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Volume and temperature dependence of the 1f noise parameter α in Si
Physica B 154 (1989) 214-224 North-Holland, Amsterdam
VOLUME AND TEMPERATURE DEPENDENCE OF THE l/f NOISE PARAMETER
CLEVERS
Eindhoven
University of Technology,
Eindhoven,
The Netherlands
Received 28 March 1988 Revised 24 September 1988
We present resistivity, Hall and noise measurements for p- and n-Si between 77 and 300 K. A l/f noise parameter a between low6 and lo-’ at 300 K and between lo-’ and IO-’ at 77 K is found in n- and p-Si. The a-value is independent of the effective volume. It is shown that 1 lf noise sources are located in the bulk. Hooge’s empirical relation is confirmed in the case where an a-value of 10e6 is observed. The temperature dependence of the a-value is measured for n- and p-Si with doping concentrations between 10” and 10” cm-‘. The magnitude of the a-value and its temperature dependence are related to the manufacturing process. Our measurements give no experimental support for the quantum l/f noise.
1. Introduction It is common practice to present experimental results on l/f noise in homogeneous semiconductors and in metals in terms of cu-values defined by PI S,/V’
= alfN,
(1)
where S, is the spectral density of the ac opencircuit voltage fluctuations, V the applied voltage, f the frequency and N the number of free charge carriers. The l/f noise parameter cy is found in the range from lo-* to lo-’ in semiconductors [l-lo]. This broad range of a-value is not yet understood. (i) An effort is made to explain this broad range of a-values by assuming that it is the contribution of lattice scattering to the mobility that causes mobility fluctuations. It is then found that
where p is the mobility and h and (Y,the mobility and the 1 lf noise parameter due to lattice scattering, respectively. However, also a broad range of a,-values is observed (lo-’ < (Y,< 10m3 [4-61).
(ii) Bisschop [2] h as suggested that the l/f noise parameter (Ydepends on the effective noise volume. His collection of a-values did not give a decisive answer. (iii) According to Peng et al. [7], the quantum l/f noise theory predicts a transition from the incoherent state with (Y- lo-’ for short devices, to the coherent state quantum l/f noise with (Y- 10m3 for long devices. The temperature dependence of (Yis still ambiguous. An a-value decreasing with decreasing temperature is observed in Ge [3], Si [3, 81 and InP [9]. A temperature-independent (Yeis found for Si [6, lo]. Here we present the results of resistivity, Hall and noise measurements on homogeneously doped n- and p-Si samples with various impurity concentrations and geometries in the temperature range from 77 to 300 K. The magnitude of the 1 lf noise parameter cx is investigated as a function of the length of the sample, effective noise volume, temperature and impurity concentration. 2. Conductivity data 2.1. Sample description In
our
geometries.
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
we used various device Table I gives the device geometry of
experiments
R. H. M. Clevers / Volume and temperature dependence
Table I Characteristic
of the 1 if noise parameter a
215
data of the samples used in this study at 300 K
Sample
Mcm)
n, p(cm-‘)
VHD En-% 1105 En-% 3500 En-% 06 Ep-Si 709 Ep-Si 100 Ep-Si 703 Ep-Si 4001 p-Si 1 p-Si 2 p-Si 3
4.5 4.4 63 1.6 24 2.2 0.82 0.088 30 15 1
1x 1x 7x 3x 6x 6x 2x 4x 5x 9x 2x
lOIS 1or5 10’) 1015 1014 lOI 1o16 10” 10” lOI lOI
p(cm”/Vs)
1420 1310 470 450 410 200 460 460 420
the samples. The planar structure of sample EnSi 1105 provided with a large gate electrode has been described elsewhere [ 111. The samples with the prefix E are manufactured by EFFIC (see Acknowledgements). The VHD structure (Vertical Hall Device) has been described in the litera-
Geometry
4,
VHD planar
1.1 x lo+ 10-‘0 - 10-7
circular contact
lo-” lo+
planar ring ring
(cm3)
1.2 x lo-’ 3.3 x 1o-6 2.5 x 1O-6
(I <9x lo-’ 6~10-~ 2x10+ 3 x lo+ 2 x 1o-6 2 x 1o-6 2 x 1o-6 8 x 1o-6 6 x 1O-5 2 x 1o-6 2 x 1o-5
+ A 0 n l v A 0 n X
+
ture [12] [fig. l(a)]. The circular contact geometry is shown in fig. l(b). On a chip of 3.5 x 3.5 x 0.4 mm3 circular contacts are present with radii between 1 pm and 100 pm. For p-Si the ohmic contacts were made by means of a boron implant and for n-Si by means of a phosphorous diffu-
Si
a
Fig. 1. (a) Vertical Hall device; (b) circular contact geometry; (L = 31).
(c) ring structure (L = rr(rl + r2)), (d) Greek-cross
structure
216
R. H. M. Clevers I Volume and temperature
sion. The depth of implantation is 1 pm and the depth of the diffusion, determined from the underdiffusion, is between 1 and 5 pm. The ring geometry of samples p-Si 2, 3 is shown in fig. l(c) (see also [2]). More details about the planar sample p-Si 1 are given in ref. [13].
dependence
of the 1 If noise parameter
p- Si
p(Q cm1 10 _I
lv
dependence
of the resistivity
-
,cJ(~OOK)R(T)/R(~OOK) ,
(3)
where R(T) is the resistance at temperature T. The resistivity at 300 K of the planar sample p-Si 1 is related to the resistance by p=RAIL,
(4)
where L and A are the contact spacing and the area of the sample, respectively. For the samples p-Si 2, 3 and the Vertical Hall Device the resistivity at 300 K is given by the manufacturer (see table I). The resistance calculated with the resistivity at 300 K agreed with the measured resistance to within 10%. Greek-cross structures [see fig. l(d)] are available on En-Si 1105 chips. Here the resistivity can be determined from the sheet resistance R, with the equation [14]
v.
ihru
-* .
16'
p(T)=
ne
px
2.2. Resbtivity The temperature p is given by
.
.
.
2)(V,,lZ,,)
,
.
*
l
i
t
,6zL
200
300
Fig. 2. Temperature dependence of the resistivity for p-Si (W: p-Si 1; +: Ep-Si 709, X: p-Si 2; v: Ep-Si 100; +: p-Si 3; A: Ep-Si 703; 0: Ep-Si 4001).
The temperature dependence of the resistivity of the devices in table I is shown in figs. 2 and 3. The radius a of a circular contact will be greater than the mask radius a,, due to underdiffusion. The radius Q can be determined from the resistance and the resistivity using the follow-
IO2
I
I
I
I
n- Si p(R cm)
R, = p/t = (r/In
(I
I
.
.
.
(5)
. .
where t is the thickness of the sample, V,, the voltage measured across two adjacent contacts and Z,, the current passed through the other two adjacent contacts. The resistivity can be determined with fourprobe resistivity measurements for the samples with circular geometry. For the device shown in fig. l(b) the resistivity can be determined with the help of the following equation (s 4 d) [15] P = (V,*lL>[~sl(l
-
WY
7
(6)
10 -
I . . l’
‘*
the contacts
l
-I
1‘,““’ ,I
jy 100
where s is the distance between (s=50I*m, 4OO
1
.
200
300
Fig. 3. Temperature dependence on the resistivity for n-Si (0: En-Si 3500; A: En-Si 1105; n : En-Si 06; +: VHD).
R. H. M. Clevers
I Volume and temperature dependence of the 1 if noise parameter a
ing equation [M] (7)
Z?=j$-{arcta”($)-arctan($
where d is the thickness of the substrate, b the diffusion or implantation depth and E = (l(bla)2)1’2. Fig. 4 shows radius a versus a, as expected from mask dimensions for En-Si 3500, 06 and Ep-Si 100. For the implanted contacts Ep-Si 100, the radius a is calculated taking an implantation depth of 1 km. The radius a is found to be almost equal to the radius a,. For the diffused contacts the underdiffusion is taken equal to the diffusion depth. An underdiffusion of 1.3 pm is observed for En-Si 06 and of 4 Km for En-Si 3500. The underdiffision increases with increasing ratio between dope concentration in the contact and dope concentration in the substrate. 2.3. Currier concentration The samples En-Si 1105 have a Greek-cross structure [17]. Therefore Hall measurements can be performed and the density of free charge carriers rz (or p) can be determined. Also, the Vertical Hall Device is suitable for Hall measurements [12]. For both structures the Hall voltage V, is given by VH = r,GZBlent
,
1
(10)
10
lo2
Fig. 4. The radius a of the circular contact as de:ermined from the resistance and the resistivity at 300K versus the radius a, from the mask dimensions (0: En-S 3500; m: En-5 06; V: Ep-Si 100).
217
where Z is the current, B the magnetic induction, rH the Hall factor and G the geometry factor close to unity. For the wafers En-Si 06, 3500 and Ep-Si 709, 100, 703, 4001 the carrier concentration n (or p) at 300 K has been determined from the resistivity p with the help of p versus n curves given in literature [18, 191. The results are given in table I. Van der Pauw structures have been made from the wafers. The product Gr, at 300 K has been determined from the carrier concentration and the measured Hall voltage using eq. (10). The temperature dependence of the carrier concentration can be determined if we assume that Gr, is temperature independent. Hall measurements are not possible with the planar sample p-Si 1 and with the ring structures p-Si 2, 3. The carrier concentration has been computed for these samples by assuming an impurity level E, meV above (for acceptors) or below (for donors) the band edge. For the density of free charge carriers p it is found that [20] P = NI , p =
for T > T, ,
(11)
(NINB/g1)“2 exp[ - E,I2kT]
,
for T < T, , (12)
T, = E,lk ln[N,lg,N,]
,
(13)
where NB is the density of states in the band, Nr the density of impurities and g, the degeneracy of the impurity state. Donor states are twofold degenerate. For acceptor states the situation is more complex. Then the structure of the valence band (light and heavy hold bands degenerate at k = 0 and the spin-orbit split-off band) must be accounted for in the effective mass approach. It is shown that the ground state has total angular momentum 3/2 and thus is four-fold degenerate [21]. Here we neglect the contribution of the excited states of the donors and the acceptors. Figs. 5 and 6 show the measured carrier concentration versus T for p- and n-Si, respectively. In figs. 5 and 6 the product r,G is assumed to be temperature independent. We have G = 0.8 [12] for the VHD and G = 0.95 [17] for En-Si 1105.
218
R. H. M. Clevers
300 200
18
‘0
!!
l.
150
I Volume and temperature
100
’
dependence
of the 1 If noiseparameter
(Y
donors E, = 44 meV [22]. For Nr we take the carrier concentration at 300 K as determined from the resistivity [18, 191 since all donors and acceptors are ionized at 300 K. The calculated carrier concentration is shown in figs. 5 and 6 for p- and n-Si, respectively.
II
1 (K) c_
2.4. Mobility The mobility can be determined from the resistivity and the carrier density with p = l/pen.
I 5
I
I
I
I
I
13
9,
Fig. 5. Temperature dependence of carrier concentration for p-Si (for symbols see fig. 2). Solid lines give the calculated carrier concentration for p-Si 1, 2 and 3.
Since the Hall measurements cannot be trusted for En-Si 3500 we compute the carrier concentration for this sample with eqs. (ll)-(13). For the samples p-Si l-3 we also compute the carrier concentration. The energy level for boron acceptors EA = 43.8 meV and for phosphorus 100
(6 300 200
10
II
(14)
Figs. 7 and 8 show the results for p- and n-S& respectively. The lattice mobility (taken from the literature [18, 191) is also given. The mobility is determined from the calculated carrier density for the samples p-Si l-3 and En-Si 3500. The calculated mobilities are in agreement with the measured mobilities of the other p-Si samples with the same dope. As expected for the low doped En-Si 3500 the mobility approximately equals the lattice mobility.
I
I
I
I
p-Si
1' 1
I
I
T(K) I -9 .==.
,$-A
'
.
A
-.
9.
.
,
'A
nkd) n-Si
II
IO"
4
I 5
I
-
. ‘. 103rr (KY
I
1
I
Id 9 II Fig. 6. Temperature dependence of carrier concentration for n-Si (for symbols see fig. 3). Solid line gives the calculated carrier concentration for En-Si 3500.
I 11
I
T IK)
150
. I 300
Fig. 7. Temperature dependence of the mobility for p-Si (for symbols see fig. 2). Solid lines are for p-Si 1, p-Si 2 and p-Si 3. The dashed line gives the lattice mobility. Carrier concentrations (cm-‘) are given.
R. H. M. Clevers
I Volume and temperature dependence
Fig. 8. Temperature dependence of the mobility for n-Si (for symbols see fig. 3). Solid line is for En-Si 3500. The dashed line gives the lattice mobility. Carrier concentrations (cm-‘) are given,
3. Results of noise measurements
We consider three types of noise: thermal noise (4kTR), generation-recombination (gr) noise and l/f noise. For the voltage noise spectrum we have S,=4kTR+$C cXv2 +fN
Tici i
1 + (2TfTi)2
(1%
where the Ci’s are constants. The gr term accounts for several gr processes. Fig. 9 shows some spectra obtained after subtracting the 4kTR noise. In fig. 9 three spectra show 1 lf noise only (a, +, A). One spectrum (0) can be explained by assuming a gr component with r = 7 ms and a 1 lf noise component. Also shown is a spectrum (x) that can be explained in terms of gr processes only with a distribution of characteristic times. This spectrum can be explained by assuming characteristic times in the range from 20 )LS to 30ms. We estimate the low-frequency plateau of the gr component with 7 = 3 ms. The spectral density at
of the 1 If noise parameter a
219
Fig. 9. Five measured spectra. The 4kTR noise is subtracted and its level is given by the dashed line. The solid line gives the l/f noise level and the dashed and dotted line the gr noise level. Vertical arrows give the inverse of the characteristic time 1/2m of the gr noise component. (0: En-% 3500, T= 118K, V=30.2mV, W: En-Si 06, T= 175K, V= 84.5 mV, X: p-Si 2, T=222K, V=O.453; a: Ep-Si 709, T= 178K, V=80.3mV, A: Ep-Si 703, T=295K, V= O.llOV).
low frequencies (f <1/27rT=50Hz) S,= lo-i6V2/Hz (see fig. 9). Then, with eq. (15) we find CilN2 = 5 x 10-i’. With the data from table I and fig. 7 we find N=nLZ=3x109 and Ci= 4 X lo5 4 N. For gr noise with one time constant we have Ci = ( (AN)2) -+ N. For gr noise with a distribution of characteristic times T the Ci’s are complicated functions of the trap properties. However, from Ci G N we conclude that we can describe the spectrum (X) with eq. (15) assuming a trap density low compared to the density of free charge carriers. Some authors [5] prefer to describe the spectrum ( x ) with the equation S, = /3V21fyN,
(16)
where p has the dimension [fl] = HzY-‘. For the spectrum under consideration y s 0.7 is found. If we describe our spectra with eq. (15), only an
R. H. M. Clevers
220
I Volume and temperature
upper limit for (Ycan then be determined for the spectrum (x). This is indicated in the figs. 13, 14 and 15 with a vertical arrow. We describe the results of our noise measurements with the empirical relation eq. (1). We deal only with homogeneously doped samples. For samples with a non-homogeneous current pattern we have to replace iV by IV,,, in eq. (1)
where Q.,, is the effective noise volume given by
[31
(j-d3rE2(r)12,/{
j- d3rE4(r)} .
(18)
L?,,, has been calculated for all the geometries used in this study [12, 13, 161. The effective noise volume for circular contacts is given by [16] 0&, = (5p3/47r2R3)(r5/s) )
(19)
where r and s are factors accounting for the eccentricity of the geometry. The factors r, s and r5/s are given in literature [6, 161. The Q,, for the ring structure is given by [13] n
= L{(r eff
of the 1 If noise parameter
LY
Fig. 10. (Yversus the effective noise volume L&, at 300 K (a: Ep-Si 709; V: Ep-Si 100; n : En-% 06) and at 77 K (V: Ep-Si 100; q : En-Si 06).
(17)
N,,, = n.n,,, 3
&,, =
dependence
- 25)~ arcsinh(dla)}2 ln[(tan( [))-‘I
7r
’
(20)
where d is the thickness of the substrate and 6 a factor between 0 and 1 that is related to the diffusion depth of the contacts. The meaning of a is explained in fig. l(c) and L is given by L = 4r1 + r2). With the effective noise volume given by eqs. (19), (20) and the empirical relation eq. 1, LYvalues are calculated.
noise sources are located at the surface predicts S” - 0;;‘3 [23]. We present the 1 lf noise parameter (Y as a function of the channel length for En-% 1105 (without gate electrode) at 77 K. Fig. 11 shows the results for one chip of 4 x 2.5 x 0.4 mm’. We observe an a-value of 1O-5 which is independent of the length of the channel (o - L6 with 0 < 6 < 0.3). This behaviour is expected for noise sources distributed homogeneously either at the surface or in the bulk of the sample. These results are in contradiction to the experimental results of Peng et al. obtained with p-MOSFETs [7]. There it was observed that (Yincreases from 3 X 1O-7 for devices with channel length L =I 6.3 pm to 3 X lob4 for devices with L = 196 km. We return to this matter in section 4.4 3.2. Temperature dependence of (Y Fig. 12 shows the temperature dependence of the 1 lf noise parameter (Yof Ep-Si 4001 obtained on three circular contacts from one chip. We observe that the o-values scatter by a factor of 2 from contact to contact. Some of the detailed variations in the a-values with temperature are
3.1. (Yversus volume and length We present (Yat 77 K and 300 K as a function of the effective noise volume in fig. 10. We observe that (Yis independent of Q.,, and of the order of 10m6 (a - Ozff with -0.1 < y
,i6t
1
I
I
10
1
~$t-.~ 10
lo3
Fig. 11. (Y as a function of the length L of the device for En-Si 1105 at 77 K.
R. H. M. Clevers I Volume and temperature dependence
lo*\
I
I
221
o!
I
EpSi 4001
a I . 16
of the 1 If nobe parameter
*
.
..A' .rn' & l
:
I
-T(K)
6
a
lo-
I
+
l
Fig. 12. Temperature dependence of a for three circular contacts fromEp-Si 4001 (A: a,, = 2 pm; n : a,, = 3 wm; 0: a, = 5 Km).
t 1Ci'
not reproducible. Therefore we concentrate on the broad variations with temperature. Scattering of (~-values is also observed by other authors ]3, 81. In figs. 13 and 14 we present the temperature dependence of (Yfor the samples in table I. The a-values are averaged over at least two circular contacts. The cr-values at 77 and 300 K are averaged over circular contacts from different chips.
4
1G7- l
4
4
100
--T(K)
I
I
200
I
I
300
Fig. 13. Temperature dependence of a for p-Si (for symbols see fig. 2). Upper limits for (I are indicated by a vertical arrow. Carrier concentrations (cmm3) are given.
T
n- Si
I
100
I
T (K)
I 200
Fig. 14. Temperature dependence see fig. 3). Carrier concentrations
I
I
1
30 0
of a for n-S (for symbols (cm-‘) are given.
4. Discussion 4.1. Location of the noise sources From the results in fig. 10 we concluded that 1 lf noise is a bulk effect. However, Jintsch [23] suggested that l/f noise is caused by noise sources located at grain boundaries. Jantsch stated that our results are no proof of a bulk effect (part of the results of fig. 10 had already been published in ref. [6]). Therefore we now state our conclusion more carefully: S, - 0,: means that the noise sources are distributed homogeneously over the sample. Thus, the model proposed by Jantsch is not in contradiction with the results of fig. 10, provided the grain boundaries are distributed homogeneously over the sample and the l/f noise sources are distributed homogeneously over the grain boundaries. In the past, the validity of the empirical relation eq. (1) was already shown for a-values of the order of 10e3 [l]. Here, the validity of eq. (1) is also shown for cx-values of lo-‘? These results do not support Bisschop’s suggestion [2] that samples with a small effective volume a,,, show lower a-values than samples with a large n eff.
222
4.2.
R. H. M. Clevers
I Volume and temperature dependence
of the 1 If noise parameter
Mobility fluctuations
(I
104
An (Y,value of 10m3 independent of PI& was found for Ge [l]. It was concluded that it is only the contribution by lattice scattering to the mobility that fluctuates. Fig. 15 shows the (Yvalues found for the samples in table I at 77 K. It can be seen that the a-value is not proportional to (p lo)* for Si at 77 K, in contradiction to the experimental results for Ge. 4.3. Temperature dependence
T(K)---
of the llf noise
Different temperature dependences are reported here and in the literature. An a-value declining to low temperatures is observed in p-Ge [3], n-Si [3], p-Si [3, 81 and n-InSb [9] (see fig. 16). This temperature dependence of (Ycan be described by .
a=Aexp(-EIkT)+B
(25)
(for Si and Ge for T < 300 K and for n-InSb for T < 200 K). Eq. (25) describes a thermally activated process. An activation energy E g 0.1 eV and a B between 10e5 and 10m3 are observed in Ge, Si and n-InSb.
IO3
I
I
I
I
l
1d I
103/Td, I
5
I
.’ I
I
9
I
I
I
13
Fig. 16. Temperature dependence of CYfound in the literature (A: Luo [8]; n : Bisschop [3]; 0: Alekperov [9]). Solid line [eq. (25)] with A = 0.33, E = 0.1 eV and B = 4 x lOA and dashed Iine [eq. (26)] are taken from Luo [8]. Dashed and dotted line shows eq. (27) fitted to Bisschop’s data (y = 4, A = 3 x lo-“, B = 4 x 10~‘).
I
A
Luo et al. [8] found that his data could also be described with the following equation (see fig. 16) a(T)
i
Fig. 15. a as a function of (PI&~ for p- and n-Si at 77 K (a: Ep-Si 709; X: p-Si 2; ‘I: Ep-Si 100; +: p-Si 3; A: Ep-Si 703; 0: Ep-Si 4001; 0: En-Si 3500; 0: En-Si 1105; Cl: En-Si 06; 0: VHD). Upper limits for (Y are indicated by a vertical arrow.
= &/p(T)&(T).
(26)
Luo et al. derived eq. (26) by assuming noise sources in a surface layer. However, Alekperov et al. [9] found noise sources homogeneously distributed through the sample. They found experimentally that the a-value is independent of surface treatment and thickness of the sample in the temperature range where the exponential term dominates over B. For our sample En-Si 1105 it was shown [l l] that noise sources near the semiconductor-oxide interface dominate the l/f noise. However, our a-value depends weakly on temperature. Thus,
R. H. M, Clevers
I Volume and temperature dependence
eq. (26) cannot be applied to Alekperov’s and our (En-Si 1105) data. It is also possible to fit the experimental results for Ge, Si and n-InSb with (see fig. 16) ol(T)=ATY+B.
(27)
Then, y s 4 for Bisschop’s data, y s 5 for Luo’s data and y z 6 for Alekperov’s data. We do not know of any physical mechanism from which eq. (27) follows. Eq. (27) does not point to a thermally-activated process. Luo found an o-value of lo-* at 300 K decreasing with decreasing temperature for his p-Si sample treated at 950°C only. The p-Si samples treated at a low temperature (T = 550°C) show a low u-value (a - 10m4) weakly dependent on temperature. The processing temperature is thought to be the crucial parameter in explaining this behaviour. Although processed at a low temperature (contacts evaporated at 9O’C) p-Ge shows the same temperature dependence of (Yas p-Si treated at 950°C. Our samples do not show this particular temperature dependence (see figs. 13 and 14), although processed at high temperatures (T > 900°C). Noise sources showing a decreasing (Y with decreasing temperature are observed when (Y> low3 at 300 K. Perhaps these noise sources can be avoided by manufacturing samples in a modem IC laboratory. 4.4. Quantum 1 If noise The validity of the quantum l/f noise theory has been questioned more than once [24]. Here, we will not be concerned with the theory itself but simply compare experimental results with the predictions of this theory. Peng et al. [7] observe that the a-value increases from 5 X 10e7 to 3 X low4 with channel length L increasing from 6.3 to 194 km (a - L*) for their p-MOSFETs. Peng et al. [7] conjecture that they observed the transition from incoherent-state quantum 1 /f noise (ff - 10e8) for short devices to coherent-state quantum 1 /f noise (o - 10e3) for long devices. According to van der Ziel’s generalized semiclassical quantum 1 /f noise theory [25], 1 /f noise in short semiconductor devices is described by colli-
of the 1if noise parameter a
223
sion 1 /f noise and 1 /f noise in long semiconductor devices by acceleration 1 /f noise theory. The acceleration l/f noise theory predicts a l/f noise parameter (Y depending on device length L as LY- L*. Thus, Peng et al. [26] state that their experimental results confirm van der Ziel’s acceleration 1 lf noise theory. Peng’s experimental results [7] are in contradiction with our results for En-Si 1105, where an o-value of lo-’ at 77 K independent of channel length (5 urn< L < 300 pm) is observed (see fig. 11). Kousik et al. [27] assume various scattering mechanisms in both n-Si and n-GaAs. With the help of the quantum 1 /f noise theory they calculate a-values as a function of temperature and impurity concentration. The o-values shown in figs. 13, 14 and 15 are higher by a factor of 10 at least than the a-values calculated by Kousik. Kousik compares his calculations with experimental results from the literature. It is worth while to commend on these experimental results quoted by Kousik. The a-value of 2 X 10s7 found in p-Si (that in our p-Si 3) at 300 K by Bisschop is based on his erroneous calculation of Q,,,. A recalculation of J2,,, based on eq. (20) gives an a -value of lo-“. There is no a-value of 7 X lo-’ for Si at 300 K in Bisschop’s thesis [2]. That value was used in fig. 16 of ref. [27]. For polysilicon resistors with doping 1.8 x 10” cme3 Bisschop reported C-values (C = (I!/N) between 10e9 and lo-! Kousik erroneously quoted these values as a-values. Schmidt et al. [28] reported an a-value of 6 x low8 for GaAs at 300 K instead of the 2 x 10e8 reported by Kousik. Thus, only three data points for Si at 300 K in fig. 16 of ref. [27] remain after elimination of erroneous data. There is not enough experimental confirmation of the temperature dependence of (Ypredicted by the quantum 1 lf noise theory.
5. Conclusions The experimental results can be described with noise sources located in the bulk. The validity of the empirical relation eq. (1) is also shown with a 1 /f noise parameter (Yof the order of 10-4 The wide range of a-values ( 10m8< (Y< lo-‘) re-
224
R. H. M. Clevers
I Volume and temperature dependence
mains unexplained. The order of magnitude of (Y does not depend on the effective volume or on the length of the device. Various trends in the temperature dependence of (Yhave been observed. A strong temperature dependence of a (a - T” with 4 < y < 6) is observed for (Y> 10e3 at 300 K. A weaker temperature dependence of LY(o - TY with -3 < y < 2) is observed for (Y< 10e4 at 300 K. The magnitude and the temperature dependence of (Yare possibly related to the manufacturing process. In this case, the processing temperature is not the only parameter of importance. Experimental results presented here and in the literature do not support the quantum 1 lf noise theory. Acknowledgements The author thanks T.G.M. Kleinpenning and L.K.J. Vandamme for stimulating discussions, EFFIC (Eindhovense Fabricage Faciliteit voor ICs) for the preparation of all the E-samples and J.G.M. Delissen for his contribution to the experimental work. References [I] F.N. Hooge, T.G.M. Kleinpenning
and L.K.J. Vandamme, Rep. Prog. Phys. 44 (1981) 479. [2] J. Bisschop, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven (1983). (31 J. Bisschop and J.L. Cuypers, Physica B 123 (1983) 6. [4] L.K.J. Vandamme, Proc. 7th Int. Conf. on Noise in Physical Systems (Elsevier, Amsterdam, 1983) p. 183.
of the 1 If noise parameter
Q
PI G.S. Bhatti and B.K. Jones, J. Phys. D.: Appl. Phys. 17 (1984) 2407. Clevers, Proc. 8th Int. Conf. on Noise in Physical Systems (Elsevier, Amsterdam, 1986) p. 411. [71 Q. Peng, A. Birbas and A. van der Ziel, Proc. 9th Int. Conf. on Noise in Physical Systems (World Scientific, Singapore, 1987) p. 400. PI J. Luo, W.F. Love and SC. Miller, J. Appl. Phys. 60 (1986) 3196. 191S.A. Alekperov, N.Ya. Guseiiiov, Ch.0. Kadzhar and E.Yu. Salaev, Sov. Phys. Semicond. 20 (1986) 973. WI V Palenskis and Z. Shobhtzkas, Solid State Commun. 43 (1983) 761. (111 R.H.M. Clevers, Physica B 147 (1987) 305. t121 L.K.J. Vandamme, Journal de Physique Colloque C4, supplement au no 9, vol. 49 (1988) p. (X-157. P31 R.H.M. Clevers, Ph.D. Thesis, University of Technology Eindhoven (1988). 1141W. Versnel, Solid-State Electron. 22 (1979) 911. WI J.D. Wasscher, Philips Res. Rep. 16 (1961) 301. WI G.W.M. Coppus and L.K.J. Vandamme, Appl. Phys. 20 (1979) 119. (171 W. Versnel, J. Appl. Phys. 52 (1981) 4659. @I S.S. Li and W.R. Thurber, Solid-State Electron. 20 (1977) 609. 1191S.S. Li, Solid-State Electron. 21 (1978) 1109. 1201R.A. Smith, Semiconductors (Cambridge University Press, Cambridge, 1978). WI S.T. Pantelides, Rev. Mod. Phys. 50 (1978) 797. P21 W. Ktimicz, Solid-State Electron. 29 (1986) 1223. [231 0. Jlntsch, IEEE ED-34 (1987) 1100. 1241L.B. Kiss, Proc. 9th Int. Conf. on Noise in Physical Systems (World Scientific, Singapore, 1987) p. 373. P51 A. van der Ziel, J. Appl. Phys. 64 (1988) 903. WI Q. Peng, A.N. Birbas, A. van der Ziel, A.D. van Rheenen and K. Amberiadis, J. Appl. Phys. 64 (1988) 907. v71 G.S. Kousik, C.M. van Vliet, G. Bosman and P.H. Handel, Adv. Phys. 34 (1985) 663. P81 R.R. Schmidt, G. Bosman and C.M. van Vliet, SohdState Electron. 26 (1983) 437.
161R.H.M.
VOLUME AND TEMPERATURE DEPENDENCE OF THE l/f NOISE PARAMETER
CLEVERS
Eindhoven
University of Technology,
Eindhoven,
The Netherlands
Received 28 March 1988 Revised 24 September 1988
We present resistivity, Hall and noise measurements for p- and n-Si between 77 and 300 K. A l/f noise parameter a between low6 and lo-’ at 300 K and between lo-’ and IO-’ at 77 K is found in n- and p-Si. The a-value is independent of the effective volume. It is shown that 1 lf noise sources are located in the bulk. Hooge’s empirical relation is confirmed in the case where an a-value of 10e6 is observed. The temperature dependence of the a-value is measured for n- and p-Si with doping concentrations between 10” and 10” cm-‘. The magnitude of the a-value and its temperature dependence are related to the manufacturing process. Our measurements give no experimental support for the quantum l/f noise.
1. Introduction It is common practice to present experimental results on l/f noise in homogeneous semiconductors and in metals in terms of cu-values defined by PI S,/V’
= alfN,
(1)
where S, is the spectral density of the ac opencircuit voltage fluctuations, V the applied voltage, f the frequency and N the number of free charge carriers. The l/f noise parameter cy is found in the range from lo-* to lo-’ in semiconductors [l-lo]. This broad range of a-value is not yet understood. (i) An effort is made to explain this broad range of a-values by assuming that it is the contribution of lattice scattering to the mobility that causes mobility fluctuations. It is then found that
where p is the mobility and h and (Y,the mobility and the 1 lf noise parameter due to lattice scattering, respectively. However, also a broad range of a,-values is observed (lo-’ < (Y,< 10m3 [4-61).
(ii) Bisschop [2] h as suggested that the l/f noise parameter (Ydepends on the effective noise volume. His collection of a-values did not give a decisive answer. (iii) According to Peng et al. [7], the quantum l/f noise theory predicts a transition from the incoherent state with (Y- lo-’ for short devices, to the coherent state quantum l/f noise with (Y- 10m3 for long devices. The temperature dependence of (Yis still ambiguous. An a-value decreasing with decreasing temperature is observed in Ge [3], Si [3, 81 and InP [9]. A temperature-independent (Yeis found for Si [6, lo]. Here we present the results of resistivity, Hall and noise measurements on homogeneously doped n- and p-Si samples with various impurity concentrations and geometries in the temperature range from 77 to 300 K. The magnitude of the 1 lf noise parameter cx is investigated as a function of the length of the sample, effective noise volume, temperature and impurity concentration. 2. Conductivity data 2.1. Sample description In
our
geometries.
0921-4526/89/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
we used various device Table I gives the device geometry of
experiments
R. H. M. Clevers / Volume and temperature dependence
Table I Characteristic
of the 1 if noise parameter a
215
data of the samples used in this study at 300 K
Sample
Mcm)
n, p(cm-‘)
VHD En-% 1105 En-% 3500 En-% 06 Ep-Si 709 Ep-Si 100 Ep-Si 703 Ep-Si 4001 p-Si 1 p-Si 2 p-Si 3
4.5 4.4 63 1.6 24 2.2 0.82 0.088 30 15 1
1x 1x 7x 3x 6x 6x 2x 4x 5x 9x 2x
lOIS 1or5 10’) 1015 1014 lOI 1o16 10” 10” lOI lOI
p(cm”/Vs)
1420 1310 470 450 410 200 460 460 420
the samples. The planar structure of sample EnSi 1105 provided with a large gate electrode has been described elsewhere [ 111. The samples with the prefix E are manufactured by EFFIC (see Acknowledgements). The VHD structure (Vertical Hall Device) has been described in the litera-
Geometry
4,
VHD planar
1.1 x lo+ 10-‘0 - 10-7
circular contact
lo-” lo+
planar ring ring
(cm3)
1.2 x lo-’ 3.3 x 1o-6 2.5 x 1O-6
(I <9x lo-’ 6~10-~ 2x10+ 3 x lo+ 2 x 1o-6 2 x 1o-6 2 x 1o-6 8 x 1o-6 6 x 1O-5 2 x 1o-6 2 x 1o-5
+ A 0 n l v A 0 n X
+
ture [12] [fig. l(a)]. The circular contact geometry is shown in fig. l(b). On a chip of 3.5 x 3.5 x 0.4 mm3 circular contacts are present with radii between 1 pm and 100 pm. For p-Si the ohmic contacts were made by means of a boron implant and for n-Si by means of a phosphorous diffu-
Si
a
Fig. 1. (a) Vertical Hall device; (b) circular contact geometry; (L = 31).
(c) ring structure (L = rr(rl + r2)), (d) Greek-cross
structure
216
R. H. M. Clevers I Volume and temperature
sion. The depth of implantation is 1 pm and the depth of the diffusion, determined from the underdiffusion, is between 1 and 5 pm. The ring geometry of samples p-Si 2, 3 is shown in fig. l(c) (see also [2]). More details about the planar sample p-Si 1 are given in ref. [13].
dependence
of the 1 If noise parameter
p- Si
p(Q cm1 10 _I
lv
dependence
of the resistivity
-
,cJ(~OOK)R(T)/R(~OOK) ,
(3)
where R(T) is the resistance at temperature T. The resistivity at 300 K of the planar sample p-Si 1 is related to the resistance by p=RAIL,
(4)
where L and A are the contact spacing and the area of the sample, respectively. For the samples p-Si 2, 3 and the Vertical Hall Device the resistivity at 300 K is given by the manufacturer (see table I). The resistance calculated with the resistivity at 300 K agreed with the measured resistance to within 10%. Greek-cross structures [see fig. l(d)] are available on En-Si 1105 chips. Here the resistivity can be determined from the sheet resistance R, with the equation [14]
v.
ihru
-* .
16'
p(T)=
ne
px
2.2. Resbtivity The temperature p is given by
.
.
.
2)(V,,lZ,,)
,
.
*
l
i
t
,6zL
200
300
Fig. 2. Temperature dependence of the resistivity for p-Si (W: p-Si 1; +: Ep-Si 709, X: p-Si 2; v: Ep-Si 100; +: p-Si 3; A: Ep-Si 703; 0: Ep-Si 4001).
The temperature dependence of the resistivity of the devices in table I is shown in figs. 2 and 3. The radius a of a circular contact will be greater than the mask radius a,, due to underdiffusion. The radius Q can be determined from the resistance and the resistivity using the follow-
IO2
I
I
I
I
n- Si p(R cm)
R, = p/t = (r/In
(I
I
.
.
.
(5)
. .
where t is the thickness of the sample, V,, the voltage measured across two adjacent contacts and Z,, the current passed through the other two adjacent contacts. The resistivity can be determined with fourprobe resistivity measurements for the samples with circular geometry. For the device shown in fig. l(b) the resistivity can be determined with the help of the following equation (s 4 d) [15] P = (V,*lL>[~sl(l
-
WY
7
(6)
10 -
I . . l’
‘*
the contacts
l
-I
1‘,““’ ,I
jy 100
where s is the distance between (s=50I*m, 4OO
1
.
200
300
Fig. 3. Temperature dependence on the resistivity for n-Si (0: En-Si 3500; A: En-Si 1105; n : En-Si 06; +: VHD).
R. H. M. Clevers
I Volume and temperature dependence of the 1 if noise parameter a
ing equation [M] (7)
Z?=j$-{arcta”($)-arctan($
where d is the thickness of the substrate, b the diffusion or implantation depth and E = (l(bla)2)1’2. Fig. 4 shows radius a versus a, as expected from mask dimensions for En-Si 3500, 06 and Ep-Si 100. For the implanted contacts Ep-Si 100, the radius a is calculated taking an implantation depth of 1 km. The radius a is found to be almost equal to the radius a,. For the diffused contacts the underdiffusion is taken equal to the diffusion depth. An underdiffusion of 1.3 pm is observed for En-Si 06 and of 4 Km for En-Si 3500. The underdiffision increases with increasing ratio between dope concentration in the contact and dope concentration in the substrate. 2.3. Currier concentration The samples En-Si 1105 have a Greek-cross structure [17]. Therefore Hall measurements can be performed and the density of free charge carriers rz (or p) can be determined. Also, the Vertical Hall Device is suitable for Hall measurements [12]. For both structures the Hall voltage V, is given by VH = r,GZBlent
,
1
(10)
10
lo2
Fig. 4. The radius a of the circular contact as de:ermined from the resistance and the resistivity at 300K versus the radius a, from the mask dimensions (0: En-S 3500; m: En-5 06; V: Ep-Si 100).
217
where Z is the current, B the magnetic induction, rH the Hall factor and G the geometry factor close to unity. For the wafers En-Si 06, 3500 and Ep-Si 709, 100, 703, 4001 the carrier concentration n (or p) at 300 K has been determined from the resistivity p with the help of p versus n curves given in literature [18, 191. The results are given in table I. Van der Pauw structures have been made from the wafers. The product Gr, at 300 K has been determined from the carrier concentration and the measured Hall voltage using eq. (10). The temperature dependence of the carrier concentration can be determined if we assume that Gr, is temperature independent. Hall measurements are not possible with the planar sample p-Si 1 and with the ring structures p-Si 2, 3. The carrier concentration has been computed for these samples by assuming an impurity level E, meV above (for acceptors) or below (for donors) the band edge. For the density of free charge carriers p it is found that [20] P = NI , p =
for T > T, ,
(11)
(NINB/g1)“2 exp[ - E,I2kT]
,
for T < T, , (12)
T, = E,lk ln[N,lg,N,]
,
(13)
where NB is the density of states in the band, Nr the density of impurities and g, the degeneracy of the impurity state. Donor states are twofold degenerate. For acceptor states the situation is more complex. Then the structure of the valence band (light and heavy hold bands degenerate at k = 0 and the spin-orbit split-off band) must be accounted for in the effective mass approach. It is shown that the ground state has total angular momentum 3/2 and thus is four-fold degenerate [21]. Here we neglect the contribution of the excited states of the donors and the acceptors. Figs. 5 and 6 show the measured carrier concentration versus T for p- and n-Si, respectively. In figs. 5 and 6 the product r,G is assumed to be temperature independent. We have G = 0.8 [12] for the VHD and G = 0.95 [17] for En-Si 1105.
218
R. H. M. Clevers
300 200
18
‘0
!!
l.
150
I Volume and temperature
100
’
dependence
of the 1 If noiseparameter
(Y
donors E, = 44 meV [22]. For Nr we take the carrier concentration at 300 K as determined from the resistivity [18, 191 since all donors and acceptors are ionized at 300 K. The calculated carrier concentration is shown in figs. 5 and 6 for p- and n-Si, respectively.
II
1 (K) c_
2.4. Mobility The mobility can be determined from the resistivity and the carrier density with p = l/pen.
I 5
I
I
I
I
I
13
9,
Fig. 5. Temperature dependence of carrier concentration for p-Si (for symbols see fig. 2). Solid lines give the calculated carrier concentration for p-Si 1, 2 and 3.
Since the Hall measurements cannot be trusted for En-Si 3500 we compute the carrier concentration for this sample with eqs. (ll)-(13). For the samples p-Si l-3 we also compute the carrier concentration. The energy level for boron acceptors EA = 43.8 meV and for phosphorus 100
(6 300 200
10
II
(14)
Figs. 7 and 8 show the results for p- and n-S& respectively. The lattice mobility (taken from the literature [18, 191) is also given. The mobility is determined from the calculated carrier density for the samples p-Si l-3 and En-Si 3500. The calculated mobilities are in agreement with the measured mobilities of the other p-Si samples with the same dope. As expected for the low doped En-Si 3500 the mobility approximately equals the lattice mobility.
I
I
I
I
p-Si
1' 1
I
I
T(K) I -9 .==.
,$-A
'
.
A
-.
9.
.
,
'A
nkd) n-Si
II
IO"
4
I 5
I
-
. ‘. 103rr (KY
I
1
I
Id 9 II Fig. 6. Temperature dependence of carrier concentration for n-Si (for symbols see fig. 3). Solid line gives the calculated carrier concentration for En-Si 3500.
I 11
I
T IK)
150
. I 300
Fig. 7. Temperature dependence of the mobility for p-Si (for symbols see fig. 2). Solid lines are for p-Si 1, p-Si 2 and p-Si 3. The dashed line gives the lattice mobility. Carrier concentrations (cm-‘) are given.
R. H. M. Clevers
I Volume and temperature dependence
Fig. 8. Temperature dependence of the mobility for n-Si (for symbols see fig. 3). Solid line is for En-Si 3500. The dashed line gives the lattice mobility. Carrier concentrations (cm-‘) are given,
3. Results of noise measurements
We consider three types of noise: thermal noise (4kTR), generation-recombination (gr) noise and l/f noise. For the voltage noise spectrum we have S,=4kTR+$C cXv2 +fN
Tici i
1 + (2TfTi)2
(1%
where the Ci’s are constants. The gr term accounts for several gr processes. Fig. 9 shows some spectra obtained after subtracting the 4kTR noise. In fig. 9 three spectra show 1 lf noise only (a, +, A). One spectrum (0) can be explained by assuming a gr component with r = 7 ms and a 1 lf noise component. Also shown is a spectrum (x) that can be explained in terms of gr processes only with a distribution of characteristic times. This spectrum can be explained by assuming characteristic times in the range from 20 )LS to 30ms. We estimate the low-frequency plateau of the gr component with 7 = 3 ms. The spectral density at
of the 1 If noise parameter a
219
Fig. 9. Five measured spectra. The 4kTR noise is subtracted and its level is given by the dashed line. The solid line gives the l/f noise level and the dashed and dotted line the gr noise level. Vertical arrows give the inverse of the characteristic time 1/2m of the gr noise component. (0: En-% 3500, T= 118K, V=30.2mV, W: En-Si 06, T= 175K, V= 84.5 mV, X: p-Si 2, T=222K, V=O.453; a: Ep-Si 709, T= 178K, V=80.3mV, A: Ep-Si 703, T=295K, V= O.llOV).
low frequencies (f <1/27rT=50Hz) S,= lo-i6V2/Hz (see fig. 9). Then, with eq. (15) we find CilN2 = 5 x 10-i’. With the data from table I and fig. 7 we find N=nLZ=3x109 and Ci= 4 X lo5 4 N. For gr noise with one time constant we have Ci = ( (AN)2) -+ N. For gr noise with a distribution of characteristic times T the Ci’s are complicated functions of the trap properties. However, from Ci G N we conclude that we can describe the spectrum (X) with eq. (15) assuming a trap density low compared to the density of free charge carriers. Some authors [5] prefer to describe the spectrum ( x ) with the equation S, = /3V21fyN,
(16)
where p has the dimension [fl] = HzY-‘. For the spectrum under consideration y s 0.7 is found. If we describe our spectra with eq. (15), only an
R. H. M. Clevers
220
I Volume and temperature
upper limit for (Ycan then be determined for the spectrum (x). This is indicated in the figs. 13, 14 and 15 with a vertical arrow. We describe the results of our noise measurements with the empirical relation eq. (1). We deal only with homogeneously doped samples. For samples with a non-homogeneous current pattern we have to replace iV by IV,,, in eq. (1)
where Q.,, is the effective noise volume given by
[31
(j-d3rE2(r)12,/{
j- d3rE4(r)} .
(18)
L?,,, has been calculated for all the geometries used in this study [12, 13, 161. The effective noise volume for circular contacts is given by [16] 0&, = (5p3/47r2R3)(r5/s) )
(19)
where r and s are factors accounting for the eccentricity of the geometry. The factors r, s and r5/s are given in literature [6, 161. The Q,, for the ring structure is given by [13] n
= L{(r eff
of the 1 If noise parameter
LY
Fig. 10. (Yversus the effective noise volume L&, at 300 K (a: Ep-Si 709; V: Ep-Si 100; n : En-% 06) and at 77 K (V: Ep-Si 100; q : En-Si 06).
(17)
N,,, = n.n,,, 3
&,, =
dependence
- 25)~ arcsinh(dla)}2 ln[(tan( [))-‘I
7r
’
(20)
where d is the thickness of the substrate and 6 a factor between 0 and 1 that is related to the diffusion depth of the contacts. The meaning of a is explained in fig. l(c) and L is given by L = 4r1 + r2). With the effective noise volume given by eqs. (19), (20) and the empirical relation eq. 1, LYvalues are calculated.
noise sources are located at the surface predicts S” - 0;;‘3 [23]. We present the 1 lf noise parameter (Y as a function of the channel length for En-% 1105 (without gate electrode) at 77 K. Fig. 11 shows the results for one chip of 4 x 2.5 x 0.4 mm’. We observe an a-value of 1O-5 which is independent of the length of the channel (o - L6 with 0 < 6 < 0.3). This behaviour is expected for noise sources distributed homogeneously either at the surface or in the bulk of the sample. These results are in contradiction to the experimental results of Peng et al. obtained with p-MOSFETs [7]. There it was observed that (Yincreases from 3 X 1O-7 for devices with channel length L =I 6.3 pm to 3 X lob4 for devices with L = 196 km. We return to this matter in section 4.4 3.2. Temperature dependence of (Y Fig. 12 shows the temperature dependence of the 1 lf noise parameter (Yof Ep-Si 4001 obtained on three circular contacts from one chip. We observe that the o-values scatter by a factor of 2 from contact to contact. Some of the detailed variations in the a-values with temperature are
3.1. (Yversus volume and length We present (Yat 77 K and 300 K as a function of the effective noise volume in fig. 10. We observe that (Yis independent of Q.,, and of the order of 10m6 (a - Ozff with -0.1 < y
,i6t
1
I
I
10
1
~$t-.~ 10
lo3
Fig. 11. (Y as a function of the length L of the device for En-Si 1105 at 77 K.
R. H. M. Clevers I Volume and temperature dependence
lo*\
I
I
221
o!
I
EpSi 4001
a I . 16
of the 1 If nobe parameter
*
.
..A' .rn' & l
:
I
-T(K)
6
a
lo-
I
+
l
Fig. 12. Temperature dependence of a for three circular contacts fromEp-Si 4001 (A: a,, = 2 pm; n : a,, = 3 wm; 0: a, = 5 Km).
t 1Ci'
not reproducible. Therefore we concentrate on the broad variations with temperature. Scattering of (~-values is also observed by other authors ]3, 81. In figs. 13 and 14 we present the temperature dependence of (Yfor the samples in table I. The a-values are averaged over at least two circular contacts. The cr-values at 77 and 300 K are averaged over circular contacts from different chips.
4
1G7- l
4
4
100
--T(K)
I
I
200
I
I
300
Fig. 13. Temperature dependence of a for p-Si (for symbols see fig. 2). Upper limits for (I are indicated by a vertical arrow. Carrier concentrations (cmm3) are given.
T
n- Si
I
100
I
T (K)
I 200
Fig. 14. Temperature dependence see fig. 3). Carrier concentrations
I
I
1
30 0
of a for n-S (for symbols (cm-‘) are given.
4. Discussion 4.1. Location of the noise sources From the results in fig. 10 we concluded that 1 lf noise is a bulk effect. However, Jintsch [23] suggested that l/f noise is caused by noise sources located at grain boundaries. Jantsch stated that our results are no proof of a bulk effect (part of the results of fig. 10 had already been published in ref. [6]). Therefore we now state our conclusion more carefully: S, - 0,: means that the noise sources are distributed homogeneously over the sample. Thus, the model proposed by Jantsch is not in contradiction with the results of fig. 10, provided the grain boundaries are distributed homogeneously over the sample and the l/f noise sources are distributed homogeneously over the grain boundaries. In the past, the validity of the empirical relation eq. (1) was already shown for a-values of the order of 10e3 [l]. Here, the validity of eq. (1) is also shown for cx-values of lo-‘? These results do not support Bisschop’s suggestion [2] that samples with a small effective volume a,,, show lower a-values than samples with a large n eff.
222
4.2.
R. H. M. Clevers
I Volume and temperature dependence
of the 1 If noise parameter
Mobility fluctuations
(I
104
An (Y,value of 10m3 independent of PI& was found for Ge [l]. It was concluded that it is only the contribution by lattice scattering to the mobility that fluctuates. Fig. 15 shows the (Yvalues found for the samples in table I at 77 K. It can be seen that the a-value is not proportional to (p lo)* for Si at 77 K, in contradiction to the experimental results for Ge. 4.3. Temperature dependence
T(K)---
of the llf noise
Different temperature dependences are reported here and in the literature. An a-value declining to low temperatures is observed in p-Ge [3], n-Si [3], p-Si [3, 81 and n-InSb [9] (see fig. 16). This temperature dependence of (Ycan be described by .
a=Aexp(-EIkT)+B
(25)
(for Si and Ge for T < 300 K and for n-InSb for T < 200 K). Eq. (25) describes a thermally activated process. An activation energy E g 0.1 eV and a B between 10e5 and 10m3 are observed in Ge, Si and n-InSb.
IO3
I
I
I
I
l
1d I
103/Td, I
5
I
.’ I
I
9
I
I
I
13
Fig. 16. Temperature dependence of CYfound in the literature (A: Luo [8]; n : Bisschop [3]; 0: Alekperov [9]). Solid line [eq. (25)] with A = 0.33, E = 0.1 eV and B = 4 x lOA and dashed Iine [eq. (26)] are taken from Luo [8]. Dashed and dotted line shows eq. (27) fitted to Bisschop’s data (y = 4, A = 3 x lo-“, B = 4 x 10~‘).
I
A
Luo et al. [8] found that his data could also be described with the following equation (see fig. 16) a(T)
i
Fig. 15. a as a function of (PI&~ for p- and n-Si at 77 K (a: Ep-Si 709; X: p-Si 2; ‘I: Ep-Si 100; +: p-Si 3; A: Ep-Si 703; 0: Ep-Si 4001; 0: En-Si 3500; 0: En-Si 1105; Cl: En-Si 06; 0: VHD). Upper limits for (Y are indicated by a vertical arrow.
= &/p(T)&(T).
(26)
Luo et al. derived eq. (26) by assuming noise sources in a surface layer. However, Alekperov et al. [9] found noise sources homogeneously distributed through the sample. They found experimentally that the a-value is independent of surface treatment and thickness of the sample in the temperature range where the exponential term dominates over B. For our sample En-Si 1105 it was shown [l l] that noise sources near the semiconductor-oxide interface dominate the l/f noise. However, our a-value depends weakly on temperature. Thus,
R. H. M, Clevers
I Volume and temperature dependence
eq. (26) cannot be applied to Alekperov’s and our (En-Si 1105) data. It is also possible to fit the experimental results for Ge, Si and n-InSb with (see fig. 16) ol(T)=ATY+B.
(27)
Then, y s 4 for Bisschop’s data, y s 5 for Luo’s data and y z 6 for Alekperov’s data. We do not know of any physical mechanism from which eq. (27) follows. Eq. (27) does not point to a thermally-activated process. Luo found an o-value of lo-* at 300 K decreasing with decreasing temperature for his p-Si sample treated at 950°C only. The p-Si samples treated at a low temperature (T = 550°C) show a low u-value (a - 10m4) weakly dependent on temperature. The processing temperature is thought to be the crucial parameter in explaining this behaviour. Although processed at a low temperature (contacts evaporated at 9O’C) p-Ge shows the same temperature dependence of (Yas p-Si treated at 950°C. Our samples do not show this particular temperature dependence (see figs. 13 and 14), although processed at high temperatures (T > 900°C). Noise sources showing a decreasing (Y with decreasing temperature are observed when (Y> low3 at 300 K. Perhaps these noise sources can be avoided by manufacturing samples in a modem IC laboratory. 4.4. Quantum 1 If noise The validity of the quantum l/f noise theory has been questioned more than once [24]. Here, we will not be concerned with the theory itself but simply compare experimental results with the predictions of this theory. Peng et al. [7] observe that the a-value increases from 5 X 10e7 to 3 X low4 with channel length L increasing from 6.3 to 194 km (a - L*) for their p-MOSFETs. Peng et al. [7] conjecture that they observed the transition from incoherent-state quantum 1 /f noise (ff - 10e8) for short devices to coherent-state quantum 1 /f noise (o - 10e3) for long devices. According to van der Ziel’s generalized semiclassical quantum 1 /f noise theory [25], 1 /f noise in short semiconductor devices is described by colli-
of the 1if noise parameter a
223
sion 1 /f noise and 1 /f noise in long semiconductor devices by acceleration 1 /f noise theory. The acceleration l/f noise theory predicts a l/f noise parameter (Y depending on device length L as LY- L*. Thus, Peng et al. [26] state that their experimental results confirm van der Ziel’s acceleration 1 lf noise theory. Peng’s experimental results [7] are in contradiction with our results for En-Si 1105, where an o-value of lo-’ at 77 K independent of channel length (5 urn< L < 300 pm) is observed (see fig. 11). Kousik et al. [27] assume various scattering mechanisms in both n-Si and n-GaAs. With the help of the quantum 1 /f noise theory they calculate a-values as a function of temperature and impurity concentration. The o-values shown in figs. 13, 14 and 15 are higher by a factor of 10 at least than the a-values calculated by Kousik. Kousik compares his calculations with experimental results from the literature. It is worth while to commend on these experimental results quoted by Kousik. The a-value of 2 X 10s7 found in p-Si (that in our p-Si 3) at 300 K by Bisschop is based on his erroneous calculation of Q,,,. A recalculation of J2,,, based on eq. (20) gives an a -value of lo-“. There is no a-value of 7 X lo-’ for Si at 300 K in Bisschop’s thesis [2]. That value was used in fig. 16 of ref. [27]. For polysilicon resistors with doping 1.8 x 10” cme3 Bisschop reported C-values (C = (I!/N) between 10e9 and lo-! Kousik erroneously quoted these values as a-values. Schmidt et al. [28] reported an a-value of 6 x low8 for GaAs at 300 K instead of the 2 x 10e8 reported by Kousik. Thus, only three data points for Si at 300 K in fig. 16 of ref. [27] remain after elimination of erroneous data. There is not enough experimental confirmation of the temperature dependence of (Ypredicted by the quantum 1 lf noise theory.
5. Conclusions The experimental results can be described with noise sources located in the bulk. The validity of the empirical relation eq. (1) is also shown with a 1 /f noise parameter (Yof the order of 10-4 The wide range of a-values ( 10m8< (Y< lo-‘) re-
224
R. H. M. Clevers
I Volume and temperature dependence
mains unexplained. The order of magnitude of (Y does not depend on the effective volume or on the length of the device. Various trends in the temperature dependence of (Yhave been observed. A strong temperature dependence of a (a - T” with 4 < y < 6) is observed for (Y> 10e3 at 300 K. A weaker temperature dependence of LY(o - TY with -3 < y < 2) is observed for (Y< 10e4 at 300 K. The magnitude and the temperature dependence of (Yare possibly related to the manufacturing process. In this case, the processing temperature is not the only parameter of importance. Experimental results presented here and in the literature do not support the quantum 1 lf noise theory. Acknowledgements The author thanks T.G.M. Kleinpenning and L.K.J. Vandamme for stimulating discussions, EFFIC (Eindhovense Fabricage Faciliteit voor ICs) for the preparation of all the E-samples and J.G.M. Delissen for his contribution to the experimental work. References [I] F.N. Hooge, T.G.M. Kleinpenning
and L.K.J. Vandamme, Rep. Prog. Phys. 44 (1981) 479. [2] J. Bisschop, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven (1983). (31 J. Bisschop and J.L. Cuypers, Physica B 123 (1983) 6. [4] L.K.J. Vandamme, Proc. 7th Int. Conf. on Noise in Physical Systems (Elsevier, Amsterdam, 1983) p. 183.
of the 1 If noise parameter
Q
PI G.S. Bhatti and B.K. Jones, J. Phys. D.: Appl. Phys. 17 (1984) 2407. Clevers, Proc. 8th Int. Conf. on Noise in Physical Systems (Elsevier, Amsterdam, 1986) p. 411. [71 Q. Peng, A. Birbas and A. van der Ziel, Proc. 9th Int. Conf. on Noise in Physical Systems (World Scientific, Singapore, 1987) p. 400. PI J. Luo, W.F. Love and SC. Miller, J. Appl. Phys. 60 (1986) 3196. 191S.A. Alekperov, N.Ya. Guseiiiov, Ch.0. Kadzhar and E.Yu. Salaev, Sov. Phys. Semicond. 20 (1986) 973. WI V Palenskis and Z. Shobhtzkas, Solid State Commun. 43 (1983) 761. (111 R.H.M. Clevers, Physica B 147 (1987) 305. t121 L.K.J. Vandamme, Journal de Physique Colloque C4, supplement au no 9, vol. 49 (1988) p. (X-157. P31 R.H.M. Clevers, Ph.D. Thesis, University of Technology Eindhoven (1988). 1141W. Versnel, Solid-State Electron. 22 (1979) 911. WI J.D. Wasscher, Philips Res. Rep. 16 (1961) 301. WI G.W.M. Coppus and L.K.J. Vandamme, Appl. Phys. 20 (1979) 119. (171 W. Versnel, J. Appl. Phys. 52 (1981) 4659. @I S.S. Li and W.R. Thurber, Solid-State Electron. 20 (1977) 609. 1191S.S. Li, Solid-State Electron. 21 (1978) 1109. 1201R.A. Smith, Semiconductors (Cambridge University Press, Cambridge, 1978). WI S.T. Pantelides, Rev. Mod. Phys. 50 (1978) 797. P21 W. Ktimicz, Solid-State Electron. 29 (1986) 1223. [231 0. Jlntsch, IEEE ED-34 (1987) 1100. 1241L.B. Kiss, Proc. 9th Int. Conf. on Noise in Physical Systems (World Scientific, Singapore, 1987) p. 373. P51 A. van der Ziel, J. Appl. Phys. 64 (1988) 903. WI Q. Peng, A.N. Birbas, A. van der Ziel, A.D. van Rheenen and K. Amberiadis, J. Appl. Phys. 64 (1988) 907. v71 G.S. Kousik, C.M. van Vliet, G. Bosman and P.H. Handel, Adv. Phys. 34 (1985) 663. P81 R.R. Schmidt, G. Bosman and C.M. van Vliet, SohdState Electron. 26 (1983) 437.
161R.H.M.