Interface phonon modes of dual-gate MOSFET system

Solid-State Electronics 94 (2014) 72–81

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Interface phonon modes of dual-gate MOSFET system Nanzhu Zhang a,⇑, Mitra Dutta a,b, Michael A. Stroscio a,b,c a

Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607, United States Departments of Physics, University of Illinois at Chicago, Chicago, IL 60607, United States c Bioengineering, University of Illinois at Chicago, Chicago, IL 60607, United States b

a r t i c l e

i n f o

Article history: Received 25 September 2013 Received in revised form 12 December 2013 Accepted 30 January 2014 Available online 15 March 2014 The review of this paper was arranged by Prof. E. Calleja

a b s t r a c t Herein, analytical expressions are derived for the interface phonon modes of the dual-gate MOSFET system. These analytical results are essential for studies of phonon scattering in MOSFET structures which will affect the performance of the device. We consider selected cubic systems within the framework of macroscopic dielectric continuum model. A principal finding of this paper is that the normally-dominant and unwanted carrier scattering caused by interface phonon interactions can be strongly suppressed through the appropriate placement of the two gates. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: High-k dielectrics Dual-gate MOSFET Interface phonon modes

1. Introduction In recent years, considerable research has been devoted to means of increasing the gate capacitance so as to increase the drive current of a MOSFET (metal–oxide-semiconductor field-effect transistor). As is well known, silicon dioxide (SiO2) is a widely used insulator separating the gate and the semiconductor. However, due to leakage current increases the thickness of SiO2 cannot be decreased below the 1.5–1.0 nm range which is required by device scaling. This drawback of SiO2 has motivated efforts to find new materials to replace it. Recently, a considerable amount of effort has been devoted to high-k dielectrics whose dielectrics constant k is larger than that of SiO2 in order to increase the overall capacitance and then increase the drive current. However, there is still a problem that needs to be solved if SiO2 is replaced with a high-k dielectric. A high-k dielectric is a material with a high dielectric constant k which results from both the ionic and the electronic polarization. For an insulator, a higher dielectric constant can only come from a large ionic polarization because the bandgap of the insulator is too large to increase the electronic polarization. We know that large ionic response dominates at low frequency which result in a large static dielectric constant while large static dielectric constant lead to large scattering strength because it is propor  tional to  hxso e11  e10 . As a result of the large scattering strength

⇑ Corresponding author. Tel.: +1 3123408868. E-mail addresses: [email protected] (N. Zhang), [email protected] (M.A. Stroscio). http://dx.doi.org/10.1016/j.sse.2014.01.007 0038-1101/Ó 2014 Elsevier Ltd. All rights reserved.

associated with low frequencies, the effective electron mobility in the inversion layer of the MOS system is reduced [1]. It is this disadwantage of high-k dielectrics that prompts us to consider using a dual-gate structure to avoid the scattering caused by the phonon modes existing in the system. The principal goal of this paper is to show that all of the interface phonon modes – which are generally dominant in the electron–phonon scattering processes – make considerably reduced contributions when metallic boundaries are selected appropriately. This is of particular importance in nanoscale structures where the interface phonon modes generally dominant in the scattering of carrier from phonon modes. Indeed, in Ref. [2], it is demonstrated that establishing metal–semiconductor interfaces at the heterojunctions of semiconductor quantum wells with the semiconductor–metal boundary conditions dramatically reduces or eliminates unwanted carrier energy loss caused by interactions with interface longitudinal-optical (LO) phonon modes. Moreover, in Ref. [3], study of the Hamiltonian describing the interaction of both confined longitudinal-optical and surface-optical photons with charge carriers, demonstrates that the interaction by the surface-optical phonon modes is very strong and may dominate over other scattering processes, especially with dimensions of about 100 A or less. Ref. [6] provides additional examples of the dominance of carrier—interface-phonon scattering among the phonon scattering mechanisms. Because of the typically dominant role of the interface phonon modes among the phonon scattering processes in nanostructures, and because of the frequent dominance of phonon scattering processes in nanodevices, it is important to

N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

explore ways to reduce scattering associated with carrier-interface phonon interactions. This paper will demonstrate that these interface modes – and the associated unwanted carrier energy loss from carrier—interface-phonon scatter – can be significantly reduced through the appropriate replacement of a single-gate transistor with a double-gate transistor having appropriately placed metallic gates. The paper is organized as follows. In Part 2, the secular equation of the interface optical phonon modes and the normalization condition in the one-gate MOSFET structure are derived. Secondly, an analysis is given of how to get the secular equation of the interface optical phonon modes and electrostatic potentials produced by the phonon (phonon potential) in the dual-gate MOSFET structure. These solutions are applied to special cases of Cu/SiO2/Si/SiO2/Cu, Cu/HfO2/Si/HfO2/Cu, Metal/SiO2/Si/SiO2/Metal and Metal/HfO2/Si/

73

HfO2/Metal; these results and a discussion are given in Part 3. Finally, a summary and conclusion are given in Part 4. 2. Theory and calculation First we consider the one-gate MOSFET case. The screening length of the metal in the structure under study is d and the thickness of oxide is d, which is 10 nm in our calculation (from 0 to d) (as shown in Fig. 1(a)). From the macroscopic dielectric continuum model, we know that the field associated with the optical phonon modes should satisfy the electrostatic equation, which is E = 5U, where E is the electric field and D is the displacement. According to Maxwell’s equation: 5D = 0, then we plot E = 5U in we can get the phonon potential of the system are solutions of Laplace equation:

Fig. 1. Structure of dual-gate MOSFET system.

(a)

(b)

Fig. 2. Phonon energy of Metal/SiO2/Si/SiO2/Metal system. (a) In the range of 58–63 meV; and (b) in the range of 140–150 meV.

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N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

!

e

@2  q2 U ¼ 0 @z2

ð1Þ

where e is the dielectric function of the semiconductor in the structure, and U is the phonon potential. The phonon potentials will be the solution of this equation. For the given structure the phonon potentials can be defined as follow [5]:

8 < U ¼ AeqðzdÞ U ¼ Beqzz þ Ceqz : U ¼ Ded

when z > d when 0 6 z < d when z < 0

when z > d when 0 6 z < d when z < 0

U1 ðZÞ ¼ U2 ðZÞ

e1

ð4Þ

@ U1 @ U2 ¼ e2 @z @z

ð5Þ

Substituting Eq. (4) into Eq. (2):

(

ð2Þ

A ¼ Beqd þ Ceqd

at z ¼ d

BþC ¼D

at z ¼ 0

ð6Þ

Substituting Eq. (5) into Eq. (3):

where A, B, C, D are constants to be determined by the boundary conditions and normalization conditions. From these phonon potentials, it follows that:

8 0 < U ¼ AqeqðzdÞ U0 ¼ qBeqz  qCeqz : 0 D z U ¼ d ed

At the heterointerface of region 1 and region 2, the following two conditions must be satisfied [5]:

ð3Þ

(

eqA ¼ eox ðBqeqd  Cqeqd Þ at z ¼ d

eox ðBq  CqÞ ¼ em Dd

at z ¼ 0

ð7Þ

where em is the dielectric function of metal, eox is the dielectric function of insulator while e is the dielectric function of semiconductor. Solving these equation sets and eliminating the constants we can get the following secular equation of this system:

Fig. 3. Phonon energy of Cu/HfO2/Si/HfO2/Cu. (a) Phonon energy when both of the two gates are non-ideal metal in the range 15–25 meV; (b) phonon energy when both of the two gates are non-ideal metal in the range 50–60 meV; (c) phonon energy when one of the two gates is ideal metal and the other gate is non-ideal in the range 15– 25 meV; and (d) phonon energy when one of the two gates is ideal metal and the other gate is non-ideal in the range 50–60 meV.

N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

 1

e eox

  eqd þ 1 þ

e eox

  eox eqd ¼ dq 1 

em

e eox



 eqd  1 þ

e eox

  eqd ð8Þ

In the secular equation, the dielectric function for the metal is:

em ¼ 1 

x2p;m s2 1  x2 s 2

75

ð9Þ

where xp,m and s are the plasma frequency and the dielectric relaxation time for the metal [5].

Fig. 4. Phonon potential of Metal/SiO2/Si system. (a) Anti-symmetric phonon potentials in Cu/SiO2/Si system with screening length equal to 5  1011 m and wave vector equal to 108 m1; (b) Cu/SiO2/Si system with screening length equal to 5  1011 m and wave vector equal to 3  108 m1; (c) Metal/SiO2/Si system with screening length equal to 5  1012 m and wave vector equal to 1  108 m1; and (d) Metal/SiO2/Si system with screening length equal to 5  1020 m and wave vector equal to 1  108 m1.

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N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

Fig. 4 (continued)

The dielectric function for the inverted substrate is:

e ¼ e1 1 

x

2 p;s

!

1  x2

ð10Þ

where xp,s is the plasma frequency of the 2DEG (2 dimensional electron gas), and e1 is the optical permittivity of the semiconductor. The dielectric function for the insulator is:

eox ¼ e1 ox

ðx2LO2  x2 Þðx2LO1  x2 Þ ðx2TO2  x2 Þðx2TO1  x2 Þ

ð11Þ

where e1 ox is the optical permittivity of the insulator; xLO1 and xLO2 are the longitudinal-optical phonon modes; xTO1 and xTO2 are the angular frequencies of the phonon modes [1]. By using the dielectric functions in the secular equation, the solution of the equation corresponds to the interface phonon modes for one-gate MOSFET system. Solving these equation sets, we can also get the following relationship between these constants:

  8 A e eqd > B ¼ 1  > e 2 ox > > <   C ¼ A2 1 þ eeox eqd > >       > > : D ¼ eox dq A 1  e eqd  A 1 þ e eqd 2 2 em eox eox

2xL2

¼

  ! X 1 1 @ ei ðxÞ Z @ Ui ðq; zÞ2  dz q2 jUi ðq; zÞj2 þ  4p 2x @ x @z  Ri

ð13Þ

where L2 is the area of the structure. By substituting potentials of the three regions into the normalization equation can be written as:

  @ em ðxÞ 2 d 1 @ eox ðxÞ 1þ 2 þ D qðB2 ðe2qd  1Þ @x 2 @x d @ eðxÞ 2 4ph qA ¼ 2 þ C 2 ð1  e2q ÞÞ þ @x L

8 U¼0 > > > > > > U ¼ AeqðztÞ þ BeqðztÞ > > < U ¼ Ceqz þ Deqz > > > > > U ¼ EeqðzþtÞ þ FeqðzþtÞ > > > : U¼0

when z > d when t 6 z < d when  t 6 z < t

ð15Þ

when  d 6 z < t when  z < d

ð12Þ

The completeness and orthonormality conditions of the phonon wavefunctions need to be satisfied to normalize the phonon potential. For cubic material, the normalization condition is given by [6]

h

D. Now combine the relationship with the normalization equation we can get a equation with only one unknown A which can be easily solved. As long as we have the constant A we can normalize the phonon potential. Then we consider the dual-gate MOSFET system. The structure of the dual-gate MOSFET system is shown in Fig 1(b). The thickness of semiconductor is 2t (from t to t) and the two insulator layers are from t to d and d to t. And here in our structure, the thickness of semiconductor is equal to 20 nm and the thickness of the two insulator layers is 2 nm [7]. First of all, we consider the case that the two gates are all ideal metal with zero Thomas–Fermi screening length. Also from the macroscopic dielectric continuum model, we use the same derivation as the one-gate case previously, let the phonon potentials, U, for the given structure be defined as follow [4]:

ð14Þ

There are constants A, B, C, and D in this equation. From the previous result we have the relationship between the constants A, B, C,

where A, B, C, D, E, F are constants to be determined by the boundary conditions and normalization conditions. If the phonon potentials of both gates were zero, all the phonon modes in this structure will be eliminated. As a result, the scattering caused by the typically-important interface phonon modes will be avoided. However, in reality the metal will not be ideal and it is necessary to consider the Thomas–Fermi screening length of the nonideal metal, d. First we suppose both gates have the same screening length. Then the phonon potentials become:

8 ðzdÞ > U ¼ Ae d > > > > > > U ¼ BeqðztÞ þ CeqðztÞ > > < U ¼ Deqz þ Eeqz > > > > > U ¼ FeqðzþtÞ þ GeqðzþtÞ > > > > ðzþdÞ : U ¼ Ae d

when z > d when t 6 z < d when  t 6 z < t

ð16Þ

when  d 6 z < t when  z < d

From these phonon potentials, the following equations are easily derived:

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N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

Fig. 5. Phonon potential of Metal/HfO2/Si system. (a) Cu/HfO2/Si system with screening length equal to 5  1011 m and wave vector equal to 108 m1; (b) Cu/HfO2/Si system with screening length equal to 5  1011 m and wave vector equal to 3  108 m1; (c) Metal/HfO2/Si system with screening length equal to 5  1012 m and wave vector equal to 1  108 m1; and (d) Metal/HfO2/Si system with screening length equal to 5  1020 m and wave vector equal to 1  108 m1; as expected some of these modes cross the zero-potential line.

8 ðzdÞ 0 A > > >U ¼ de d > > > > > U0 ¼ BqeqðztÞ  CqeqðztÞ > > < U0 ¼ Dqeqz  Eqeqz > > > > > U0 ¼ FqeqðzþtÞ  GqeqðzþtÞ > > > > > 0 A ðzþdÞ : U ¼ de d

when z > d when t 6 z < d when  t 6 z < t

ð17Þ

when  d 6 z < t when  z < d

At the heterointerface of region 1 and region 2, we use the same boundary conditions as the previous case (Eqs. (3) and (4)) [5]. Substituting Eq. (4) into Eq. (16):

8 qðdtÞ > þ CeqðdtÞ > > A ¼ Be > < B þ C ¼ Deqt þ Eeqt > Deqt þ Eeqt Þ ¼ F þ G > > > : FqeqðdþtÞ þ GqeqðdþtÞ ¼ A

at z ¼ d at z ¼ t at z ¼ t

ð18Þ

at z ¼ d

Substituting Eq. (5) into Eq. (17):

8 e ð AÞ ¼ eox ðBqeqðdtÞ  CqeqðdtÞ Þ > > > m d > < e ðBq  CqÞ ¼ eðDqeqt  Eqeqt Þ ox > eðDqeqt  Eqeqt Þ ¼ eox ðFq  GqÞ > > > : eox ðFqeqðdþtÞ  GqeqðdþtÞ Þ ¼ em Ad

at z ¼ d at z ¼ t at z ¼ t at z ¼ d

ð19Þ

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N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

Fig. 5 (continued)

where em is the dielectric function of metal, eox is the dielectric function of insulator while e is the dielectric function of semiconductor. Solving these equation sets and eliminating the constants we can get the following secular equation of this system:

  em 1 e ð1þe2qt þ2e2qð2tdÞ e2qð2tþdÞ þ ðe2qd 2e2qð2tdÞ þ1ÞÞ 1 eox dq eox   em 1 2qðdtÞ 2qd 2qt ðe þe e þ2e4qt þ 1þ eox dq þ

e 2qd ðe þ2e2qd 3e2qðd2tÞ e4qt þe2qt ÞÞ8 ¼ 0 eox

ð20Þ

In the secular equation, the dielectric functions for the metal, inverted substrate and insulator have been introduced in the previous case. Next we consider a more complicated case in which the two gates that have different screening lengths d1 and d2, respectively. The phonon potentials become the following:

8 ðzdÞ d > > > U ¼ Ae 1 > > qðztÞ > > U ¼ Be þ CeqðztÞ < U ¼ Deqz þ Eeqz > > > > U ¼ FeqðzþtÞ þ GeqðzþtÞ > > > ðzþdÞ : U ¼ Ae d2

when z > d when t 6 z < d when  t 6 z < t

ð21Þ

when  d 6 z < t when  z < d

Then we can get the following equations:

8 ðzdÞ > U0 ¼  dA1 e d1 > > > > > qðztÞ 0 > >  qCeqðztÞ < U ¼ qBe

when z > d when t 6 z < d

U0 ¼ qDeqz  qEeqz when  t 6 z < t > > qðzþtÞ qðzþtÞ 0 > >  qGe when  d 6 z < t > > U ¼ qFe > > :

ðzþdÞ d2

U0 ¼ dA2 e

ð22Þ

when  z < d

Solving these equations following the same steps, we can get the secular equation for this case:

      em 1 e eox e eox e2qd 2 þ þ eqð4t2dÞ 2  1 þ  eox d1 q eox e eox e     e e e e ox ox  e2qt þ   e2qt 2  eox e eox e       em 1 e eox eox e e2qt þ e2qt   þ 1þ eox d1 q e eox eox e     e eox e eox em 8 qð4t2dÞ 2qd e ¼ 2  2þ þ ð23Þ e eox d2 q eox e eox e

The dielectric functions of the metal, semiconductor and insulator are still the same as the previous case. Also, we need to substitute these functions into this secular equation to get the interface phonon modes of this system.

In the process of applying the boundary conditions for the phonon potentials we can get the relationship between the constants A, B, C, D, E, and F as follows:

8   1 > B ¼ A2 1  eeoxm dq eqðtdÞ > > > >   > > > 1 > C ¼ A2 1 þ eeoxm dq eqðtdÞ > > > >    > > > e 1 1 > ÞeqðtdÞ eqðtdÞ þ ð1 þ eeoxm dq > D ¼ A4 eqt 1  eoxm dq > > >       > > > 1 1 > þ eeox 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >     > > 1 1 > > E ¼ A4 eqt 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >     > > 1 1 > ÞeqðtdÞ  eeox 1  eeoxm dq eqðtdÞ  ð1 þ eeoxm dq > > > >      > > > 1 1 > F ¼ A8 e2qt 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >       > > > 1 1 > þ eeox 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >     > > > 1 1 > þe2qt 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >       > > 1 1 > > eeox 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >      > > 1 1 > < þe2qt eeox 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ      > 1 1 > þ 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >      > > > 1 1 >  e2qt eeox 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >      > > > 1 1 >  1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >      > > 1 1 > > G ¼ A8 e2qt 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >       > > 1 1 > þ eeox 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >     > > > 1 1 > þe2qt 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >       > > > 1 1 >  eeox 1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ > > > >      > > > 1 1 > e2qt eeox 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >   > > 1 1 > > ÞeqðtdÞ  ð1 þ eeoxm dq ÞeqðtdÞ þ ð1  eeoxm dq > > > >      > > 1 1 >  e2qt eeox 1  eeoxm dq eqðtdÞ þ 1 þ eeoxm dq eqðtdÞ > > > >       > > > 1 1 :  1  eeoxm dq eqðtdÞ  1 þ eeoxm dq eqðtdÞ

ð24Þ

In this equation set we have written constants B, C, D, E, and F in terms of A. By doing this we can get the phonon potential in terms of only one constant A which is determined by the normalization condition of this system.

N. Zhang et al. / Solid-State Electronics 94 (2014) 72–81

Here we use the same normalization condition for cubic material as the previous case — that is, Eq. (13). By substituting potentials of the five regions into the normalization equation can be written as:

  @ e1 ðxÞ 2 2 d2 1 @ e2 ðxÞ 2 F 2 þ A q þ q ð1  e2qðtdÞ Þ @x @x 2 2d2 q !  G2 2qðtdÞ @ e3 ðxÞ 2 1 2  1Þ þ 2FG þ q ðD þ E2 Þ þ ðe @x q q

@ e4 ðxÞ 2 B2 2qðdtÞ q ðe  1Þ  e2qt  e2qt Þ þ 2DE þ @x q !   C2 @ e5 ðxÞ 2 d1 1 4ph ¼ 2 þ ð1  e2qðdtÞ Þ þ A2 q þ @x q 2 2d1 L

ð25Þ

There are constants A, B, C, D, E and F in this equation. From the previous result we have the relationships between the constants A, B, C, D, E, and F. Combining the relationship with the normalization equation we can get an equation with only one unknown A which can be easily solved. As long as we have the constant A we can normalize the phonon potential. 3. Results and discussion As demonstrated previously [1], in the SiO2 conduction band, LO-phonon scattering will dominate high-field electron transport if the field is below 3  106 V/cm, and if the field is higher than that number acoustic phonons will dominate in turn [8]. So, here in the field is taken to be below 3  106 V/cm since we want to study regime where LO-phonon effects dominate carrier scattering processes. Eq. (20) determines the interface phonon modes of dual-gate MOSFET system. For the purpose of illustrating the characteristics of how the phonon energies change with the screening length of the metal, we consider a Metal/SiO2/Si/SiO2/Metal system. Fig. 2 shows the phonon energy of this system in the range of 58– 63 meV and 140–150 meV. In this case, we have taken the wave vector to a constant equal to 108 m1 for illustrative purposes. From Fig. 2 we find that the energies of the phonon modes are reduced as the screening length gets smaller. Also, the phonon energy changes dramatically before the screening length goes below 1010 m and it change slightly when the screening length is smaller than 1011 m. The carrier density is 1013 cm2. Fig. 3 shows the comparison between two cases and the structure here is Cu/HfO2/Si/HfO2/Cu. Fig. 3(a) shows the phonon energy when both of the two gates are non-ideal metal in the range 15– 25 meV and Fig. 3(b) shows he phonon energy when both of the two gates are non-ideal metal in the range 50–60 meV. Fig. 3(c) shows the phonon energy when one of the two gates is ideal metal and the other gate is non-ideal in the range 15–25 meV and Fig. 3(d) shows the phonon energy when one of the two gates is ideal metal and the other gate is non-ideal in the range 50– 60 meV. We can see the two higher phonon modes and the lower two phonon modes get close faster in the one-gate ideal case than the both gates non-ideal case. The carrier density is 1013/cm2 and the screening length of copper is 0.5 Å. Fig. 4 shows the potential of Metal/SiO2/Si one-gate MOSFET system with different wave vectors and screening lengths. Now we focus on the potential at the interface of insulator and semiconductor where the electron channel located. From Fig. 4(a) and (b) we can conclude that the potential will decrease if we increase the wave vector. And by comparing Fig. 4(a)–(d), we can see that the phonon potentials are reduced as the screening length gets smaller. Also, the smaller the screening length is, the slighter the phonon potential changes with the change of the screening length.

79

The carrier density is 1013 cm2. This figure highlights the case of anti-symmetric modes that cross the zero-potential vale near the center of the structure. In mobility calculations, it is necessary to construct fully symmetric and fully anti-symmetric phonon modes (given this inversion symmetry within the channel, cosh and sinh functions centered about the channel center). This symmetrization has not been exploited here since out goal has been to demonstrate that unwanted carrier-interface-phonon scattering can be reduced greatly by use of dual-gate structures instead of single-gate structures. Fig. 5 shows the potential of Metal/HfO2/Si one-gate MOSFET system with different wave vectors and screening lengths. From Fig. 5 we can get a similar conclusion with Fig. 4. While HfO2 is a high-k dielectric material, now we compare Figs. 4(a) and 5(a). From these two figures we can see the potential is much smaller in the Metal/HfO2/Si system than in Metal/SiO2/Si system near the metal/dielectric inerface. And at the dielectric/channel interface, the highest potential is quiet similar while the lowest potential and the average potential of all the three modes of the Metal/HfO2/Si system is smaller than Metal/SiO2/Si system. Fig. 6 shows the potential of Metal/SiO2/Si/SiO2/Metal dualgate MOSFET system with different wave vectors and screening lengths. Same as the previous one-gate case we focus on the potential at the interface of insulator and semiconductor where the electron channel located. From Fig. 6(a) and (b) we can conclude that the potential will decrease if we increase the wave vector. And by comparing Fig. 6(a) and (c) whose screening length is larger than Fig. 6(a), we can see that the phonon potential gets higher as the screening length gets larger. The carrier density is 1013 cm2. For all of the figures above (Figs. 4–6), the first screening length we used is 0.5 Å which is the screening length of copper. Then, we smaller the screening length to 1020 m to make it close to zero to model a ideal metal. According to Ref. [9] (metal–semiconductor interface) and Ref. [10] (screening length of 2-D electron gas), we can see that the screening occurs over very short distances (within several Å) in the metal. As the phonon modes in the metal disappear, for the dual-gate MOSFET system, because of its symmetric structure the modes will be killed no matter where they located in the system. Fig. 7 shows the potential of Cu/HfO2/Si/HfO2/Cu dual-gate MOSFET system. Here the wave vector is 108 m1 and the screening length of copper is 0.5 Å. Now we first compare Fig. 7 with Fig. 6(a) we can also see that potential is much smaller if we replace SiO2 with a high-k dielectrics like HfO2. Then we look into the different between the one-gate and dual-gate structures. In Figs. 4(a) and 6(a) we used the same material while one is a onegate system and the other one is a dual-gate system. Since the dual-gate system has more phonon modes here we calculate the average potential at the interface of insulator and semiconductor. In the one-gate system, the average potential is 11.44 meV (result of Fig. 5(a)) and in the dual-gate system the average potential is 5.81 meV (result of Fig. 7) which is about just half of the one-gate system. For the material HfO2 the result is the same. In the onegate system the average potential is 10.18 meV while in the dual-gate system is 2.35 meV. Then ratio of dual-gate potential to one-gate potential is about 23% indicating a significant reduction in unwanted carrier–interface-phonon scattering. Here in our paper we found 3 optical IF phonon modes for the one-gate system and 6 optical IF phonon modes in total. According to Ref. [6], for a system with one barrier there will be 4 IF phonon modes and for a system with 2 barriers there will be 8 IF phonon modes. Our one-gate system corresponding to the 1 barrier case while dual-gate system corresponding to the 2 barrier case. Comparing to the reference, we have 1 mode missing for the first case

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Fig. 6. Phonon potential of Metal/SiO2/Si/SiO2/Metal system. (a) Cu/SiO2/Si/SiO2/Cu system with screening length equal to 5  1011 m and wave vector equal to 108 m1; (b) Cu/SiO2/Si/SiO2/Cu system with screening length equal to 5  1011 m and wave vector equal to 2  108 m1; and (c) Metal/SiO2/Si/SiO2/Metal system with screening length equal to 5  1010 m and wave vector equal to 1  108 m1; as expected some of these modes cross the zero-potential line.

and two modes missing for the latter case. The reason may come from the materials we used here for our calculation. For the references, all the materials are semiconductor while here we used metal as gate material and set the screening length to a small number which may lead to the vanish of modes.

4. Summary and conclusions A considerable number of theoretical studies have been performed on the interface phonon modes heterostructures; as example, see Refs. [11–18]. Also the dual-gate MOSFET system has

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Fig. 7. Phonon potential of Cu/HfO2/Si/HfO2/Cu system with screening length equal to 5  1011 m and wave vector equal to 108 m1.

attracted a lot of interest because of its benefits: (1) reduce the Ioff; (2) eliminates intrinsic parameters; (3) minimizes impurity scattering; (4) higher current drive capability; (5) better control of short channel effects [16]. Here we have examined the use of high-k dielectrics in these dual-gate MOSFETs from the point of view of reducing unwanted carrier–interface-phonon scattering for dual-gate MOSFET system. Indeed, using the dual-gate MOSFET system, it is found that the interface phonon mode potential in the structure can reduced significantly by exploiting the near-vanishing interface-phonon potentials potential at symmetrically placed metallic gates. Indeed, unwanted interface phonon potentials are reduced by about a factor of four for designs considered herein. References [1] Fischetti Massimo V, Neumayer Deborah A, Cartier Eduard A. J Appl Phys 2001;90:4587–608. [2] Reduction of inelastic longitudinal-optical phonon scattering in narrow polarsemiconductor quantum wells. In: Proc SPIE 1675, quantum well and superlattice physics IV, vol. 237. September 3, 1992.

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