• Sonuç bulunamadı

Electronic structure of Ge-Si superlattices grown on Ge (001)

N/A
N/A
Protected

Academic year: 2021

Share "Electronic structure of Ge-Si superlattices grown on Ge (001)"

Copied!
5
0
0

Yükleniyor.... (view fulltext now)

Tam metin

(1)

Semiconductor Science and Technology

Electronic structure of Ge-Si superlattices grown

on Ge (001)

To cite this article: O Gulseren and S Ciraci 1991 Semicond. Sci. Technol. 6 638

View the article online for updates and enhancements.

Related content

Structure, properties and applications of GexSi1-x strained layers and superlattices S C Jain and W Hayes

-Electronic structure of (111)Si/Ge superlattices

J M Bass and C C Matthai

-Ultrathin SimGen strained layer superlattices-a step towards Si optoelectronics

H Presting, H Kibbel, M Jaros et al.

-Recent citations

Some aspects of the physics of man-made semiconductor heterosystems: superlattice electronic structure and related topics Pierre Masri

(2)

Semicond. Sci. Technol. 6 (1991) W 1 . Printed in me UK

I

I

Electronic structure

of Ge-Si

superlattices grown on Ge

(001)

0

GIilseren and S

Ciraci

Department of Physics, Bilkent University, Bilkent 06533, Ankara, Turkey Received21 January 1991, accepted for publication 13 February 1991

Abstract. We have studied the electronic energy strudure of pseudomorphic GedSi, superlattices by using the empirical tight-binding method. Effects of the band offset. sublanice periodicity and the lateral lattice constant on the transition energies have been investigated. We found that GedSi, superlattices grown on Ge (001) can have a direct band gap, if m + n = 10 and m = 6. However, optical matrix elements for in-plane and perpendicular polarized light are negligible for the transition from the highest valence band to the lowest conduction band state

at

the centre of the superlatlice Brillouin zone.

The novel properties discovered in the electron systems of lower dimensionality have created a major impact in solid state electronics leading to new concepts in electronic de- vices. In this context, it is hoped that pseudomorphic Si-Ge superlattices can compensate for the shortcomings

of

silicon and CM open new horizons in the applications of Si-based

devices in photonics. In

an

effort to fabricate a high car- rier mobility direct-band semiconductor, the Si-Ge superlat- tices grown on the (001) substrate have been the focus of at- tention 11.21. While the electroreflectance measurements of SidGe, grown on Si (001) have indicated new optical transi- tion [l], theoretical studies [3-61 showed that the difference between the direct and indirect band gap, 6E =

-

E&,,,,

decreases with increasing n, but the gap is still indirect. For example, it was found [3,4] that 6 E is

as

small

as

0.07 eV for the largest superlattice periodicity (2n = 12) one can obtain for SiJGe, grown on Si (001). Moreover, optical matrix el- ements of the transition

to

the lowest zone folded states were found U) be small [7].

In view of evidence indicating that the strained Si,/Ge, laterally restricted

to

Si (001) has

an

indirect band gap, attention has been drawn to the Si-Ge superlattices with different structural parameters [8,9]. In fact,

as

ar-

gued earlier

a

pseudomorphic superlattice represented by {Sii-xGex)n,all /{Si,-yGey]m,o,, provides several degrees of freedom for controlling the electronic properties. For exam- ple, in the superlattice with x = 0 and y

=

1 the lateral lattice constant a and the superlattice periodicity, n

+

m ,

(also n and m itsel0

are

important parameters for controlling the electronic structure. Since the lateral lattice constant of asubstrateSi,-,GeZ(001) can vary between U? (equilibrium

latticeconstant of Si, i.e. 5.43

&J")

and

or,

the snblattices grown in registry with this substrate undergo a lateral com- pressive (expansive) strain, while the lattice constant in the

perpendicular direction expands (contracts). This gives rise to a uniform tetragonal distortion if the misfit dislocations are prevented from forming.

In this paper we first investigate the effect of the struc- tural paramekrs on the direct and indirect bmd g a p of the pseudomorphic Si-Ge superlattices. Guided by these find- ings and based on the electronic structure calculations, which were performed for several supercells by using the empirical tight-binding method, we determine the structural parame-

ters which make the band gap direct.

It is known that Si-Ge superlattices generally make type- I1 band alingments, such that the edge of the conduction band and the top of the valance band

occur

in the Si and Ge sublat- tices, respectively. To examine the lowest conduction band states

of

the Si,,/Ge,, we concentrate on the Si-sublattice. The minima of the conduction band of bulk Si occur along the six equivalent

[Ool]

directions. These directions

are

labelled as

A in the (fcc) Brillouin zone (BZ)

of

the bulk Si. In the su-

perlatticegrown on the (001) plane, four A directions, which are labelled by A , , coincide with the r M direction of the su-

perlattice BZ

(see

inset of figure 1). The remaining two

are

along thesuperlattice direction [001] and are labelled by A,. Bands along A, are folded. In the absence of strain in the Si- sublattice the minimum

of

the lowest conduction band stale along the r M direction of the SBZ occurs at relatively lower energy than the lowest folded states along the

TZ

direction. Moreover, the lowest conduction band state was found to be itinerant [3,6]. This is the situation for the strained Si,/Ge,

(n

5

6) superlattices grown on the Si (001). If, however,

a

tetragonal strain is induced by forcing the Si sublattice to have all

>

a? the bands along A,, and A,

are

expected to experience different shifts.

We explored this situation by examining the bands of Si and Ge supercells under tetragonal strain. The variation of

(3)

Electronic structure of Ge-Si superlanices earlier in several calculations. For Si, in figure 1 the IOW-

est conduction band state Z* along the TZ direction and the minimum A; of the lowest conduction band along All are both lowered

as

all + o r . Moreover, the state

Z*

is low-

ered below the A i state, and the energy difference

E A .

-E=*

increases with increasing all

-

asi

we

note that the Z* state band along A,, which is folded to

TZ

upon the superlath formation. Similary, the splitting of the valance bands in- creases as all

-

a y , so that the p,-like state raises towards Z*, butp, states

are

lowered.

II

corresponds to the minimum AL

!'

of the lowest conduction

-0 O0: 4

....

%o

:---::---i

10 - 0.6 - 0 2 ~ -0.2 - W Re 0 4 - 5 L O 5 4 5 5 % 5 55 5 60 5 65 a,( A')

Figure 1. Variation of the energy of the lowest conduction band states

(c,

Aic along

TZ

and r M directions) and

rc

state and highest valance band states

(rv,z

and

rv,xy)

of

Si, and Gem supercells as a fundion of the lateral lattice constant all. The symmetry points and directions of the superlattice Brillouin zone are shown in the inset.

the lowest conduction and highest valance bands of Si, and Gem superlattices are calculated

as

a function of all. Our re- sults

are

presented in figure 1 for m = 20. In these calcu- lations the perpendicular lattice parameter a, corresponding to a given all are calculated by using elastic constants. Ear- lier it was shown that aL obtained from the Poisson ratio is in compliance with that obtained by using ab initio

cal-

culations [lo]. The results of the fully optimized structure calculations on the Si-Ge superlattices show that in general

aL

differs slightly from the valueobtained by using thePois- son ratio. Moreover, interlayer spacings in the sublattices

are

not homogeneous [3,41. Nevertheles, in view

of

Iheaccu- racy obtainable from the empirical tight-binding calculations the structural parameters obtained from the elastic constants suite to the objectives of our calculation. The energy param- etersgivenbyLiandChung[Illarescaled [I21 byd-? with the scaling exponent q

=

2 when the interatomic distances

of the sublattices deviate from their equilibrium values in the presence

of

the strain. Since splitting of the top of the valence band state at the point, &E,,,,

,

is related to the de- formation potential b, and strain components ea and czz as

SE,, = %(ezz

-

ea), the exponent q can be fitted lo the deformation potential. Expressing the deformation poten- tial [13] in Lerms of the strain components, first and second nearest-neighbour tight-binding energy parameters and q , we find q

-

1.8. This justifies the value of exponent 7 = 2 used

I i

0 2 4 6 8 10

m

Figure 2. Variation of the relevant transition energies

of GedSi, with the number of the Ge layers m. The superlattice is laterally restricted to the Ge (001) surface

so that all =

a r

and m + n = 20. (A€, = 0:31 eV).

In view of the above discussion one concludes that the lowest conduction band state, A;,

can

be folded to the cen- tre of superlattice BZ to yield a direct band gap for a pseudo- morphic GeJSi, superlattice grown on Ge (001). This

can

be achieved for certain superlattice periodicity. For exam- ple, if

the

lowest conduction band state A; were occured exactly at k,

=

0,

k, = 0, and k, = 8 ~ / 5 a for a strain-free Si, superlattice, this state A; would fold to

r

for 1 = 20 (which h a s

G,

=

2x/5a for the magnitude

of

lhe shortest re- ciprocal vector). For a strained GeJSi, superlattice grown on Ge (001) A; does not appear exactly at

k,

= 8a/5a.

However, small deviations from the value 8a/5a can ac- counted by varying lattice strain and sublattice periodicity

but by keeping 2(m

+

n ) = 20 so that the minimum value

A*, may occur at I' point upon folding. Figure 2 shows the

variation of various transition energies

as

a function of m for the strained Ge,/Si, superlattice grown on Ge (001). For 5

5

m

5

6 the lowest conduction band state occurs at

r,

at- tributing directness to the band. The superlattice with m = 7 lies at the borderline and beyond m = 7 the Ge character dominates and thus the lowest conduction band state at R dips below that at

r.

Form

<

5 the lowest conduction band state starts to occur either at 2 or along

rZ. These

results are in agreement with the band structure of Gem/Si, which are calculated by using the self-consistent pseudopotential cal- culation (91

as

well

as

experiment [21. Note that the calcu- lated transition energies

as

a functionof thenumber of the&

(4)

0 Gulseren and S Ciraci

layers in the Ge,,,/Si, superlattice is not continuous, but dis- play wiggles in figure 2. This is due to the discrete variation

of

m, which leads to discrete changes in the geometrical pa- rameters and hence to the discontinuous variation in (folded) band energies.

We note that the transition energies in figure 2 are ob- tained from the band structure calculations of the Ge,,,/Si, superlattice. Because of the emprical nature of the method the elements

of

the Hamiltonian matrix (i.e. self-energies and hopping energies)

are

inputs in OUT calculations. In this context, the value

of

the band offset is implemented in our calculations by upshifting the self-energies (or the diagonal elements of the Hamiltonian matrix) of the Ge orbitals by

AE,. The value of the band offset we used in our calcula- tions wasobtainedfrom theearlierab initiocalculations [IO]. Since the band offset is the crucial parameter which influ- ences the conduction band structure of the superlattice, we investigated whether the value of AE, affects our conclu- sions regarding the directness of the band gap. In figure 3 we present the variation of the transition energies

as

a

func- tion of AE, calculated for GedSi, and Ge,/Si, both hav- ing all = a;” The band gap

of

Ge6/Si, is direct

as

long

as AE,

5

1

dV, but the difference between the direct and indirect gap decreases

as

AE, increases. In contrast, the di- rectness of the band gap of Ge,/Si, is sensitive to the value of the band offset. For example, the R,r, transition energy becomes smaller than the lowest direct transition energy and thus the gap becomes indirect if AE,

5

0.3 eV. Note that the band offset AE, is affected due to the upshifting of the

A E v ( e V )

Figure 3. Variation of the transition energies with respect to the band offset, A€, for (top) Ge6/Si, and (bonom)

Ge7/Si3. Superlattices are restricted to the Ge (001).

maximum valance band state by the one-third of the spin- orbit coupling energy A,. The value

of

A, is negligible for Si (A,,

=

0.04 eV) but significant for Ge (A, = 0.3 eV). Consequently, owing to the spin-orbit coupling AE, is in- creased by 0.1 eV for a GdSi superlattice [lo]. The tran- sition energies illustrated in figure 2 are calculated by AE,

which does not include the spin-orbit coupling. As

a

matter

of

fact, this effect is neglected in several studies of Si/Ge su-

perlattices. However, this effect is implicit in the discussion

of

figure

3

in which AE, is taken

as

a parameter.

Table 1. Calculated optical transition energies (eV) from p l h highest valenm band state to the qth lowest

conduaion band state (i.e. Evp

-

ECp) ?I the zone centre. All optical matrix elements for the in-plane

(11)

and perpendicular (I) light polarization are negligible, except those indicated by stars have small but finile values

(- in au). ( A & = 0.31 eV).

Ge&/Ge (001) Ge&/Ge (001)

Energy

11

Pol.

I

Pol. Energy

11

Pol.

I

Pol.

0.62 €.I -€,I 0.57 €,I - E a 0.87 0.84 * €,I - Ev3 0.89 0.85 * 0 66 0.88 0.89

From rhe above discussion it becomes clear that Ge6/Si, grown

on

Ge (001) is a direct band gap semiconductor. How- ever, the directness of the band gap can have i m p o m t tech- nological implications only if the lowest direct transitions

are

allowed and the value of the related uansition matrix ele- ments are subtantial. To this end we calculate optical matrix elements for in-plane and perpendicular polarizations for di-

rect

transitions at

r.

In the emprical tight-binding method the optical matrix element can be approximated [I41 by

T*-,,,,

I

ci;,(k)c:”(w

D I ’ d . 2 ”

where C:,(k) is the

coefficient

of the Blwh sum,

n,

is the Bravais lartice vector and T””, is the distance between the

nuclei, v and U’, oi labels the atomic orbitals. The energy in-

tegral,EkI: = (#DI(~)lH~#DI,(r

-

R,

-

T ” ” , ) ) , corresponds to the energy parameter in our calculation, and

t

is the p- larization vector

of

the light. The optical matrix elements calculated from the above expresion and summarized in lil-

ble 1 are crude but indicate that the direct optical transitions of Ge,/Si, and Ge,/Si,

are

not significant

as

far

as

photonics is concerned. For the light polarized along the superlattice axis (perpendicular polarization), all optical matrix elements ofGe6/Si, andGe,/Si, arenegligibleexcept for the transition from the highest valance band to the second lowest conduc- tion band state (Ev, i Ec2) at the centre of BZ. In the case Of

(5)

Electronic structure of Ge-Si superlatlices [2] Pearsdl T P, Vandenberg J M, Hull R and Bonar J M [3] Froyen S, Wood D M and Zunger A 1987 Phys. Rev.

[4] Ciraci S and Batra I P 1988 Phys. Rev. B 38 1835

Ciraci S, Baratoff A and Batra I P 1990 Phys. Rev. B [5] People R and Jackson S A 1987 Phys. Rev. B 36 [6] Ciraci S, Giilseren 0 and Ellianioglu S 1988 Solid [7] Hyberstsen M S and Schlijter M 1987 Phys. Rev. B

Wong K B, Jams M, Morrison I and Hagon J P 1988 [8] Gel1 M A 1988 Phys. Rev. B 38 7538

191 SatDathv S. Martin R M and Van de Walk C G 1988 1989 Phys. Rev. Left 63 2104

B 36 4547; 1988 Phys. Rev. B 37 6893

41 6063 1310

State Commun. 65 1285

36 9683

Phys. Rev. Lett. 60 2221

transition from the highest valance band to the lowest con- duction band state (Ev,

-

.Ec,) at the centre of the BZ

are

also negligible. In Ge,/Si, strained superlattices only tran- sitions

from

the second and third valance band to the lowest conduction band state

(Ea

-+ E,, and Ev3 -* E e l ) have non-

zerooptical matrix elements. For Ge,/Si, system, transitions from second and third valance band states to the second con- duction band state

(Ea

-

&) and

Ea

-+ Ec3) have small but finite optical matrix element at the zone centre.

In conclusion, both GeJSi, and Gq/Si, strained super- lattices have direct band gaps but optical matrix elements for corresponding transitions are negligible. This is in agree- ment with experiment, which fails to observe these transi-

tions. Possible direct transitions are from the second and third valance band states to the first (for GeJSi,) and to the second (for Ge,/Si,) conduction band states.

References

[ l ] Pearsall T P, Bevk J. Feldmann L C, Bonar J, Mannaerts J P and Ourmazd A 1987 Phys. Rev. Len. 58 729

Pearsall T P, Bevk J, Bean J C, Bonar J. Mannaerts J P and Ourmazd A 1989 Phys. Rev. B 39 3741

..

Phys'Rev. B 38 13237

I101 Van de Wale C G and Martin R M 1986 Phys. Rev. B

. .

34 5621

[ l l ] Li Y and Lin-Chuq

P

J 1983 Phys. Rev. B 273465 [12] Harrison W A and Ciraci S 1974 Phys. Rev. B 10

1516

Harrison W A 1980 Nectronic Structure and Properlies of Solids (San Fransdsco: Freeman} p 253

[13] Potz W and Vogl P 1981 Phys. Rev. B 24 2025 [14] Ren Y and Harrison W A 1981 Phys. Rev. B 23 762

Brey L and Tejedor J 1983 Solid State Commun. 48 403

Şekil

Figure  1.  Variation  of  the  energy  of  the lowest conduction  band states  (c, Aic  along  TZ  and  r M   directions) and  rc
Table  1.  Calculated optical transition energies  (eV)  from  p l h  highest  valenm  band state to the  qth  lowest  conduaion band  state (i.e

Referanslar

Benzer Belgeler

of the parse tree, the system checks the results coming from the traversal of its child nodes and it decides to form a single Prolog query or to create a connector query node

Nonalkolik yağlı karaciğer hastalığı grubunda hiperlipidemisi ve hipertrigliseridemisi olan ve olmayan hastalar ACE düzeyleri açısından

Bitki örtüsü özellikleri olarak; toprağı kaplama oranı, örtü materyali ve toprakaltı bioması miktarı ile mera kalite derecesi, toprak özellikleri ise; kil, silt ve kum oranı,

Consequently, although in recent years Turkey has taken important steps towards consolidating its democracy, it failed to completely democratize its political regime.. Thus,

“A reorientation of the organizational logic [of] national politics and policy making” (Ladrech 1994) will be the result of European transformation in security

Turkey’s another concrete reaction was in the form of using hard power capacity in northern Syria following the increasing level of threat from several VNSAs both inside and

A re- stricted NP decision rule aims to maximize the average detection prob- ability under the constraints on the worst-case detection and false-alarm probabilities, and adjusts

Normalized gain-bandwidth product as a function of nor- malized membrane radius or thickness for receiver cMUTs without tuning.. For this choice, we use the f 1 curve