Impurity induced vibrations in light doped silicon

Solid State Communications, Vol. 62, No. 3, pp. 163-167, 1987. Printed irl Great Britain.

0038-1098/87 $3.00 + .00 © 1987 Pergamon Journals Ltd.

IMPURITY INDUCED VIBRATIONS IN LIGHT DOPED SILICON Fu Ying and Xu Wenlan Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Academia Sinica, 420, Zhong San Bei Yi Road, Shanghai, China and Zheng Zhaobo Physics Department, University of Science and Technology of China (Received 9 September 1986 by A.A. Maradudin)

The local densities of states (LDOSs) are calculated by use of the recursion method for the lithium or phosphorous-compensated boron-doped crystalline silicon. The impurity induced local and quasi-local vibrational modes are studied. Good agreement between experiments and calculation is achieved. THE OPTICAL PROPERTIES related to phonon processes for doped, mixed and amorphous materials have been studied vastly and successfully in experiments. Intrared and Raman scattering spectra have been used most widely for this purpose. Particular attention has been given to lithium or phosphorous-compensated and boron-doped crystalline silicon. Cardona et al. [1] have found experimentally that there exist not only the local modes outside the main band of the host silicon but also the so-called quasi-local mode at frequency 225 cm -~ in this system [ 1]. With the calculation and analysis of the local densities of states (LDOSs) of impurity atoms by use of the recursion method which has been developed recently, the impurity induced lattice vibrations are discussed in detail in this paper. Good agreement between the calculation and experiments has been achieved over full frequency range for the lattice vibrations.

between k-th and k'-th atoms, m k is the mass of the k-th atom, w be the frequency. The Green's function is the defined as (4)

G(w 2)(lw 2 -- H) = I,

here I is the unit matrix. The recursion method constructs a new set of normalized orthogal basis in which a general matrix H is turned into a tri-diagonal form. The diagonal elements of the corresponding Green's function matrix, which is called the local Green's functions, can be expressed as the continued fractions by the diagonal and off-diagonal elements (ai, bi) of the tri-diagonized matrix H, the LDOSs then are obtained over full frequency range from the local Green's function: - b~/w 2 -a2 --b~/ ....

G l l ( w 2) = 1/w ~ - a ~

g(w) = - - 2 w Lim(Im(Gll (w 2 + ie)))/Ir.

(5)

(6)

e"~O

1. RECURSION METHOD, PHYSICAL MODELS AND INTERACTIONS We start from the eigenequation of the lattice vibration: H l u ) = w 2 lu),

(1)

where the elements u ~ of the eigenfunction lu) are related to the components Akx of the vibration amplitude in x direction of the k-th atom by the relation: Ukx = ~/mkxA~x ,

E = (w 2 -- a -- ((w 2

--

a) 2

--

4b 2)1/2)/2 ,

(7)

where a and b are determined experimentally as WZmin = a - - 2b,

(8)

2

(2)

wrenx = a + 2b,

(3)

where Wmin = O, wmax is the highest frequency value of the host silicon. In S i : B - L i and S i : B - P systems, boron and phosphorous are substitutional, lithium atoms are inter-

and Hkk'xx' of the matrix H is defined as Hkk'xx' = ¢bkk'xx' /~'-mkmk' ,

With the method of the atomic structure construction developed by Wu and Zhaobo [2], we have calculated the first 18 pairs of a,b coefficients, the remaining part of the continued fraction beyond the 18th recursion cycle is replaced by an asymtotic form:

rbkk'**' is the xx' element of the interaction matrix

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iMPURITY INDUCED VIBRATIONS 1N LIGHT DOPED SILICON

Vol. 62, No. 3

for silicon atoms, the parameters (Asisi , Bsisi ) ~re obtained using the equation 2

A sisi = m si Wmax/8,

(10)

and Bsisi/Asisi = 0.65 is selected to give a best fit to the observed frequencies at F, X and L points. Figure 2(d) shows the calculated density of the states which is in good agreement with those obtained by K-space method and other ones for the crystalline silicon. The disturbance induced by the introduction of impurities are in many aspects, such as the changes of Fig. 1. Interstitial impurity configuration in a diamond the symmetry, the ionic or covalent properties and the structure. length of interaction bonds. So besides the change of mass for the vibrations, variations of interaction parastitial in a tetrahedral site [3, 4]. The number of silicon meters are also vital to the impurity and impurity pair atoms per cubic centimeter is about 5 x 1022 , the boron induced modes. M. Vandevyer et al. show that the interconcentration is about 10 TM c m - 3 , and those of lithium action parameters decrease when the bond ionicity and and phosphorous are the same. The impurities exist length increase [5]. This is exactly the case in our mainly in the form of B-Li and B - P as a result of com- calculation. pensation, in addition, the probabilities for B,P and Li The interaction between the single isolated impurity atoms existing isolately in silicon are much larger than atom and its nearby silicon atoms has also the bond those of B-B, P - P and Li-Li pairs for the remained symmetry of C3o, so the interaction matrix is in the uncompensated impurity atoms. Figure 1 shows the form of (9) with y = B,P,Li and z = Si. Since the interpossible impurity configuration in a diamond structure. stitial lithium atom interacts with host silicon atoms via The interaction is in 3 x 3 matrix form for the four 'pseudo' bonds, the interactions between them are vibrational behaviour. For the interaction between atoms weaker as compared with B-Si and P-Si interactions. with Car bond symmetry, the matrix has the form of Due to the compensation of the oppositely charged atoms, the interactions between the paired impurity atoms must be different from those between isolated Ayz Byz By!] impurity atoms and host silicon atoms. The most importBYz Ay, Byz , (9) ant point here we believe, is the Coulomb force in addition to the bond interaction. That is especially BYz Byz Ay important in the B-Li case. Since interstitial lithium atom interacts with its neighbours by four weak 'pseudo' bonds, further more the B - L i bond is weakened by the increase of B-Li bond ionicity, the net interaction between boron and lithium therefore is attractive. The nearby lattice will be relaxed, the most effected positions for such relaxation are along the B - L i line (see Fig. 1), the distance betwen boron and lithium is shortened and i the one between boron and silicon-1 in B - L i line is b increased. The interaction between boron and silicon-1 is weakened. The interactions between boron and other three silicon atoms and those between lithium and Si ¢ atoms may also be weakened, but our calculations shows that further weakening are not enough to influence the I LDOSs of the boron atom, we simply keep those interactions the same as in isolated cases. B - P pair is considered similarly. For the strong i 60O b 200 400 bond interaction of B - P bond, the net interaction Wovenurnber ( e r a -~) between them is still repulsive, and the lattice relaxation is small. Table 1 gives out the interaction parameters of Fig. 2(a) LDOS of boron: (b) LDOS of first nearest Si atom: (c) LDOS of second nearest Si atom: (d) Density- various situations and the comparison between calculation and experiment results. of-states of crystalline silicon. -

-

i

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IMPURITY INDUCED VIBRATIONS IN LIGHT DOPED SILICON

Table 1-1. About the single isolated impurities Impurity

Interactions

X°B 11B 6Li 7Li 3x p

5.135

2.791

1.675

1.088

5.024

1.675

w ~

wt

Ref.

ca~c.

exper,

calc.

exper.

226.13 226.13

225 225

639.79 617.73 535.00 523.97

644 620 539 522

444.00

440.4

[4] [1] [3]

Table 1-2. About isolated compensated impurity pairs Impurity l°B-6Li

Interactions between paired atoms

Interactions with Si atoms

wt calc.

0.000 --2.512

B-Si(1) 4.577 2.484 B-Si(2, 3, 4) 5.135 2.791 Li-Si 1.675 1.088

588.94 584 677.19 683 569.64 566 652.37 659 586.19 585 674.43 680 566.88 565 649.62 654 622.04 622 649.61 653 599.97 601 624.79 628

11B-6Li 1°B-7 Li ix B - 7 Li 1°B-31P 11B

5.080

2.233

31 p

B-Si 5.135 P-Si 5.024

2.791 1.675

Ref. exper.

[1] [4]

[3]

With interaction be (Aye, Bye), A s i s i = 5.582, B s i s i = 4.466, interaction parameters in 104 dyn/cm, frequencies 1 in cm- , wql is the quasi-local mode frequency, wl is for the local mode. 2. RESULTS AND DISCUSSIONS We plot in Fig. 2 the LDOSs of substitutional boron and its first and second nearest neighbour Si atoms. It is clear from the figure that the mode outside the main band of host silicon is the local mode. We calculated the mode degeneracy using the following way [6] :

f = ~ gkxkx(Wl) '

(11)

kx

which in our case should be written as

f = ~_ f Dgkxkx(W) dw'

(12)

lex

because of the finite imaginary part we added when we calculated the local Green's function, k runs over all atoms and x all three directions, D is the range the mode expands. The result is f = 2.978 when we only consider the contributions from boron and those Si atoms up to the second nearest neighbours. Physically the mode induced by an impurity at the site with symmetry Ta is triply degenerated. This coincidence shows the local mode is very local, our calculation is believable. The in-band TA-like mode at 225 cm -x in the LDOS of the boron is the so-called quasi-local mode

found by Cardona et al. experimentally [ 1]. Here we see the physical meaning of 'quasi'. For the equations:

gkxkx(W) = lU~x(W)12g'(w), I~kx(w)l 2 =

~

iui~x(w)12/r,

(13) (14)

j =l

where ukx(w)/X/~mk is the component of the j-th eigenfunction for the k-th atom in x direction, r is the degeneracy at frequency w, Ukx(W)2/mk is then the square of the amplitude, g'(w) is the density-of-states. So that the LDOS for different k reflects the vibrational intensity at different' atom relatively. We can see from Fig. 2 that the mode intensity decreases when the distance between the corresponding silicon atom and the central boron atom increases for the second or the third nearest neighbours, and then goes to a constant, in contrast to the local mode. Figure 3 shows the LDOSs of phosphorous and its nearby silicon atoms. Because phosphorous is an impurity with similar mass as Si, but the interaction between it and its neighbours are weakened as compared with the interactions among silicon atoms due to the increase of bond ionicity, there is no local mode outside the main band, only the quasi-local mode at 444.0 cm -l appears.

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IMPURITY INDUCED VIBRATIONS IN LIGHT DOPED SILICON

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5 .6

Cl

e(3

o ~o O C~

(3

tJ~ w 0 ..J

b 400

200

Wovenumber

200

6O0

(cm-~)

Fig. 3.(a) LDOS of phosphorous: (b) LDOS of first nearest Si atom. The LDOSs of the interstitial lithium atom and its nearby host atoms are shown in Fig. 4. Obviously the behaviors of lithium are different from those of the substitutional impurities. Firstly, the decay of the local mode intensity is slow, the second nearest host atoms still have a certain intensity while they are almost zero in the substitutional case. Secondly, there appears a strong vibrational mode in the main band at the region of LO or between LO and LA bands. The first result is quite easy to understand since the distance between the central lithium atom and the second nearest host silicon atoms is shorter than that in the substitutional case (see Fig. 1). The in-band strong mode is due to the topoligic change of the lattice structure [7]. But since it locates in the two-phonon process region, it is probably hard to resolve it in experiment.

tvl

600

400

Wavenumber

( c m -=)

Fig. 5. (a) LDOS of boron in B - L i pair: (b) LDOS of lithium in B - L i pair. The LDOSs of B - L i pair are plotted in Fig. 5. The introduction of the lithium atom lowers the site symmetry of boron from T a to C3o, the triply degenerated local mode is split into a single longitudinal and a double degenerated transverse mode. By equation (12), the degeneracy for the lower-frequency mode is 1.002 and 1.964 for the higher-frequency mode up to the second nearest silicon atoms. Figure 6 is for the B - P pair. Next we will see how good are the single isolated impurity or impurity pair approximation for the light doped crystal. We put another boron atom near the central one to see the effect on the LDOS of the central boron due to the introduction of the nearby one. We can see from our calculations the effect gores to zero when the distance between two boron atoms increases to a certain value. The concentration corresponding to that value is about 102° cm -3. In our S i : B - L i and S i : B - P systems, the impurity concentrations are about 1019 cm -a. So The approximation is quite good here.

O3 0 C3 .J

O _.1

0

200

400

Wovenumber

600

( c m -~}

Fig. 4. (a) LDOS of lithium: (b) LDOS of first nearest Si atom: (c) LDOS o f second nearest Si atom: (d) LDOS of third nearest Si atom.

0

200

400

Wovenumber

600

(cm -t )

Fig. 6. (a) LDOS of boron in B - P pair: (b) LDOS of phosphorous in B - P pair.

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IMPURITY INDUCED VIBRATIONS IN LIGHT DOPED SILICON REFERENCES

3. CONCLUSIONS The recursion method is successfully used to study the characteristics of the local and quasi-local modes in crystalline silicon due to the electrically active impurities: boron, lithium and phosphorous, for the first time to our knowledge.

1. 2. 3. 4.

Acknowledgement - We wish to thank Professor Shen Xuechu (S.C. Shen) for helpful discussions.

167

5. 6. 7.

M. Cardona, S.C. Shen & S.P. Varma, Phys. Rev. 23, 5329 (1980). S.Y. Wu & Zheng Zhaobo, Acta Physica Sinica 32, 46 (1983). J.F. Angress, A.R. Goodwin & S.D. Smith, Proc. Roy. Soc. 287A, 64 (1965). M. Balkanski & W. Nazarewicz, J. Phys. Chem. Solids, 27,671 (1965), M. Vardevyer, D.N. Talwer, P. Plumelle, K. Kunc & L.M. Zigore, Phys. Status Solidi. (bJ 99, 727 (1980). E.N. Economous, in Green's Function in Quantum Physics, (Edited by M. Cardona et al.), SpringerBerlin, p. 42 (1979). Fu Ying & Xu Wenlan, Chin. J. Semicond. to be published.