Investigation of dielectric modulated (DM) double gate (DG) junctionless MOSFETs for application as a biosensors

Superlattices and Microstructures 85 (2015) 557–572

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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices

Investigation of dielectric modulated (DM) double gate (DG) junctionless MOSFETs for application as a biosensors Ajay a, Rakhi Narang b, Manoj Saxena c, Mridula Gupta a,⇑ a

Semiconductor Device Research Laboratory, Department of Electronic Science, University of Delhi South Campus, New Delhi 110021, India Department of Electronics, Sri Venkateswara College, University of Delhi, New Delhi 110021, India c Department of Electronics, Deen Dayal Upadhyaya College, University of Delhi, New Delhi 110015, India b

a r t i c l e

i n f o

Article history: Received 14 February 2015 Received in revised form 28 April 2015 Accepted 29 April 2015

Keywords: Analytical Dielectric modulation Drain current Junctionless Modeling MOSFET Nanogap Simulation Threshold voltage TCAD

a b s t r a c t In this paper, an analytical model for Junctionless (JL) Metal–Oxide–Semiconductor Field-Effect Transistor (MOSFET) based biosensor for label free electrical detection of biomolecules like enzyme, cell, DNA etc. using the Dielectric Modulation (DM) technique has been developed. The analytical results are validated with the help of ‘‘Sentaurus’’ device simulation software. For the biomolecule immobilization, nanogap cavity is formed in the JL MOSFET by etching gate oxide layer from both source as well as drain end of the channel. As a result, the surface potential in the channel underneath the nanogap cavity region is affected by the neutral and charged biomolecules that binds to SiO2 adhesion layer in the cavity. The surface potential solution is obtained by solving a 2-D Poisson’s equation assuming parabolic potential profile in the channel. The shift in threshold voltage and drain current of the device has been considered as the sensing metric for detection of biomolecules under dry environment condition. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction In recent time, MOSFET dimensions have continuously been scaled down giving rise to short channel effects (SCEs) which seriously affect the device behavior [1–3]. At nanoscale level, the impact of SCEs on the characteristics of conventional MOSFETs cannot be ignored. In order to reduce this impact, numerous multi-gate devices have been proposed such as Double Gate (DG) MOSFET, Surrounding-Gate/Gate All Around (SG/GAA) MOSFET and Fin-FETs, which can suppress the SCEs and improve gate controllability [4–7]. However, at nanoscale level, realization of an ultrasharp doping profile between an n- or p-type source/drain (S/D) region and a p- or n-type body region still poses a great difficulty in the fabrication of short channel devices. Many novel MOSFET designs have been proposed to overcome this technological problem. Junctionless (JL) MOSFET being one of them which has no abrupt junctions between source/drain (S/D) and body channel region which can be seen as n-n-n-type (n-channel) or p-p-p-type (p-channel) [8–11]. Moreover, Junctionless MOSFET makes manufacturing simpler because it needs neither abrupt doping nor high thermal budget. Therefore, Junctionless MOSFET is one of the most promising alternative architecture for CMOS technology due to its immunity to SCEs and outstanding characteristics [9].

⇑ Corresponding author. E-mail addresses: [email protected] (Ajay), [email protected] (R. Narang), [email protected] (M. Saxena), [email protected] (M. Gupta). http://dx.doi.org/10.1016/j.spmi.2015.04.040 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.

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Colinge et al. [1,9] have already demonstrated the fabrication feasibility of Nanowire Junctionless MOSFET. JL-MOSFETs are highly immune from Short Channel Effects (SCEs) i.e. lower DIBL and deliver improved on-state and transfer characteristics than conventional MOSFETs [11,12]. But in the present work, Junctionless DG MOSFET has been demonstrated for its application as a biosensor for the label free electrical detection of the biomolecules. Moreover dry environment condition has been assumed for detection [13]. Although, there are previous reports in which biosensors are characterized under the aqueous state, but few biosensors are also characterized under dry environment. In this work, dry environment condition is assumed for the detection purpose because of the following advantages of dry environment over the watery environment (a) the dry environment can provide high degree of freedom for various structures that may improve the sensors characteristics [14], (b) the electrical signal of the biosensor does not depend on the Debye length, which is the function of ionic concentration of the sample solution [15], which is the function of ionic concentration of the sample solution, (c) the electrical signal of FET biosensors changes significantly by the various ionic concentrations of the sample [16], and it is not easy to control the ionic concentration accurately of any real human sample, such as blood serum, urine or saliva [17], and (d) the interaction potential between the receptors and analytes that causes the conductance change in the FET sensor is partially screened by the strong ionic strength of the buffer solution. This screening directly depends on the Debye–Huckel length [17]. Therefore, Debye-screening-free sensing is another advantage of the biosensors which are working under the dry environment. Basically, biosensor is an analytical device used to detect biological elements, harnessing the exquisite sensitivity and binding specificity of biology such as proteins, enzymes, antibodies, and nucleotides. In biosensor regime, field effect transistor (FET) based biosensors have been studied extensively owing to their various advantages over other methods including direct transduction, high sensitivity, mass production, miniaturization and compatibility with the standard complementary metal–oxide–semiconductor (CMOS) technology [18,19]. The first FET based biosensor was investigated by Bergveld in 1970 i.e., ion sensitive field-effect transistor (ISFET) [20,21]. The basic structure of an ISFET differs from conventional metal–oxi de–semiconductor field-effect transistor (MOSFET) in that the metal gate is replaced by an ion-sensitive membrane, an electrolyte solution and a reference electrode. The working principle of ISFET is the change in electrical property i.e. conductance, threshold voltage or current, due to presence of charged biomolecules between the gate dielectric and the ionic solution. An ISFET gives good performance with high sensitivity for charged biomolecules [22–26], but it has serious limitations like poor detection capability of neutral biomolecules and incompatibility with the standard CMOS process [27]. In order to overcome these problems, a modified architecture of FET based sensor was demonstrated i.e. dielectric modulated field-effect transistor (DM-FET), where the insulator layer is etched to carve out a vertical nanogap underneath the gate material, DM-FET is capable of detecting the neutral biomolecules and shows great compatibility with standard CMOS process [28–31]. In the present works, the performance of Junctionless MOSFET for the label free electrical detection of biomolecules like enzyme, cell, DNA etc. has been investigated with the help of an analytical model. The impact of neutral and charged biomolecules on the electrical characteristics of both p-type and n-type JL-DM-DG-MOSFET has been analyzed under dry environment situation. The change in the threshold voltage has been used as the sensing metric to detect the sensitivity of the prepared device architecture for biomolecule detection. 2. Device architecture and simulation The device architecture for n-type and p-type Junctionless Dielectric Modulated Double Gate MOSFET based biosensors used in this work is depicted in Fig. 1. Here, L1 and L3 are the length of the nanogap cavity (25 nm), L2 is the length of the gate oxide Al2O3 (50 nm). tbio, tsi, tox1 are the thickness of the nanogap cavity, channel and gate oxide respectively. A SiO2 layer of thickness 1 nm is also considered to account for the growth of native oxide in the nanogap cavity region, whenever silicon substrate is exposed to air ambient [32] which act as the adhesion layer for the immobilization of biomolecules. These nanogap cavity regions serve as sensing sites, in which the target biomolecules are immobilized. The simulations were carried out with help of device simulation tool, ‘‘Sentaurus’’, which is commonly used to characterize the electrical properties of the semiconductor devices [44]. The presence of the neutral biomolecules in the nanogap cavity is simulated by introducing material having dielectric constant (K > 1) corresponding to biomolecules (for e.g. streptavidin = 2.1 [33], protein = 2.50, biotin = 2.63 [34], and APTES = 3.57 [35]) [36] in the nanogap cavities (assuming that the cavities are completely filled with biomolecules). In order to simulate the effect of charged biomolecules, negative or positive interface fixed charge (Nf = ±4  1016 m2) (for e.g. DNA [37]) at the SiO2–Air interface of the device are considered. In simulation constant mobility model, field-dependent mobility (FLDMOB), Shockley–Read–Hall (SRH) recombination model and Boltzmann transport model have been used to simulate the current and the electrostatics of the device. Quantum effects have not been considered in the simulation and model. Before biomolecule immobilization, the underlap region is empty and thus filled with air (dielectric constant K = 1). To simulate the presence of biomolecules in the nanogap cavity region, an oxide layer with height of tbio = 9 nm is defined and its dielectric constant is varied as K = 2, 3, 4, 5, 7, 10. The height/thickness of the layer is so chosen such that the varying height of the biomolecules [33,35]. 3. Working principle of device The nanogap cavity regions formed in the gate oxide in the device serve as sensing sites in which the target biomolecules are immobilized. When the target biomolecules are absent in the nanogap cavity region it means the cavities are filled with

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y

x Source

Drain

Source

(a)

Drain

(b) Gate 1

Gate 1 Source

Source

Drain

Drain Gate 2

(c)

Gate 2

(d)

Fig. 1. (a) Schematic initial structure of Junctionless DM-DG-MOSFET, (b) schematic structure after forming cavity region and filled with biomolecules, (c) schematic simulated structure of p-type Junctionless DM-DG-MOSFET and (d) schematic simulated structure of n-type Junctionless DM-DG-MOSFET. Other parameters are considered such as, tbio = 9 nm, tox1 = 1 nm, tsi = 10 nm. Doping in source, drain and channel is 1  1025 m3.

air, so dielectric constant of the cavity region (i.e., region I is considered to be) is different from region II. Therefore the gate capacitance of the cavity regions is also different from region II and thus the threshold voltage shifts in the positive and negative direction with respect to its initial value (when the cavities were not formed) for p-type and n-type JL-DM-DG-MOSFET respectively. When the target biomolecules (like streptavidin, protein, biotin, enzyme, cell, DNA and APTES) are present and immobilized at the sensing site the dielectric constant changes and the gate capacitance of device also changes. Consequently, electrical characteristics of the device, such as threshold voltage and drain current also changes according to the dielectric constant or charge of the target biomolecules. 4. Two dimensional analytical model To obtain analytical expressions for potential distribution and drain current, the channel is divided into three regions as follows: Region I: 0 6 x 6 t si ; 0 6 y 6 L1 . Region II: 0 6 x 6 t si ; L1 6 y 6 L1 þ L2 . Region III: 0 6 x 6 t si ; L1 þ L2 6 y 6 L1 þ L2 þ L3 . Potential distribution is obtained by solving the Poisson’s equation separately in each region as follows:

@ 2 /i ðx; yÞ @ 2 /i ðx; yÞ qN a þ ¼ @x2 @y2 esi @ 2 /i ðx; yÞ @ 2 /i ðx; yÞ qN þ ¼ d esi @x2 @y2

For p-type JL DM DG MOSFET

For n-type JL DM DG MOSFET

ð1aÞ

ð1bÞ

where i = 1, 2, 3 for region 1, 2, and 3, respectively. /i(x, y) is the 2-D potential distribution in the Silicon channel, Na and Nd are the doping in the silicon channel, q is the electron charge, and esi is the dielectric permittivity of Silicon. Using parabolic approximation [38], /i(x, y) is given by

/i ðx; yÞ ¼ P0i ðyÞ þ P1i ðyÞx þ P2i ðyÞx2

ð2Þ

where P0i(y), P1i(y), and P2i(y) are coefficients and obtained by using following boundary conditions.

/i ð0; yÞ ¼ /fsi ðyÞ

ð3aÞ

/i ðt si ; yÞ ¼ /bsi ðyÞ

ð3bÞ

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/i

  t si ; y ¼ /ci ðyÞ 2

ð3cÞ

 @/i ð0; yÞ C i  ¼ / ðyÞ  V gs þ V fbi @x esi fsi @/i

t

 ;0

si

2

ð3dÞ

¼0

@x

ð3eÞ

 @/i ðt si ; 0Þ Ci  ¼ / ðyÞ  V gs þ V fbi @x esi bsi

ð3fÞ

/fsi ðyÞ is the front gate surface potential and /bsi ðyÞ is the back gate surface potential, /ci ðyÞ is the central potential, Vgs is the gate to source voltage and Vfbi is the flat band voltage which is given by

V fb2 ¼ /m  /si

ð4aÞ

/si ¼ vsi þ Eg =2

ð4bÞ

V fb1 ¼ V fb3 ¼ V fb2 

C gap ¼

ebio

qNf C gap

ð4cÞ

ðEffective capacitance of cavity regionsÞ

tbio

ð4dÞ

Ci is the gate capacitance per unit area of the gate dielectric of JL-DM-DG-MOSFET.

C 1 ¼ C 3 ¼ C eff C eff ¼

ð5aÞ

ebio eox1 ebio tox1 þ eox1 tbio

ð5bÞ

C 2 ¼ C ox C ox ¼

ð5cÞ

eox

ð5dÞ

t ox

where N f is the charge density (unity m2) of biomolecules, eCavity is the permittivity of the biomolecules present in cavity and eox1 is the permittivity of the SiO2 layer. Substituting constants P0i(y), P1i(y) and P2i(y) value in Eq. (2) to obtain surface potential

P0i ðyÞ ¼ /fsi ðyÞ P1i ¼

Ci

esi

P2i ¼ 

ð6aÞ

ð/fsi ðyÞ  V gs þ V fbi Þ Ci 

esi tsi

/fsi ðyÞ  V gs þ V fbi

/i ðx; yÞ ¼ /fsi ðyÞ þ

Ci 

esi

ð6bÞ 

ð6cÞ

  Ci  /fsi ðyÞ  V gs þ V fbi x  / ðyÞ  V gs þ V fbi x2 esi tsi fsi

ð7Þ

Since /ci ðyÞ should be relevant to the punch through current, we obtained the relation between /fsi ðyÞ and /ci ðyÞ from (above equation) by substituting x = tsi/2 as

/fsi ðyÞ ¼

  C i t si / ðyÞ þ ðV  V Þ gs fbi ci 4esi 1 þ C4ietsi 1

ð8Þ

si

/i ðx; yÞ ¼

   C i tsi Ci Ci 2 Ci Ci 2 1 þ  xðV gs  V fbi Þ þ / ðyÞ þ ðV  V Þ x  x x ðV gs  V fbi Þ gs fbi ci 4esi esi esi tsi esi esi tsi 1 þ C4ietsi 1

ð9Þ

si

@ 2 /ci ðyÞ /ci ðyÞ  V gs þ V fbi qNa  ¼ @y2 esi g2i

For p-type JL DM DG MOSFET

ð10aÞ

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@ 2 /ci ðyÞ /ci ðyÞ  V gs þ V fbi qN  ¼ d @y2 esi g2i 1

g2i

¼

For n-type JL DM DG MOSFET

8C i 4esi tsi þ C i t 2si

ð10bÞ

ð11Þ

The general solution of Eq. (10a) or (10b) is given as y

y

g

/ci ðyÞ ¼ Ai egi þ Bi e

ri ¼ g2i ri ¼ g2i

qNd

esi

qN a

esi

þ ri

i

ð12Þ

 ðV gs  V fbi Þ For n-type JL DM DG MOSFET

 ðV gs  V fbi Þ For p-type JL DM DG MOSFET

ð13aÞ

ð13bÞ

Constant Ai and Bi are obtained by using the boundary conditions at the source and drain junctions such as

/ci ð0Þ ¼ V bi

ð14Þ

/ci ðL1 þ L2 þ L3 Þ ¼ V bi þ V ds

ð15Þ

  kt Na V bi ¼ ln q ni

ð16Þ

L1

ðV bi  r1 Þeg1  ðw1  r1 Þ  B1 ¼ ; 2 sinh gL1

A1 ¼ V bi  r1  B1

ð17Þ

1

L2

B2 ¼

ðw1  r2 Þeg2  ðw2  r2 Þ ;  L1  2 sinh gL2 e g2

w1  r2  B2 e e

2

L3

B3 ¼

L

g1

A2 ¼

ðw2  r3 Þeg3  ðV bi þ V ds  r3 Þ ;  ðL1 þL2 Þ  2 sinh gL3 e g3

A3 ¼

2

ð18Þ

L1

g2

w2  r3  B3 e

3

e

ðL1 þL2 Þ g3

ðL þL Þ

 1g 2 3

ð19Þ

Vbi is the built-in potential, w1 and w2 are the intermediate potentials, obtained by maintaining continuity of the potential and lateral electric field at the interface of each region. Continuous surface potential for complete channel length can be obtained from Eq. (2).

/1 ðx; yÞ ¼ P 01 ðyÞ þ P 11 ðyÞx þ P21 ðyÞx2

ð20aÞ

/2 ðx; yÞ ¼ P 02 ðyÞ þ P 12 ðyÞx þ P22 ðyÞx2

ð20bÞ

/3 ðx; yÞ ¼ P 03 ðyÞ þ P 13 ðyÞx þ P23 ðyÞx2

ð20cÞ



/1 ðx; yÞ for 0 6 y 6 L1



/ðx; yÞ ¼ /2 ðx; yÞ for L1 6 y 6 L1 þ L2

/ ðx; yÞ for L1 þ L2 6 y 6 L1 þ L2 þ L3 3

ð21Þ

The threshold voltage is defined as the gate voltage where the minimum surface potential reaches /s, min = 2/f (where /f is the Fermi potential) [39] and expressed as:

pffiffiffiffiffiffiffiffi /s;min ¼ 2 Ai Bi þ ri

V th;i ¼

Z 2i þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 22i  4Z 1i Z 3i 2Z 1i

ð22Þ

ð23Þ

The threshold voltage is chosen from the largest among V th;i . Where Z1i, Z2i and Z3i are summarized in the Appendix-A. Drain current in the subthreshold region is obtained by using the expression of potential (Eq. (20)), and is given by

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Idsub ¼ R L 1 0

R tsi 0

ni exp

  ds q W lkT 1  exp  VkT R R 1 1 / ðx;yÞq dy þ 0L2 R tsi / ðx;yÞq dy þ 0L3 R tsi 1 2 kT

dx

ni exp

0

kT

dx

0

ni exp

1 

/3 ðx;yÞq kT

 dy

ð24Þ

dx

Here, ni is the intrinsic carrier concentration and l is the carrier mobility. The linear and saturation region drain current for the four-gate MOSFET is determined using threshold voltage expression (24) and three transistor modeling approach [40,41]. For the linear region, current in each region is calculated separately. It can be assumed as three transistors having different properties are connected in series. The drain current in region 1, i.e., Idlinear1 can be treated as a drain current of p-type and n-type JL-DM-DG-MOSFETs with channel length Lcavity1 and applied drain to source voltage equal to Vp1. 2

Idlinear1

V p1 C eff W l ¼ ðV gs  V th ÞV p1  L1 2

! ð25Þ

The drain current in region 2, i.e., Idlinear2 can be treated as a drain current of p-type and n-type JL-DM-DG-MOSFETs with channel length Lg and applied drain to source voltage equal to (Vp2–Vp1). 2

Idlinear2

2

ðV p2  V p1 Þ C ox W l ¼ ðV gs  V th ÞðV p2  V p1 Þ  L2 2

! ð26Þ

The drain current in region 3, i.e., Idlinear3 can be treated as a drain current of p-type and n-type JL-DM-DG-MOSFETs with channel length Lcavity2 and applied drain to source voltage equal to (Vds–Vp2). 2

Idlinear3 ¼

2

ðV ds  V p2 Þ C ox W l ðV gs  V th ÞðV ds  V p2 Þ  L3 2

! ð27Þ

To obtain value of Vp1 and Vp2 voltages, all three currents are equated to each other leading to two quadratic equations of the form

V 2p1 R1 þ V p1 R2 þ R3 ¼ 0

ð28aÞ

V 2p2 S1 þ V p2 S2 þ S3 ¼ 0

ð28bÞ

where R1, R2, R3, S1, S2, and S3 are given by

R1 ¼ 

R2 ¼

R3 ¼

  C eff C ox ðV gs  V th Þ þ ðV gs  V th Þ L1 L2 2 C ox V p2 C ox  ðV gs  V th ÞV p2 L2 2 L2

S1 ¼ 

S2 ¼

S3 ¼

  1 C eff C ox þ 2 L1 L2

  1 C ox C eff þ 2 L2 L3

  C ox C eff ðV gs  V th Þ þ ðV gs  V th Þ L2 L3 2 C eff C eff V 2ds C ox V p2 C ox ðV gs  V th ÞV ds   þ ðV gs  V th ÞV p1 L3 L3 2 L2 2 L2

ð29Þ

ð30Þ

ð31Þ

ð32Þ

ð33Þ

ð34Þ

Solution of quadratic equation gives the value of Vp1 and Vp2, which are then substituted back in any of Eqs. (25), (26) or (27) to obtain current of the device. Drain current in the saturation region is calculated by replacing the Vds by saturation drain to source voltage (Vdssat), which is given by

V dssat ¼

ðV gs  V th Þ 1þ

lefld ðV gs V th Þ ðL2 þL1 Þmsat

ð35Þ

where l is the maximum low field mobility and msat is the saturation velocity for electron. Mobility reduction effects at higher electric field lefld are incorporated using [42]

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lefld ¼

leff

Where

V ds l 1 þ d ðL2 þL 1 Þmsat

leff ¼

l

ð36Þ

1  hðV gs  V th Þ

h is a fitting constant whose value used in this paper is h = 0.57 and

 d¼

V ds l ðL2 þ L1 Þmsat

  1 V ds l 1:5 þ ðL2 þ L1 Þmsat

ð37Þ

5. Results and discussion 5.1. Surface potential Fig. 2 shows the surface potential distribution along the channel length for the initial structure (without cavity) and after the formation of the cavity in which neutral biomolecules and charged biomolecules are present. Fig. 2(a) shows the surface potential profile for p-type JL-DM-DG-MOSFET and (b) shows the potential profile for n-type JL-DM-DG-MOSFETs. Deformation of the potential profile appears under the cavity regions, whereas no deformation is observed in the potential profile when the cavities are not formed in the n- and p-type Junctionless DM-DG-MOSFETs. When the biomolecules are not immobilized in the cavity region means that cavities are filled with the air, then the surface

(a)

(b)

p-type JL DM DG MOSFET

0.4

Line : Model 0.2 Symbol: TCAD

1.4

0

Line : Model Symbol: TCAD

Vds = 1 V tsi = 10 nm 1 tox = 10 nm tbio = 9 nm 0.8 Lnano = 25 nm Lch = 100 nm 0.6 1.2

Surface Potential (V)

Surface Potential (V)

n-type JL DM DG MOSFET 1.6

-0.2 -0.4 -0.6 Vds = -1 V -1 tsi = 10 nm tox = 10 nm -1.2 t = 9 nm bio Lnano = 25 nm -1.4 L = 100 nm ch

Without Cavity K=1 K=2 K=3 K=4 K=5 K=7 K = 10

-1.6 -20

40

-0.8

0

20

60

0.4 0.2 0 -0.2

80

100

-0.4 -20

120

0

Position along channel (nm)

Line : Model Symbol: TCAD

0.2

Surafec Potential (V)

(d)

p-type JL DM DG MOSFET 0.4

1.4 1.2

-0.2

1

-0.4 -0.6 -0.8 -1 -1.2 -1.4

Nf = -3e12 16 m -2 Nf = -3x10 Nf = -2x10 -2e12 16 m -2 -1e12 16 m -2 Nf = -1x10 Vds = -1 V Nf = 0 tsi = 10 nm 1e12 16 m -2 Nf = 1x10 tox = 10 nm 2e12 16 m -2 Nf = 2x10 tbio = 9 nm 3e12 16 m -2 Nf = 3x10 Lnano = 25 nm 4e12 16 m -2 Nf = 4x10 Lch = 100 nm

-1.6 -20

0

20

40

m -2

60

40

60

80

100

120

n-type JL DM DG MOSFET 1.6

0

-4e12 16 -4x10

20

Position along channel (nm)

Surafec Potential (V)

(c)

Without Cavity K=1 K=2 K=3 K=4 K=5 K=7 K = 10

Nf Nf == -4x10 -4e1216 m -2 Nf Nf == -3x10 -3e1216 m -2 Nf Nf == -2x10 -2e1216 m -2 Nf Nf == -1x10 -1e1216 m -2 Nf Nf == 00 Nf Nf == 1x10 1e1216 m -2 Nf Nf == 2x10 2e1216 m -2 Nf Nf == 3x10 3e1216 m -2 Nf Nf == 4x10 4e1216 m -2

Vds = 1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lnano = 25 nm Lch = 100 nm

0.8 0.6

Line : Model Symbol: TCAD

0.4 0.2 0 -0.2

80

Position along channel (nm)

100

120

-0.4 -20

0

20

40

60

80

100

120

Position along channel (nm)

Fig. 2. Surface potential along the device length from source to drain. (a) Effect of the presence of neutral biomolecules on p-type DM-DG-MOSFET. (b) Effect of the presence of neutral biomolecules on p-and n-type DM-DG-MOSFETs. (c) Effect of positively charged biomolecules and (d) effect of the negatively charged biomolecules on both p-and n-type DM-DG-MOSFET. For Vgs = 0 V.

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potential in the cavity regions is increased (decreased) by 0.5 V in source side of channel whereas 1.35 V by drain side of the channel for n-type (p-type) JL-DM-DG-MOSFETs. When the neutral biomolecules are present in cavity regions the dielectric constant of the cavity changes from unity to higher value (K > 1) which results is increase (decrease) of the surface potential for p-type (n-type) JL-DM-DG-MOSFET. Fig. 2(c) and (d) shows the surface potential distribution when the charged biomolecules are present in the cavity region. If the cavity regions are filled with the positively (negatively) charged biomolecules, the minimum surface potential increases (decreases) for n-type JL-DM-DG-MOSFET but it decreases (increases) for p-type JL-DM-DG-MOSFET in comparison to the case when the cavity regions are filled by the neutral biomolecules. As can be seen from Fig. 2(c) and (d) at the source side of channel the minimum surface potential decreases (0.60 V) for the negatively charged biomolecules, but slightly increases (0.1 V) for the positively charged biomolecules for n-type JL-DM-DG-MOSFET whereas the minimum surface potential slightly increases (0.1 V) for the negatively charged biomolecules and decreases (0.6 V) for the positively charged biomolecules for p-type JL-DM-DG-MOSFET in comparison to the neutral biomolecules which have the dielectric constant K = 5.

5.2. Threshold voltage The threshold voltage is an important characteristic for FET based biosensor and is used as the sensing parameter to detect the sensitivity after the biomolecules interacts with device. Fig. 3 shows the impact of neutral biomolecules (such as streptavidin, protein, biotin, and APTES) and charged biomolecules (such as DNA) on the threshold voltage for both p-type and n-type JL-DM-DG-MOSFETs. Fig. 3(a) and (b) shows the

(a)

(b)

p-type JL DM DG MOSFET -0.5

Line : Model Symbol: TCAD

Line : Model Symbol: TCAD

Vds = -1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

-0.7

-0.8

-0.9

0.8

V ds = 1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

0.7

0.6

0

2

4

6

8

10

0.5

12

0

2

Dielectric Constant (K)

(c)

(d)

p-type JL DM DG MOSFET -0.5

Line : Model Symbol: TCAD

= nm 9nm tTbio bio = 9 tTbio nm bio = 14 = 14nm

6

8

10

12

n-type JL DM DG MOSFET 1

Line : Model Symbol: TCAD

tbio = =9 9nm nm Tbio tbio = =1414nm nm Tbio

0.9

Threshold Voltage (V)

V ds = -1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

-0.7

-0.8

-0.9

-1

4

Dielectric Constant (K)

-0.6

Threshold Voltage (V)

tTbio 9 nm 9nm bio = = tTbio 1414nm nm bio = =

0.9

Threshold Voltage (V)

Threshold Voltage (V)

-0.6

-1

n-type JL DM DG MOSFET 1

tTbio = nm 9nm bio = 9 tTbio nm bio = 14 = 14nm

0.8

Vds = 1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

0.7

0.6

-6

-4

-2

0

2

Charge of Biomolecules (X10

4 16

m -2 )

6

0.5

-6

-4

-2

0

2

4

6

Charge of Biomolecules (X 10 16 m -2 )

Fig. 3. Variation of threshold voltage for p-type and n-type DM-DG-MOSFETs at different cavity height (tbio), (a) and (b) neutral biomolecules, (c) and (b) charged biomolecules.

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Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

impact of neutral biomolecules for both types of devices for different height of cavity. Since biomolecules have certain dielectric constant, therefore, the dielectric constant of cavity region is changed to investigate the effect of neutral biomolecules on both n and p-type JL-DM-DG-MOSFETs. When the neutral biomolecules are immobilized in the cavity region, the dielectric constant change from unity to higher value (K = 2, 3, 4, 5, 7, 10), the threshold voltage decreases for p-type JL-DM-DG-MOSFET and increases for n-type JL-DM-DG-MOSFET. As the cavity height increases the threshold voltage decreases (increases) for p-type (n-type) JL-DM-DG-MOSFET at particular dielectric constant (for e.g. K = 5) but trend of the threshold voltage remains same. Fig. 3(c) and (d) shows the impact of charged biomolecules on the threshold voltage. When positively charged biomolecules are immobilized in the cavity regions of both p-type and n-type JL-DM-DG-MOSFET, the threshold voltage decreases for p-type device but n-type device shows negligible change in the threshold voltage with increase in the positively charged biomolecules density. When negatively charged biomolecules are immobilized in the cavity regions of both devices, for p-type device there is negligible change in threshold voltage but for the n-type device the change is appreciable. When the height of cavity region (tbio) increases the threshold voltage increases (decreases) for n-type (p-type) JL-DM-DG-MOSFET. At 14 nm cavity height filled with negatively (positively) charged biomolecules in n-type (p-type) JL-DM-DG-MOSFET devices do not show drastic change in the threshold voltage as shown for tbio = 9 nm case. Fig. 4(a) and (b) shows the impact of variation of nanogap cavity length on the threshold voltage for neutral biomolecules. When the length of nanogap cavity increases from 10 nm to 25 nm then the threshold voltage increases (decreases) for p-type (n-type) JL-DM-DG-MOSFET. The change in the threshold voltage is 12 mV when dielectric constant change from K = 5 to K = 10 for 10 nm cavity length whereas the change in the threshold voltage is 61 mV when dielectric constant change from K = 5 to K = 10 for 25 nm nanogap cavity length. Fig. 4(c) and (d) shows the impact of charged biomolecules on the threshold voltage when the nanogap cavity length is increased. When length of cavity region is 10 nm, there is negligible

(a)

(b)

p-type JL DM DG MOSFET

-0.58 Line : Model Symbol: TCAD

Line : Model Symbol: TCAD

V ds = -1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

-0.62

-0.64

-0.66

0.66

Vds = 1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

0.64

0.62

0.6

-0.68

0.58

0

2

4

6

8

10

12

0

2

(d)

p-type JL DM DG MOSFET -0.5

Line : Model Symbol: TCAD

= 10nm LLnano nm nano = 10 LLnano nm = 25nm nano = 25

6

-0.7

V ds = -1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

-0.8

8

10

12

n-type JL DM DG MOSFET 1

Line : Model Symbol: TCAD

Lnano Lnano = =1010nm nm Lnano = =2525nm nm Lnano

0.9

Threshold Voltage (V)

Threshold Voltage (V)

-0.6

-0.9

-1

4

Dielectric Constant (K)

Dielectric Constant (K)

(c)

Lnano = =1010nm nm Lnano Lnano = =2525n nmm Lnano

0.68

Threshold Voltage (V)

Threshold Voltage (V)

-0.6

-0.7

n-type JL DM DG MOSFET 0.7

LLnano 1010n nmm nano = = LLnano 2525n nmm nano = =

V ds = 1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

0.8

0.7

0.6

-6

-4

-2

0

2

Charge of Biomolecules (X10

4 16

m -2 )

6

0.5

-6

-4

-2

0

2

4

6

Charge of Biomolecules (X10 16 m -2 )

Fig. 4. Variation of threshold voltage for p-type and n-type DM-DG-MOSFETs at different nanogap length, (a) and (b) neutral biomolecules, (c) and (d) charged biomolecules.

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Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

change in the threshold voltage for both types of device but at 25 nm, when cavities are filled with positively (negatively) charged biomolecules then the p-type (n-type) JL-DM-DG-MOSFET shows a large change in the threshold voltage. 5.3. Drain current Transistor characteristics such as transfer curve as a function of applied gate voltage are known to be sensitive to changes in the environment around the channel including the presence of neutral and charged biomolecules. The impact of the dielectric constant of neutral biomolecules on transfer curve of both devices is shown in Fig. 5(a) and (b). When the neutral biomolecules are immobilized in the cavity regions, both devices exhibit a sufficient change in the off-current (Ioff) and almost no change in the on-current (Ion). With increase in the dielectric constant of cavity the turn on voltage (threshold voltage) of both devices also increases. The impact of positively/negatively charge of biomolecules on transfer curve of both devices is shown in Fig. 5(c) and (d). When positively charged biomolecules are immobilized in the cavity region then both on-current and off-current decreases for p-type JL-DM-DG-MOSFET whereas both on current and off current increases for n-type JL-DM-DG-MOSFET in comparison to the case when neutral biomolecules interact with the device having dielectric constant K = 5. Whereas due to the presence of negatively charged biomolecules in cavity region, the on and off current decreases for n-type JL-DM-DG-MOSFET while both on- and off-current increases for p-type JL-DM-DG-MOSFET. This is due to the change in surface potential as a result of change in flat band voltage (DVfb) in the cavity region. The flat band voltage in turn depends on the dielectric constant of the biomolecules and the charge of biomolecules (Nf) as:

V fb1 ¼ V fb3 ¼ V fb2 

(a)

qNf C gap

Where DV fb ¼ qN f =C gap

(b)

p-type JL DM DG MOSFET 1.E-04 1.E-05 1.E-06 Line : Model Symbol : TCAD

1.E-08

Drain Current (A)

Drain Current (A)

1.E-07

1.E-09 Without Cavity K=1 K=2 Vds = -1 V K=3 tsi = 10 nm K=4 tox = 10 nm tbio = 9 nm K=5 Lnano = 25 nm K=7 Lch = 100 nm K=10

1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 1.E-15 1.E-16

-1

-0.8

-0.6

-0.4

-0.2

n-type JL DM DG MOSFET 1.E-04

Vds = 1 V Line : Model 1.E-05 tsi = 10 nm Symbol : TCAD t = 10 nm 1.E-06 ox tbio = 9 nm 1.E-07 Lnano = 25 nm Lch = 100 nm 1.E-08 1.E-09

Without Cavity K=1 K=2 K=3 K =4 K=5 K=7 K = 10

1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 1.E-15 1.E-16

0

0

0.2

Gate Voltage (V)

Line : Model Symbol : TCAD

1.E-05

Drain Current (A)

(d)

p-type JL DM DG MOSFET 1.E-04

1.E-06 1.E-07

1.E-08 1.E-09 Nf Nf = = 4x10 4e12 16 m -2 Nf Nf = = 3x10 3e12 16 m -2 16 -2 Nf Nf = = 2x10 2e12 m Nf = 1x10 1e12 16 m -2 Nf = 0 Nf = -1x10 -1e12 16 m -2 Nf = -2x10 -2e12 16 m -2 Nf = -3x10 -3e12 16 m -2 Nf = -4x10 -4e12 16 m -2

1.E-11 1.E-12 1.E-13 1.E-14 1.E-15 1.E-16

-1

-0.8

-0.6

Vds = -1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lnano = 25 nm Lch = 100 nm -0.4

Gate Voltage (V)

0.8

1

-0.2

Line : Model Vds = 1 V tsi = 10 nm Symbol : TCAD tox = 10 nm tbio = 9 nm Lnano = 25 nm Lch = 100 nm

1.E-05

1.E-07

1.E-10

0.6

n-type JL DM DG MOSFET

1.E-04

1.E-06

Drain Current (A)

(c)

0.4

Gate Voltage (V)

1.E-08 1.E-09

Nf == 4x10 4e1216 m -2 Nf Nf == 3x10 3e1216 m -2 Nf Nf == 2x10 2e1216 m -2 Nf Nf == 1x10 1e1216 m -2 Nf Nf == 00 Nf Nf ==-1x10 -1e12 16 m -2 Nf Nf ==-2x10 -2e12 16 m -2 Nf Nf ==-3x10 -3e12 16 m -2 Nf Nf ==-4x10 -4e12 16 m -2 Nf

1.E-10 1.E-11 1.E-12 1.E-13 1.E-14 1.E-15

0

1.E-16

0

0.2

0.4

0.6

0.8

1

Gate Voltage (V)

Fig. 5. Variation of transfer characteristics with gate-to-source voltage (Vgs) for p-type and n-type DM-DG-MOSFETs, (a) and (b) neutral biomolecules, (c) and (d) charged biomolecules.

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Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

5.4. Sensitivity The mathematical formulation which has been considered to check the sensitivity of both devices when the neutral and charged biomolecules are immobilized in the cavity is expressed as:

DVth = Vth (K = 1)  Vth (K > 1) for neutral Biomolecules. DVth = Vth (Neutral Biomolecule)  Vth (Charged Biomolecule) for charged biomolecules. Fig. 6(a) and (b) shows the sensitivity when neutral biomolecules are immobilized in the cavity regions of p-type and n-type JL-DM-DG-MOSFET. As can be seen from Fig. 6(a) and (b), as the dielectric constant of the biomolecules present in the cavity region increases, the sensitivity (DVth) for both p-type and n-type JL-DM-DG-MOSFET linearly increases. Fig. 6(c) and (d) shows the impact of charged biomolecules on the sensitivity of both devices. When positively charged biomolecules (having charge density 4  1016 m2) are immobilized in the cavity regions then the sensitivity factor i.e., DVth changes by 132 mV for p-type JL-DM-DG-MOSFET whereas it is changed by 27 mV for n-type JL-DM-DG-MOSFET in comparison to the case when neutral biomolecules which have the dielectric constant K = 5 are immobilized in the cavity region. When negatively charged biomolecules (having charge density 4  1016 m2) are immobilized in the cavity regions then the sensitivity factor DVth changes by 29 mV for p-type JL-DM-DG-MOSFET whereas it changes by 138 mV for n-type JL-DM-DG-MOSFET in comparison to case when neutral biomolecule which have the dielectric constant K = 5 immobilized in the cavity region. The shift into the DVth is depend on the dielectric constant and charge effect, these two factors can be considered to compete toward an increases or decreases in threshold voltage [43].

Line : Model 0.09 Symbol : TCAD

ΔVth = Vth (K=1)-Vth (K>1) (V)

(b)

p-type JL DM DG MOSFET 0.1

Tbio 9nm tbio = = 9 nm 1414nm nm tbio = = Tbio

ΔVth = Vth (K = 1) - Vth (K > 1) (V)

(a)

Vds = -1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

2

4

6

8

10

12

n-type JL DM DG MOSFET 5E-17

Line : Model Symbol : TCAD

-0.01

= nm 9nm tTbio bio = 9 nm tTbio bio = 14 = 14nm

-0.02 -0.03 -0.04

Vds = 1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

-0.05 -0.06 -0.07 -0.08 -0.09 -0.1

0

2

Dielectric Constant (K)

0.3 0.25

(d)

p-type JL DM DG MOSFET Line : Model 9nm tTbio 9 nm bio = = Symbol : TCAD 1414nm nm tTbio bio = =

0.2 0.15 0.1

ΔVth = Vth (Neutral Biomolecule) - Vth (Charged biomolecules) (V)

ΔVth = Vth (Neutral Biomolecule) - Vth (charge Biomolecules) (V)

(c)

Vds = -1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

0.05 0 -0.05 -0.1 -6

-4

-2

0

2

Charge of Bilomolecules (X10

4

6

8

10

12

Dielectric Constant (K)

4 16

6

m -2 )

n-type JL DM DG MOSFET 0.1 0.05

Line : Model Symbol : TCAD

= nm 9nm tTbio bio = 9 nm tTbio bio = 14 = 14nm

0 -0.05

Vds = 1 V tsi = 10 nm tox = 10 nm Lnano = 25 nm Lch = 100 nm

-0.1 -0.15 -0.2 -0.25 -0.3 -6

-4

-2

0

2

4

6

Charge of Biomolecules (X10 16 m -2 )

Fig. 6. Variation of sensitivity factor DVth for p-type and n-type DM-DG-MOSFETs at different height of biomolecules (a) and (b) neutral biomolecules, (c) and (d) charged biomolecules.

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Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

p-typeJL DM DG MOSFET

(b)

0.1

Line : Model 0.09 Symbol: TCAD

L 1010nm nm Lnano nano = = L 2525nm nm nano = = Lnano

0.08

V ds = -1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

0.07 0.06 0.05

n-typeJL DM DG MOSFET 0

Line : Model Symbol: TCAD

-0.01

Vth = Vth(K=1)-Vth(K>1) (V)

ΔVth = Vth (K=1) -V th (K>1) (V)

(a)

0.04 0.03 0.02 0.01

-0.02 -0.03 LLnano nm = 10nm nano = 10 LLnano nm = 25nm nano = 25

-0.04 -0.05 -0.06

Vds = 1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

-0.07 -0.08 -0.09

0

0

2

4

6

8

10

-0.1

12

0

2

Dielectric Constant (K)

(d)

p-typeJL DM DG MOSFET 0.3

Line : Model Symbol: TCAD

0.25 0.2

Lnano L 1010nm nm nano = = L 2525n nmm nano = = Lnano

ΔVth = Vth (Neutral Biomolecule)Vth (Charged Biomolecules) (V)

ΔVth = Vth (neutral Biomolecule) Vth (Charged Biomolecules) (V)

(c)

V ds = -1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

0.15 0.1 0.05 0 -0.05

-6

-4

-2

0

2

4

6

8

10

12

Dielectric Constant (K)

4

n-typeJL DM DG MOSFET 0.05

= 10nm LLnano nano = 10 nm LLnano 2525nm nm nano = =

0

V ds = 1 V tsi = 10 nm tox = 10 nm tbio = 9 nm Lch = 100 nm

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3

6

Line : Model Symbol: TCAD

-6

-4

-2

0

2

4

6

Charge of Biomolecules (X10 16 m -2 )

Charge of Biomolecules (X10 16 m -2 )

Fig. 7. Variation of sensitivity factor DVth for p-type and n-type DM-DG-MOSFET at different nanogap lengths, (a) and (b) neutral biomolecules, (c) and (d) charged biomolecules.

(a) 1.E+10

n-type JL DM DG MOSFET Line : Model Symbol: TCAD

p-type JL DM DG MOSFET n-type JL DM DG MOSFET

1.E+09

I on /I off

1.E+09

I on /I off

(b) 1.E+10

p-type JL DM DG MOSFET

1.E+08

1.E+08

1.E+07

1.E+07

Line : Model Symbol: TCAD 1.E+06

1.E+06 0

2

4

6

8

Dielectric Constant (K)

10

12

-5

-4

-3

-2

-1

0

1

2

3

4

5

Charge of Biomolecules (X10 16 m -2 )

Fig. 8. Variation of sensitivity factor Ion/Ioff ratio for p-type and n-type DM-DG-MOSFET, (a) neutral biomolecules and (b) changed biomolecules.

Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

569

Fig. 7(a) and (b) shows the impact of dielectric constant of neutral biomolecules on sensitivity factor of both devices at two different cavity lengths (Lnano). As the cavity length increases the sensitivity factor increases for both type of devices. As can be observed from Fig. 7(a) at Lnano = 10 nm the change in sensitivity factor is only 18 mV when the dielectric constant of neutral biomolecules is changed from K = 1 to K = 10 but at Lnano = 25 nm the change in sensitivity factor is 88 mV for both p-type and n-type JL-DM-DG-MOSFET. Fig. 7(c) and (d) shows the impact of charged biomolecules on the sensitivity factor at two different cavity lengths for both types of devices. When positively charged biomolecule (having charge density 4  1016 m2) interacts with the device the sensitivity factor shows a change of 26 mV (9 mV) for p-type (n-type) JL-DM-DG-MOSFET whereas for negatively charged biomolecule (having charge density 4  1016 m2) the sensitivity factor shows a change of 8 mV (29 mV) for p-type (n-type) JL-DM-DG-MOSFET in comparison to the neutral biomolecules having the dielectric constant K = 5 at cavity length 10 nm. But at Lnano = 25 nm length of cavity region, for positively charged biomolecules, sensitivity is 263 mV (31 mV) whereas for negatively charged biomolecule it is 30 mV (270 mV) for p-type (n-type) JL-DM-DG-MOSFET in comparison to case when the neutral biomolecules interacts with devices which have the dielectric constant K = 5. Thus, it can be concluded that both devices shows good sensitivity for the longer and elevated nanogap cavity regions. Ion/Ioff ratio is also considered to check the sensitivity of both devices. Fig. 8(a) shows the impact of dielectric constant of neutral biomolecules on the Ion/Ioff ratio of the devices. As can be observed, with increase in the dielectric constant in cavity region the Ion/Ioff ratio increases by an equal amount for both types of devices. Fig. 8(b) shows the impact of charged biomolecules on the Ion/Ioff ratio of both devices. When positively charged biomolecules are immobilized in the cavity region Ion/Ioff ratio increases for p-type JL-DM-DG-MOSFET but it decreases for n-type JL-DM-DG-MOSFET. When negatively charged biomolecules interact with devices the Ion/Ioff ratio increases for n-type JL-DM-DG-MOSFET but it decreases for p-type JL-DM-DG-MOSFET. Thus, the results reveal that for neutral biomolecules both devices show same trends but for charged biomolecules both devices show opposite trends for Ion/Ioff ratio. Therefore, it can be beneficial to fabricate complementary JL-DM-DG-MOSFET on same chip in an array fashion to achieve improved sensitivity for both types of charged biomolecules. 6. Conclusion In this work, p-type and n-type JL-DM-DG-MOSFET have been proposed for the biosensing application. An analytical Drain current model is developed for both devices and results are verified with Senstaurus Device Simulation tool. In this work, p-type JL-DM-DG-MOSFET shows a threshold voltage change of about 103 mV for positively charged biomolecules in comparison to n-type JL-DM-DG-MOSFET. The threshold voltage of n-type JL-DM-DG-MOSFET changes by an amount of 109 mV for negatively charged biomolecules in comparison to p-type JL-DM-DG-MOSFET. The p-type JL-DM-DG-MOSFET shows good change in sensitivity factor i.e., DVth for positively charged biomolecules and for negatively charged biomolecules n-type JL-DM-DG-MOSFET shows good change in DVth. From the above results it can be concluded that for the negatively (positively) charged biomolecules n-type (p-type) JL-DM-DG-MOSFET shows better sensitivity. Acknowledgments Authors would like to thank Ministry of Science and Technology, Department of Science and Technology, Government of India (SR/S3/EECE/0063/2012) and University of Delhi. Ajay would like to thank University Grants Commission (4009/(NET-JUNE 2013)), Government of India, for providing the necessary financial assistance during the course of this research work. Appendix A

Z 11 ¼ a22  4a28 a26 Z 21 ¼ 2a2 a3  2uf a2  4a28 a27  4a29 a26 Z 31 ¼ 4u2f þ a23  2uf a3  4a29 a27 Z 12 ¼ a210  4a32 a30 Z 22 ¼ 2a10 a11  2uf a10  4a32 a31  4a33 a30 Z 32 ¼ 4u2f þ a211  2uf a11  4a33 a31 Z 13 ¼ a28  4a36 a34

570

Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

Z 23 ¼ 2a8 a9  2uf a8  4a36 a35  4a37 a34 

a37 ¼ ða23  a9 Þe 2

a35 ¼ 6 4

ðL1 þL2 Þ g3



 a35 e

2ðL1 þL2 Þ g3





3

L3

ðL1 þL2 Þ g3

a36 ¼ ða22  a8 Þe 2

a33 ¼ 4



g3

ða23  a9 Þe þ ðV bi þ V ds  a9 Þ7 6ða22  a8 Þe  a8 7 5 a34 ¼ 4  ðL1 þL2 Þ  ðL1 þL2 Þ 5  g  L3 3 2 sinh gL3 e g3 2 sinh g e g3

3

3

L1

2

a25  a11  a31 eg2 5

a32 ¼ 4

L1

L1

3

a24  a10  a30 eg2 5 L1

eg2

eg2

2

a31 ¼ 6 4

2ðL1 þL2 Þ



3

L3

g3

3

2

 a34 e

3

L2

2

3

L2

ða25  a11 Þe  a23 þ a11 7 6ða24  a10 Þe  a22 þ a10 7 5 a30 ¼ 4 5  L1  L1 g  L2 2 sinh g e 2 2 sinh gL2 e g2 g2

g2

2

2

a29 ¼ V bi  a3  a27 a28 ¼ ða2  a26 Þ 2

a27

3 2 3 L1 L1 g1 g1 ðV  a Þe  a þ a  a e  a þ a 3 25 3 2 24 2 bi 5 a26 ¼ 4 5   ¼4 2 sinh gL1 2 sinh gL1 1

a25 ¼ a23 ¼



a23 Q 4 þ a15  a13



a21



a22 ¼

Q 10 

2

a18

a24 ¼

Q2

a21 ¼ a19  a17 

a19

1





a20



a22 Q 4 þ a14  a12 Q2



Q 10

a15 Q 5 Q2

a13 Q 5

þ



Q2



a20 ¼ a18  a16 



Q2

3

3

3

2 L 3 L2 2   L     L   g2 g2 2 2 e 1 L 1 L e 1 L 1 L 2 2 2 2 5 a16 ¼ a10 4 5 eg2  eg2  ¼ a11 4 þ coth coth þ coth coth

g2

g2

1 ¼ a11 4 þ

g2

2

g2

g2

L2

eg2

g2 sinh

  L2

g2

1

g2 sinh

g2

g2

3

2 1 5  a14 ¼ a10 4 þ

g2

L2

g2

3 L1      g1 e L a L 1 3 1 5  ¼ 4ðV bi  a3 Þ 1  coth coth

g1

2

a12

a12 Q 5

3 V þ V 1 bi ds  þ    5 ¼ a9 4 þ g3 g sinh L3 g3 sinh gL33 g3 sinh gL33 3 g3 2 3 L3 1 eg3 V bi þ V ds 1 4  þ    5 ¼ a8  þ L3 L3 g3 g sinh L3 g sinh g sinh 3 3 3 g g g

2

a13

Q2

þ

eg3

3

a15

a14 Q 5

L3

1

2

a17



e ¼ 4a2

L1

g1

g1

g1

g1

g1

3      L1 a2 L1 5 1  coth coth 

g1

g1

g1

a11 ¼ a7 g22 a10 ¼ a6 g22 a9 ¼ a5 g23 a8 ¼ a4 g23 a7 ¼ 

qN a=d

esi

þ

V fb2

g22

a6 ¼ 

1

g22

a5 ¼ 

qNa=d

esi

þ

V fb3

g21

a4 ¼ 

1

g23

g2

L2

eg2

g2 sinh

  L2

g2

g2

g2

1

g2 sinh

3  5 L2

g2

g2

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Ajay et al. / Superlattices and Microstructures 85 (2015) 557–572

a3 ¼ a1 g21 a2 ¼ a0 g21 a1 ¼  2

  a2 L1 1  þ Q 2 ¼ 4 coth

g1

g1

L2

Q5 ¼

eg2

g2 sinh

  L2

g2

g2

1

g1

coth

qNa=d

esi

g2 sinh 

L2

g2

V fb1

g21

3

L2

eg2

þ

 5 Q 4 ¼

 L 2 eg2

L2

g2

Q 10 ¼

a0 ¼  1

g2 sinh

Q 4Q 5  Q7 Q2

1

g21  L2

g2

Q7 ¼

1

g3

L3



eg3

g3 sinh

  L3

g3

1

g2

coth



L2



g2

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