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Effective electron mass in compound semiconductor films
Vacuum/volume 44/number Printed in Great Britain
8/pages 803 to 804/l
0042-207X/93$6.00+.00 0 1993 Pergamon Press Ltd
993
Effective electron mass in compound semiconductor films D Bhattacharyya. the Cultivation
R Pal, S Chaudhuri
of Science,
Calcutta
and A K Pal, Department
of Materials
Science,
Indian
Association
for
700 032, India
received 26 May 1992
The effective mass of electrons was evaluated for compound semiconductor films of CdS,Te,_, and Cd&Se,_, which showed bowing phenomena similar to those for optical bandgaps for the above alloy films.
1. Introduction
&f(m*,x)
Values of the effective mass of the electron for compound semiconductors for all compositions of alloys are frequently needed to generate information on most physical properties. Data analyses from the experimental observations also require information on the effective mass of the electron. It appears that the data for the above, determined by cyclotron resonance or evaluated from band structure calculations for all the compositions of semiconductor alloy films, are not readily available. So it will be a worthwhile exercise to obtain a meaningful derivation for the effective mass from the composition dependence of optical bandgaps of these materials. 2. Theory The composition dependence of the bandgap Eg(x) of ternary system ABC, ox or A,B, _,C is generally seen to fit an expression of the form’- 3
Eg(x) = E,+[E*-E,-b]x+bx2,
(1)
where E, and E, are the bandgaps of materials with x = 0 and 1, respectively, and b is the bowing parameter. Now, the band-edge effective mass of the electron (me*) can be expressed as“
1/m* = l+p
P2 2mEg ’
m* 2m (1 -m*)
= &
f(m*),
where f (m*) = m*/(l -m*). (l), we get
+
&f(mZ)--
&f(m:)-h
[
1
+bx2
or f(m*,x)
= f(m:)+[f(mt)-f(mT)-h,]x+b,x’,
(5)
where 6, = b/(p2/2m) and rn: and mf correspond to materials with x = 0 and 1, respectively. Thus, the function f(m*) of a ternary compound semiconductor should show a bowing behaviour with the bowing parameter b, which is related to b by b, = b/(p2/2m). It is to be noted that we have assumed p to be constant which is only true for materials having almost similar band structures. Since p - hG, and G is the same for all semiconductors with the gap at the zone centre or at the zone edge, the quantity p can be taken as constant for all such materials. There are variations in p for some materials like HgTe but this quantity is remarkably uniform among various semiconductors as is indicated in ref 3. Thus, if the effective masses of the constituent semiconductors (m: and m$) and bandgap bowing parameter, b, are known, one can compute the effective mass of the semiconductor alloy for any required composition by using expression (5).
3. Experimental results
where m* = me*/m, m being the free electron mass and p is the momentum matrix element and is given by p = hG. G is the smallest reciprocal lattice vector so that p becomes equal to b/a, a being the lattice constant. Now, from equation (2) we get
,d!-
= &f(m:)
Putting
equation
(4) in equation
CdS,Te , _ ~and Cd&Se , ox films have been prepared by co-evaporating the constituent compounds from a two-zone evaporation jig. The details of the experimental techniques are given elsewhere5,6. The variations of the bandgaps determined from reflectance data’ are plotted in Figure 1. The bowing parameters for CdS,Te, ~, and CdS,Se, _ , films’.’ were found to be 1.7 and 0.38 eV, respectively. The values of f (m*) for the above alloy films were computed by using expression (5) and are also shown in Figure 2. The b values generated from the fit (Figure 2) off (m*) vs x plots are 1.4 and 0.43 eV for CdS,Te, ~I and CdS,Se, li films, respectively. These values agree extremely well with those obtained from Eg(x) vs x plots. The m* values for the above films generated from f (m*) are also plotted in Figure 2. It can be 803
D Bhattacharyya
eta/:
Compound
semiconductor
films
2.6
I
2.2 s---
+ E
l.Ol 0 Figure
CdS,Se,
1. Variation
0.2
0.4
0.6
of bandgap (Q)
0.8
with composition
1.0
LY) for (0) Figure 2. Variations of f(m*) and IX* with composition (.L): ( 1 -) f (M*) of CdS,Te, \ films: (--0 -~-I f(m*) of CdS,Se, ~~ films; ( m* values of CdS,Te, m* values of Cd&Se, ~ films : and (-A--) iihns.
j films ; and (A) CdS,Te,_ j films.
that the effective mass of an electron in the above compound semiconductor alloy films also shows a bowing behav-iour. For materials with no* c< 1, f (m*) - PI*. and hence the same bowing behaviour will be reflected in the variation of m* with .Xwhich is also evident from Figure 2. It is interesting to note that in such cases. the n?* values can be fitted well with an empirical cxprcssion : wz* = mT+ [nf - WIT~ C].r + C’.?. The C‘ values generated from the fit tally extremely well with the numerical value seen
Table 1. Values of different CdS,Sc, I films
parameters
determined
h (eV)
Film _ CdS CdSe CdTe CdS,Te, CdS,Sc,
804
from Eg
/ I
1.7
0.38
for CdS,Te,
I and
h
pL;2m (eV)
(eV) h,
from h,
m*
13.2
0.153 (ref 8)
11.5
0.130 (ref 8) 0.110(ref 8)
13.1 13.1 13.2
0.106 0.0x2
I .40 0.43
of h, The values of different parameters calculations are shown in Table 1.
used for the above
Acknowledgements Two of’ us (DB and RP) wish to thank the UGC, India, for providing fellowships to them for carrying programme.
Govt ot out this
References
’ R Hill and D Richardson, Thin Solid Film.v, 18, 25 (1973). ’ R Hill, J Phys C: Solid St Phys. 7, 52 I (1972). ’ A Zunger, Proceedings 11th Internarional C’ot~~mvwc~ wz the Plly.sic.v Srm~onduc~or.u, San Francisco, CA. 6 10 August 1984 (Edited by J D Chadi and WA Harrison). Springer. New York (1985). ‘D K Ferry, Senziconductor.v, Chap 5. Macmillan, New York (IYY I )_ ’ R Pal. J Dutta, S Chaudhurl and A K Pal, J Mrrrcr SC,;. submitted. “.I Dutta, R Pal, S. Chattopadhyay. S C‘haudhuri and A K Pal. P/I>,\ SIUILLYSolidi, submitted. ’ D Bhattacharyya, S Chaudhuri and A K Pal, P’u~rrum. 43, 3 I3 (1992). ” B Ray, II VI Compoz~ndS~mi~ondu~tors. 83. Pergamon, Oxford (1969).
8/pages 803 to 804/l
0042-207X/93$6.00+.00 0 1993 Pergamon Press Ltd
993
Effective electron mass in compound semiconductor films D Bhattacharyya. the Cultivation
R Pal, S Chaudhuri
of Science,
Calcutta
and A K Pal, Department
of Materials
Science,
Indian
Association
for
700 032, India
received 26 May 1992
The effective mass of electrons was evaluated for compound semiconductor films of CdS,Te,_, and Cd&Se,_, which showed bowing phenomena similar to those for optical bandgaps for the above alloy films.
1. Introduction
&f(m*,x)
Values of the effective mass of the electron for compound semiconductors for all compositions of alloys are frequently needed to generate information on most physical properties. Data analyses from the experimental observations also require information on the effective mass of the electron. It appears that the data for the above, determined by cyclotron resonance or evaluated from band structure calculations for all the compositions of semiconductor alloy films, are not readily available. So it will be a worthwhile exercise to obtain a meaningful derivation for the effective mass from the composition dependence of optical bandgaps of these materials. 2. Theory The composition dependence of the bandgap Eg(x) of ternary system ABC, ox or A,B, _,C is generally seen to fit an expression of the form’- 3
Eg(x) = E,+[E*-E,-b]x+bx2,
(1)
where E, and E, are the bandgaps of materials with x = 0 and 1, respectively, and b is the bowing parameter. Now, the band-edge effective mass of the electron (me*) can be expressed as“
1/m* = l+p
P2 2mEg ’
m* 2m (1 -m*)
= &
f(m*),
where f (m*) = m*/(l -m*). (l), we get
+
&f(mZ)--
&f(m:)-h
[
1
+bx2
or f(m*,x)
= f(m:)+[f(mt)-f(mT)-h,]x+b,x’,
(5)
where 6, = b/(p2/2m) and rn: and mf correspond to materials with x = 0 and 1, respectively. Thus, the function f(m*) of a ternary compound semiconductor should show a bowing behaviour with the bowing parameter b, which is related to b by b, = b/(p2/2m). It is to be noted that we have assumed p to be constant which is only true for materials having almost similar band structures. Since p - hG, and G is the same for all semiconductors with the gap at the zone centre or at the zone edge, the quantity p can be taken as constant for all such materials. There are variations in p for some materials like HgTe but this quantity is remarkably uniform among various semiconductors as is indicated in ref 3. Thus, if the effective masses of the constituent semiconductors (m: and m$) and bandgap bowing parameter, b, are known, one can compute the effective mass of the semiconductor alloy for any required composition by using expression (5).
3. Experimental results
where m* = me*/m, m being the free electron mass and p is the momentum matrix element and is given by p = hG. G is the smallest reciprocal lattice vector so that p becomes equal to b/a, a being the lattice constant. Now, from equation (2) we get
,d!-
= &f(m:)
Putting
equation
(4) in equation
CdS,Te , _ ~and Cd&Se , ox films have been prepared by co-evaporating the constituent compounds from a two-zone evaporation jig. The details of the experimental techniques are given elsewhere5,6. The variations of the bandgaps determined from reflectance data’ are plotted in Figure 1. The bowing parameters for CdS,Te, ~, and CdS,Se, _ , films’.’ were found to be 1.7 and 0.38 eV, respectively. The values of f (m*) for the above alloy films were computed by using expression (5) and are also shown in Figure 2. The b values generated from the fit (Figure 2) off (m*) vs x plots are 1.4 and 0.43 eV for CdS,Te, ~I and CdS,Se, li films, respectively. These values agree extremely well with those obtained from Eg(x) vs x plots. The m* values for the above films generated from f (m*) are also plotted in Figure 2. It can be 803
D Bhattacharyya
eta/:
Compound
semiconductor
films
2.6
I
2.2 s---
+ E
l.Ol 0 Figure
CdS,Se,
1. Variation
0.2
0.4
0.6
of bandgap (Q)
0.8
with composition
1.0
LY) for (0) Figure 2. Variations of f(m*) and IX* with composition (.L): ( 1 -) f (M*) of CdS,Te, \ films: (--0 -~-I f(m*) of CdS,Se, ~~ films; ( m* values of CdS,Te, m* values of Cd&Se, ~ films : and (-A--) iihns.
j films ; and (A) CdS,Te,_ j films.
that the effective mass of an electron in the above compound semiconductor alloy films also shows a bowing behav-iour. For materials with no* c< 1, f (m*) - PI*. and hence the same bowing behaviour will be reflected in the variation of m* with .Xwhich is also evident from Figure 2. It is interesting to note that in such cases. the n?* values can be fitted well with an empirical cxprcssion : wz* = mT+ [nf - WIT~ C].r + C’.?. The C‘ values generated from the fit tally extremely well with the numerical value seen
Table 1. Values of different CdS,Sc, I films
parameters
determined
h (eV)
Film _ CdS CdSe CdTe CdS,Te, CdS,Sc,
804
from Eg
/ I
1.7
0.38
for CdS,Te,
I and
h
pL;2m (eV)
(eV) h,
from h,
m*
13.2
0.153 (ref 8)
11.5
0.130 (ref 8) 0.110(ref 8)
13.1 13.1 13.2
0.106 0.0x2
I .40 0.43
of h, The values of different parameters calculations are shown in Table 1.
used for the above
Acknowledgements Two of’ us (DB and RP) wish to thank the UGC, India, for providing fellowships to them for carrying programme.
Govt ot out this
References
’ R Hill and D Richardson, Thin Solid Film.v, 18, 25 (1973). ’ R Hill, J Phys C: Solid St Phys. 7, 52 I (1972). ’ A Zunger, Proceedings 11th Internarional C’ot~~mvwc~ wz the Plly.sic.v Srm~onduc~or.u, San Francisco, CA. 6 10 August 1984 (Edited by J D Chadi and WA Harrison). Springer. New York (1985). ‘D K Ferry, Senziconductor.v, Chap 5. Macmillan, New York (IYY I )_ ’ R Pal. J Dutta, S Chaudhurl and A K Pal, J Mrrrcr SC,;. submitted. “.I Dutta, R Pal, S. Chattopadhyay. S C‘haudhuri and A K Pal. P/I>,\ SIUILLYSolidi, submitted. ’ D Bhattacharyya, S Chaudhuri and A K Pal, P’u~rrum. 43, 3 I3 (1992). ” B Ray, II VI Compoz~ndS~mi~ondu~tors. 83. Pergamon, Oxford (1969).