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Optical constants of pure and heavily doped silicon and germanium: Electronic interband transitions
Phymca 117B & 118B (1983) 356-358 North-HollandPubhslungCompany
356
OPTICAL CONSTANTS OF PURE AND HEAVILY D(~ED SILICON AND GERMANIUM: ELECTRONIC IN~ERBAND TRANSITIONS
Luls Vlha and Manuel Cardona Max-Planck-Instltut f~r Festkorperforschung, Helsenbergstrasse I, 7000 Stuttgart 80, Federal Republlc o f Germany The technique of Fourxer analysls elllpscmetry has been applled to pure and heavlly doped n- and p-type Ge and S1 (doping up to =1020 cm -3) to obtain accurate spectra of E l and E 2. We have studied the shift and broadenzng of the E l (3.4 eV in SI, 2.1 - 2.3 eV in Ge) and E 2 (~4.3 eV in both) wlth doplng. All these peaks shlft to lower photon energles and broaden w l t h doplng. These shlfts amount to ~O.O5 eV for dopings of 1020 cm -3 . Tne results are dlscussed in terms of the perturbation produced b y the lonlzc.d impurltles.
The effects of temperature on the fundamental electronlc spectra of sen~conductors have been profusely studled. [1,2] Slngularltles (crltzcal polnts) in these spectra shlft and broaden wlth increaslng temperature. The dlsorder introduced by doplng wlth "hydrogenic" impurltles produces slm~lar effects• These effects, however, are smaller• They have been m u c h less studled [3-6] than those o f temperature: the range of doplng possible is severely llmlted by the solublllty of the Imperltles (~<1%). W e present here the optlcal constants of pure a n d heavxly doped Ge and Sl as determlned wlth ~n automatlc rotatlng analyzer ell~psometer.[ 7] The positlon of the E l and E 2 critlcal polnts and thelr broadenlng have been determlned from the second derivatlve spectra. They all shift to lower energles wlth doping.[3] The real (e;) and imaglnary (E 2) parts of the dlelectrlc ~onstant of pure and p-type (Bdoped, 4 x I ~ 0 holes/cm 3) S1 are shown in Flg. i. The samples were pollshed w i t h Syton and a Br-methanol solutlon.[8] They were etched in the N2-flushed measurement chamber, followlng the prescrlptlon in Ref. 8. Thls procedure was repeated until the hlghest values of E 2 at E 2 (Flg. i) were obtalned. The treatment was found to be reproducible to wlthin Ae 2 = +2 at the E 2 peak. The red shlft o f E 2 wl5h doplng, and its broadenlng, is clearly seen in Flg. i, also the broadenxng of E I . In order to see the red shlft of the cr~tlcal frequency E 1 we m u s t look at the second der~vatlve spectrum of ep ~n Fig. 2. We recognlze by slght a red s ~ f t of E. of ~O.O5 eV. A llne 1 wldth analysls of thls crltlcal polnt ylelds a Lorentz~an broadening parameter F = 0.06 e V for the pure materlal and 0.09 eV for the heavlly doped one. Slmxlar analys~s ylelds a red sh~ft of 0•05 eV for E_. The Lorentzzan broadenlngs are 0.05 e V an~ 0.065 eV in the pure and heavlly doped case, respectively. The spectra of e I and £2 for a pure and a heavlly doped Ge sample, treated as dlscussed in [ 8] , are shown in F~g. 3. The red shlfts for the heavily doped sample (~o.o4 eV for El, O 03 eV
0 378..4363/83/0000-0000/$03.00 © 1983 North-Holland
for E I + A and O.O4 eV for E 2) are seen xn Flg. 4. 1Thelr corresponding broadenlng parameters are F(E.) = F (E. + A I) = 0.055 eV, F(E 2) = O. IO eV ~or pure~Ge, and F(E I) = F(E 1 + A I) = 0.08 eV, F(E 2) = O.11 eV for the heavily doped sample.
[E2
50
L
5C
¢ 2! -(2
{:2 i ,4...~__J i "%~, 2
I.
3 s,
[,v]
--purl - - - - B doped p , & x l O 20
Fxgure I :
0
v
\/-"
i
25
S
. -25
£i and e 2 for undoped and heavily doped (4 x 1020 holes x cm -3) sxl~con at room temperature.
The crxtlcal energxes E l , E l + A 1 and E 2 found from measurements for many samples are shown in Figs. 5 and 6. These shifts agree wxth the early results of [ 3] and wlth slmllar results for SaAs.[ 5]
357
L Vz~a, M Cardona /Opttcal constants o f pure and heavdy doped Szhcon and Germamum
d2~2 |
S, pure . . . . B doped p=/, x lO20
.
1200 d2~2
pure
Ge
p= l ~ c m "3
(eV.2)
5
VE1
E2
-1200
El
-3500
Figure 2 :
Second derivative spectra of the c2 in Flg. I. The derivatives were calculated w l t h a mesh of O.O1 eV.
Flgure 4 :
~0 A0
-¢%
Second derlvatlve spectra of the samples i n Fig. 3. The derivatives were calculated wlth a mesh of 0.02 eV.
3~ S, p - type • n- type.
['r El(.V] I ~ . ~ Z 20
El/" 2C-
--:I 52' i
i Go-doped
p= 1020cm-3
, 0
, "'""i'\\ \
pure .201Ge
33
""
I 2xl0Z0
I N,
/'x1020
/.3 \
//
', /
_ -20
~T
Si p -type •
"-"
E2(eV) Figure
3
E 1 and E 2 for undoped and heavlly doped (1020 holes x cm -3) germanxum at r o o m temperature.
~,2 0 Figure 5 :
[
2xi02°
I
N~
/'x102°
Energies of the E I and E 2 critical points of Si at room temperature vs. doping. The dashed line is a linear fit while the solid line represents a fit = N~1/3.-- The data for both n- and p-type samples were fitted simultaneously.
L Vz~a,M Cardona / Opncal constants of pure and heavdy doped Sd~con and Germamum
358
231 ~LT~. E 1• A1 2 2~
}
Ge
ded anto two categoraes, terms wath large qtransfer to the antermedaate state and wlth small q-transfer. The former yaeld, a shlft proportlonal to Na, the latter a N I/3 contrabutlon (at low temperatures so that akT < EF) be-
p-type •
(eV) 2 0~ El 2 07 2 05 0
5xi019
Ni
1020
cause of screenang by the free carraers. We have thus fltted Fags. 519~d 6 wath curves proportaonal to N and to N ~ . The fat is nearly equally good a~ both casels. In the absence of more heavaly doped samples due, in part, to the solubllaty of the impurataes, the type of dependence on N must be clarafaed through calcula1 tlons~/3Our data also agree, wathln error, wath the N - dependence of the E gap of GaAs after l removang the Bursteln-Moss s~ift whach as not present an our case.[ll]
i,35 We have not seen the broadenang of the E l edge of As-doped Sa reported in [6] for dopangs up to 5 x 1019 . Our results look the same, up to thas concentrataon, for As as for P or B. We suspect the results of [6] are due to peculaarataes of the aon amplantatlon.
Ge
EE: E2{IV)
ACKNOWLEDGEMENTS
t,3
I
5x!019
N,
1020
Thanks are due to D.E. Aspnes for hls advlce, to H.J. Mattausch for has help at the early stage of the pro3ect, and to A. Barkner and M. Bleder for technacal help an bualdang the ellapsometer. REFERENCES
Figure 6 :
Energaes of the El, E 1 + 41, and E 2 cratlcal poants of Ge at room temperature vs. dopang. The dashed line as a linear fat whale the solad lane Is a fat = N I/3.
The lowest order contrabutaons to the shifts of Fags. 5 and 6 arase from the change in the "vartual crystal" potentaal (farst order perturbataon) and from second order terms anvolvang a vartual antermedaate state.[9] The former, proportaonal to ampuraty concentrataon N , has been estamated by pseudopotentaal calclulataons usang for the perturbataon potentaal the antlsymmetrac potentaal of GaAs.[10] We fand at to be an order of magnltude smaller than the data of Figs. 5 and 6. Note that tb_ts contrlbutlon should change sagn in golng from n- to the ptype case. There as no evadence for that in Figs. 5 and 6 where data for n- and p-type material have been plotted together vs. N . We are an the process of calculating the selcond order terms whach are also responsable for the ancrease an F. The contrabutions of these terms to the energy shafts can be roughly dlvl-
[I] Landoldt-Bornstean Tables, Vol. 17, ed. by O. Madelung, M. Schulz, H. Welss (Spranger Verlag, New York, 1982). [2] Allen, P B. and Cardona, M., Phys. Rev. B23, 1495 (1981). [3] Cardona, M. and Sommers, H.S., Jr., Phys. Rev. 122, 1382 (1961), Cardona, M., Shaklee, K.L., and Pollak, F.H., Phys Rev. 154, 696 (1967). [4] Aspnes, D.E., in "Laser and Electron Beam Processang of Electronac Materaals" (Electrochemacal Soclety, Pranceton, N.J., 1980), p. 414. [5] Vagal, E., Rodriguez, J.A., P~rez-Alvarez, R , Phys. Stat. Sol. (b) 90, 409 (1978). [6] Jelllson, G.E., Jr., Modlne, F.A., Whate, C.W., Wood, R F., and Young, R.T., Phys. Rev. Letters 46, 1414 (1981) [7] Aspnes, D.E., In "Spectroscopac Ellapsometry of Solads, New Developments", ed. by B.O. Seraphan (North Holland, Amsterdam, 1 976) [8] Aspnes, D.E , Appl. Phys. Letters 39, 316 (1981). [9] Allen, P.B., Phys. Rev. BI8, 5217 (1978). [iO] Cohen, M.L. and Bergstresser, T.K., Phys. Rev. 141, 789 (1966). [Ii] Casey, H.C. and Stern, F., J. Appl. Phys. 47, 631 (1976).
356
OPTICAL CONSTANTS OF PURE AND HEAVILY D(~ED SILICON AND GERMANIUM: ELECTRONIC IN~ERBAND TRANSITIONS
Luls Vlha and Manuel Cardona Max-Planck-Instltut f~r Festkorperforschung, Helsenbergstrasse I, 7000 Stuttgart 80, Federal Republlc o f Germany The technique of Fourxer analysls elllpscmetry has been applled to pure and heavlly doped n- and p-type Ge and S1 (doping up to =1020 cm -3) to obtain accurate spectra of E l and E 2. We have studied the shift and broadenzng of the E l (3.4 eV in SI, 2.1 - 2.3 eV in Ge) and E 2 (~4.3 eV in both) wlth doplng. All these peaks shlft to lower photon energles and broaden w l t h doplng. These shlfts amount to ~O.O5 eV for dopings of 1020 cm -3 . Tne results are dlscussed in terms of the perturbation produced b y the lonlzc.d impurltles.
The effects of temperature on the fundamental electronlc spectra of sen~conductors have been profusely studled. [1,2] Slngularltles (crltzcal polnts) in these spectra shlft and broaden wlth increaslng temperature. The dlsorder introduced by doplng wlth "hydrogenic" impurltles produces slm~lar effects• These effects, however, are smaller• They have been m u c h less studled [3-6] than those o f temperature: the range of doplng possible is severely llmlted by the solublllty of the Imperltles (~<1%). W e present here the optlcal constants of pure a n d heavxly doped Ge and Sl as determlned wlth ~n automatlc rotatlng analyzer ell~psometer.[ 7] The positlon of the E l and E 2 critlcal polnts and thelr broadenlng have been determlned from the second derivatlve spectra. They all shift to lower energles wlth doping.[3] The real (e;) and imaglnary (E 2) parts of the dlelectrlc ~onstant of pure and p-type (Bdoped, 4 x I ~ 0 holes/cm 3) S1 are shown in Flg. i. The samples were pollshed w i t h Syton and a Br-methanol solutlon.[8] They were etched in the N2-flushed measurement chamber, followlng the prescrlptlon in Ref. 8. Thls procedure was repeated until the hlghest values of E 2 at E 2 (Flg. i) were obtalned. The treatment was found to be reproducible to wlthin Ae 2 = +2 at the E 2 peak. The red shlft o f E 2 wl5h doplng, and its broadenlng, is clearly seen in Flg. i, also the broadenxng of E I . In order to see the red shlft of the cr~tlcal frequency E 1 we m u s t look at the second der~vatlve spectrum of ep ~n Fig. 2. We recognlze by slght a red s ~ f t of E. of ~O.O5 eV. A llne 1 wldth analysls of thls crltlcal polnt ylelds a Lorentz~an broadening parameter F = 0.06 e V for the pure materlal and 0.09 eV for the heavlly doped one. Slmxlar analys~s ylelds a red sh~ft of 0•05 eV for E_. The Lorentzzan broadenlngs are 0.05 e V an~ 0.065 eV in the pure and heavlly doped case, respectively. The spectra of e I and £2 for a pure and a heavlly doped Ge sample, treated as dlscussed in [ 8] , are shown in F~g. 3. The red shlfts for the heavily doped sample (~o.o4 eV for El, O 03 eV
0 378..4363/83/0000-0000/$03.00 © 1983 North-Holland
for E I + A and O.O4 eV for E 2) are seen xn Flg. 4. 1Thelr corresponding broadenlng parameters are F(E.) = F (E. + A I) = 0.055 eV, F(E 2) = O. IO eV ~or pure~Ge, and F(E I) = F(E 1 + A I) = 0.08 eV, F(E 2) = O.11 eV for the heavily doped sample.
[E2
50
L
5C
¢ 2! -(2
{:2 i ,4...~__J i "%~, 2
I.
3 s,
[,v]
--purl - - - - B doped p , & x l O 20
Fxgure I :
0
v
\/-"
i
25
S
. -25
£i and e 2 for undoped and heavily doped (4 x 1020 holes x cm -3) sxl~con at room temperature.
The crxtlcal energxes E l , E l + A 1 and E 2 found from measurements for many samples are shown in Figs. 5 and 6. These shifts agree wxth the early results of [ 3] and wlth slmllar results for SaAs.[ 5]
357
L Vz~a, M Cardona /Opttcal constants o f pure and heavdy doped Szhcon and Germamum
d2~2 |
S, pure . . . . B doped p=/, x lO20
.
1200 d2~2
pure
Ge
p= l ~ c m "3
(eV.2)
5
VE1
E2
-1200
El
-3500
Figure 2 :
Second derivative spectra of the c2 in Flg. I. The derivatives were calculated w l t h a mesh of O.O1 eV.
Flgure 4 :
~0 A0
-¢%
Second derlvatlve spectra of the samples i n Fig. 3. The derivatives were calculated wlth a mesh of 0.02 eV.
3~ S, p - type • n- type.
['r El(.V] I ~ . ~ Z 20
El/" 2C-
--:I 52' i
i Go-doped
p= 1020cm-3
, 0
, "'""i'\\ \
pure .201Ge
33
""
I 2xl0Z0
I N,
/'x1020
/.3 \
//
', /
_ -20
~T
Si p -type •
"-"
E2(eV) Figure
3
E 1 and E 2 for undoped and heavlly doped (1020 holes x cm -3) germanxum at r o o m temperature.
~,2 0 Figure 5 :
[
2xi02°
I
N~
/'x102°
Energies of the E I and E 2 critical points of Si at room temperature vs. doping. The dashed line is a linear fit while the solid line represents a fit = N~1/3.-- The data for both n- and p-type samples were fitted simultaneously.
L Vz~a,M Cardona / Opncal constants of pure and heavdy doped Sd~con and Germamum
358
231 ~LT~. E 1• A1 2 2~
}
Ge
ded anto two categoraes, terms wath large qtransfer to the antermedaate state and wlth small q-transfer. The former yaeld, a shlft proportlonal to Na, the latter a N I/3 contrabutlon (at low temperatures so that akT < EF) be-
p-type •
(eV) 2 0~ El 2 07 2 05 0
5xi019
Ni
1020
cause of screenang by the free carraers. We have thus fltted Fags. 519~d 6 wath curves proportaonal to N and to N ~ . The fat is nearly equally good a~ both casels. In the absence of more heavaly doped samples due, in part, to the solubllaty of the impurataes, the type of dependence on N must be clarafaed through calcula1 tlons~/3Our data also agree, wathln error, wath the N - dependence of the E gap of GaAs after l removang the Bursteln-Moss s~ift whach as not present an our case.[ll]
i,35 We have not seen the broadenang of the E l edge of As-doped Sa reported in [6] for dopangs up to 5 x 1019 . Our results look the same, up to thas concentrataon, for As as for P or B. We suspect the results of [6] are due to peculaarataes of the aon amplantatlon.
Ge
EE: E2{IV)
ACKNOWLEDGEMENTS
t,3
I
5x!019
N,
1020
Thanks are due to D.E. Aspnes for hls advlce, to H.J. Mattausch for has help at the early stage of the pro3ect, and to A. Barkner and M. Bleder for technacal help an bualdang the ellapsometer. REFERENCES
Figure 6 :
Energaes of the El, E 1 + 41, and E 2 cratlcal poants of Ge at room temperature vs. dopang. The dashed line as a linear fat whale the solad lane Is a fat = N I/3.
The lowest order contrabutaons to the shifts of Fags. 5 and 6 arase from the change in the "vartual crystal" potentaal (farst order perturbataon) and from second order terms anvolvang a vartual antermedaate state.[9] The former, proportaonal to ampuraty concentrataon N , has been estamated by pseudopotentaal calclulataons usang for the perturbataon potentaal the antlsymmetrac potentaal of GaAs.[10] We fand at to be an order of magnltude smaller than the data of Figs. 5 and 6. Note that tb_ts contrlbutlon should change sagn in golng from n- to the ptype case. There as no evadence for that in Figs. 5 and 6 where data for n- and p-type material have been plotted together vs. N . We are an the process of calculating the selcond order terms whach are also responsable for the ancrease an F. The contrabutions of these terms to the energy shafts can be roughly dlvl-
[I] Landoldt-Bornstean Tables, Vol. 17, ed. by O. Madelung, M. Schulz, H. Welss (Spranger Verlag, New York, 1982). [2] Allen, P B. and Cardona, M., Phys. Rev. B23, 1495 (1981). [3] Cardona, M. and Sommers, H.S., Jr., Phys. Rev. 122, 1382 (1961), Cardona, M., Shaklee, K.L., and Pollak, F.H., Phys Rev. 154, 696 (1967). [4] Aspnes, D.E., in "Laser and Electron Beam Processang of Electronac Materaals" (Electrochemacal Soclety, Pranceton, N.J., 1980), p. 414. [5] Vagal, E., Rodriguez, J.A., P~rez-Alvarez, R , Phys. Stat. Sol. (b) 90, 409 (1978). [6] Jelllson, G.E., Jr., Modlne, F.A., Whate, C.W., Wood, R F., and Young, R.T., Phys. Rev. Letters 46, 1414 (1981) [7] Aspnes, D.E., In "Spectroscopac Ellapsometry of Solads, New Developments", ed. by B.O. Seraphan (North Holland, Amsterdam, 1 976) [8] Aspnes, D.E , Appl. Phys. Letters 39, 316 (1981). [9] Allen, P.B., Phys. Rev. BI8, 5217 (1978). [iO] Cohen, M.L. and Bergstresser, T.K., Phys. Rev. 141, 789 (1966). [Ii] Casey, H.C. and Stern, F., J. Appl. Phys. 47, 631 (1976).