Sum rules and universality in electron-modulated

PHYSICAL REVIEW B 79, 235328 共2009兲

Sum rules and universality in electron-modulated acoustic phonon interaction in a free-standing semiconductor plate Shigeyasu Uno* Department of Electrical Engineering and Computer Science, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

Darryl Yong Department of Mathematics, Harvey Mudd College, 301 Platt Boulevard, Claremont, California 91711, USA

Nobuya Mori Department of Electronic Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, Japan 共Received 22 April 2009; published 24 June 2009兲 Analysis of acoustic phonons modulated due to the surfaces of a free-standing semiconductor plate and their deformation-potential interaction with electrons are presented. The form factor for electron-modulated acoustic phonon interaction is formulated and analyzed in detail. The form factor at zero in-plane phonon wave vector satisfies sum rules regardless of electron wave function. The form factor is larger than that calculated using bulk phonons, leading to a higher scattering rate and lower electron mobility. When properly normalized, the form factors lie on a universal curve regardless of plate thickness and material. DOI: 10.1103/PhysRevB.79.235328

PACS number共s兲: 72.10.Di

II. THEORY

I. INTRODUCTION

Acoustic phonons in nanoscale devices differ from bulk phonons due to mechanical mismatch, as reported for freeGaAs/AlAs and GaN/AlN standing plates,1–3 heterostructures,4–8 Si/ SiO2 structures in advanced MOSFETs,9–11 and other structures.12 Incorporating those modulated phonons in electron-transport calculation is becoming important for device characteristic predictions. Among such nanoscale devices, free-standing 共FS兲 plate is the simplest and most fundamental structure. A good understanding of the physics of electron-modulated acoustic phonon interactions in a free-standing plate will help the analysis of other heterostructures. Donetti et al. reported that modulated-acoustic-phonon-limited electron mobility in a FS Si plate is smaller than that calculated using bulk phonons.9 However, despite the simplicity of the structure, mechanisms of such reduction are not yet clear. Deeper analysis is required for better understanding of the phenomena and underlying physics. In this work, insights into modulated acoustic phonons and their deformation-potential interaction with electrons in an ultrathin FS plate are reported. Electron-acoustic phonon interaction is formulated within the framework of Fermi’s golden rule and elastic approximation. Modulated-acousticphonon wave functions are formally encapsulated into the form factor, so that the scattering rate and electron mobility are described in the same manner as the bulk phonon case. The form factor is studied both by analytical and numerical approaches. The analytical study of the form factor exhibits that sum rules hold at least for zero in-plane phonon wave number, and the numerical study reveals a universality in the form factor. The formulations and analytical study of the form factor are described in Sec. II, and the numerical study and discussions are presented in Sec. III. Finally, the conclusion follows in Sec. IV. 1098-0121/2009/79共23兲/235328共7兲

A. Modulated acoustic phonons and electron phonon interaction

Assuming an isotropic and continuous material, modulated acoustic phonons in a free-standing plate are described by the Navier’s equation in three dimensions,

␳

⳵ 2S = 共␭ + 2␮兲 ⵜ 共ⵜ · S兲 − ␮ ⵜ ⫻ 共ⵜ ⫻ S兲, ⳵ t2

共1兲

where S is the displacement vector, ␳ is the mass density, ␭ and ␮ are Lamé constants. Assuming a three-dimensional Cartesian coordinate system with z axis normal to the plate, S can be written as superposition of phonon normal modes S共r,t兲 = 兺 Cqe−i␻qteiQ·Rv共z兲,

共2兲

q

where ␻q is the phonon energy, Cq is a constant, and R = 共x , y兲 and Q = 共qx , qy兲 are position and phonon wave vectors in x-y plane, respectively. A three-dimensional vector v共z兲 represents the z dependence of the normal modes. The traction-free boundary condition and symmetry consideration give an equation for determining allowed phonon modes. Resulting normal modes fall in two categories: shearhorizontal 共SH兲 modes, which have only y component assuming that the phonon wave travels in the x direction, or mixed pressure-shear vertical 共mixed P-SV兲 modes, which have x and z components and zero y component. Only the mixed P-SV modes contribute to the acoustic deformationpotential 共ADP兲 scattering, and they can be written as

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©2009 The American Physical Society

PHYSICAL REVIEW B 79, 235328 共2009兲

UNO, YONG, AND MORI

ω Lz (ρ/(λ+2µ))1/2

25

longitudinal sound velocity. Although the deformationpotential interaction has anisotropy in some semiconductors such as Si, it can be treated as isotropic problem by introducing an effective deformation-potential constant.14 Such approximation has also been used for studies of electronmodulated acoustic phonons in Si in a previous report.9 The quantity In,n⬘ in Eq. 共4兲 is the form factor defined as

20 15

7 6

10

5 4

5

3 2

0 0

In,n⬘共Q兲

Silicon σ = 1 (dilatational)

1

5

10

QLz

15

20

FIG. 1. Modulated phonon frequency ␻ as a function of in-plane phonon wave vector calculated for a free-standing Si plate. The vertical and horizontal axes are normalized so that the plot gives dispersion relations for arbitrary plate thickness, Lz. Numbers 1 through 7 are added to the lowest seven branches for explanation. The meaning of ␴ is given in the text.

vPSV共z兲 =

Cl

冑q2x + q2l +

Ct

冦 冢 冣 冢 冣冧 冦 冢 冣 冢 冣冧 e

冑q2x + q2t

iqlz

qx qx −iqlz 0 0 + ␴e − ql ql − qt

e

iqtz

0

qx

− ␴e

−iqtz

qt 0

LxLyv2l ␳

⬅兺

,

2 ␻⫾q

␴,ql

具n⬘兩 iqxvx共z兲 + iqyvy共z兲 +

where vx共z兲, vy共z兲, and vz共z兲 denote x, y, and z components of v共z兲, respectively, Q is given by Q = K⬘ − K, the bra and ket vectors represent electron wave functions in the confinement 共z-兲 direction, and the summation over ql is taken for all allowed discrete values. Note that Eq. 共5兲 is invariant to the choice of the x and y axes. As long as form-factor calculation is concerned, the x axis can then be reset as parallel to Q without any loss of generality. Using Eq. 共3兲 with qx = Q, Eq. 共5兲 reads In,n⬘共Q兲 = 兺 ␳LxLy兩Cl兩2兩具n⬘兩共eiqlz + ␴e−iqlz兲兩n典兩2

共3兲

␴,ql

⬅ 兺 Kn,n⬘共Q,ql兲. ␴

where ql and qt denote the z components of longitudinal 共pressure兲 and transverse 共shear兲 phonon wave vectors, respectively. The coefficients Cl and Ct are related to phonon amplitude, and ␴ takes either +1 共dilatational兲 or −1 共flexural兲. The origin of the z axis is set at the middle of the plate thickness. The squared coefficients 兩Cl兩2 and 兩Ct兩2 can be written explicitly.3 However, allowed values of ql and qt for a given qx are determined by an implicit transcendental equation,1 so numerical calculation is needed. Allowed phonon modes have either real or pure imaginary ql values. Modes with real ql have longitudinal waves that are sinusoidal in z direction, and those with pure imaginary ql have longitudinal waves that exponentially decay from the plate surfaces. Figure 1 shows the phonon-dispersion relations with ␴ = 1 calculated for a free-standing silicon plate. Values of ␳ = 2330 kg/ m3, ␭ = 93.4 GPa, and ␮ = 51.5 GPa were used.13 The horizontal and vertical axes are normalized so that the dispersion relation is valid for any plate thickness, Lz. The allowed phonon energy for a given in-plane phonon wave vector Q = 兩Q兩 is discrete due to phonon confinement, and therefore constitute the branches. At room temperature, acoustic phonon energy is small compared to the thermal energy, so ADP scattering is approximated as elastic scattering. The transition probability is then given by Fermi’s golden rule as

共6兲

␴,ql

Thus, the coefficient 兩Cl兩2 of the mixed P-SV modes plays an important role in electron-phonon interaction, as expected. For bulk phonons, the form factor reads In,n⬘,bulk =

冕

1 2␲

⬁

兩具n⬘兩eiqlz兩n典兩2dql =

−⬁

冕

Lz/2

−Lz/2

兩␾n⬘共z兲兩2兩␾n共z兲兩2dz, 共7兲

where ␾n共n⬘兲共z兲 denote electron wave functions before 共after兲 scattering. B. Sum rules in the form factor

It is instructive to analyze phonon modes and the form factor at Q = 0. In such case, the lowest branch in Fig. 1 represents the phonon modes with pure imaginary ql, while the other branches give real ql. Real ql phonons have 兩Cl兩2 = 共2␳LxLyLz兲−1 when ql = 关2p − 共␴ + 1兲 / 2兴␲ / Lz with p = 1 , 2 , 3 , . . ., and otherwise 兩Cl兩2 = 0. Therefore, the form factor calculated only for real ql reads

兺

␴=⫾1,ql苸real

␴

Kn,n⬘共Q → 0,ql兲 ⬁

=

共4兲

where 共n , n⬘兲 are electron quantum numbers of confinement before/after the scattering, 共K , K⬘兲 are in-plane electron wave vectors, Dac is the ADP constant, LxLy plate area, kB the Boltzmann constant, TL the lattice temperature, and vl the

冎冏

2 ⳵ vz共z兲 兩n典 , ⳵z

共5兲

qx

2 ␲Dac k BT L T共nK,n⬘K⬘兲 = In,n ␦关En⬘共K⬘兲 − En共K兲兴, បLxLyv2l ␳ ⬘

冏 再

1

兺 兺 兩具n⬘兩共eiz␲共2p−共␴+1兲/2兲/L + ␴e−iz␲共2p−共␴+1兲/2兲/L 兲 ␴=⫾1 p=1 2Lz z

z

⬁

⫻兩n典兩2 =

2 兺 兵兩具n⬘兩cos关z␲共2s + 1兲/Lz兴兩n典兩2 Lz s=0

+ 兩具n⬘兩cos关z␲共2s + 2兲/Lz − ␲/2兴兩n典兩2其,

共8兲

where s = p − 1. From the Appendix, assuming that the wave

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SUM RULES AND UNIVERSALITY IN ELECTRON-…

function has a value only within −Lz / 2 ⬍ z ⬍ Lz / 2, the following equation holds

1

2ρLxLyLz|Cl|2

兩具n⬘兩cos关z␲共2s + a兲/Lz + b兴兩n典兩2 兺 s=0

冋

1 = Lz e2bi 4 +

冕

⬁

−⬁

冕

⬁

−⬁

ⴱ

dz␾n⬘共z兲␾n共z兲␾ⴱn共− z兲␾n⬘共− z兲

册

l

⬁

−⬁

1

共10兲

l

As ql → 0 when Q → 0,

兺

␴=−1

Kn,n⬘ 共Q → 0,ql兲 = lim 关␳LxLy兩Cl共Q,ql共Q兲兲兩2兩具n⬘兩 Q→0

⫻共e

iql共Q兲z

− e−ql共Q兲z兲兩n典兩2兴.

冉

冊

c−1 2 2 L Q Q, 6c2 z

共14兲

where c ⬅ 2 + ␭ / ␮. Using the asymptotic form, 兩Cl共Q , ql兲兩2 can be written for Q Ⰶ 1 / Lz as 兩Cl共Q,ql兲兩2 ⯝

20

共16兲

12 z

l

共17兲 Therefore, from Eqs. 共12兲 and 共17兲, ␴ 共Q = 0,ql兲 兺 Kn,n 兺␴ q 苸imag ⬘ l

=

冉

冊

12 1 ␦n,n⬘ + 2 兩具n⬘兩z兩n典兩2 . 共1 + ␭/␮兲Lz Lz

共18兲

The total form factor at Q = 0 is finally given by the sum of Eqs. 共10兲 and 共18兲, leading to In,n⬘共0兲 = In,n⬘,bulk +

冉

冊

12 1 ␦n,n⬘ + 2 兩具n⬘兩z兩n典兩2 . 共1 + ␭/␮兲Lz Lz 共19兲

Thus, the form factor at Q = 0 is larger than that obtained using bulk phonons due to the modes with longitudinal surface waves. Note that Eqs. 共10兲–共19兲 hold for any electron wave functions.

共13兲

Unlike the case of ␴ = 1, care must be taken for the limit calculation. To further calculate, the following asymptotic expression for the lowest phonon branch with ␴ = −1 is used: ql = i 1 −

15

1

共12兲

where 兩具n⬘ 兩 n典兩2 = ␦n,n⬘ is used. Similarly, the form factor calculated for imaginary ql and ␴ = −1 is written as ql苸imag

10 QLz

␴=−1 2 Kn,n⬘ 共Q → 0,ql兲 = 兺 2 兩具n⬘兩z兩n典兩 . L 共1 + ␭/ ␮ 兲 L z q 苸imag

l

共11兲 1 ␦n,n⬘ , Lz共1 + ␭/␮兲

5

Combining Eqs. 共15兲 and 共16兲, we obtain

l

l

0.4

FIG. 2. Squared longitudinal phonon amplitude as a function of QLz plotted for each dispersion branch on Fig. 1.

␴=1 兩具n⬘兩共eiq z + e−iq z兲兩n典兩2 . Kn,n⬘共Q → 0,ql兲 = 兺 4Lz共1 + ␭/␮兲 q 苸imag

兺 q 苸imag

3

兩具n⬘兩共eiql共Q兲z − e−ql共Q兲z兲兩n典兩2 ⯝ 4␪2兩具n⬘兩z兩n典/Lz兩2 .

兩␾n⬘共z兲兩2兩␾n共z兲兩2dz

On the other hand, there is only one pure imaginary ql allowed at Q = 0, which tends to zero as Q does. For such modes, 兩Cl兩2 = 兵4␳LxLyLz共1 + ␭ / ␮兲其−1 for ␴ = 1, and ⬁ for ␴ = −1. The infinite value results from the fact that the vectors in the first braces in Eq. 共3兲 tends to zero as Q 共and therefore ql兲 does. The form factor calculated for imaginary ql and ␴ = 1 is written as

␴=1

2

0 0

共9兲

= In,n⬘,bulk .

Kn,n⬘共Q → 0,ql兲 =

7

5

0.2

where a = 1 and b = 0 for the summation of the first term in Eq. 共8兲, and a = 2 and b = −␲ / 2 for the second term. Consequently,

冕

0.6

6

4

1

dz兩␾n⬘共z兲兩2兩␾n共z兲兩2 ,

␴ 共Q → 0,ql兲 = 兺 Kn,n ⬘ ␴=⫾1,q 苸real

Silicon σ = 1

0.8

⬁

关共1 − 2c兲␪4 − 12c3共␪2 − 6兲 + c2␪2共␪2 + 12兲兴2 , 4␳LxLyLz共c − 1兲␪2共12c2 + 共1 − c兲␪2兲共6c2 + 共1 − c兲␪2兲2 共15兲

where ␪ ⬅ QLz. The squared matrix element in Eq. 共13兲 is approximated for Q Ⰶ 1 / Lz as

III. NUMERICAL RESULTS AND DISCUSSIONS

Figure 2 shows 兩Cl兩2 values along each branch shown in Fig. 1, as a function of QLz. The vertical axis is normalized by the amplitude of bulk phonons, 共2␳LxLyLz兲−1. The longitudinal phonon amplitude varies with QLz because some of the longitudinal phonon amplitude is converted to/from that of transverse phonons, which is referred to as mode conversion.15 The amount of mode conversion varies with incident angle of the phonon waves at the interfaces. Figure 3 shows an example of the form factor numerically calculated for a special case that electron wave functions are those in an infinite square-well potential. Such simplification is valid for the silicon layer thickness less than about 5 nm and the vertical electric field less than 0.1 MV/cm.16,17 More realistic wave functions can be used, but it is sufficient to use the simplest ones to demonstrate the validity of the equations

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In,n' / In,n',bulk

UNO, YONG, AND MORI 1.6 1.4 1.2 1 0.8 2 0.6 0.4 0.2 0 0

Silicon Infinite square well potential n = n' = 1

Total

3

4

6

5

7

1

5

10 QLz

15

20

FIG. 3. Form factor of intra-lowest-subband scattering calculated for modulated phonons 共denoted as “Total”兲. The vertical axis is the form factor calculated using modulated phonons, divided by that obtained using bulk phonons. Solid lines with indices show contributions from branches numbered in Fig. 1. The figure is valid for any plate thickness due to normalization of the in-plane phonon wave vector Q.

3(1+
) In,n / 2In,n,bulk

In,n / In,n,bulk

shown above. The solid line denoted as “Total” shows the form factor plotted for intra-lowest-subband scattering 共n = n⬘ = 1兲 as a function of QLz. The form factor is normalized by that of bulk phonons, and the horizontal axis is QLz so that the figure is valid for any plate thickness. The lines with indices show contributions from each phonon branch shown in Fig. 1. The curves have similar shapes as 兩Cl兩2 shown in Fig. 2, proving that the longitudinal phonon amplitude plays a major role in electron-phonon interaction. The total form factor approaches that calculated for bulk phonons at large QLz. At small QLz, on the other hand, the normalized formfactor value is larger than unity. From Eq. 共19兲, the form factor increase at Q = 0 divided by bulk phonon form factor is written as ⌬I1,1共0兲 / I1,1,bulk = 2 / 兵3共1 + ␭ / ␮兲其. Note that the angle brackets in Eq. 共19兲 vanishes due to the symmetry of the electron wave functions. For silicon, ⌬I1,1共0兲 / I1,1,bulk gives 0.236. Figure 4共a兲 shows the total form factor plotted for different channel materials. The form factor at Q = 0 will have different values because of the difference in ␭ / ␮. Figure 4共b兲 shows ⌬I1,1 / I1,1,bulk normalized by ⌬I1,1共0兲 / I1,1,bulk, exhibiting a universal curve regardless of the material as well 1.5 1.4 1.3 1.2 1.1 1 1 0.8 0.6 0.4 0.2 0 0

n=1 n' = 1

as thickness. This universality indicates that the sum rules similar to Eqs. 共10兲–共19兲 also exist for nonzero QLz values, although the formulation might be much more complicated. We confirmed the same universality for n ⱖ 2, as well as for intersubband scattering. The universal form factors are shown by open circles in Figs. 5共a兲 and 5共b兲. For intersubband scattering with odd 兩n − n⬘兩, the form factor increase at Q = 0 comes from the second term in the round bracket in Eq. 共19兲. Considering that the bulk phonon form factor for intersubband scattering is given by In,n⬘,bulk = Lz−1, we obtain 3共1 + ␭ / ␮兲⌬In,n⬘共0兲 / 2In,n⬘,bulk = 18兩具n⬘兩z兩n典兩2 / Lz2, which gives 0.584 and 3.74⫻ 10−3 for 共n , n⬘兲 = 共1 , 2兲 and 共1,4兲, respectively. Such universality is beneficial for obtaining analytical expressions for electron-phonon scattering rates and mobility. For intrasubband scattering, the universal curve has been shown to be fitted well by10 f n,n共␰兲 ⬅

共20兲 where ␰ = 0.22QLz, which gives good agreement for 1 ⱕ n ⱕ 5. When the assumption of electric quantum limit is valid, analytical formulae of scattering rate and electron mobility calculated using Eq. 共20兲 and electron wave functions in an infinite square-well potential are given by 2 ⴱ Dac m dk BT L 1 1 = nv ␶n,intra共Exy兲 v2l ␳ប3 Lz

⫻

␮n,intra =

共21兲

2 ⴱ ⴱ nvDac m c m dk BT L

冋再

4n4␲2 tanh共␰0兲 1 1 3 + 4 Lz 2 1 + ␭/␮ ␰0共␰0 + 5n2␲2␰20 + 4n4␲2兲

10 QLz

(b)

15

冎册

−1

,

共22兲

f n,n共␰兲 = 5

冎

4n4␲2 tanh共␰兲 1 3 , + 4 2 1 + ␭/␮ ␰共␰ + 5n2␲2␰2 + 4n4␲2兲

where n = 1, ␰ = 0.22⫻ 2冑mⴱc Exy / ប2Lz, and ␰0 = 0.22 ⫻ 2冑mⴱc kBTe / ប2Lz. The mobility formula demonstrated reasonable agreement with numerical results for a plate thickness at which the assumption of electric quantum limit holds.10 A more compact expression than Eq. 共20兲 is

Si (
 = 1.82) GaAs (
 = 0.895) GaN (
 = 3.99)

Si GaAs GaN

再

ev2l ␳ប3

⫻

(a)

=1

冉 冊

3 4n4␲2 tanh共␰兲 ␭ ⌬In,n共Q兲 , 1+ = 4 2 ␮ In,n,bulk ␰共␰ + 5n2␲2␰2 + 4n4␲2兲

20

FIG. 4. 共a兲 Normalized form factor of intra-lowest-subband scattering, plotted for different channel materials. 共b兲 Form factor increase from that obtained using bulk phonons, normalized by the values at Q = 0, showing a universal curve independent of plate material and thickness.

2n − 1 , 2␰ + 2n − 1 5/2

共23兲

which gives good agreement for 1 ⱕ n ⱕ 3, but less accuracy than Eq. 共20兲 for n ⱖ 4. Similarly, for intersubband scatterings, the following fitting expressions can be used:

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f 1,2共␰兲 = c1

3 , 2␰5/2 + 3

共24兲

PHYSICAL REVIEW B 79, 235328 共2009兲

3(1+
) In,n'/ 2In,n',bulk

SUM RULES AND UNIVERSALITY IN ELECTRON-…

1 0.8 0.6 0.4 0.2 0 0.6

Intra-subband scattering

(3,3) (n,n') = (1,1)

form factor increase is attributed to the surface phonon generation and the mode conversion. It is interesting to note that the form factor does not increase for optical phonon scattering in polar materials.18 For optical phonon scattering, dielectric continuum model gives good approximation, which is equivalent to ignoring the transverse components of phonon vibration. As only the longitudinal phonons constitute the orthonormal set of functions, summed contribution to the form factor is unchanged even with phonon modulation due to material interfaces. On the other hand, for acoustic phonon scattering, a mixture of longitudinal and transverse phonons constitutes a normal mode, while only longitudinal components contribute to the electron-phonon scattering. Summed contribution to the form factor can thus be changed due to phonon modulation in the case of acoustic phonon scattering.

(a)

Numerical Analytical

(2,2)

(b)

Inter-subband scattering

0.4

(n,n') = (1,2)

0.2 0 0

(1,3) (1,4)

5

10 QLz

15

20

FIG. 5. Universal form factor increase for 共a兲 intra- and 共b兲 intersubband scatterings. Open circles represent numerically obtained universal curves for Si, GaAs, and GaN. Solid lines show the analytical compact formulae 关Eqs. 共23兲–共26兲兴.

f 1,3共␰兲 = c2

冉

f 1,4共␰兲 = c3 + c4

冊

5 ␰3 + 3 −1 , 3 ␰ +2 ␰ +5

冉

共25兲

冊

5 ␰3 + −1 , 3 4 ␰ + 2 c 5␰ + 5

共26兲

where c1 = 0.584, c2 = 0.38, c3 = 3.74⫻ 10−3, c4 = 0.048, and c5 = 0.05. Comparison with numerical results is shown in Fig. 5, exhibiting excellent agreement. The increased interaction between electrons and acoustic phonons is due to increase in number of phonon modes having longitudinal phonon vibration. One mechanism is generation of surface modes. Modulated acoustic phonons are either sinusoidal or surface modes, and at near Q = 0, the form factor increase is merely due to the surface modes as demonstrated in the sum rule, as well as in Fig. 3. Such surface phonons are not included in bulk acoustic phonons because there is no interface of material assumed. Another mechanism is mode conversion.15 In three-dimensional bulk phonons, there are two transverse 共shear兲 modes per one longitudinal 共pressure兲 mode due to the degree of freedom. In a free-standing plate, however, such transverse phonons undergo the mode conversion at the interfaces; some of the transverse phonon amplitude is converted to longitudinal phonon amplitude. As a result, virtually all phonon modes can have some longitudinal phonon amplitude, increasing the number of phonon modes contributing to the electronacoustic phonon interaction. Such mode conversion does not occur when the incident angle is zero, which explains the fact that the form factor increase at Q = 0 is due only to the surface mode. For QLz → ⬁, phonon waves are not affected by the surfaces at all, so that the form factor becomes identical to that of bulk phonons. The above discussion is also supported by the following argument. Elastic wave theory states that both the surface phonon generation and mode conversion do not occur when ␮ = 0, no shear stress case. As the form factor increase in Eq. 共19兲 also vanishes when ␮ = 0, the

IV. CONCLUSION

In conclusion, insights into modulated acoustic phonons and their interaction with electrons in an ultrathin freestanding semiconductor plate have been presented. The form factor at large QLz is identical to that calculated for bulk phonons, whereas it is larger at small QLz. The form factor increase is due to increase in number of phonon modes having longitudinal component, which leads to the reduction in the electron mobility. The sum rules of the form factor at Q = 0 have been presented. When properly normalized, the form factors lie on a universal curve regardless of plate thickness and material. ACKNOWLEDGMENTS

This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists 共B兲, Grants No. 18760251 and No. 20760221, as well as the Global COE Program “Electronic Devices Innovation.” The authors would also like to thank Tatematsu Foundation for financial support. APPENDIX

In this appendix, a proof of the following equation is presented: ⬁

兩具n⬘兩cos关z␲共2s + a兲/Lz + b兴兩n典兩2 兺 s=0

冋

1 = Lz e2bi 4 +

冕

⬁

−⬁

冕

⬁

−⬁

ⴱ

dz␾n⬘共z兲␾n共z兲␾ⴱn共− z兲␾n⬘共− z兲

册

dz兩␾n⬘共z兲兩2兩␾n共z兲兩2 ,

共A1兲

where the wave function has a value only within −Lz / 2 ⬍ z ⬍ Lz / 2, and 共a , b兲 = 共1 , 0兲 or 共2 , −␲ / 2兲. By changing the order of the summation and integration in the left-hand side 共LHS兲 of Eq. 共A1兲, we obtain

235328-5

PHYSICAL REVIEW B 79, 235328 共2009兲

UNO, YONG, AND MORI x

⬁

兩具n⬘兩cos关z␲共2s + a兲/Lz + b兴兩n典兩 兺 s=0 =

冕

Lz/2

ⴱ dz␾n⬘共z兲␾n共z兲

−Lz/2

冕

Lz/2

−Lz/2

x

new argument

2

s =1

s =1

dz⬘␾ⴱn共z⬘兲␾n⬘共z⬘兲

2s + 1

⬁

⫻ 兺 cos关z␲共2s + a兲/Lz + b兴cos关z⬘␲共2s + a兲/Lz + b兴.

new argument

old argument

s=0

s=0 0

s = −1

s=0

s = −2

s =1

2s + 1 2s + 2 − ( 2s + 1)

old argument

s =1

s =1

s=0

s=0

s = −1

2s + 2

0

s = −2

s=0

s = −3

s =1

− ( 2s + 2 )

s=0

共A2兲 The summation over s is rewritten as

(a) a = 1

FIG. 6. Schematic illustration of rearranging the summation over the argument s of in Eq. 共A5兲 for 共a兲 a = 1 and 共b兲 a = 2.

⬁

cos关z␲共2s + a兲/Lz + b兴cos关z⬘␲共2s + a兲/Lz + b兴 兺 s=0

⬁

1 兺 cos关共z + z⬘兲␲共2s + a兲/Lz + 2b兴 2 s=0

⬁

=

1 兺 cos关共z + z⬘兲␲共2s + a兲/Lz + 2b兴 2 s=0 ⬁

1 + 兺 cos关共z − z⬘兲␲共2s + a兲/Lz兴. 2 s=0

(b) a = 2

=e

2bi 1

4

共A3兲 =e

The first term of the right-hand side 共RHS兲 of Eq. 共A3兲 is calculated as follows: ⬁

2bi 1

4

= e2bi

1 兺 cos关共z + z⬘兲␲共2s + a兲/Lz + 2b兴 2 s=0

1 4

冋兺 冋 冋兺 ⬁

ei共z+z⬘兲␲共2s+a兲/Lz − ␦a,2

s=−⬁

e

i␲a共z+z⬘兲/Lz ⬁

册

⬁

兺 ei共z+z⬘兲␲2s/L − ␦a,2 z

s=−⬁

册

册

ei␲a共z+z⬘兲/Lz␦关共z + z⬘兲/Lz − s兴 − ␦a,2 ,

s=−⬁

共A6兲

⬁

1 = 兺 关ei兵共z+z⬘兲␲共2s+a兲/Lz+2b其 + e−i兵共z+z⬘兲␲共2s+a兲/Lz+2b其兴 4 s=0

where the following formula is used: ⬁

兺e

⬁

= e2bi

1 兺 关ei共z+z⬘兲␲共2s+a兲/Lz + e−i共z+z⬘兲␲共2s+a兲/Lz兴, 4 s=0

共A4兲

where the relation e−4bi = 1 has been used. Now, with the help of Fig. 6, the following equation holds for a = 1 or 2 and arbitral function f关x兴: ⬁

兺 兵f关共2s + a兲兴 + f关− 共2s + a兲兴其 = s=0

共A7兲

s=−⬁

Similarly, the second term of the RHS of Eq. 共A3兲 becomes ⬁

1 兺 cos关共z − z⬘兲␲共2s + a兲/Lz兴 2 s=0

f关共2s + a兲兴 − ␦a,2 f关0兴.

=

s=−⬁

共A5兲 Therefore, Eq. 共A4兲 leads to

兺 ␦共x − s兲.

=

s=−⬁

⬁

兺

⬁

i2␲sx

1 4

再

⬁

ei␲a共z−z⬘兲/L ␦关共z − z⬘兲/Lz − s兴 − ␦a,2 兺 s=−⬁ z

冎

. 共A8兲

Therefore, Eq. 共A3兲 is rewritten using Eqs. 共A6兲 and 共A8兲 as

⬁

cos关z␲共2s + a兲/Lz + b兴cos关z⬘␲共2s + a兲/Lz + b兴 兺 s=0 1 = 4

冋兺 ⬁

册

兵e2biei␲a共z+z⬘兲/Lz␦关共z + z⬘兲/Lz − s兴 + ei␲a共z−z⬘兲/Lz␦关共z − z⬘兲/Lz − s兴其 − ␦a,2共e2bi + 1兲 .

s=−⬁

共A9兲

The last term in the square bracket vanishes for either case of 共a , b兲 = 共1 , 0兲 or 共2 , −␲ / 2兲. Thus, Eq. 共A2兲 can be written as 235328-6

PHYSICAL REVIEW B 79, 235328 共2009兲

SUM RULES AND UNIVERSALITY IN ELECTRON-… ⬁

兩具n⬘兩cos关z␲共2s + a兲/Lz + b兴兩n典兩2 兺 s=0 =

1 4

冕

⬁

ⴱ

−⬁

冕 冕

dz␾n⬘共z兲␾n共z兲

⬁

冋

1 = Lz 兺 ei␲as e2bi 4 s=−⬁

⬁

−⬁

⬁

−⬁

⬁

dz⬘␾ⴱn共z⬘兲␾n⬘共z⬘兲 兺 兵e2biei␲a共z+z⬘兲/Lz␦关共z + z⬘兲/Lz − s兴 + ei␲a共z−z⬘兲/Lz␦关共z − z⬘兲/Lz − s兴其 s=−⬁

ⴱ

dz␾n⬘共z兲␾n共z兲␾ⴱn共sLz − z兲␾n⬘共sLz − z兲 +

冕

⬁

−⬁

ⴱ

册

dz␾n⬘共z兲␾n共z兲␾ⴱn共z − sLz兲␾n⬘共z − sLz兲 ,

共A10兲

where the integrals are formally extended over 共−⬁ , ⬁兲. As the electron wave functions have a value only in 共−Lz / 2 , Lz / 2兲, the summation over s in the RHS has a value only at s = 0, leading to Eq. 共A1兲.

*[email protected] 1 N.

Bannov, V. Aristov, V. Mitin, and M. A. Stroscio, Phys. Rev. B 51, 9930 共1995兲. 2 N. Bannov, V. Aristov, and V. Mitin, J. Appl. Phys. 78, 5503 共1995兲. 3 N. Bannov, V. Mitin, and M. Stroscio, Phys. Status Solidi B 183, 131 共1994兲. 4 B. A. Glavin, V. I. Pipa, V. V. Mitin, and M. A. Stroscio, Phys. Rev. B 65, 205315 共2002兲. 5 M. Stroscio, J. Appl. Phys. 80, 6864 共1996兲. 6 S. M. Komirenko, K. W. Kim, A. A. Demidenko, V. A. Kochelap, and M. A. Stroscio, Phys. Rev. B 62, 7459 共2000兲. 7 E. P. Pokatilov, D. L. Nika, and A. A. Balandin, Appl. Phys. Lett. 85, 825 共2004兲. 8 E. P. Pokatilov, D. L. Nika, and A. A. Balandin, J. Appl. Phys. 95, 5626 共2004兲. 9 L. Donetti, F. Gámiz, J. B. Roldán, and A. Godoy, J. Appl. Phys.

100, 013701 共2006兲. Uno and N. Mori, Jpn. J. Appl. Phys. 46, L923 共2007兲. 11 S. Uno and N. Mori, Silicon Nanoelectronics Workshop, 4–23, 2007. 12 A. Soukiassian, W. Tian, D. A. Tenne, X. X. Xi, D. G. Schlom, N. D. Lanzillotti-Kimura, A. Bruchhausen, A. Fainstein, H. P. Sun, X. Q. Pan, A. Cross, and A. Cantarero, Appl. Phys. Lett. 90, 042909 共2007兲. 13 M. V. Fischetti, IEEE Trans. Electron Devices 38, 634 共1991兲. 14 M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 共1993兲. 15 Karl F. Graff, Wave Motion in Elastic Solids 共Dover Publications, Mineola, 1991兲, Fig. 6.3 on p. 318 and Fig. 8.7 on p. 447. 16 B. Majkusiak, T. Janik, and J. Walczak, IEEE Trans. Electron Devices 45, 1127 共1998兲. 17 M. Shoji and S. Horiguchi, J. Appl. Phys. 85, 2722 共1999兲. 18 N. Mori and T. Ando, Phys. Rev. B 40, 6175 共1989兲. 10 S.

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