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Prakash, Gyan

Sharma, Ashutosh

Verma, Mahendra P

Sodha, Mahendra Singh

English

Nonlinear Electromagnetic Propagation Parameters of the Ionosphere

Shrivastava, S K

Kailash

Raju, K M

352 S K Shrivastava, Kailash and KM Raju 
23]. It is also reported by the previous workers that ultrasonic shear wave velocity is more 
sensitive for the study of the basic properties [14]. 
In the present investigation, ultrasonic attenuation as a function of temperature due to 
phonon-phonon interaction is studied in Copper (Cu), Silver (Ag) and Gold (Au) at different 
temperatures, viz. 100, 200,300 and 400 K, using repulsive parameter (q), interatomic 
distance (r) and Born-Mayer [24] potentials following Mason's approach [13]. The thermal 
relaxation time, average Gruneisen parameters, non-linearity constants and attenuation 
coefficients for longitudinal and shear waves are evaluated along <100> direction of propa-
gation. 
2. Theory 
As present study is concerned with attenuation coefficients at different temperatures, the 
third order elastic constants are required at these specific temperatures. Third order elastic 
constants (TOECs) are evaluated using Born-Mayer potential, 0(r) = A exp(-r/g) and Cou-
lomb potential, ±e2/r, (r,q and e being interatomic distance, hardness parameter and 
electronic charge, respectively). According to Brugger's [25] definition for elastic constants, 
one obtains the three expressions for TOECs at absolute zero temperature, given in 
Table 1. 
Table 1. Expressions for the third order elastic constants at OK. 
C?„ = W2639G-GZ-2GA, 
C?12 = C?ee - 1.208625G-G^ 
Cfzj = C?* - C^e = 0.678375G. 
Expressions for G's 
G = e*/r/f Gf ~ (1/r0 + 1/q) O (or)/qr0, 
G2 = (r2/2r0+1/q)Q(rj2)/qr0$ 
G^ (3/r0* + 3/qr0 + 1/cf) Q (rj/q, 
G4= (3f2/rf + 6/qr0 + 2 fc/q*) Q (rQ f2)/4q, 
G s - (Wrf + 15/qrf + 6/q*r0 * 1/<?) r0 Q (rj/q, 
Gd« (15f2/4r0°+15/2qr0* + 3f2/cfr0 + 1/q°) rft (rj2)/2q. 
Seeger and Mann [26] has already studied these crystals and obtained the second 
order elastic constants (SOECs) using the same potentials. The temperature dependence 
of elastic constants is determined using anharmoniclty theory of crystal dynamics devel-
oped by Leibfried and Hahn [27] or Leibfried and Ludwig [28] given below : 
^Pr(T)*Ppr+CpT, (D 
Ultrasonic study of highly conducting metals 353 
where 
/ is a function of hardness parameter and interatomic distance given by 
/, = - Q { ( 2 + 2Q0 -qlp(r0) + 2j2(n2q0 -ql)O(r0J^)}/l2 (3) 
and 
k -2{(«fc-2)0(ro) + 2 ( ( f c - ^ r o ^ } x { ( q b - 2 ) 0 ( r 0 ) + 4(gD-^)o(r0V5)} (4) 
Table 2. Expressions for vibrational contribution to the third order elastic constants. 
CJf2 = gf? + fcF,(2Fs + F2) + g,F6, 
Table 3. Expressions for g„ for highly conducting metals. 
ffi = JfoS; 
fc-flb[(X/S,) + S]/2; 
0o=/xt>o/8i£; 
flf3 = Po[(2XaS / 3S,)+(X / S,) + SJ / 48; 
X = oa>0/2KT; 
04 - - f l o ^ ' S 2 / 3S,) + (X3 / 6Sf)+(X*S / S,) + (5X / 45,) + (5S / 4)] /144; 
a)0=(l/M+ + W-)/QroF0; 
5 = CothX, S,=Sinti*X 
and cj? are the contributions due to vibrational energy of the crystal and these values are 
given in Table 2. The expressions of different G„*s and F„'s are shown in Table 3 and 4. 
Mason [13] has shown that the Gruneisen numbers are related with the second and third 
order elastic constants. Therefore, the Gruneisen numbers y'f are evaluated using 
Gruneisen tables [13], and second and third order elastic constants are thus obtained. The 
354 S K Shrivastava, Kailash and KM Raju 
non-linearity constant [13] 'D ' is related to Gruneisen numbers as 
D^9(Yfy[3CT(Y)f)/Et (5) 
where C, T and E are specific heat, absolute temperature and energy of the crystal, 
respectively; <yf > and <Y',> a r e square average and simple average of Gruneisen 
( a / / 2 L^^ 2 £ D T ^ / 3 d V ' 3 ( 1 + £ £ > 2 ^) ' <6> 
numbers. 
One gets the following relations for phonon viscosity and thermoelastic losses [13] 
using Gruneisen numbers and non linearity constants. 
{<xlf\=(2n)2(YtfKTI2dV\ (7) 
where 'K" is the thermal conductivity, '<f the density of the substance, V (=2nf) the 
angular frequency of the ultrasonic wave and *Tm* the thermal relaxation time for the 
exchange of acoustic and thermal energies, given by 
jT,=T,=Tlh=3K/CV2 (8) 
V represents V (longitudinal wave velocity) or V (shear wave velocity) and V is the 
Debye average velocity. These are computed according to the relations : 
V,=(C„/d)5 . 
3/73=1/V,3 + 2/V;3 . (9) 
As the whole evaluation is based upon elastic constants and these elastic constants 
are obtained using repulsive parameter and interatomic distance of the crystal, one may 
successfully conclude that attenuation is one of the basic properties of substance. 
Table 4. Expressions for Fn for highly conducting metals. 
F. ' 1/Kq0-2)Q(rJ + 2(q0-f2)Q(rj2) 
qc » r/q; 
F, = 2[(2 + 2qe-q*) Q (rj+ 2 (f2*2q0 -f2q») Q (r/2)]F0; 
F3 = 2(-6-6qB-qe* + qe*)Q(rJFt + Fs; 
F3 = 2(-30-30qB-9qo* + qo3-q,<)Q(ro)F0 + Fe; 
F4 = 2(-210-210q0-75qg*.5q03 + 4qo< + qo')Q(ro)F0 +FT. 
F5 - (•3f2-6q0-f2qJ> + 2q03)Q(r0f2)Fe, 
F, = K1S/f2)+1Sq9-(9/f2)q/-9q03-f2q<;]Q(ref2)F0; 
FT = [-(105/2 f2) - 105/2) qe - (75/ 2&) q/ - (5/2) q3 + 2& qo" 
+ q05]Q(rj2)F0. 
Ultrasonic study of highly conducting metals 355 
3. Evaluation 
According to Tables 1-4, the third order elastic constants are evaluated at different tem-
peratures using hardness parameter [26] and nearest neighbour distance [29]. Taking 
second order elastic constants from literature [29, |0] and calculated third order elastic 
constants, average Gruneisen numbers <yf> and <yiI> are evaluated along <100> 
direction of propagation for longitudinal and shear vfcves using Manson's Gruneisen [13] 
Tables (Table 5). The thermal relaxation time ' i / is^computed using thermal conductivity 
16 i 
14 
10-1 
8 
»H 
4 
2 
0 
Cu 
i 
300 100 200 
" i 
400 
Ag 
J* 
^ 
i 
141 
12-
10-6-
6-
4-
2-
A « 
u i 
20 
16 
10 
5 
0 
100 
^ 
4 
_l , 
100 
200 
— • - » 
200 
• - — i — 
300 
"t? 
- - - ' • • 
300 
• - - - ' — • - 1 
400 
Au 
o 
400 
TEMPERATURE (K) 
Figure 1. Temperature dependence of thermal relaxation time in <100> highly conducting metals. 
S K Shrivastava, Kailash and K M Raju 
Longitudinal wave 
Shear wave Cu 
o-
i 
100 200 
• O-
1
 i " 
300 
11
 i' 
400 
14< 
12 
10-
8 
a-
4-
2-
0« 
Ag 
o-
11
 "1 
100 
—"I 
300 
t'• 
400 200 
Au 
•"T 
100 
o 
— r - • 
•O-
T 
200 300 400 
TEMPERATURE (K) 
RBure 2. Temperature dependence of nonlinearity constant along <100> direction in highly conducting 
metals. 
Ultrasonic study of highly conducting metals 357 
Cu 
f 
100 
I 
200 
1
 i " 
300 
i 
400 
Ag 
J 
% 
£ 
«0 
(a 
• * • 
4 
> 
3 
2.6 
2 
1.S 
1 
04 
0 I 
100 
I 
200 
i ' 
300 400 
All 
I 
100 
I 
200 300 
i 
400 
TEMPERATURE (K) 
figure 3. Temperature dependence of thermoelastic loss for longitudinal waves along <100> direction In 
highly conducting metals. 
358 S K Shrivastava, Kailash and KM Rap 
-o— Longitudinal wave 
o • • Shear wave 
Cu 
Ag 
8 
-v 
£ 
5 
^ 
*• •* 
t c s 
40 
35 
30 
25 
20 
15 
10 
5 
0 
Au 
801 
25-
20-
15-
10* 
5-
o -
« 
^ ^ ° 
r ^ ^ ^ -
c r
' 
fr^rrrr^-" 
• 1 - • • T - L l J — J L . L - . ....
 T .-
100 200 300 400 
TEMPERATURE (K) 
» 
Figure 4. Temperature dependence of phonon viscosity loss along <100> direction In highly 
conducting metals. 
Ultrasonic study of highly conducting metals 359 
Table 5. Average Gruneisen numbers and nonlinearity constants ratio along <100> direction of 
propagation in highly conducting metals. 
Average Gruneisen 
numbers & nonlinearity 
constants ratio 
<Y >i 
<Y2>i 
<Y>, 
D,/Ds 
Substances 
Cu 
Ag 
Au 
Cu 
Ag 
Au 
Cu 
Ag 
Au 
Cu 
Ag 
Au 
r 
100 
-0.0203* 
-0.4973; 
-0.4001* 
0.4479 
1.1366 
0.6449 
0.0809 
0.2202 
0.1162 
5.53 
4.41 
4.50 
Temperatures (K) 
200 
-.00332 
-0.5179 
-0.4318 
0.5176 
1.2340 
0.7388 
0.0945 
0.2214 
0.1247 
5.47 
4.98 
5.22 
300 
-0.0279 
-0.5248 
-0.4589 
0.5637 
1.2362 
0.8199 
0.1101 
0.2248 
0.1381 
5.11 
4.96 
5.32 
400 
-0.0187 
-0.5837 
-0.4818 
0.7621 
1.4044 
0.9183 
0.1274 
0.2286 
0.1584 
5.98 
5.73 
5.23 
[29-31], density [29-31] and eq. (8) and (9) and is shown in Figure 1. From eq. (5), the 
non linearity constants '£7 are obtained taking specific heat [30] and energy [30] of the 
crystal as a function of temperature (Table 5) and are shown in Figure 2. At last, phonon 
viscosity and thermoelastic losses,(a/f2)m and {a,f2)p_p are calculated from eqs. (6) and 
(7) and are presented in Figure 3 and 4. 
4. Results and discussion 
Comparison of calculated and experimental values of thermal relaxation time, Gruneisen 
numbers, nonlinearity constants, ultrasonic attenuation due to phonon-phonon interaction 
and thermoelastic loss, with the values for similar substances is the best way to check 
the validity of present theory. 
The value of T,„IS of the order of 10~"s[33]. From Figure 1, it is clear that at low 
temperature, its value is very high and decreases as temperature is increased, which is 
also expected and seen in other types of crystals [32, 33]. After a certain temperature, 
it achieves, constant value. The expression for temperature variation of thermal relaxation 
time is 
Tt-tmmxp(-Ta). (10) 
where r0 and A are the constants and Tis the absolute temperature. 
It can be concluded that non linearity constants increase as temperature increases as 
shown in Figure 2. This behaviour can be related with atomic mass or atomic number. As 
atomic number or atomic mass increases, nonlinearity constant will increase in metallic 
crystals. It is also clear from Table 5 that the ratio of D,/D9 lies between 3 and 16 which 
360 S K Shrivastava, Kailash andKMRaju 
is also seen in other types of substances [32, 33]. D, is greater than D, along <100> 
direction of propagation. 
In all the three noble metals Cu, Ag and Au, (a/f2)th\s negligible in comparison with 
p-p interaction. The thermoelastic attenuation due to the thermal conduction between the 
compressed and expanded parts of acoustic waves increases as temperature is increased 
and shown in Figure 3. 
The values of {a/f2)p_p have been calculated using second and third order elastic 
constants data at 293 K by Tondon et al [33]. Attenuation for both longitudinal and shear 
waves is increasing as the temperature is increased (Figure 4). Ultrasonic attenuation for 
the shear wave is less that that for the longitudinal wave. The values are in good agreement 
with the literature [28, 30]. The attenuation varies with temperature as 
a = a0T\ (11) 
where a and n are constants. 
From the above results, one may successfully conclude that ultrasonic attenuation is a 
fundamental property of the substance. 
References 
[I] C Falter Phy. Stat Sol. (b), 242 30 (2005) 
[2] Z H Fang Physica 0357 433 (2005) 
[3] B Voleidiene and D Ruf kuneine Ultragarsas 42 46 (2002) 
[4] J C Wendoloski J. Acoust Soc. Am., 110 724 (2001) 
[5] E S R Gopal and B J Thangaraju Pure Appl. Ultrasonics, 22 29 (2000) 
[6] P Kielczynski, W Pajewski and M Szalew IEEE Trans. Ultrason Ferroelect. Freq. Control 46 233 
(1999) 
[7] R R Reddy, Y Ahmad Nazeer, M Ravi Kumar and P M Reddy Acoust Letts. 18 31 (1994) 
[8J P Schrey J. Phys. (France) 42 C5-671 (1981) 
[9] J P Bonnet M Boisser C Vedei and R Vacher J. Phys. Chem. Solids, 44 515 (1983) 
[10] J Kruatkramer and H Kruatkramer Ultrasonic Study of Materials (New York : Academic) 
(3rd edn) (1983) 
[II] T Soma and H M Kagaya Phys. Stat Sol (b) 126 91 (1984) 
[12] M S Anderson and C A Swenson Phys. Rev. B31 668 (1985) 
[13] W P Mason Physical Acoustics 111 B (New York : Academic) (1965) 
[14] H M Ledbetter Elastic Properties in Materials at Low Temperatures (Metals Park, Ohio: American 
Society for Metals) p1 (1983) 
[15] S K Kor, R R Yadav and Kailash J. Appl. Phys. 46 467 (1975) 
[16] A Akheizer J. Phys. (USSR) 1 277 (1939) 
[17] P G K\emensSoffd State Phys. (USA) 7 82 (1956) 
[18] V V Lemanov, V Ya Awdonin and A V Patiw Zh. Phys. JETP Lett. (USSR) 12 360 (1970) 
[19] LGMetkvfowSov. Phys. Solid State (USA) 11 2241(1970) 
[20] H J Maris Physical Acoustics Vol. VHIB (New York : Academic) (1971) 
Ultrasonic study of highly conducting metals 
[21] L I Bitenskii and Yu A Logachev Sov. Phys. Solid Stale (USA) 17 2345 (1976) 
[22] R Nava and J Romero J. Acoust. Soc. Am. (USA) 64 529 (1978) 
[23] J Trivisonno, A R Slotwinski and M B Johnson J. Phys. (Franca) 42 C5-983 (1981) 
[24] M Bom and J M Mayer Z Phys. (Germany) 75 1 (1938) 
[25] K Brugger Phys. Rev. A1331611 (1964) 
[26] A Seeger and E Mann J. Phys. Chem. Solids (GB) l £ 314 (1960) 
[27] G Leibfried and A Hahn Z Phys. (Germany) 150 314 (1958) 
[28] G Leibfried and W Ludwig Solid State Physics Vol. tfi (New York : Academic) (1961) 
[29] C Kittel Introduction to Solid State Physics (New York: John Wiley, (5th edn.) (1961) 
[30] AIP Handbook (New York : Mc Graw-Hill), (2nd edn.) (1963) 
[31] Handbook of Chemistry and Physics (Florida: CRC Press), (58th edn.) (1977-78) 
[32] S K Kor, R R Yadav and Kailash J. Phys. Soc. Jpn. 15 207 (1986) 
[33] U S Tondon, S K Kor and G Rai Phys. Rev. B 6 2195 (1972) 


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Front Title

Singh, Raman K

Sarma, Manabendra

Mishra, Manoj K

Approximate Construction of Local Complex Potentials for a Time Dependent Wave Packet Based Treatment of Vibrational Excitation Cross-Sections in Resonant e-N<SUB>2</SUB> e-Co and e-H<SUB>2</SUB> Scattering

Krishna, C B

Dhall, R

Chopra, S

Banerjee, S

Indian J. Phvs.81 (12), 1315-1322(2007) O fete 
Futuristic memory device: a theoretical modeling 
C B Krishna1, R Dhall2, S Chopra^ and S Banerjee3* 
'Department of Electronics and Instrumentation Engineennq 
Birla Institute of Technology and Science, Pilani-333 031, Rajasthan India 
department of Electrical and Electronics Engineering, 
Birla Institute of Technology and Science Pilani 333 031, Rajasthan, India 
department of Physics Birla Institute of Technology and Science 
Pilani-333 031, Rajasthan, India 
E mail sound bits-pilani ac in 
Received 14 August 2007, accepted 7 September 2007 
Abstract The paper presents a theoretical modelmg of a memory device that may be treated as an 
alternative to conventional flash memory devices based on the principle that proximity induces electric field 
modulation leading to a high ON-OFF current ratio, which can be used to distinguish betweent the two binary 
states 0 and 1 The present device is expected to operate with synthetic bilayer membrane opening up the 
possibility of a biocompatible memory device for the future A first order calculation of parameters, relevant 
fabrication steps and challenges in commercialization of this device are also presented The device has an edge 
over other flash and quantum-dot memories with respect to non-volatility and endurance 
Keywords Flash-memory, synthetic bilayer membrane, charge storage, large on-off ratio 
PACS NOS. 81 07.b, 81 16 -c, 73 43 Fj 
Conventional flash memory devices [1] base their operation on the storage of charges in 
a floating gate layer and the modulation of the electric field because of the stored charges 
leading to a threshold voltage shift that results in a distinction between the two binary 
states. Most devices use a poly-Si layer as the floating gate. However this enhances 
charge leakage and degradation of the oxide layer with repeated use. The problem occurs 
because the charge that is stored on the gate can leak away because of the presence of 
local defects in the oxide layer. To overcome this problem, it was proposed that the 
charge could be stored in localized quantum dots embedded in an oxide layer [2]. But it 
was found that even such a quantum dot memory device failed to solve the problem of 
Corresponding Author 
©2007IACS 
1316 C B Krishna, R Dhall, S Chopra and S Banerjee 
charge retention satisfactorily [3]. With shrinking device sizes, scaling of these devices 
has also become a problem due to easy tunneling with reduced tunnel barriers. The most 
logical step to overcome charge leakage from the storage node of a memory device is to 
cut out the leakage path which can be done by replacing a conventional floating gate by 
a suspended floating gate, similar to a MEMs structure. Application of silicon 
micromachining technologies to the nonvolatile memory device by using mechanical 
bistability of the object was proposed by Peterson [4] and later developed by Halg [5]. 
They are expected to be compatible to conventional Si-technology and at the same time 
yield high-speed operation [6]. Tsuchiya et al were first to show that Nano Electro-
mechanical systems based memory devices are viable with a layer of Si-nanocrystals as 
suspended floating gate and they are compitable with existing Si-micro-fabrication 
techniques [7]. 
However, with organic/biological nano-systems being continuously investigated for their 
potential MEMs application as sensors or in nanoelectronics [8,9], the need of the day is 
to fabricate a device, which can dwell with Silicon and with other technologies as well 
Based on the idea of a suspended membrane memory device [7], in this letter, we present 
a realistic theoretical modeling of such a structure using MATLAB that is expected to be 
more versatile in its application. This would also throw light on solving the long-standing 
issues of flash or quantum-dot memories. 
The schematic of the device is shown in Figure 1. The device consists of a thin 
membrane capable of storing charges suspended over a conventional Si-MOSFET structure 
and the memory operation is subject to the two positions of the charged membrane and 
it is important that the membrane be adequately charged (explained later in text). The 
likely candidates for the purpose are a poly-silicon layer or a layer of nano-crystalline 
dots [2,10]. There are several reports which claim that nano-crystalline dots embedded in 
a matrix behave as effective charge storage centers and they can be easily charged by 
contact Atomic Force Microscope (AFM) [11,12]. However, there are quite a few issues, 
which should be addressed in the context of nano-crystalline quantum dots as charge 
storage centers. First, charging in quantum dots is self-limiting due to the Coulomb blockade 
effect [13] and they may not store adequate charge for successful operation of the device. 
Second, the charging of nano-crystalline dots is very sensitive to humidity and charge 
VOLTAGE GATE (V0) 
Figure 1. Schematic representation of the memory device. 
Futuristic memory device: a theoretical modeling 1317 
leakage is enhanced in moist condition [12]. And finally, other charging methods such as 
application of large gate voltage as prescribed by Tsuchiya et al [7] are not user friendly 
The alternative would be to use synthetic bilayer membranes such as lipid bilayers as 
they are easily available in various charged states depending on their structure [14] It 
has three-fold advantages : (1) As synthetic bilayers are available in charged state, there 
is no need for additional charging ; (2) The total charge contained in a typical a 100 nm 
x 100 nm x 50 nm synthetic bilayer membrane is much above the threshold value needed 
for switching operation of the device [15] and, (3) The bilayers can be made to bend on 
application of electnc field much more easily than a layer containing nanocrystalline quantum 
dots to achieve bistability [16]. 
The fabrication process involves deposition of an oxide/nitnde layer on top of the substrate 
followed by the deposition of a photo-resist by spinning Etch windows are then defined 
on the resist. A combination of anisotropic and isotropic dry etching in proper sequence 
is then carried out to provide the release of the structure as suggested by Cleland and 
Roukes [17]. Figure 2 depicts the Scanning Electron Micrograph image of a Si02 structure 
in floating condition 
Figure 2. Air gaps are clearly observed in a SiO? structure indicated by arrows 
Instead of conventional memory devices, which differentiate between the two binary 
states based on a threshold voltage shift between the two states, here we use the ratio 
of the ON current (strong inversion) and the OFF current (weak inversion) to separate the 
two states. This is achieved by having the membrane in two stable states (bi-stability, 
shown in Figure 3) wherein state 1 is a rest state (stable equilibrium) whereas the second 
one is that of unstable equilibrium wherein an external electric field has to be applied to 
hold the membrane in this position. The proximity of the membrane to the substrate 
leads to electric field modulation in the substrate resulting in strong and weak inversion 
respectively that can be differentiated by the currents in the two cases. 
The electric field in an MOS structure is given by 
4 = (Sv/Sx) = (j2kT/qLD) F(p\(f, n^jp^) 0) 
where 
1318 C B Krishna, R Dhall, S Chopra and S Banerjee 
STATE 1
 w „ STATE 2 
weak inversion 
strong inversion 
Figure 3. The two stable states of the device (CM = charged Membrane; VG = Gate Voltage, Et = Fermi 
Level; Et = Intrinsic Energy Level). 
W: defined as zero in the bulk and measured with respect to the intrinsic Fermi level, 
npo*Ppo'> equilibrium carrier densities in the bulk, LD : extrinsic Debye length, k 
Boltzmann constant, T: Absolute temperature 
F(Pv,npo/Ppo) = [(e~p* +p¥-^ npo/Ppo el* - j 8 y - l ] 1 ' 2 . (2) 
Considering 
Strong inversion : y/$ = 2\j/F, 
Weak inversion : \j/s = 05yF , 
and 100% ionization (ppo = A/a, npo = nf /A/a) in the bulk. 
Considering the background concentration to be variable, we plot the charge on 
membrane required for strong and weak inversion as a function of the substrate 
concentration (Figure 4). The difference in the electric field leads to the high ratio between 
the ON/OFF current, which can be detected. 
One can also get an estimate about the lon/loff ratio using similar plots of ld (drain 
current) vs. VDS (Source-to-drain voltage) for MOSFET operation [1] in the normal and in 
the sub-threshold regions of operation. As weak inversion lies in the sub-threshold region 
Futuristic memory device: a theoretical modeling 
1319 
of operation an intuitive feel of the order of magnitude of difference between the two 
currents can be obtained. As is Ulustrated by the graphs, this has an o r f e ? Z £ t S 
difference of at .east (normal MOSFET current having values around mA where
 u b 
threshold currents falling in the pA region) [1]. 
*Ki 
mm-
.«*» 
'
 r
~" 1 1—***-—i , , ,,, 
^ * • * strong inversion 
»-• 1 
„ 0 1 2 3 4 5 6 7 8 9 
substrate concenlftfiwi (/cm3) tr~ 
Figure 4. Electric field required for strong and weak inversion with respect to the substrate concentration 
Assuming the membrane is located at a distance, d from the substrate, we can 
estimate the charge required on the membrane by Coulomb's models using a first order 
approximation. Even if the membrane is of nano-dimension, the classical Coulomb model, 
which is purely electrostatic in nature, remains valid for further calculations and has been 
used by several authors [7]. Three such values of d(10nm, 50 nm, 100 nm) are assumed 
and the charges required for bi-stable memory operation has been plotted as a function of 
the substrate concentration (Figure 5). 
Knowing the field required to cause weak inversion, we can again use Coulomb's 
model to get the value of distance d2, where d2 represents the distance between the 
membrane and the substrate for the second bi-stable state. For the same three values of 
dv the plots of the bi-stable distances vs. the substrate concentration are obtained and 
they are shown in Figure 6. 
As is evident from the plots, a high substrate concentration leads to a higher difference 
between the two field values but requires considerably higher charge values. Therefore the 
1320 C B Krishna, R Dhall, S Chopra and S Banerjee 
design of the membrane involves a tradeoff between factors such as electnc field separation 
and charging of the membrane Moreover, as argued before, a thin suspended membrane 
4 6 
«ubfitrart» cons*tfr»tion {/cm3) 
Figure 5 Substrate concentration vs charge on membrane for different d values 
i , 
n 
b 
*» 
A4 
ft£ 
0 
**• 
• ^ " ^ " ^
-
* *
, < T ™ 
-
-
I i 
D 1 
mmmmtmmmmfmtmmtm 
i, 
2 
i 
3 
mmmmmpmmtm 
1 
4 
i 
5 
i 
6 • 
1 
7 
, 
lOOnm I 
-J-
-j 
J 
SOnm 1 
J 
H 
J 
lOfwn J 
l i t ' 
k 4 ii 
•ubltrtft « « t t M M t f » » ( & n 6 ) *«T 
Figure 6. Plot of bi-stable distance 'd2' as a function of substrate concentration 
Futuristic memory device a theoretical modeling 
1321 
with enhanced charge trapping capabilities would be more appropriate for the purpose 
However, the mechanism to get the membrane into the second bi-stable state could be 
through an application of an electric field The calculation of the field is a simple force 
balance equation between the force exerted by the electric field on the membrane and 
gravity This may be used as a suitable starting point for more refined calculations 
As estimate of the frequency of operation of the membrane has been done by calculating 
the resonance frequency of the beams, which are given as [17] 
where 
a) -resonance frequency, 
E - Young's Modulus 
P - density 
w - beam width in the direction of motion and 
L - beam length 
Taking beam dimensions being (100 nm x 100 nm x 50 nm), we arrive at a value greater 
than 25 GHz that is close to the value obtained for a nanometer-scale mechanical resonator 
[17] Though it establishes the suitability of this memory for high frequency applications 
there are several problems still persisting associated with the fabrication of the device and 
integrating it into Si-IC manufacturing processes The fabrication steps involved to arrive at 
the memory structure are intricate and challenging The mechanical and the elastic 
properties of the membrane should also be thoroughly investigated for a successful operation 
the device It is to be noted that the resonance frequency for a charged membrane is 
different from that of an uncharged one and hence suitability of a charged membrane for 
high frequency operations needs to be reassessed 
If we compare the two memory structures, conventional flash memory and the 
theoretically modeled memory that is expected to operate with organic/biological system, 
we find that in case we use the new structure as a random access memory (RAM), it is 
expected to provide much faster writing time as compared to flash memories where both 
0 and 1 need to be written with suitable voltage application between the source and the 
gate Also this process provides a lesser power consumption especially if the data is 
biased towards one bit and we exploit this fact Problems before commercialization involve 
the fabrication of several membranes on a chip, pre-charging them, determining the ideal 
combination of substrate concentration, membrane distance and the electric field for bi-
stability Investigations are already underway to address the above-mentioned issues as a 
suspended floating gate memory device may solve many long-standing problems of a 
flash memory 
1322 C B Krishna, R Dhall, S Chopra and S Banerjee 
Acknowledgment 
One of the authors (CBK) acknowledges the Practice School Division, Birla Institute of 
Technology and Science, Pilani for their constant support. SC thanks CEERI for supporting 
fabrication. 
References 
[1] S M Sze Physics of Semiconductor Devices (2nd Edition) (New York • Wiley) Chapter 7 (1981) 
[2] S Banerjee, S Huang, T Yamanaka and S Oda J Vac Sc Technol B20 1135 (2002) 
[3] S Banerjee, S Huang and S Oda IEEE Transactions, Nanotech 2 88 (2005) 
[4] K E Pterson IBM Tech Disci Bull 20 5309 (1978) 
[5) B Halg IEEE Trans Electron Devices 37 2230 (1990) 
[6] Q Zheng and Q Jiang Phys Rev Lett 88 045503 (2002) 
[7] Y Tsuchiya, K Takai, N Momo, T Nagami, H Mizuta and S Oda J Appl Phys 100 094306 (2006) 
[8] S Kumagai, S Yoshn, K Yamada, N Matsukawa, I Fujiwara, K Iwahori and I Yamashita Appl Phys 
Lett 88 153103 (2006) 
[9] J Fragakis, S Chatzandroulis D Papadmitrtou and C Tsaonis J Physics, Conference Series 10 305 
(2005) 
[10] L Guo, E Leobandang and S Y Chou Appl Phys Lett 70 850 (1997) 
[11] S Banerjee, M A Salem and S Oda Appl Phys Lett 83 3788 (2003) 
[12] T A Boer, L D Bell and H A Atwater Appl Phys Lett 79 791 (2001) 
[13] D K Ferry and S M Goodnick Transport in Nanostructures (Cambridge University Press) (1997) 
[14] J Fuhrhop and W Helfnch Chem Rev 93 1565 (1993) 
[15] V Kumaran Phys Rev Lett 85 4996 (2000) 
[16] T M Fischer Biophys J 63 1328 (1992) 
[17] A N Cleland and M L Roukes Appl Phys Lett 69 2653 (1996) 


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