and tilted with respect to each other and studied them by transmission electron microscopy. The DNs differ greatly from the mere superposition of a square screw DN and of a one-dimensional mixed DN accommodating, respectively twist and tilt. We present a detailed quantitative analysis of these DNs and explain how their properties relate to the crystal disorientations. Dislocation interaction generates a surface pattern of dilatational and compressive strain favouring the ordered growth of (Ga)InAs nanostructures. r 2004 Elsevier B.V. All rights reserved. PACS: 61.72.Lk; 61.72.Mm; 68.37.Lp; 68.35.Gy; 68.65.k Keywords: A1. Interfaces; A1. Line defects; A1. Nanostructures; A1. Transmission electron microscopy; A3. Epitaxial wafer bonding; B2. Semiconducting III–V materials
1. Introduction Quantum dots (QDs) may be formed spontaneously when a semiconductor is deposited epitaxially on a mismatched substrate [1], but the sizes, shapes, compositions and spatial distribution of the QDs usually remain fairly uncontrolled. However, by homogenizing these quantities, better properties could be achieved for existing devices Corresponding
author. Tel.: +33 1 69 63 60 79; +33 1 69 63 60 06. E-mail address: [email protected] (F. Glas).
fax:
and, by tailoring then, we could fabricate new devices. A promising way to order QDs is to use the strain field induced at the growth front by a shallowly buried periodic dislocation network (DN), which affects the surface potentials of the deposited species and the nucleation sites of the QDs. By choosing appropriate DNs, it might become possible to order laterally QDs having identical shapes an sizes. Such DNs can be obtained by epitaxial wafer bonding [2,3]. Elasticity calculations show that edge dislocations should be most efficient because they induce
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.11.219
ARTICLE IN PRESS e1648
F. Glas et al. / Journal of Crystal Growth 275 (2005) e1647–e1653
dilatational surface strains [4,5]. However, their periods, governed by the misfit between the two bonded crystals, are not easily adjusted. On the other hand, the period of a screw DN can be selected by simply choosing the twist angle between the two crystals, but such DNs induce pure shear strains at the surface, which affect only weakly the deposited species [4]. So far, lateral organization of QDs mediated by DNs has been reported for metals [6] and for Ge on Si [7] but not for III–V materials. In this paper, we present a detailed analysis of DNs fabricated for growing (Ga)InAs QDs on GaAs. We obtain shallowly buried DNs by bonding epitaxially a GaAs host substrate to a GaAs substrate on which several lattice-matched layers have been grown. Details are given elsewhere [8]. The final assembly is composed of a 20 nm thick GaAs layer tilted and twisted with respect to the host substrate to which it is bonded. The DNs lie in the interface, which is a grain boundary (GB). Tilt is achieved by using vicinal substrates with normals slightly and equally disoriented from [0 0 1] around a /1 0 0] direction. We bond them with their one-dimensional (1D) sets of surface steps nearly orthogonal, so that the resultant tilt axis [3] is close to a /1 1 0S direction, hereafter noted [1 1 0]. A small additional twist is set up by rotating the crystals by up to a few degrees around the normal to the interface. We study the DNs by transmission electron microscopy (TEM).
2. Structure and first analysis of the DNs In these materials, the Burgers vectors (BVs) are of type a=2h1 1 0i: Taken independently, twist may be accommodated by a square two-dimensional (2D) network of screw dislocations with ‘in-plane’ a=2h1 1 0 BVs, whereas tilt accommodation requires a BV edge component normal to the GB [9]. From the previous studies of Si/Si [10] and GaAs/ InP bonding [3], we expect the latter to be provided by a 1D network of mixed dislocations oriented along the tilt axis (601 dislocations if this axis is /1 1 0]) (Table 1). For these two ideal DNs, the dislocation period is D ¼ 12 b0 =sinðy=2Þ; with y
Table 1 Moduli of the BV components of ideal screw and 601 mixed dislocations Dislocations
Screw 601 mixed
Burgers vector component Screw
Edge, in GB
Edge, normal to GB
pffiffiffi a 2=2 pffiffiffi a 2=4
0 pffiffiffi a 2=4
0 a/2
Fig. 1. TEM weak beam plan-view images of the GB: (a) g ¼ 2 2 0; showing mainly screw subnetwork #2; (b) g ¼ 2¯ 2 0; showing screw subnetwork #1 (full arrow) and mixed DN (dashed arrow).
Table 2 Characteristics of the DNs of the sample studied
DN
Index j
Nearest /1 1 0S direction oj ð1Þ
Mixed Screw #1 Screw #2 Ideal screw
m 1 2
[1 1 0] [1 1 0] ½1¯ 1 0
2.870.8 2.171.7 0
Dj (nm) 27.972.8 21.070.6 15.070.3 13.972.1
All values are measured, except those on the last line (period calculated from ytwist ).
the crystal disorientation and b0 the BV component accommodating it (Table 1). In this section, we give a general analysis of the DNs and illustrate it for a typical sample (Fig. 1 and Table 2). We might expect the actual DNs to be a mere superposition of the two abovementioned ideal DNs. Indeed, TEM analysis of the GB (Fig. 1) shows a 2D DN with in-plane a=2h1 1 0i BVs (reminiscent of the ideal screw DN) and a 1D set of mixed dislocations oriented close
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to the twist axis, having both in-plane and normal BV components. However, the major differences are: (i) the periods of the two subnetworks of the 2D DN are different; (ii) these subnetworks are slightly disoriented with respect to the /1 1 0] directions. We note o the disorientation of a DN with respect to its neighbouring /1 1 0] direction and D its period; indices m, 1 and 2 specify, respectively, the mixed DN and the two subnetworks of the 2D DN oriented close to [1 1 0] and ½1¯ 1 0 (Fig. 2, drawn for om o0). We measure o1 ; o2 and all periods by TEM (Table 2), and the actual twist angle ytwist (here equal to 1.6570.251), but not the tilt ytilt ; by electron diffraction. Moreover (iii) each mixed dislocation interacts with the nearly orthogonal dislocations of subnetwork #2 and becomes a sequence of segments disoriented from the average line direction. Each interaction shifts the dislocation of subnetwork #2 by half a period (Figs. 1(a) and 3). Point (iii) manifests energy-minimizing shifts and rotations already observed when mixed dislocations interact with screw or edge DNs [3,9–11]. Our mixed dislocations have eight possible BVs, each having two components of modulus a=2; respectively along an in-plane /1 0 0] direction and [0 0 1] (Table 1). We assume that all
screw #1
010
110
110
1 2
m b2
b1
screw #2
mixed
100
100
010 Fig. 2. Orientations of the interfacial DNs. All directions are relative to a given crystal. Dashed lines: crystallographic directions. Full and dotted lines: average dislocation directions. Note the rotation of the figure with respect to Fig. 1.
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screw #2
110
screw #2 110
110 110 m D1
011 101
101 011 011
101
011 101
Dm D2 (a)
110
110
(b) screw #2
screw #2
Fig. 3. (a) Schematics of the DNs of Fig. 1, oriented accordingly. Full lines: screw dislocations; dotted lines: mixed dislocations. (b) Interaction between mixed dislocations (m) and screw subnetwork #2. Arrows give line orientation. BVs are indicated for each segment with factor a/2 omitted. BV sequences ‘a’ and ‘b’ appear, respectively above and below the mixed dislocation line. Dashed line: average mixed line.
normal components have the same sign (arbitrarily taken as positive along [0 0 1]); this amounts to assuming that there are no more mixed dislocations than necessary for tilt accommodation. This leaves four possible BVs. Nodal conservation rules [9] imply that each segment of mixed dislocation has its own BV and that only two ‘a’ and ‘b’ sequences are possible (Fig. 3(b)). Any sequence is fully determined once one segment is known, but sequences corresponding to two dislocations are a priori independent. If the segments of the mixed dislocations are of equal lengths, the in-plane component of the average BV is along [1 1 0] and ½1¯ 1¯ 0 for ‘a’ and ‘b’ dislocations, respectively. Point (i) can be understood if ‘a’ sequences outnumber ‘b’ sequences. Then, the in-plane component of the average BV of the whole mixed DN is along [1 1 0], i.e. parallel to the BV of subnetwork #1. Since the mixed dislocations are also oriented close to [1 1 0], twist accommodation is thus shared between the mixed DN and subnetwork #1. No such sharing occurs with subnetwork #2, which has orthogonal BVs. Hence, less #1 dislocations are needed than expected, D1 is larger than expected and D2 as expected. This is exactly what we observe (Table 2). Finally, point (ii) implies that the dislocations of the 2D DN have small edge components. Since the two bonded substrates have identical lattice parameters, these must compensate edge components associated with the
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mixed DN. For simplicity, we shall call ‘screw’ the dislocations of the 2D DN, although this is not strictly appropriate.
3. Quantitative analysis of the DNs These conclusions are supported by a quantitative analysis of the GB. In the sample studied, o1 and o2 have the same sign (Table 2). Since twist is accommodated chiefly by the screw dislocations, the projections of the BVs of the latter along their lines must be of the same sign; pffiffiffi we arbitrarily choose these BVs to be b ¼ ða= 2Þu1 1 0 and b2 ¼ 1 pffiffiffi ða= 2Þu1¯ 1 0 ; respectively, where uhkl is a unit vector along [hkl] (Fig. 2). We note p1 0 ; p0 1 ; p1¯ 0 and p0 1¯ pthe four ffiffiffi probabilities pffiffiffi of the p ffiffiffi possible BVspffiffiffiða= 2Þu1 0 1 ; ða= 2Þu0 1 1 ; ða= 2Þu1¯ 0 1 and ða= 2Þu0 1¯ 1 of the mixed dislocations, which verify 0ppij p1 and p1 0 þ p0 1 þ p1¯ 0 þ p0 1¯ ¼ 1: The geometry of all combinations of periodic linear interfacial DNs compatible with given disorientations between two crystals is prescribed by Frank’s formula [9]. If vector V of one crystal transforms into V0 (having the same coordinates) in the other, this formula gives the ‘closure defect’ BðVÞ ¼ V V0 in terms of the geometry and BVs of the dislocations. Taking n as having a (large) positive component along u0 0 1 and applying Frank’s formula to vectors u1 1 0 and u1¯ 1 0 and to our three DNs (#1, #2 and mixed), we get
where a ¼ p1 0 p1¯ 0 þ p0 1 p0 1¯ and b ¼ p1¯ 0 p1 0 þ p0 1 p0 1¯ characterize the projection pffiffiffi in the GB of the average BV b¯ m ¼ 14 a 2ðau1 1 0 þ bu1¯ 1 0 Þ þ 12 u0 0 1 of the mixed dislocations. We retain only the terms of first order in angles o1, o2, om, ytwist and ytilt ; which are all small. If ytwist 40 corresponds to a rotation from u1 1 0 towards u1¯ 1 0 ; in terms of crystal disorientations, the closure defects are then simply Bðu1 1 0 Þ ¼ ytwist u1¯ 1 0 þ om ytilt u0 0 1 and Bðu1¯ 1 0 Þ¼ ytwist u1 1 0 ytilt u0 0 1 : Using these formulas in Eqs. (1) and (2), projecting on u0 0 1 ; u1 1 0 and u1¯ 1 0 and setting d 0 ¼ pffiffiffi a 2 yields: ytilt ¼ a=2Dm ;
(3)
aom ¼ 2o1 Dm =D1 ;
(4)
bom ¼ 2Dm ð1=D2 ytwist =d 0 Þ;
(5)
a ¼ 2Dm ð1=D1 ytwist =d 0 Þ;
(6)
b ¼ 2o2 Dm =D2 :
(7)
Eq. (3) simply expresses tilt accommodation by the mixed DN and yields ytilt ¼ 0:58 0:06 (compatible with substrate vicinalities of 0:5 0:1 ). From (4)–(7) and the experimentally determined quantities and their uncertainties (Table 2), we want to get om, a and b. Since the DN periods are measured more precisely than the disorientations, we first calculate a refined value of ytwist rather than using our measurement. From (4)–(7), we get om ¼
Bðu1 1 0 Þ sin o1 a cos o2 a pffiffiffi u1 1 0 pffiffiffi u1¯ 1 0 D1 D2 2 2 " pffiffiffi # sin om a 2 a ðau1 1 0 þ bu1¯ 1 0 Þ þ u0 0 1 ; 4 2 Dm
¼
ð1Þ Bðu1¯ 1 0 Þ cos o1 a sin o2 a pffiffiffi u1 1 0 pffiffiffi u1¯ 1 0 ¼ D1 D2 2 2 " pffiffiffi # cos om a 2 a ðau1 1 0 þ bu1¯ 1 0 Þ þ u0 0 1 ; þ 4 2 Dm ð2Þ
o1 1 ytwist D2 =d 0 ¼ : 1 ytwist D1 =d 0 o2
(8)
Hence, ytwist verifies a second order equation, of which only one solution is compatible with the experiments, namely ytwist ¼ d 0 =D2 þ o1 o2 d 0 = ðD1 D2 Þ: This value depends only weakly on o1 and o2 and can be used to obtain accurate values of the other parameters. Inserting the value so calculated (ytwist ¼ 1:52 0:04 ; compatible with but more precise than our electron diffraction measurement) and the measurements of D1 and Dm in (6) leads to 0:78pap1:34: Since jajp1; we get: 0:78pap1 and, from (7) and the values of D2, Dm and o2 ; 0:023pbp0:277: Since p1 0 þ p0 1 ¼ 1 1 2 ð1 þ aÞ and p1¯ 0 þ p0 1 ¼ 2 ð1 þ bÞ; the final constraints on the pij are 0:89pp1 0 þ p0 1 p1;
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p0 1 X0:4; p1 0 X0:6 and 0pp1¯ 0 þ p0 1¯ p0:11: This confirms that b¯ m is oriented close to [1 1 0]. Moreover, from (8), om ¼ 7:9 3:5 ; so that om ; difficult to measure, is non-zero and has a sign of opposite to that of o1 and o2 : The results confirm and develop the conclusions of Section 2. To summarize, a 2D network of ‘screw’ dislocations accommodates mainly the twist; this DN is neither square nor exactly along the /1 1 0S directions. Tilt is accommodated by a 1D network of mixed dislocations. Their average BV has a large component a along [1 1 0], so that collectively the mixed dislocations have a large screw component and accommodate twist significantly, which increases the period of the screw subnetwork oriented close to [1 1 0]. Such BV selection must be driven by the reduced interface energy accompanying this increased periodicity. The disorientation of the mixed dislocations with respect to [1 1 0] and the component b of their BV along ½1¯ 1 0 produce an edge component cancelled by the disorientations of the screw dislocations with respect to /1 1 0S. More details are given elsewhere [12].
4. Designing DNs for the ordered growth of QDs We now explain how this analysis allowed us to design DNs adapted to the ordered growth of QDs. Fig. 4 details a unit cell of the hexagonal DN formed by the interaction of the mixed dislocations with those of subnetwork #2. The only arbitrary choice is that of the sign of the BV common to all #2 screw lines, here called 12 ½1 1¯ 0a (i.e. the opposite of Fig. 3(b)). Recall that two sequences ‘a’ and ‘b’ are possible for each mixed dislocation. From Fig. 4, which shows the screw and edge component of each BV, two conclusions can be drawn. First, if the segments of each mixed dislocation are of equal length, their edge components cancel. Second, a ‘b’ sequence produces large screw BV components; the sum of two consecutive components is along ½1¯ 1¯ 0 and induces a twist in the same rotational direction as subnetwork #2; conversely, an ‘a’ sequence produces a smaller twist in the opposite direction. Hence, subnetwork #1 can be totally eliminated, provided ‘b’ se-
_
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m2
011 N2
+
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110 100 a1 m1
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_
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m1
011
101
010
a2
101
+
b2 1 10
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110
_ 011
101 _
N1 011
+ +
101
b1 screw
Fig. 4. Unit cell of the DNs used for growth. Inset: crystalline directions. m1, m2: mixed dislocations. Large arrows as in Fig. 2(b). Near each segment are given the possible BVs (with factor a/2 omitted) and the screw and edge components of their inplane projection (small arrows). Full (resp. dashed) ellipses indicate segments strongly contrasted in TEM images formed with diffraction vectors along 200 (resp. 020). ‘+’ and ‘’ give the type of strain in the region above the interface (see text).
quences dominate and the balance between ‘b’ and ‘a’ sequences produces exactly the same twist as subnetwork #2. This is possible when the density of mixed dislocations is larger than twice the nominal screw density (Table 2, bottom line) by a factor between 1 and 2 (depending on the angle between segments). This is the ultimate stage of the mechanism explained in Section 3, the partial ‘replacement’ of subnetwork #1 by the screw components of the mixed DN now becoming total. Then, the DN consists solely of hexagonal cells. Such DNs were fabricated by selecting smaller crystal disorientations. The twist was set as close to zero as possible but we kept enough tilt to have D2 bDm : TEM confirms the absence of subnetwork #1. As mentioned above, this requires a given proportion of dominant ‘b’ and minority ‘a’ sequences. This balance can be visualized by TEM by selecting a reflection g of 200 type (Fig. 5). The rules governing dislocation contrast [13] imply that every second segment of each mixed dislocation should be strongly contrasted for g ¼ 200; the other half of the segments being weakly contrasted. The pattern should reverse for g ¼ 020: All screw segments should be strongly contrasted for both g (Figs. 4 and 5(b)). Thus, in Fig. 5(a), the segments of mixed dislocations in contrast for
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Fig. 5. (a) TEM plan-view 200 dark field image of the GB. Some hexagonal cells are highlighted and steps s, s’ and nonstandard quadruplet of hexagons (P) indicated. (b) Segments of screw and mixed dislocations contrasted strongly (full lines) and weakly (dashed lines) for diffraction vector along 200. Directions and sequence types as in Fig. 4. Dotted line: average orientation of the dark lines of (a).
g ¼ 200 have BV 12 ½1¯ 0 1a (resp. 12 ½1 0 1a) if they belong to sequences ‘b’ (resp. ‘a’). Moreover, upon going from a mixed dislocation to the next one (m1 to m2), the pattern of contrasted and faint segments should shift laterally by 12 D2 if and only if the sequences of the two dislocations are of the same type. In Fig. 5(a), we observe parallel and equidistant dark lines. Each line has steps (e.g., s and s0 ) shifting the lines in the same direction. This can be interpreted as follows (Fig. 5(b)). The portion of line between two steps is constituted by ‘b’ segments linked by barely visible short screw segments. A step denotes the absence of lateral shift between the segments in contrast for two consecutive mixed dislocations and corresponds to the insertion of a single ‘a’-type segment between two ‘b’-type series. Since the remaining mixed
segments appear faintly, each hexagon can be reconstituted. Between steps, the strongly contrasted lines should thus be parallel and equidistant. This is always verified (Fig. 5(a)). Moreover, their steps should be correlated, with two nonstandardly contrasted hexagons (h1 and h2 in Fig. 5(b)) between steps belonging to two neighbouring lines. This is frequently the case, but we also find sets of four non-standardly contrasted hexagons (e.g., around P in Fig. 5(a)); such phase shifts could be due to disruptions of the sequences by cavities or to missing screw segments. Constituting such patterns is essential for the subsequent growth of nanostructures. Indeed, the edge component of two consecutive segments of each mixed dislocation have opposite directions (Fig. 4). Hence, the strains induced on a given side of the interface by such a pair also alternate between dilatation and contraction (normal to the line), as indicated by ‘+’ and ‘’ in Fig. 4. These signs remain arbitrary as long as we do not specify relatively to which crystal Fig. 4 is drawn. Let us assume that ‘+’ corresponds to an expansion in the crystal on which growth will proceed. Fig. 6 shows several hexagons of the DN in a ‘step-free’ area and the dilatational surface strain induced by each mixed segment. Screw segments remain neutral, since they only induce shear and, for symmetry, dilatation cancels at the centre of the hexagons. Thus, dislocation interaction and elimination of subnetwork #1 generates a periodic 2D pattern of alternating compressed and dilated areas at the surface. Such a pattern should affect the deposition of both lattice-matched and lattice-mismatched material. We expect GaAs to grow preferentially in unstrained areas (dashed ovals in Fig. 6) and InAs
-
+
-
+
-
+
+
-
+
-
+
-
-
+
-
+
-
+
+
-
+
-
+
-
Fig. 6. Hexagonal cells of the DN. ‘+’ and ‘’ indicate dilatation and contraction above the interface. Full (resp. dashed) ellipses indicate regions where (Ga)InAs (resp. GaAs) grows preferentially.
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to grow preferentially in dilated areas (‘+’ in Fig. 6). Our first experiments confirm that this is the case. Indeed, we observe that during a standard growth sequence, the thicknesses of both the GaAs buffer layer and the InAs layer vary. Moreover, the periods, orientations and positions of these variations are correlated with those of the underlying DN. More details are given elsewhere [12,14]. To summarize, we studied in detail the dislocation networks generated at the interface obtained by bonding two slightly twisted and tilted GaAs crystals. Dislocation interaction leads to the formation of a periodic pattern of alternating dilatational and compressive strain above the interface. Since the geometrical parameters of this pattern can easily be adjusted by selecting appropriate twist and tilts, the method offers a large flexibility for controlling the geometrical parameters of the surface strain field, and hence, hopefully, of the subsequently grown nanostructures. Acknowledgements This work was partly supported by Re´gion Ile de France, SESAME project No 1377 and Conseil Ge´ne´ral de l’Essonne.
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