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Concentration dependent spin-lattice relaxation in n-type silicon
J. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 1099-l 110.
Printed in Great Britain.
CONCENTRATION DEPENDENT SPIN-LATTICE RELAXATION IN n-TYPE SILICON KO SIJCIHARA Wireless Research Laboratory, Matsushita Electric Industrial Co. Ltd., Kadoma,
Osaka,Japan
(Received 5 October 1967; in revisedform 22 December 1967) Abstract-Spin-lattice relaxation mechanisms for the donor electrons in uncompensated silicon are presented. Major isolated spins transfer their excitation energies via the spin-diffusion process to the fast-relaxing centers. For lightly doped samples, N1, 2 10’Ycm3, exchange-coupled donor pairs act as the fast-relaxing centers. Theory provides the correct order of magnitude for the relaxation rate l/T,. However, the calculated relaxation rate l/T, for this process is field independent, while the observed rate shows a weak field dependence. For more heavily doped samples, ND > 10i6/cms, the relaxation rate can be explained by assuming the presence of a small concentration of neutral-ionized donor pairs. The relaxation process for these pairs is the resultant of two different mechanisms, a field dependent mechanism and a field independent one. The former depends strongly on the donor concentration and the latter shows relatively weak dependence on ND. 1. INTRODUCTION
SPIN-LATTICE relaxation
of shallow donors in silicon assumes various aspects according to the magnitudes of the four parameters, magnetic field H, temperature T, donor concentration ND and compensation degree K. In the case of lightly doped samples in which donors are su~ciently isolated, relaxation mechanisms have been well established by Hasegawa[l], Roth[2] and Castner[3]. With increasing donor concentration a new feature, which strongly depends on impurity concentration, appears[4]. For these samples the Ii- and T-dependence of l/T, are entirely different from the isolated donor’s case. The concentration dependent regions are grouped into the three cases according to the impurity concentration as follows: (1) ND < 10’7/cm3
and
(2) ND < 1017/cm3 and
K 4 1 K > 0.1
(3) ND > 10%m3. Case (3) corresponds to the impurity band conduction range and the mechanisms of the transport phenomena and the magnetic resonance are not so well understood[5].
Igo[6], Jerome[7] and Yang-Honig[8],t performed detailed me~urements of l/T, for the case (2) samples. We proposed a theoryE related to Igo’s work and succeeded in explaining his results qualitatively except for the highly compensated samples. The validity of our mechanism has been ascertained by the recent measurement of Y-H [S]. It has been reported that the relaxation rate is enhanced by application of an electric field for case (2) samples[lO]. A strong field exictes the donor electrons into the conduction band by impact ionization and the localized spins relax rapidly through interaction with the free carriers. However, it is not clear why the relatively weak field accelerates the relaxation rate. In this paper we shall treat case (1). HonigStupp[ll], Jerome171 and Y-H[8] carried out detailed measurements related to these samples. It is worth noticing that Y-H have succeeded in separating different mechanisms by filling up the ionized donor sites using the optical pumping technique. In this article our theory is manly compared with the Jkrome’s tHereafter
1099
we shall refer this paper Y-H.
1100
K. SUGIHARA
work which covers a fairly wide concentration range. We shall review the main observed results: (1). l/T, decreases with increasing magnetic field and its dependence is stronger for the more concentrated samples. At high fields the H-dependence approaches asymptotically an H4-law, which corresponds tc the dependence of the isolated donor. (2). At low temperatures l/T, shows weak variation with T but at high temperatures it obeys a P-dependence, which appears in the Raman process for the isolated donor’s case. (3). The concentration dependence of l/T, is nearly linear for the samples ND < 10%m3 and rises sharply with ND for the more highly concentrated ones. Honig-Stupp [ 1 l] suggested a mechanism for the concentration dependent spin-lattice relaxation: the excitation energies of the major part of the donor spins are transferred via the spin-diffusion process to the fastrelaxing centers? which are few in number but can relax very rapidly. In compensated samples neutral-ionized pairs (n-i pairs) assume the role of f.r.c. [6, 8, 91 but in case (1) importance of the exchange coupled pairs was pointed out [8, 111. As ND increases, resonance lines of exchange coupled clusters consisting of two or more donors are observed[ 121 besides the hyperfine lines. Resonance lines of donor pairs in silicon have been investigated by Jerome and Winter] 131 with the aid of the observation of the ENDOR spectrum and the line shape of the spectrum has given a determination of the sign of the exchange integral J which turns out to be antiferromagnetic. The relaxation mechanisms presented in this article are based on the Honig-Stupp’s suggestion. According to the impurity concentrations relaxation mechanisms of f.r.c. are classified into three different types. Exchange coupled neutral pairs (n-n pairs) play an important role in the relaxation process for the samples ND < 101%m3. Triplet and THereafter we shall abbreviate f.r.c.
singlet states of a n-n pair are admixed each other via the hyperfine interaction and the electron-phonon interaction induces the triplet + singlet transition being accompanied with the Zeeman energy change. This process does not depend on the magnetic field strength and the relaxation rate on the whole is also field independent. This feature is in contradiction to the observed results [7, 8, 111. However, there is a possibility to remove this discrepancy in the high field region by assuming a g-factor difference (- 10e3) between isolated donors and n-n pairs[ll]. At high fields the Zeeman energy difference will exceed the inhomogeneous broadening, which is 2.5 Oe in the phosphorous doped silicon, and then energy exchange between the isolated donors and the n-n pairs becomes very difficult. But it would not explain the observed field dependence in the low field region. In our calculation variation of the f.r.c. number with H is very small and so at this stage we are unable to interpret the observed field dependence. Y-H [ 81 adopted the relaxation mechanisms of the n-n pairs induced by modulation of the dipolar interaction[ 141 or the hyperfme and exchange couplings [ 151 due to the lattice vibration. However, these mechanisms are both ineffective for the donor spins in silicon, while our mechanism provides a correct order of magnitude of l/T, without any special assumptions for the samples 1015/cm3 and For the 1.1 X 101g/cm3 used by Jerome. samples N,, < 101’%m3 l/T, is proportional to ND. This is consistent with our calculation. However, the above-mentioned mechanism fails in the interpretation of l/T, for the samples 3.2 X 10%m3, 6 X 1016/cm3 and 8 x 1016/cm3. For these samples the observed concentration dependence of l/T, is very strong and the field dependence is steeper than for the lightly doped samples. If these samples are slightly compensated, n-i pairs behave like the f.r.c. Consider an n-i close pair i-j. Admixture of different spin states is introduced into the wave functions localized at i orj sites
SPIN-LATTICE
RELAXATION
through the hyperfine interaction [7] or through the spin-orbit interaction of the impurity atom[9]. Phonon-assisted hopping from an occupied level to a vacant one of the i-j system, accompanied by a reversal of spin direction, occurs frequently, if the interdonor distance R is not too large and the energies Et and Ej are close to the Fermi level 5. The process related to the hyperfine interaction is proportional to He2 and strongly dependent on R, while the one arising from the spin-orbit interaction of the impurity atom is field independent and weakly dependent on R [9]. It is very difficult to formulate the spindiffusion problem correctly in the case of a random distribution of impurity centers. Therefore, we shall introduce an appropriate spin-diffusion time for each sample instead of entering into the details of the spin-diffusion process. The observed features of the experimental results can be explained qualitatively by use of the above-mentioned three mechanisms. In Section 2 calculation of the relaxation rate related to the n--)2 pairs is presented. In Section 3 we shall discuss the two relaxation mechanisms caused by the n-i pairs. Finally we summarize the results obtained in this paper. 2. RELAXATION PROCESS RELATED TO THE EXCHANGE COUPLED DONORS
For definiteness we shall confine our discussion to the phosphorus doped silicon. Now suppose an n-n pair whose exchange energy is sufficiently greater than the Zeeman splitting of the triplet state. Hyperfine coupling associated with donors A and B is given
IN n-TYPE SILICON
1101
it by a spherical Is-function 4 - exp(-r/a*). Triplet and singlet states of an n-n pair are represented by
-Hrl-
RBM(r2- RA)IxW)
1 @P,= V12( 1 + A”) 3 { $(rr - R&(r2 -
RB)
+ddV-Rd~(W--,d~xs
(2.2)
where xt (M) and x8 are spin functions and A denotes the overlap integral. @,1&l)and @‘r are admixed with each other through the hyperfine coupling (2.1). After a simple calculation we get yp
=
@,I”’
A
f
2&[2(1-A4)]
(zBT--zAS)@~
A v\Ir,= @‘+ 2JV[2(1 -A4)] X
[(IA+-ZB+)@jl)-
(ZA--ZB-)@/-l)]
(2.3) where A =.$#~(0)1~= 39.2X 10-4cm-*[16]. Electron-phonon interaction Re-, causes the transitions between W>‘l) and W,. Non-zero matrix elements are given by
A = 5~2JV[2( 1 - A’)]
UM%PI~)
--‘w&o,-A~)~9
(w,c1)(7,~)~~-Pw,(-,-) ) = (w:“(-,-)~~_,Y,(~,7)) = (Yt’-“(T,f)~~_,Y~(+,+)) = (Y’ll’(+,+)~~_,Y,(~,7)) (2.4)
where g, and CL,,denote the nuclear g-factor and the nuclear magneton. For simplicity we shall disregard the many-valley character of the donor wave function and approximate
Here, we put the nuclear spin states in the form (ntAt mB). Since the donor wave function is approximated by the superposition of wave functions in the vicinity of the bottom of the conduction band, we can choose the electronphonon interaction as usual in the form
K. SUGIHARA
1102
St’+, = iEl (2.5) where El, p, V and s are the deformation potential constant, density, volume and velocity of longitudinal sound; b, and b,* are annihilation and creation operators for the phonon of wave vector q. Using the expressions (2.4) and (2.5) and under the conditions A2 + 3, (qa*)’ 4 I, the square of the matrix elements (2.4) becomes I(*IIIYm‘4, mL3)Xe-, W&i’
%‘)I”
we can indicate easily that the transition = I_..~in the O-A process is limited only ik the process (tl, (+,+)I C= {s, (m,-m)} ;t where (s, (m,-HZ)} indicates if-ll t-t->I, the intermediate state. Then the matrix element of the O-A process is given by [ 191 (b,
(-7-I;
N,,
N,+
1, Nqr-
lIsK$,Jf*,
(+,-t-j;
N,)
=
z
(f-1,
c-,-k
b9
h,--m>;
N,t
N,S-
-
1 I~e-*l~,
(mm);
N,)
1 ]~,+xrIt*, (-I,+); N,)
X (E,,-- E,-fssq+$iM’,)-l 2( 1 - cos qR)
(2.6)
where fL
B
0
l/2
iE1 OS
=
AA2 2Jt/( I- A4) *
(i-,-t);
N&,
(4-m);
N,S-
2?r v = -fL (27r)”
x wt, If the condition (2.7) becomes
1)
(2.7)
J % Et1 - Et+ is satisfied,
PI (k (+,+); Nqls, (w-m) =- El2 A2JA4 -(iv,-+ 27Fps (hS)4
; iv,+
(2.7)’
where J = fisq
-hq+tisq’).
and
f(qR)
to
(2.9)
WI4 p2= ------+N,(N,t(7&%)2
IIN{&;
m,m’
l)f(qJ?).
Iwe~
-N{t-1;
(2.7)”
Next we shall consider the Orbach-Aminov process [ 17, 18]t (O-A process). From (2.4) tOrbach and Aminov developed their theories of the
so called ‘Orbach process’ independently. Hence we shall callit the Orbach-Aminov process.
l)f2(qR)rs-’ (2.9)’
(2.9)’ is just equal to (2.7)‘. The Z-component of the exchange system is written as S, = z
sin qR = 1 --a~.
co~es~nding
dqdq’l(2.8)126(E,,-_E;-,
p2=333/[
= _E,’ A=(Np+ 27rps (6s)’
1)
l)f(qR)
The transition probability the matrix element (2.8) is
Disregarding the Zeeman energy E,, - Et_, in comparison with J, (2.9) is expressed as follows using (2.6), (2.8) and (2.8)‘:
dq{ (2.6)) Z&- ?isq).
where Ts-’ denotes the life time of the iniermediate state s and
(2.6)’
N, or N,+ 1 in (2.6) correspond to the phonon absorption process or emission process, respectively. We get the phonon emission transition probability {tI, (+,+)} --;, {s, (m,-m)} as follows: Plfb,
(2.8)
coupled
(f%m’>Il (2.10)
where N{tltI; (m,-m’)} denotes the population in the state lt_ . cm _mtj , +1,
3
Averaging over the nuclear spin states, we find the relaxation rate of S, given by 6P(P, = Pz = P). Thus, the relaxation rate of the f.r.c. is given by
SPIN-LATTICE
RELAXATION
IN n-TYPE SILICON
1103
(2.11)
l/7,= ZAS(N,+l)f(qR).
Multiplying (2.11) by the weight of the triplet state kvt= 3 e-‘lkT/ ( 1+ 3 e-‘lkT) the effective relaxation rate of the n--n pair may be written in the form (1/7)n_n = z = ZASj-(qR)n(J)
(2.11)’
where n(J) = (2+eJlkT-
3 eVJlkT)-l. (2.11)”
J and A are complicated
functions of the effective Bohr radius a* [7, 201. Calculated curves of (l/7)“-” vs. R are shown in Figs. 1 and 2, where two values of a* are chosen as follows by use of the two radii a and b in the donor wave function.
I
IQ
I
I 40
35
I 50
45
D =I?/.¶*
I
20*7A:a
=
= 25 A, b = 14.2 A[211
Fig. 1. (l/7)“_ vs. R curves calculated by use of (2.1 I)‘.
(2.12)
16.06a:a=21.1&b=9.38,[22].
If all fast-relaxing centers are identical, the relaxation rate of this system is 1
l/T, = Td+tl
+N/N,)T,
Nd (NT,) =
1 +TdN,/(
NT/)
(2.13) where N, N, and rf denote the total spin density, concentration and relaxation time of the f.r.c., respectively. Here we have represented the spin-diffusion process by a time constant TV As has been shown ,in (2.11) or (2.1 l)‘, the relaxation rate of the f.r.c. depends strongly on R through J and A and t!le present situation is very complicated. Therefore, we shall assume simply that l/T, is given by substituting the averaged (I/T),_, over the Poisson distribution into N,/(NT,) of(2.13). Thus we get
(l/T,),-* =
I P(R) (1/~),-& 1 +Td I P(R) (lh),t-ndR
167 35
(2.14)
I 40
I 45
I 50
D=R/ll'
Fig. 2. (1 /T)._,, vs. R curves calculated by use of (2.11)‘.
1104
K. SUGIHARA
where P(R)
= 4wR2ND exp (- fR3ND).
‘r (2.14)’
Since we have no mathematical theory to calculate rd in the present case, we shall mined experimentally by Honig-Stupp [ 111. By saturating continuously about 1 per cent of the total number of spins at the center of the inhomogeneously broadened line, the spin diffusion times were measured. According to their results for a 10r6/cm3 phosphorus doped sample about 1.5 min was required for the resonance signal to be diminished to one half of its original value. For the 4 x 10’5/cm3 sample about 50min was necessary. They found that these times were independent of temperature between 1.25 and 2.16”K. Strictly speaking the spectral diffusion time and 74 are not the same quantities but they should have the same order of magnitude. Inserting the two values of thetime obtained by Honig-Stupp, (l/T,),_, was calculated for the samples ND= 1015/cm3, 4 X 101Vcm3 and 101‘Vcm3.The calculated results are shown in Fig. 3 and compared with the experimental results of Jerome, where a* = 20.7 A is used. The calculated values are field independent, which is not in accord with the measurements. However, if a difference in g-factor between the 0-n pairs and isalated impurities is present to a part in thousand, the field dependence in the high field region (H > 1O3Oe) could be explained. Since the inhomogeneous broadening due to the Si2s nuclei for the phosphorus doped silicon is 2.5 Oe[23], the Zeeman energy difference is comparable to the inhomogeneous line width at H - lo3 Oe and the slowdown of the energy exchange rate between the isolated donors and the f.r.c. makes l/T, longer. However, this mechanism would fail in the interpretation of the field dependence in the low field region because the Zeeman frequency difference AgpH is much smaller than the inhomogeneous line width. Apart from the field independent result of
T = I.27’K
d
IO<
I
I
I03
104
Magnetic
field
I
105
Ii (Oe)
Fig. 3. Calculated values of (I/T,),_, and the field dependence of I/T, observed by Jtrome.
our theory, qualitative agreement with the Jerome’s data for the samples 10*“/cm3 and l-1 X 1016/cm3 supports the validity of our model. The computed values of l/T, are approximately proportional to N,, and this feature is consistent with the Y-H results. (see Fig. 4). The calculated temperature dependence of l/T, lies between T’ and T’. while the observed dependence[ 1 l] at H = 200 Oe is nearly proportional to T. Since the temperature dependence of l/T, varies with the Bohr radius a+, improved agreement might be obtained by use of the correct donor wave function. Comparison with experiment is shown in Fig. 5 and the calculated J-R curves are given in Fig. 6 where R denotes interdonor distance. As shown in Fig. 3 the present mechanism
SPIN-LATTICE
RELAXATION
1105
IN n-TYPE SlLlCON
IO
‘ii ?,
s I
I6
/ 50
100
130
R ii,
10'5 N, (cm-‘1
I
lo”
Fig. 6. Variation of the exchange integral J with the interdonor distance R.
Fig. 4. Observed concentration dependence by YangHonig and calculated result of i/T, at T = 1*25”K.
3. RRLAXA’RON PROCESS RELATED TO THE ALL-IO~ZRD DONOR PAIRS
Y I
-r/ /
I
‘,’ .;/
k
I
,fl’ I I
.
:
1.7x10’6
I
:
,., xlo’6
m :
---
?-
1.1xlo” I5
: 4x10 : 10’5
H= 200
Obr
10’5
COIC
oc
/
t
2
3
T’
fails to explain the results for the more heavily doped samples. In this connection we will suggest more effective mechanisms in Section 3.
4
5
K
Fig. 5. Temperature dependence of l/7’, as compared with the observed values at H = 200 Oe by Honig-Stupp.
The order of magnitude and the field dependence of l/T, for ND > 10Vcm3 samples can not be explained by the mechanism proposed in Section 2. In this section we shall present two different mechanisms related to the n-i pairs which are suitable for in~rpreting the experimental results for the heavily doped samples. There are two interactions which cause the admixture of the different spin states. (1). The one is the hypefine coupling[7] and (2). the other the spin-orbit interaction due to the impurity potential[9]. (1). Even in the uncompensated samples a small number of donor atoms are inevitably compensated, Thus consider an n-i pair i-j (see Fig. 7). The energies Ei and EJ are not equal because of the random dist~bution of the ionized impurities. The energy difference A comes mainly from adjacent acceptor ions. The wave functions of an electron at i and j
K. SUGIHARA
1106
=+&!yliE -2AE __;__-_____+y
9
Nb = N,+- 1
eiqwa
:Nh = N,+ 1 (3.3)
(i; -, mi = Tk N$T’e_p~j: +, mi = -1-3;N,) l’2
AE
4
1,2 eiqRifl:
Nh= N,-1
-ei,~:N~=N,+l
\
Fig. 7. Energy levels of i-j system.
sites, which include small admixture different spin states, are
of the
**to = $=(r - RJ f *J,‘s(r
- RJ
- R&,(r
I,2
-e,,%m)
+ kj
A
+
(3.3)’
where mA represents the nuclear spin state of a A site. The transition probability for the matrix element (3.3) is PI(j;+,mj=k&
N,li;-,mj=Ti;
=--2lr v h (2Tr)3 J x S(fisq -
N,,-1)
4 (3.3) I2
A) (3.4)
The transition probability for (3.3)’ is the same as (3.4). (1 W,12 ) is the angular average of 1W,12 and is given by[22]
+~{JIA- RJ
e-2Rla (3.5)
(I W,12) = $Er(s)1’2R’
where JI+(r - R,) denote the impurity wave functions localized at c site whose spin directions are specified by +, and W, the resonance energy between i and j sites[9,22]. AE is the Zeeman splitting. The matrix element for a transition from the j site to i site accompanied by the spin reversal is ( *T(*)sre_p**(‘))
(3.2)
where (Y= ( a/b)2 - 1, K and R are the dielectric constant and the interdonor distance. Next we shall consider the O-A process. The effective matrix element for the transition m = &+) is given by (j; +, +
Wlj=k*)
(j;
-,
(~;-,m,=~~;Nq-l,Nq~+l(~~~~~j;+,~~ = +i; N,, N,I) = {(j;-; X
(i; -,
N,,+ lJSY+,(i; --; N,f) ??Zj
=T+;
Nq-l(Xe-,lj;+,mj
where X=-e is the electron-phonon interaction given by (2.5). The matrix elements (3.2) explicitly wirtten are
=k&
N,) + (j;-,
(i; -,
= *t; N,,)
mj
=
3;
N~~~e-p~
j, +, ml = kk; N,)
mj
= ~4; N,, + 1I%‘C_pI i; +, mj
SPIN-LATTICE
X
(i;+; N,-
RELAXATION
l~Gi?~&;+;
IN n-TYPE SILICON
1107
and one for the upper state i
NJ)
x (-A + hsq f &ifLI’f)-l.
2 exp (A/&-) (Pt. + P2).
(3.6)
This is the mechanism proposed by Jerome [7]. Using (3.1) and (3.3) (3.6) is expressed as
In consideration of the nuclear spin states and the populations of i and j states the effective relaxation rate of the f.r.c. is given by
X (-A + hsq + )ifiTi)-l x {&%
(e-is’%
_
e-W&)
+
e-&‘&(
@R.
_
@RI
(3.10)
) }
(3.6)’ where (3.6)”
(2). In the following we shaft treat the second mechanism of the relaxation rate of the n-i pair[9]. Wave functions which are necessary in the following procedure, are
Throughout the above calculation we have assumed that A % AE. Carrying out a similar cafcufation to (2.9) we get the transition probability
‘?kfi) s Jl,(r-Ri)
w1* -,Jrdr-R,)
w** -,JIT(r-Rjl
‘I.‘,‘” s t,b,(r-Rj) fisq = A
+?$k(r--R,)
f?Jlp(r-RJ.
(3.1 f)
where F(q)
= 3-4
sin qR
-+-
qR
sin qR 2 (
qR
> *
(3.7)’
Using the wave functions (3.1 I) one can easily obtain the transition probability for the direct process (j; +) -+ (i; -) as follows:
Ti-’ is the fife time of the state i and may be written in the following form r = 2lG/* i
--&+w9+
1Y(4).
(3.8)
(hs)’
F(q)
u,f‘tY) ’ (3.9) -
(3.9) is identical with (3.4) in the limit qR + 0 because in this limit F/f tends to 2. In the limit qR >> 1 Fijf’ tends to 3 and then P,lP, approaches 3/2. Jirome disregarded the factor Flfi The same probability as (3.9) is obtained for the process with 1~~instead of IN, in (3.6). Summing up (3.4) and (3.9) one gets a refaxation rate for the lower statej 2(P,+Pz)
x&q)
(3.12)
where (f W#)
Inserting (3.8) into (3.7) we get
‘W’,l’MN
E,* (Iw,l”)A P, = 7rps (hs)’
s
(~)2(~)*(~)4(~)112e-~~),
A indicates the spin-orbit coupling due to the impurity atom and E a quantity which is nearly equal to the ionization energy of the donor electron. The transition probability in the O-A process is just twice (3.12). By taking into consideration the direct and the O-A processes simultaneously, the effective relaxation rate of the f.r.c. becomes ( ,,7Z) _, = 3E1’ (Iw~l’> Af(q) )I I G (ns)4 2 sinh (AjkT)’
(3’13)
1108
K. SUGIHARA
Finally, summing up (3.10) and (3.13), one obtains the relaxation rate of the n-i pair (l/r)n_i=
(l/TI)n-j+
(i/rc!)n-i.
5x.102 Oe
(3.14)
( l/rl),,+
which is proportional to W2, is a more effective mechanism than (l/rz)n-i at low magnetic field (H < 103Oe) for the samples ND > 101Vcm3. The concentration dependence of ( l/Tl)n-i and ( l/Tz)n_i at T = 1.25”K are shown in Figs. 8 and 9. The ND dependence of (l/rq)n-i is relatively weak but (l/rl)n_i shows a drastic change with ND. In Figs. 8 and 9 we have replaced R by the average distance R = (3/47rNC3 and put A = kT. Hitherto, we have considered the relaxation process of an n-i pair, but at this stage it remains to estimate the effective number of the n-i pairs which behave as f.r.c. This can be carried out according to a method developed in our previous paper[6, 91. Effective donor pairs exist only in the energy range in which the condition (E, - 51 - kT is satisfied (r = ij ). Now suppose a thin spherical shell around an acceptor ion such that Ir,--1-1 < Arl2
(3.15)
T=
ND(cmm3) Fig. 9. Donor concentration
dependence of (l/~,)~_+
where r, and Ar are defined by
1;=$, k+$.
IO / (I/T,)n_,,
r
C
,H=I03 Oe
T = b25*K
(3.15)’
C
An electron trapped in the donor site satisfying the condition (3.15) has the energy j,C-51 < kT.
(3.15)”
The probability of two donors being contained in the shell (3.15) is given by the Poisson distribution as
10-I
IO'5
/I N
Fig. 8. Donor
ND2(4wc2Ar)2 exp [-(87r/3)r,.“N,,].
I
I
iO'6
IO"
for the Fermi energy
5=-E,[K/ln(l-_)I”“,
D(cmm3)
concentration dependence (1 /T*)n--I.
By use of the expression ]221
of (I/T&~
and
(3.16)
R = (3/47~N[>)“”
E,=-$, (3.17)
SPIN-LATTICE
RELAXATION
and (3. IS)‘, (3.16) is rewritten in the form
(3.1 w Since the ratio of the acceptors to the isolated donors is given by K/ ( 1 - K), the relaxation rate as a whole is expressed by (ljT,j,_j
(l/T)=-ineff
=
1+ Td(llT)n-i&K
(3.19)
( l/r)n_i is the relaxation rate (3.14) with the substitutions R = (3/4rrNUP3 and A = kT . neff is related to (3.18) by the relation K bff =
2(1-K)
x (3.18).
IN n-TYPE
SILICON
1109
could shorten considerably the experimentally measured relaxation times. Y-H confirmed that the compensation dependence of their results for the samples K < O-5 was well explained using our previous formula[9] which shows nearly the same dependence on K as the present formula in the range of small compensation. Assigning the appropriate values of K and Td to the three samples 3.2 X f0%m3, 6 X 1016/cm3 and 8 x 1W6/cm3, calculated results are shown in Fig. 10 in comparison with the experimental results. K and Td values are reasonable ones. According to JCrome[7]
(3.20)
The extra factor l/2 appears in (3.20), since we have assumed that the probability of one other vacant is equal to I /2. Now it is necessary to decide the magnitude of A in (l/~~),~_.~in comparison with experiment. For this purpose we use the Y-H’s data. They obtained l/T, = le.5 X lop3 set-’ at T = 1%‘K and H = lo” Oe for the sample ND = 6 x 101”/cm3 and K = 0.33. For this sample ( l,/Tz)e-i is very much larger than ( l/?I)n_i. Assuming ?d = 300 set-this iS a reasonable diffusion time judging from the experiment by Honig-Stupp [ IX] - one gets h = 4.70 X lOemeV.
(3.21)
This value is close to the one obtained previously[9]. Using (3.21) we get (1/71)n--i= and ( 1/TJ n_i = 1.08 set-I. 1.02 X 10-r set-’ Igo obtained a very different result [6] from Y-H for almost the same condition as theirs. He reported that l/T, = 2.8 X 10v2 see-* at T = 2°K for the sample ND = 5.9 X 10i5/cm3 and K = O-32, while the relaxation rate measured by Y-H is only l/20 of Igo’s result. Y-H pointed out that one possible origin of such discrepancy is a small room temperature i.r. radiation leak into Igo’s samples which t In (4.15) in reference [9] the exponential term in (3.18) is missing.
If? 102
I
I
IO3
10-J
I05
Magneticfield H (Oe 1 Fig. IO. Calculated values of (l/T,),~.+ and the observed results of Ii?‘, at 7’ = I-27°K. Foiiowinrr values of K and 7@ are used foi each sample: II = 0.01, 7d= 40 set for the sample N, = 3.2 X 10’“/cm3; K = 0.01, 7d= 10set for ND= 6X 10LB/cm3 and K= 0.04, T== 4sec for ND= 8 X 10*B/cmy.
1110
the compensation cm3 is less than approximations results are in observed ones.
K. SUGIHARA
of the 0.05. used good
sample ND = 6 X 1OV In spite of the crude here the computed agreement with the
4. SUMMARY
In this paper we have discussed the relaxation mechanisms of the donor electron in uncompensated phosphorous doped silicon (K < 0.1). The principal results obtained in the preceeding sections are summarized as follows. (1). The spin-lattice relaxation for uncompensated silicon is classified into three mechanisms according to impurity concentration. In all the mechanisms spin excitations of the main body of the donor electrons which are sufficiently isolated, are transferred via a spin-diffusion process and finally relax at fast-relaxing centers which are few in number. (2). For lightly doped samples, ND < 1OV cm3, exchange coupled donor pairs act as the fast-relaxing centers. This mechanism is field independent and can not explain the weak field dependence of the observed relaxation rates. At this stage we have no definite explanation of such a discrepancy. Apart from this point the order of magnitude of the calculated results is reasonable. (3). For the more heavily doped samples, ND > 101s/cm3, two different mechanisms are active if the samples are slightly compensated (K - 10P2).’ In this case a fastrelaxing center is composed of a neutralionized donor pair. Mixing of different spin states is caused by the hyperfine interaction or the spin-orbit coupling due to the impurity potential. The transition rate for the first mechanism is proportional to He2 and strongly dependent on ND. On the other hand the second mechanism is field independent and weakly dependent on N,. By selecting appropriate values of K and spin-diffusion times for each sample the observed results are well explained by our theory. (4). Instead of entering into the complicated problem of analyzing the spin-diffusion pro-
cess, an appropriate spin-diffusion time has been introduced for each sample. Finally we propose an experiment which will throw further light on the relaxation mechanisms. This is the measurement of relaxation times under the application of uniaxial stress[24]. Reduction of the hyperfine coupling due to the application of such a stress may cause the spin-relaxation rate to decrease appreciably. Acknowledgement-The author wishes to thank Dr. K. Morigaki of the Institute for Solid State Physics, University of Tokyo, for his helpful discussions and for informing him of J rome’s thesis. REFERENCES 1. HASEGAWA H., Phys. Rev. l&1523 (1960). 2. ROTH L. M., Phys. Rev. 118, 1534 (1960). 3. CASTNER T. G. Jr., Phys. Rev. 130, 58 (1963); ibid. 155,8 16 (1967).
4. FEHER G. and GERE E. A., Phys. Rev. 114, 1245 (1959).
5. SASAKI
W., Proc. Inf. Conf. Semicon. Phvsics. Kyoto, 543 (1966); Maekawa S., ibid. 574. 6. IGO T., J. Dhys. Sot. Jaoan 21,874 (1966). 7. J&ROME ‘D:, Thesis, ‘University of Paris (1965).
Unpublished. 8. YANG G. and HONIG A., Private communication. 9. SUGIHARA K.,J.phys.Soc.Jupan 18,961 (1963). 10. MORIGAKI K. and WINTER J. M., Proc. Int. Conf. Semicon. Physics, Kyoto, 570 (1966). 11. HONIGA.andSTUPPE.,Phys. Rev. 117,69(1960). 12. SLICHTER C. P., Phys. Rev. 99, 479 (1955); FEHER G., FLETCHER R. C. and GERE E. A., P$ys. Rev. 100, 1784
(1955).
13. JEROME D. and WINTER J. M., Phys. Rev. 134, 1001 (1964). 14. MOROCHA A. K., Soviet Phys. solid Sr. 4, 1683 (1963). 15. GLINCHUK M. D., GRACEV V. G. and DEIGEN M. F., Soviet Phys. solid Sr. 82678 (1967). H. H., Solid 16. LUDWIG G. W. and WOODBURY Stnte Physics (Edited by Seitz F. and Turnbull D.), Vol. 13. II. 223. Academic Press. New York (1962). 17. ORBAdH R., Proc. R. Sot. A264,458 (196 i). 18. AMINOV L. K., Soviet Phys. JETP 15,547 (1962). 19. HEITLER W., The Quantum Theory of Radiation, p. 196. Clarendon Press, Oxford (1954). 20. PAULING L. and WILSON E. B., Introduction to Quantum Mechanics
with Applications
to Chemistry,
Chap. 12. McGraw-Hill, New York (1935). 21. KOHN W., So/id State Physics (Edited by Seitz F. and Turnbull D.), Vol. 5. Academic Press, New York (1957). 22. MILLER A. and ABRAHAMS E., Phys. Rev. 12% 745 (1960). 23. FEHER G., Phys. Rev. 114,1219 (1859). 24. WILSON D. K. and FEHER G., Phys. Reu. 124, 1068 (1961).
Pergamon Press 1968. Vol. 29, pp. 1099-l 110.
Printed in Great Britain.
CONCENTRATION DEPENDENT SPIN-LATTICE RELAXATION IN n-TYPE SILICON KO SIJCIHARA Wireless Research Laboratory, Matsushita Electric Industrial Co. Ltd., Kadoma,
Osaka,Japan
(Received 5 October 1967; in revisedform 22 December 1967) Abstract-Spin-lattice relaxation mechanisms for the donor electrons in uncompensated silicon are presented. Major isolated spins transfer their excitation energies via the spin-diffusion process to the fast-relaxing centers. For lightly doped samples, N1, 2 10’Ycm3, exchange-coupled donor pairs act as the fast-relaxing centers. Theory provides the correct order of magnitude for the relaxation rate l/T,. However, the calculated relaxation rate l/T, for this process is field independent, while the observed rate shows a weak field dependence. For more heavily doped samples, ND > 10i6/cms, the relaxation rate can be explained by assuming the presence of a small concentration of neutral-ionized donor pairs. The relaxation process for these pairs is the resultant of two different mechanisms, a field dependent mechanism and a field independent one. The former depends strongly on the donor concentration and the latter shows relatively weak dependence on ND. 1. INTRODUCTION
SPIN-LATTICE relaxation
of shallow donors in silicon assumes various aspects according to the magnitudes of the four parameters, magnetic field H, temperature T, donor concentration ND and compensation degree K. In the case of lightly doped samples in which donors are su~ciently isolated, relaxation mechanisms have been well established by Hasegawa[l], Roth[2] and Castner[3]. With increasing donor concentration a new feature, which strongly depends on impurity concentration, appears[4]. For these samples the Ii- and T-dependence of l/T, are entirely different from the isolated donor’s case. The concentration dependent regions are grouped into the three cases according to the impurity concentration as follows: (1) ND < 10’7/cm3
and
(2) ND < 1017/cm3 and
K 4 1 K > 0.1
(3) ND > 10%m3. Case (3) corresponds to the impurity band conduction range and the mechanisms of the transport phenomena and the magnetic resonance are not so well understood[5].
Igo[6], Jerome[7] and Yang-Honig[8],t performed detailed me~urements of l/T, for the case (2) samples. We proposed a theoryE related to Igo’s work and succeeded in explaining his results qualitatively except for the highly compensated samples. The validity of our mechanism has been ascertained by the recent measurement of Y-H [S]. It has been reported that the relaxation rate is enhanced by application of an electric field for case (2) samples[lO]. A strong field exictes the donor electrons into the conduction band by impact ionization and the localized spins relax rapidly through interaction with the free carriers. However, it is not clear why the relatively weak field accelerates the relaxation rate. In this paper we shall treat case (1). HonigStupp[ll], Jerome171 and Y-H[8] carried out detailed measurements related to these samples. It is worth noticing that Y-H have succeeded in separating different mechanisms by filling up the ionized donor sites using the optical pumping technique. In this article our theory is manly compared with the Jkrome’s tHereafter
1099
we shall refer this paper Y-H.
1100
K. SUGIHARA
work which covers a fairly wide concentration range. We shall review the main observed results: (1). l/T, decreases with increasing magnetic field and its dependence is stronger for the more concentrated samples. At high fields the H-dependence approaches asymptotically an H4-law, which corresponds tc the dependence of the isolated donor. (2). At low temperatures l/T, shows weak variation with T but at high temperatures it obeys a P-dependence, which appears in the Raman process for the isolated donor’s case. (3). The concentration dependence of l/T, is nearly linear for the samples ND < 10%m3 and rises sharply with ND for the more highly concentrated ones. Honig-Stupp [ 1 l] suggested a mechanism for the concentration dependent spin-lattice relaxation: the excitation energies of the major part of the donor spins are transferred via the spin-diffusion process to the fastrelaxing centers? which are few in number but can relax very rapidly. In compensated samples neutral-ionized pairs (n-i pairs) assume the role of f.r.c. [6, 8, 91 but in case (1) importance of the exchange coupled pairs was pointed out [8, 111. As ND increases, resonance lines of exchange coupled clusters consisting of two or more donors are observed[ 121 besides the hyperfine lines. Resonance lines of donor pairs in silicon have been investigated by Jerome and Winter] 131 with the aid of the observation of the ENDOR spectrum and the line shape of the spectrum has given a determination of the sign of the exchange integral J which turns out to be antiferromagnetic. The relaxation mechanisms presented in this article are based on the Honig-Stupp’s suggestion. According to the impurity concentrations relaxation mechanisms of f.r.c. are classified into three different types. Exchange coupled neutral pairs (n-n pairs) play an important role in the relaxation process for the samples ND < 101%m3. Triplet and THereafter we shall abbreviate f.r.c.
singlet states of a n-n pair are admixed each other via the hyperfine interaction and the electron-phonon interaction induces the triplet + singlet transition being accompanied with the Zeeman energy change. This process does not depend on the magnetic field strength and the relaxation rate on the whole is also field independent. This feature is in contradiction to the observed results [7, 8, 111. However, there is a possibility to remove this discrepancy in the high field region by assuming a g-factor difference (- 10e3) between isolated donors and n-n pairs[ll]. At high fields the Zeeman energy difference will exceed the inhomogeneous broadening, which is 2.5 Oe in the phosphorous doped silicon, and then energy exchange between the isolated donors and the n-n pairs becomes very difficult. But it would not explain the observed field dependence in the low field region. In our calculation variation of the f.r.c. number with H is very small and so at this stage we are unable to interpret the observed field dependence. Y-H [ 81 adopted the relaxation mechanisms of the n-n pairs induced by modulation of the dipolar interaction[ 141 or the hyperfme and exchange couplings [ 151 due to the lattice vibration. However, these mechanisms are both ineffective for the donor spins in silicon, while our mechanism provides a correct order of magnitude of l/T, without any special assumptions for the samples 1015/cm3 and For the 1.1 X 101g/cm3 used by Jerome. samples N,, < 101’%m3 l/T, is proportional to ND. This is consistent with our calculation. However, the above-mentioned mechanism fails in the interpretation of l/T, for the samples 3.2 X 10%m3, 6 X 1016/cm3 and 8 x 1016/cm3. For these samples the observed concentration dependence of l/T, is very strong and the field dependence is steeper than for the lightly doped samples. If these samples are slightly compensated, n-i pairs behave like the f.r.c. Consider an n-i close pair i-j. Admixture of different spin states is introduced into the wave functions localized at i orj sites
SPIN-LATTICE
RELAXATION
through the hyperfine interaction [7] or through the spin-orbit interaction of the impurity atom[9]. Phonon-assisted hopping from an occupied level to a vacant one of the i-j system, accompanied by a reversal of spin direction, occurs frequently, if the interdonor distance R is not too large and the energies Et and Ej are close to the Fermi level 5. The process related to the hyperfine interaction is proportional to He2 and strongly dependent on R, while the one arising from the spin-orbit interaction of the impurity atom is field independent and weakly dependent on R [9]. It is very difficult to formulate the spindiffusion problem correctly in the case of a random distribution of impurity centers. Therefore, we shall introduce an appropriate spin-diffusion time for each sample instead of entering into the details of the spin-diffusion process. The observed features of the experimental results can be explained qualitatively by use of the above-mentioned three mechanisms. In Section 2 calculation of the relaxation rate related to the n--)2 pairs is presented. In Section 3 we shall discuss the two relaxation mechanisms caused by the n-i pairs. Finally we summarize the results obtained in this paper. 2. RELAXATION PROCESS RELATED TO THE EXCHANGE COUPLED DONORS
For definiteness we shall confine our discussion to the phosphorus doped silicon. Now suppose an n-n pair whose exchange energy is sufficiently greater than the Zeeman splitting of the triplet state. Hyperfine coupling associated with donors A and B is given
IN n-TYPE SILICON
1101
it by a spherical Is-function 4 - exp(-r/a*). Triplet and singlet states of an n-n pair are represented by
-Hrl-
RBM(r2- RA)IxW)
1 @P,= V12( 1 + A”) 3 { $(rr - R&(r2 -
RB)
+ddV-Rd~(W--,d~xs
(2.2)
where xt (M) and x8 are spin functions and A denotes the overlap integral. @,1&l)and @‘r are admixed with each other through the hyperfine coupling (2.1). After a simple calculation we get yp
=
@,I”’
A
f
2&[2(1-A4)]
(zBT--zAS)@~
A v\Ir,= @‘+ 2JV[2(1 -A4)] X
[(IA+-ZB+)@jl)-
(ZA--ZB-)@/-l)]
(2.3) where A =.$#~(0)1~= 39.2X 10-4cm-*[16]. Electron-phonon interaction Re-, causes the transitions between W>‘l) and W,. Non-zero matrix elements are given by
A = 5~2JV[2( 1 - A’)]
UM%PI~)
--‘w&o,-A~)~9
(w,c1)(7,~)~~-Pw,(-,-) ) = (w:“(-,-)~~_,Y,(~,7)) = (Yt’-“(T,f)~~_,Y~(+,+)) = (Y’ll’(+,+)~~_,Y,(~,7)) (2.4)
where g, and CL,,denote the nuclear g-factor and the nuclear magneton. For simplicity we shall disregard the many-valley character of the donor wave function and approximate
Here, we put the nuclear spin states in the form (ntAt mB). Since the donor wave function is approximated by the superposition of wave functions in the vicinity of the bottom of the conduction band, we can choose the electronphonon interaction as usual in the form
K. SUGIHARA
1102
St’+, = iEl (2.5) where El, p, V and s are the deformation potential constant, density, volume and velocity of longitudinal sound; b, and b,* are annihilation and creation operators for the phonon of wave vector q. Using the expressions (2.4) and (2.5) and under the conditions A2 + 3, (qa*)’ 4 I, the square of the matrix elements (2.4) becomes I(*IIIYm‘4, mL3)Xe-, W&i’
%‘)I”
we can indicate easily that the transition = I_..~in the O-A process is limited only ik the process (tl, (+,+)I C= {s, (m,-m)} ;t where (s, (m,-HZ)} indicates if-ll t-t->I, the intermediate state. Then the matrix element of the O-A process is given by [ 191 (b,
(-7-I;
N,,
N,+
1, Nqr-
lIsK$,Jf*,
(+,-t-j;
N,)
=
z
(f-1,
c-,-k
b9
h,--m>;
N,t
N,S-
-
1 I~e-*l~,
(mm);
N,)
1 ]~,+xrIt*, (-I,+); N,)
X (E,,-- E,-fssq+$iM’,)-l 2( 1 - cos qR)
(2.6)
where fL
B
0
l/2
iE1 OS
=
AA2 2Jt/( I- A4) *
(i-,-t);
N&,
(4-m);
N,S-
2?r v = -fL (27r)”
x wt, If the condition (2.7) becomes
1)
(2.7)
J % Et1 - Et+ is satisfied,
PI (k (+,+); Nqls, (w-m) =- El2 A2JA4 -(iv,-+ 27Fps (hS)4
; iv,+
(2.7)’
where J = fisq
-hq+tisq’).
and
f(qR)
to
(2.9)
WI4 p2= ------+N,(N,t(7&%)2
IIN{&;
m,m’
l)f(qJ?).
Iwe~
-N{t-1;
(2.7)”
Next we shall consider the Orbach-Aminov process [ 17, 18]t (O-A process). From (2.4) tOrbach and Aminov developed their theories of the
so called ‘Orbach process’ independently. Hence we shall callit the Orbach-Aminov process.
l)f2(qR)rs-’ (2.9)’
(2.9)’ is just equal to (2.7)‘. The Z-component of the exchange system is written as S, = z
sin qR = 1 --a~.
co~es~nding
dqdq’l(2.8)126(E,,-_E;-,
p2=333/[
= _E,’ A=(Np+ 27rps (6s)’
1)
l)f(qR)
The transition probability the matrix element (2.8) is
Disregarding the Zeeman energy E,, - Et_, in comparison with J, (2.9) is expressed as follows using (2.6), (2.8) and (2.8)‘:
dq{ (2.6)) Z&- ?isq).
where Ts-’ denotes the life time of the iniermediate state s and
(2.6)’
N, or N,+ 1 in (2.6) correspond to the phonon absorption process or emission process, respectively. We get the phonon emission transition probability {tI, (+,+)} --;, {s, (m,-m)} as follows: Plfb,
(2.8)
coupled
(f%m’>Il (2.10)
where N{tltI; (m,-m’)} denotes the population in the state lt_ . cm _mtj , +1,
3
Averaging over the nuclear spin states, we find the relaxation rate of S, given by 6P(P, = Pz = P). Thus, the relaxation rate of the f.r.c. is given by
SPIN-LATTICE
RELAXATION
IN n-TYPE SILICON
1103
(2.11)
l/7,= ZAS(N,+l)f(qR).
Multiplying (2.11) by the weight of the triplet state kvt= 3 e-‘lkT/ ( 1+ 3 e-‘lkT) the effective relaxation rate of the n--n pair may be written in the form (1/7)n_n = z = ZASj-(qR)n(J)
(2.11)’
where n(J) = (2+eJlkT-
3 eVJlkT)-l. (2.11)”
J and A are complicated
functions of the effective Bohr radius a* [7, 201. Calculated curves of (l/7)“-” vs. R are shown in Figs. 1 and 2, where two values of a* are chosen as follows by use of the two radii a and b in the donor wave function.
I
IQ
I
I 40
35
I 50
45
D =I?/.¶*
I
20*7A:a
=
= 25 A, b = 14.2 A[211
Fig. 1. (l/7)“_ vs. R curves calculated by use of (2.1 I)‘.
(2.12)
16.06a:a=21.1&b=9.38,[22].
If all fast-relaxing centers are identical, the relaxation rate of this system is 1
l/T, = Td+tl
+N/N,)T,
Nd (NT,) =
1 +TdN,/(
NT/)
(2.13) where N, N, and rf denote the total spin density, concentration and relaxation time of the f.r.c., respectively. Here we have represented the spin-diffusion process by a time constant TV As has been shown ,in (2.11) or (2.1 l)‘, the relaxation rate of the f.r.c. depends strongly on R through J and A and t!le present situation is very complicated. Therefore, we shall assume simply that l/T, is given by substituting the averaged (I/T),_, over the Poisson distribution into N,/(NT,) of(2.13). Thus we get
(l/T,),-* =
I P(R) (1/~),-& 1 +Td I P(R) (lh),t-ndR
167 35
(2.14)
I 40
I 45
I 50
D=R/ll'
Fig. 2. (1 /T)._,, vs. R curves calculated by use of (2.11)‘.
1104
K. SUGIHARA
where P(R)
= 4wR2ND exp (- fR3ND).
‘r (2.14)’
Since we have no mathematical theory to calculate rd in the present case, we shall mined experimentally by Honig-Stupp [ 111. By saturating continuously about 1 per cent of the total number of spins at the center of the inhomogeneously broadened line, the spin diffusion times were measured. According to their results for a 10r6/cm3 phosphorus doped sample about 1.5 min was required for the resonance signal to be diminished to one half of its original value. For the 4 x 10’5/cm3 sample about 50min was necessary. They found that these times were independent of temperature between 1.25 and 2.16”K. Strictly speaking the spectral diffusion time and 74 are not the same quantities but they should have the same order of magnitude. Inserting the two values of thetime obtained by Honig-Stupp, (l/T,),_, was calculated for the samples ND= 1015/cm3, 4 X 101Vcm3 and 101‘Vcm3.The calculated results are shown in Fig. 3 and compared with the experimental results of Jerome, where a* = 20.7 A is used. The calculated values are field independent, which is not in accord with the measurements. However, if a difference in g-factor between the 0-n pairs and isalated impurities is present to a part in thousand, the field dependence in the high field region (H > 1O3Oe) could be explained. Since the inhomogeneous broadening due to the Si2s nuclei for the phosphorus doped silicon is 2.5 Oe[23], the Zeeman energy difference is comparable to the inhomogeneous line width at H - lo3 Oe and the slowdown of the energy exchange rate between the isolated donors and the f.r.c. makes l/T, longer. However, this mechanism would fail in the interpretation of the field dependence in the low field region because the Zeeman frequency difference AgpH is much smaller than the inhomogeneous line width. Apart from the field independent result of
T = I.27’K
d
IO<
I
I
I03
104
Magnetic
field
I
105
Ii (Oe)
Fig. 3. Calculated values of (I/T,),_, and the field dependence of I/T, observed by Jtrome.
our theory, qualitative agreement with the Jerome’s data for the samples 10*“/cm3 and l-1 X 1016/cm3 supports the validity of our model. The computed values of l/T, are approximately proportional to N,, and this feature is consistent with the Y-H results. (see Fig. 4). The calculated temperature dependence of l/T, lies between T’ and T’. while the observed dependence[ 1 l] at H = 200 Oe is nearly proportional to T. Since the temperature dependence of l/T, varies with the Bohr radius a+, improved agreement might be obtained by use of the correct donor wave function. Comparison with experiment is shown in Fig. 5 and the calculated J-R curves are given in Fig. 6 where R denotes interdonor distance. As shown in Fig. 3 the present mechanism
SPIN-LATTICE
RELAXATION
1105
IN n-TYPE SlLlCON
IO
‘ii ?,
s I
I6
/ 50
100
130
R ii,
10'5 N, (cm-‘1
I
lo”
Fig. 6. Variation of the exchange integral J with the interdonor distance R.
Fig. 4. Observed concentration dependence by YangHonig and calculated result of i/T, at T = 1*25”K.
3. RRLAXA’RON PROCESS RELATED TO THE ALL-IO~ZRD DONOR PAIRS
Y I
-r/ /
I
‘,’ .;/
k
I
,fl’ I I
.
:
1.7x10’6
I
:
,., xlo’6
m :
---
?-
1.1xlo” I5
: 4x10 : 10’5
H= 200
Obr
10’5
COIC
oc
/
t
2
3
T’
fails to explain the results for the more heavily doped samples. In this connection we will suggest more effective mechanisms in Section 3.
4
5
K
Fig. 5. Temperature dependence of l/7’, as compared with the observed values at H = 200 Oe by Honig-Stupp.
The order of magnitude and the field dependence of l/T, for ND > 10Vcm3 samples can not be explained by the mechanism proposed in Section 2. In this section we shall present two different mechanisms related to the n-i pairs which are suitable for in~rpreting the experimental results for the heavily doped samples. There are two interactions which cause the admixture of the different spin states. (1). The one is the hypefine coupling[7] and (2). the other the spin-orbit interaction due to the impurity potential[9]. (1). Even in the uncompensated samples a small number of donor atoms are inevitably compensated, Thus consider an n-i pair i-j (see Fig. 7). The energies Ei and EJ are not equal because of the random dist~bution of the ionized impurities. The energy difference A comes mainly from adjacent acceptor ions. The wave functions of an electron at i and j
K. SUGIHARA
1106
=+&!yliE -2AE __;__-_____+y
9
Nb = N,+- 1
eiqwa
:Nh = N,+ 1 (3.3)
(i; -, mi = Tk N$T’e_p~j: +, mi = -1-3;N,) l’2
AE
4
1,2 eiqRifl:
Nh= N,-1
-ei,~:N~=N,+l
\
Fig. 7. Energy levels of i-j system.
sites, which include small admixture different spin states, are
of the
**to = $=(r - RJ f *J,‘s(r
- RJ
- R&,(r
I,2
-e,,%m)
+ kj
A
+
(3.3)’
where mA represents the nuclear spin state of a A site. The transition probability for the matrix element (3.3) is PI(j;+,mj=k&
N,li;-,mj=Ti;
=--2lr v h (2Tr)3 J x S(fisq -
N,,-1)
4 (3.3) I2
A) (3.4)
The transition probability for (3.3)’ is the same as (3.4). (1 W,12 ) is the angular average of 1W,12 and is given by[22]
+~{JIA- RJ
e-2Rla (3.5)
(I W,12) = $Er(s)1’2R’
where JI+(r - R,) denote the impurity wave functions localized at c site whose spin directions are specified by +, and W, the resonance energy between i and j sites[9,22]. AE is the Zeeman splitting. The matrix element for a transition from the j site to i site accompanied by the spin reversal is ( *T(*)sre_p**(‘))
(3.2)
where (Y= ( a/b)2 - 1, K and R are the dielectric constant and the interdonor distance. Next we shall consider the O-A process. The effective matrix element for the transition m = &+) is given by (j; +, +
Wlj=k*)
(j;
-,
(~;-,m,=~~;Nq-l,Nq~+l(~~~~~j;+,~~ = +i; N,, N,I) = {(j;-; X
(i; -,
N,,+ lJSY+,(i; --; N,f) ??Zj
=T+;
Nq-l(Xe-,lj;+,mj
where X=-e is the electron-phonon interaction given by (2.5). The matrix elements (3.2) explicitly wirtten are
=k&
N,) + (j;-,
(i; -,
= *t; N,,)
mj
=
3;
N~~~e-p~
j, +, ml = kk; N,)
mj
= ~4; N,, + 1I%‘C_pI i; +, mj
SPIN-LATTICE
X
(i;+; N,-
RELAXATION
l~Gi?~&;+;
IN n-TYPE SILICON
1107
and one for the upper state i
NJ)
x (-A + hsq f &ifLI’f)-l.
2 exp (A/&-) (Pt. + P2).
(3.6)
This is the mechanism proposed by Jerome [7]. Using (3.1) and (3.3) (3.6) is expressed as
In consideration of the nuclear spin states and the populations of i and j states the effective relaxation rate of the f.r.c. is given by
X (-A + hsq + )ifiTi)-l x {&%
(e-is’%
_
e-W&)
+
e-&‘&(
@R.
_
@RI
(3.10)
) }
(3.6)’ where (3.6)”
(2). In the following we shaft treat the second mechanism of the relaxation rate of the n-i pair[9]. Wave functions which are necessary in the following procedure, are
Throughout the above calculation we have assumed that A % AE. Carrying out a similar cafcufation to (2.9) we get the transition probability
‘?kfi) s Jl,(r-Ri)
w1* -,Jrdr-R,)
w** -,JIT(r-Rjl
‘I.‘,‘” s t,b,(r-Rj) fisq = A
+?$k(r--R,)
f?Jlp(r-RJ.
(3.1 f)
where F(q)
= 3-4
sin qR
-+-
qR
sin qR 2 (
qR
> *
(3.7)’
Using the wave functions (3.1 I) one can easily obtain the transition probability for the direct process (j; +) -+ (i; -) as follows:
Ti-’ is the fife time of the state i and may be written in the following form r = 2lG/* i
--&+w9+
1Y(4).
(3.8)
(hs)’
F(q)
u,f‘tY) ’ (3.9) -
(3.9) is identical with (3.4) in the limit qR + 0 because in this limit F/f tends to 2. In the limit qR >> 1 Fijf’ tends to 3 and then P,lP, approaches 3/2. Jirome disregarded the factor Flfi The same probability as (3.9) is obtained for the process with 1~~instead of IN, in (3.6). Summing up (3.4) and (3.9) one gets a refaxation rate for the lower statej 2(P,+Pz)
x&q)
(3.12)
where (f W#)
Inserting (3.8) into (3.7) we get
‘W’,l’MN
E,* (Iw,l”)A P, = 7rps (hs)’
s
(~)2(~)*(~)4(~)112e-~~),
A indicates the spin-orbit coupling due to the impurity atom and E a quantity which is nearly equal to the ionization energy of the donor electron. The transition probability in the O-A process is just twice (3.12). By taking into consideration the direct and the O-A processes simultaneously, the effective relaxation rate of the f.r.c. becomes ( ,,7Z) _, = 3E1’ (Iw~l’> Af(q) )I I G (ns)4 2 sinh (AjkT)’
(3’13)
1108
K. SUGIHARA
Finally, summing up (3.10) and (3.13), one obtains the relaxation rate of the n-i pair (l/r)n_i=
(l/TI)n-j+
(i/rc!)n-i.
5x.102 Oe
(3.14)
( l/rl),,+
which is proportional to W2, is a more effective mechanism than (l/rz)n-i at low magnetic field (H < 103Oe) for the samples ND > 101Vcm3. The concentration dependence of ( l/Tl)n-i and ( l/Tz)n_i at T = 1.25”K are shown in Figs. 8 and 9. The ND dependence of (l/rq)n-i is relatively weak but (l/rl)n_i shows a drastic change with ND. In Figs. 8 and 9 we have replaced R by the average distance R = (3/47rNC3 and put A = kT. Hitherto, we have considered the relaxation process of an n-i pair, but at this stage it remains to estimate the effective number of the n-i pairs which behave as f.r.c. This can be carried out according to a method developed in our previous paper[6, 91. Effective donor pairs exist only in the energy range in which the condition (E, - 51 - kT is satisfied (r = ij ). Now suppose a thin spherical shell around an acceptor ion such that Ir,--1-1 < Arl2
(3.15)
T=
ND(cmm3) Fig. 9. Donor concentration
dependence of (l/~,)~_+
where r, and Ar are defined by
1;=$, k+$.
IO / (I/T,)n_,,
r
C
,H=I03 Oe
T = b25*K
(3.15)’
C
An electron trapped in the donor site satisfying the condition (3.15) has the energy j,C-51 < kT.
(3.15)”
The probability of two donors being contained in the shell (3.15) is given by the Poisson distribution as
10-I
IO'5
/I N
Fig. 8. Donor
ND2(4wc2Ar)2 exp [-(87r/3)r,.“N,,].
I
I
iO'6
IO"
for the Fermi energy
5=-E,[K/ln(l-_)I”“,
D(cmm3)
concentration dependence (1 /T*)n--I.
By use of the expression ]221
of (I/T&~
and
(3.16)
R = (3/47~N[>)“”
E,=-$, (3.17)
SPIN-LATTICE
RELAXATION
and (3. IS)‘, (3.16) is rewritten in the form
(3.1 w Since the ratio of the acceptors to the isolated donors is given by K/ ( 1 - K), the relaxation rate as a whole is expressed by (ljT,j,_j
(l/T)=-ineff
=
1+ Td(llT)n-i&K
(3.19)
( l/r)n_i is the relaxation rate (3.14) with the substitutions R = (3/4rrNUP3 and A = kT . neff is related to (3.18) by the relation K bff =
2(1-K)
x (3.18).
IN n-TYPE
SILICON
1109
could shorten considerably the experimentally measured relaxation times. Y-H confirmed that the compensation dependence of their results for the samples K < O-5 was well explained using our previous formula[9] which shows nearly the same dependence on K as the present formula in the range of small compensation. Assigning the appropriate values of K and Td to the three samples 3.2 X f0%m3, 6 X 1016/cm3 and 8 x 1W6/cm3, calculated results are shown in Fig. 10 in comparison with the experimental results. K and Td values are reasonable ones. According to JCrome[7]
(3.20)
The extra factor l/2 appears in (3.20), since we have assumed that the probability of one other vacant is equal to I /2. Now it is necessary to decide the magnitude of A in (l/~~),~_.~in comparison with experiment. For this purpose we use the Y-H’s data. They obtained l/T, = le.5 X lop3 set-’ at T = 1%‘K and H = lo” Oe for the sample ND = 6 x 101”/cm3 and K = 0.33. For this sample ( l,/Tz)e-i is very much larger than ( l/?I)n_i. Assuming ?d = 300 set-this iS a reasonable diffusion time judging from the experiment by Honig-Stupp [ IX] - one gets h = 4.70 X lOemeV.
(3.21)
This value is close to the one obtained previously[9]. Using (3.21) we get (1/71)n--i= and ( 1/TJ n_i = 1.08 set-I. 1.02 X 10-r set-’ Igo obtained a very different result [6] from Y-H for almost the same condition as theirs. He reported that l/T, = 2.8 X 10v2 see-* at T = 2°K for the sample ND = 5.9 X 10i5/cm3 and K = O-32, while the relaxation rate measured by Y-H is only l/20 of Igo’s result. Y-H pointed out that one possible origin of such discrepancy is a small room temperature i.r. radiation leak into Igo’s samples which t In (4.15) in reference [9] the exponential term in (3.18) is missing.
If? 102
I
I
IO3
10-J
I05
Magneticfield H (Oe 1 Fig. IO. Calculated values of (l/T,),~.+ and the observed results of Ii?‘, at 7’ = I-27°K. Foiiowinrr values of K and 7@ are used foi each sample: II = 0.01, 7d= 40 set for the sample N, = 3.2 X 10’“/cm3; K = 0.01, 7d= 10set for ND= 6X 10LB/cm3 and K= 0.04, T== 4sec for ND= 8 X 10*B/cmy.
1110
the compensation cm3 is less than approximations results are in observed ones.
K. SUGIHARA
of the 0.05. used good
sample ND = 6 X 1OV In spite of the crude here the computed agreement with the
4. SUMMARY
In this paper we have discussed the relaxation mechanisms of the donor electron in uncompensated phosphorous doped silicon (K < 0.1). The principal results obtained in the preceeding sections are summarized as follows. (1). The spin-lattice relaxation for uncompensated silicon is classified into three mechanisms according to impurity concentration. In all the mechanisms spin excitations of the main body of the donor electrons which are sufficiently isolated, are transferred via a spin-diffusion process and finally relax at fast-relaxing centers which are few in number. (2). For lightly doped samples, ND < 1OV cm3, exchange coupled donor pairs act as the fast-relaxing centers. This mechanism is field independent and can not explain the weak field dependence of the observed relaxation rates. At this stage we have no definite explanation of such a discrepancy. Apart from this point the order of magnitude of the calculated results is reasonable. (3). For the more heavily doped samples, ND > 101s/cm3, two different mechanisms are active if the samples are slightly compensated (K - 10P2).’ In this case a fastrelaxing center is composed of a neutralionized donor pair. Mixing of different spin states is caused by the hyperfine interaction or the spin-orbit coupling due to the impurity potential. The transition rate for the first mechanism is proportional to He2 and strongly dependent on ND. On the other hand the second mechanism is field independent and weakly dependent on N,. By selecting appropriate values of K and spin-diffusion times for each sample the observed results are well explained by our theory. (4). Instead of entering into the complicated problem of analyzing the spin-diffusion pro-
cess, an appropriate spin-diffusion time has been introduced for each sample. Finally we propose an experiment which will throw further light on the relaxation mechanisms. This is the measurement of relaxation times under the application of uniaxial stress[24]. Reduction of the hyperfine coupling due to the application of such a stress may cause the spin-relaxation rate to decrease appreciably. Acknowledgement-The author wishes to thank Dr. K. Morigaki of the Institute for Solid State Physics, University of Tokyo, for his helpful discussions and for informing him of J rome’s thesis. REFERENCES 1. HASEGAWA H., Phys. Rev. l&1523 (1960). 2. ROTH L. M., Phys. Rev. 118, 1534 (1960). 3. CASTNER T. G. Jr., Phys. Rev. 130, 58 (1963); ibid. 155,8 16 (1967).
4. FEHER G. and GERE E. A., Phys. Rev. 114, 1245 (1959).
5. SASAKI
W., Proc. Inf. Conf. Semicon. Phvsics. Kyoto, 543 (1966); Maekawa S., ibid. 574. 6. IGO T., J. Dhys. Sot. Jaoan 21,874 (1966). 7. J&ROME ‘D:, Thesis, ‘University of Paris (1965).
Unpublished. 8. YANG G. and HONIG A., Private communication. 9. SUGIHARA K.,J.phys.Soc.Jupan 18,961 (1963). 10. MORIGAKI K. and WINTER J. M., Proc. Int. Conf. Semicon. Physics, Kyoto, 570 (1966). 11. HONIGA.andSTUPPE.,Phys. Rev. 117,69(1960). 12. SLICHTER C. P., Phys. Rev. 99, 479 (1955); FEHER G., FLETCHER R. C. and GERE E. A., P$ys. Rev. 100, 1784
(1955).
13. JEROME D. and WINTER J. M., Phys. Rev. 134, 1001 (1964). 14. MOROCHA A. K., Soviet Phys. solid Sr. 4, 1683 (1963). 15. GLINCHUK M. D., GRACEV V. G. and DEIGEN M. F., Soviet Phys. solid Sr. 82678 (1967). H. H., Solid 16. LUDWIG G. W. and WOODBURY Stnte Physics (Edited by Seitz F. and Turnbull D.), Vol. 13. II. 223. Academic Press. New York (1962). 17. ORBAdH R., Proc. R. Sot. A264,458 (196 i). 18. AMINOV L. K., Soviet Phys. JETP 15,547 (1962). 19. HEITLER W., The Quantum Theory of Radiation, p. 196. Clarendon Press, Oxford (1954). 20. PAULING L. and WILSON E. B., Introduction to Quantum Mechanics
with Applications
to Chemistry,
Chap. 12. McGraw-Hill, New York (1935). 21. KOHN W., So/id State Physics (Edited by Seitz F. and Turnbull D.), Vol. 5. Academic Press, New York (1957). 22. MILLER A. and ABRAHAMS E., Phys. Rev. 12% 745 (1960). 23. FEHER G., Phys. Rev. 114,1219 (1859). 24. WILSON D. K. and FEHER G., Phys. Reu. 124, 1068 (1961).