Milo8 B. %vanov, Ljiljana D. Tivanov and Jovan Dj. Jovovic' Fakultet tehniEkih nauka, Novi Sad, Yugoslavia

- In this paper was cahlated the position of Fermi eneygy level and efective mass of electron in n-type GUA s as a finction ofdonor concentrations in the range fram 10' to IO2] em4, tahng into mxxmc upper minima in ~ ~ n ~ ~ c ~ o ~ band, at 300 K Calculations show that is necessary taken besides r a n d L minimum for concentrations grenter than Id9 cm-3 andall three minima (4 L and x) far concentrations greater thm i O Z 0 cnr? ~~~~i~~ terms: Ultra heavily doped semiconductors, Fermi level, density of states effective mass 1.

Se~iconductin~ c o ~ p o ~ n d s such as GaAs used for the construction of lasers, FE% and similar devices are usually doped in the range up to 10'9 cm-3. Modern HBTs are fabricated with p- doping of 1020 ~ m - ~ [I] and n-doping 3-10*9 cm-3 121, and doping densities above 1Q21 cme3 have been achieved 131. n-type doping in thin layers about 1030 cnr3 was shown in [4]. Despite significant a d ~ ~ ~ c ~ s k~ Gds processing many diEculties remain to be overcome. The transport properties Of materials with such doping are therefore of great technical interest.

It was shown in [5] that heavy doping distorts the band structure radically. The insens~ t~v i~ of the transport c o e ~ c ~ e ~ ~ to the band distortion appears to be due to the fact contribution of the electrons ~ ~ u ~ ~ ~ g the distorted part of the band is very small compare contribution of electrons ~ c u ~ y i n g the higher undistorted regions of conduction band.

For ~ a l c u l a t ~ o ~ s tbe physical and transport parameters in GaAs tlie most autors[5,6,7] considered only r6minimum. The dependence of Fermi energy level and eEkctive mass on concentration in n-type GaAs for cQncentration~ lower than IOl9 c w 3 were earlier observed [$I. In our previous paper[9] we also calculated Fermi level and average effective inass in n-type GaAs taking into account only ~ ~ i ~ i ~ ~ m o ~ c o n ~ u c t j o n ~ ~ ~ e band. From obtained rcsarlt it is seen that Fermi energy level enters the conduction band at donor concentrations abaut 3 1 O I 7 C I ~ I - ~ . It was shown that cha ra~ te r i~~cs (for example effective mass) would be changed from this concentration. Fermi energy level begins to increase rapidly for ~oncent r~ t io~s that are greater than concentration at which Fermi level enters the conduction band and to attain upper minima. Because, for higher concentrations it is necessary to include in the calculation and s ~ ~ ~ a t j ~ n , and other minima, particularly in ultra heavily doped ( Gds. J'G Lgand &g minima have energy gaps of 1.42 eV. 1.71 eV and 1.90 eV above the top Of valence band, respectively[6].

EL AN SULTS The total ~ ~ ~ b ~ r o ~ c o n ~ u c t i ~ ~ electrons (assuming that on 300 K all impuity atoms are

ionized) should be written as a sum ofthree contributions lev ICV

n = nr i- nL + n, = pr (E ) f ( E ) c/E + pL ( E - A CL f (E)dE + 0 A n

0 1996EEE 3 57

where nD nL and nX are number of electrons in L, Xminima, respectively, pdE) is nonparabolic density of states for I‘ ~ ~ n i m u m taken as in Altschul et al. [IO], pL(E) and &(E) axe parabolic density of states for L and X minima, ArL i s energy elevation (EL-Er)=8.284 eV, Arx is energy elevatka (erEr)=0;476 eV, f{E) ~ Fermi-Dirac distribution function of electrons.

~ Q n ~ ~ a b ~ l i c i ~ factor, a, and effective mass of electrons at the conduction band minimum C mc0, are ~ ~ c t ~ o n s of energy gap at r anhimum , Egp These connections are obtained from the

la given in [6], h healjly doped s e ~ i ~ o ~ d u c t ~ r s it was observed the band gap narrowing as a of ~~~~~~ concentrations. In the present paper we took published results for the band gap

given in [ l I ] and we substituted it into relations for a and mco. Band-gap narrowing of as a ~ n ~ ~ ~ ~ n of doping concentration has been measured using n-doped samples with

~ Q ~ C e ~ ~ r a t i o ~ $ from 3.1017 ~ m - ~ to 3.101* cm-3. Assuming that same relation is valid above this range, we c a ~ c ~ a ~ ~ that ATg = 158 meV for ND =I819 cm-3 , AEg = 340 meV for MO =1020 and AEg = 734 meV for ND =IOzr cmS3. These vatues are more applicable than the values A& = 1006 meV for ND =102O cm-3 in [12] and have better agreement wid1 experiment at ND =io19 ~n pL and px, density of states masses are mL==0.56 m, and mp0.85 m, [6]. m, is mass of free electron. Using rdation ( 1 ), we ~ ~ c ~ ~ a ~ ~ the c o ~ c ~ ~ ~ a t i o n dependence of the three-way split of conduction electrons arnong the r~ L6 and X, bands. That is shown in Fig. 1.

Fig. 1.

10

10

10

10 n y 10

E 10 0 v

10

10

i o 10

:IF ,/” /’ / /’

/’

// 14 I ~ ~ / ’ ‘ I,/’’

20

19

18

17

16

15

14

13

11

I

10 j 7 10 18 10 l 9 ’1 0 *O 10 21

12

N,( c M - 3 )

Concen fration dependence of the three-way split of conduction elm frons among the

The presented results in Fig. 1 show that the number of electrons at the L, minimum attained ~ m - ~ and that for greater

s greater. Similarly it is happ~ned with number of electrons at n ~ ~ e ~ of electrons im TG m~~ at donor concentrations (ND> of 10‘

Xe m~~ for No. > 1 Q20 car3, Numbers of electrons in L, and X, minima become same for ND %I@* “ 3 .

nuniber is several

Assuming a11 donors, ND ~ are ionized at room temperature, the total number of t h s condition, using self-c,onsistent method we obtained the

‘on donor c o n c ~ ~ ~ a t i o n . Integrating limits are taken from 0 to 1 eV: n results is in t h ~ s limit. The result for EF as a function of ND is given only rminimum and (a) all thee minima and th s results we are using

late in this W W ~ . T h g h t o account only r , minimum for calculations EF;, it reaches I eV at ~ ~ ~ c ~ ~ a t ~ o ~ about cm3 and begins to increase rapidly. bcluaing upper minima ( ~6 and &)

358

in calculations it is seen t h t EF changed much slower and reaches 1 eV at concentrations about lo2’ “3.

For parabolic isotmpic bands electron concentration, n, is given with

/

Fig.2. Variation of the Fermi energy level with donor concentrations tahng into account (a) all three minima ( r6, L6 and X& and (b) only r6 minimum.

3

Fig.3 Density of states effective mass, md in the n-type GaAs as ahnction of the donor Concentration No

where F I ~ ( q ) is Fermi-Diraa: integral and q=EF ACT. m* is effective mass at the bottom of the parabolic band. For a s ~ p l i ~ ~ d view ofthe distribution among the band minima at high concentrations, we may rewrite (1) in the form of (2) by defining a density of states effective mass given by

359

right of eqn. 1 (n= ND ). we can describe by model for one parabolic, effective mass Hid This approach is similar to that given by N%[51

ent for some kind of deliice simulations. The calculated results from ntrations are shown in Fig.3. Our results of calculations

n af constant effectitre mass up to 1019 cm-3, which, for

ts it is convenient to introduce the approximate relation,

re, qmjn =0.063 m, is eEective mass in bottom of r ~ l i ~ l ~ m u ~ , mdnlax =0.85 in, i s density of s e ~ e c ~ ~ ~ ~ mas at theX ~ ~ ~ u m md the fitting parameters are ND =1.2 1021 ctr3 and ~ 0 . 8 .

endence of EF and /?zd in UPlD n-type GaBs. This other tsanspo~t parameters in W D semiconductors. It

ed distribution of electrons can have great influence on structures of UWD n-type GaAs at this moment i s not

p ~ s § ~ b ~ ~ to realize due the t e c ~ o ~ o ~ ~ ~ a ~ problems. For that these results are interesting for prediction the ~ r O p e ~ ~ e s of such ~ ~ c ~ ~ ~ s .

dion bipolar transistors for microwave and milimetar-wave . on Electron Devices, V 0 1 . E ~ ~ 3 4 ( ~ 2 ) ~ 1 ~ ~ 7 , pp.2571-2577 : ‘ ~ A ~ ~ ~ / G d s heterojunction bipolar transtors fabricated using

ss”, HEEE Elect. Dev. Lett., ~ o ~ . E ~ ~ - $ ~ 7 ) , 1987, pp.303-305 ce rela~ed material properties of hevily doped ~ a ~ ~ i u ~ arsenide”,

de MESFET technology”,

ucthg and other major properties of G d s “ , J, Appl. Phys., Vo1.53,

functin of temperature and

masses of majority GaAs”, Proc. Cod9 CAS’95,

MEL, E. FINKMAN: “Effects of band n Q ~ ~ a ~ a b Q 1 ~ ~ ~ ~ on

wing in highly doped n- and p-type GAS studied by pl. Phys., VoI 66, 1989, pp.4381-4386 mple expression for band gap narrowing (BGN) in

.&pi.Phys., V0l.71, 1992, pp.4382-4384

strained layers”.Solid-St Electron., Vol. 34(5), 1990,

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