Ultrafast Dynamics of Coherent Phonons in Thin Films and Free-Standing Membranes

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Ultrafast Dynamics

of Coherent Phonons in Thin Films and Free-Standing Membranes

Dissertation submitted for the degree of Doctor of Natural Sciences

Presented by Chuan He

at the

Faculty of Sciences Department of Physics

Date of the oral examination: 08.12.2015 First referee: Prof. Dr. Thomas Dekorsy

Second referee: Prof. Dr. Vitalyi Gusev

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Abstract

Femtosecond pump-probe spectroscopy is an important tool for investigating ultrafast dynamics of coherent phonons in sub-µm-thick thin films and free-standing membranes.

The absorption of the pump pulses generates a strain pulse which consists of high- frequency coherent acoustic phonons propagating into sample. As the strain pulse propagates it changes the local optical constants of the sample, which in turn leads to a reflectivity change. The relative reflectivity change is then recorded in the time domain by a time-delayed probe pulse train. In this dissertation, gigahertz to terahertz coherent phonons in sub-µm-thick skutterudites, zinc-blende (cubic) GaN, and GaAs are investigated by femtosecond pump-probe spectroscopy based on high-speed asynchronous optical sampling.

Firstly, the phononic properties of thin films of CoSb₃, FeSb₃, and partially filled YbxCo4Sb12skutterudites are studied. Skutterudites are considered as interesting material for thermoelectric applications. Filling foreign atoms into the cage-like structure of a CoSb₃ skutterudite is beneficial to its thermoelectric properties by increasing phonon scattering while maintaining the electrical conductivity. We demonstrate the generation and detection of coherent acoustic phonons in thin films of CoSb₃ and partially filled YbxCo4Sb12 skutterudites using femtosecond pump-probe spectroscopy. By using a pulse-echo method, the longitudinal sound velocity of amorphous and polycrystalline CoSb3 thin films is obtained. For partially filled YbxCo4Sb12 thin films, an obvious decrease of the longitudinal sound velocity is observed at high filling fractions. Con- comitantly, the high frequency acoustic phonon modes are strongly damped as the Yb filling fraction increases, which gives direct evidence for an increase of acoustic phonon scattering processes. It is shown that the reduction of the lattice thermal conductivity after Yb filling is mainly achieved by a strong scattering of acoustic phonons. In addition, coherent optical phonons are excited and detected in thin films of Yb_xCo₄Sb₁₂ (x = 0.57 and 0.68) and Fe_yCo1−ySb₃ (y= 0.49 and 1). The observation of the optical phonons are both related to their enlarged lattice constants after filling or substitution.

Secondly, two sub-µm-thick semiconductor films, cubic GaN and GaAs, are studied by femtosecond pump-probe spectroscopy. Their phononic properties are thoroughly investigated by employing a free-standing membrane geometry, where the free-standing membranes are obtained through etching away the substrates. In the membranes, the absorption depth of the pump light, which has a wavelength of ∼800 nm for the GaAs membrane and ∼400 nm for the c-GaN membrane, is much larger than the membrane thickness, thus the strain is generated throughout the membrane. The optoexcited strain leads to a period thickness oscillation at gigahertz frequencies. A theoretical model is

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model the phonon lifetime can be extracted despite of the limitation of the measurement window. Additionally, by using a frequency-doubled light, the excitation of the GaAs membrane generates a strain pulse which travels back and forth in the GaAs membrane. The detected echo signal has a double-peak structure in the frequency domain.

Our analysis indicate that this is basically caused by the double-peak structure of the spectral transformation function of the acousto-optic conversion,K_AO(ω). This function determines the effectiveness of different acoustic frequencies to be transformed into the spectrum of the reflected probe intensity variation in the detection region of the probe light.

Keywords: picosecond ultrasonics; ultrafast dynamics; femtosecond pump-probe spectroscopy; asynchronous optical sampling; coherent phonons; thermoelectric; thin films;

semiconductor; free-standing membrane; skutterudites; cubic GaN; GaAs; CoSb₃; FeSb₃; thickness oscillation.

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Chapter 1 Picosecond Ultrasonics

1.1 Introduction

With the advent of ultrafast lasers, the interactions between ultrashort laser pulses and solids have been intensively investigated in various applications, such as picosecond ultrasonics [1–3], laser cleaning [4], laser ablation, and micro-machining [5, 6]. Of these, picosecond ultrasonics as a non-destructive technique is of interest for studying the ultrafast dynamics of solids in picosecond to nanosecond range. For a better understanding of the imaging of an ultrafast process, firstly, let us consider the image acquisition of a falling apple. There are two ways to capture a series of transient scenes of the falling apple. As illustrated in Fig. 1.1 I-A₀, the falling event begins at t = 0 and ends at t = t0. If a camera takes four frames with an equal spacing ∆t, where t0 = 3∆t, the falling event could be recorded with the same spacing ∆tin a single take, as shown in Fig. 1.1 I-a₀. An equivalent way to capture this event (depicted in Fig. 1.1 II) is to repeat the event four times (A,B,C, andD), assuming the event is repetitive. Instead of capturing the falling event in one take, the imaging process is also taken four times so that for each of the repeating event only one picture is taken. By setting the time delay between successive imaging processes (indicated by a, b, c, and d) to ∆t, the falling event imaged in Fig. 1.1 I is exactly reproduced. The latter method becomes really helpful when it comes to imaging extremely ultrafast phenomena in solids and has been employed in ultrafast pump-probe spectroscopy.

For capturing the ultrafast dynamics in solids, a train of ultrashort pulses, so-called pump, with a definite repetition rate are utilized for triggering the same ultrafast processes repeatedly, where the energy density or fluence of the laser is below the damage threshold of the excited solids. The pulse duration, typically on the picosecond or femtosecond scale, is shorter than the duration of the major relaxation processes, therefore most of the energy is stored in the electronic subsystem during the pulse duration. At

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Chapter 1. Picosecond Ultrasonics

coupling [7]. The temporal carrier and phonon dynamics after photo-excitation are detected using another pulse train, i.e. the probe pulses. Analogous to Fig. 1.1 II, in a single measurement there is an increasing time delay of the probe pulses with respect to the pump pulses, therefore the transient reflectivity [3, 8–10] and transmission [11–13]

change of solids is measured by the time-delayed probe pulses. This is called ultrafast pump-probe spectroscopy. Using this method, the ultrafast dynamics in thin films [9], supperlattices [14, 15], and multilayer films [16] has been resolved in the time domain at a sub-nanosecond time scale.

Figure 1.1: Schematic diagram of imaging of a falling apple in two different ways.

In this thesis, gigahertz to terahertz coherent phonons in sub-µm-thick skutterudites, zinc-blende (cubic) GaN, and GaAs are generated and detected using femtosecond pump- probe spectroscopy. The out-of-plane properties of thin films are characterized by gen- erating an elastic strain pulse traveling normal to the surface. Generally the laser spot size is much larger than the absorption depth of the light, so only the longitudinal waves propagating perpendicular to the surface are considered.

The thesis is structured as follows: following this introductory part, Chapter 1 shortly overviews the fundamentals of continuum mechanics, then the generation and detection processes of coherent phonons in solids are discussed in detail. Chapter 2 introduces a high-speed asynchronous optical sampling technique for femtosecond pump-probe spectroscopy and all the optical setups we used for the pump-probe measurements are listed.

In Chapter 3 the dynamics of coherent phonons in thin films of skutterudites is studied.

This work is performed in collaboration with Marcus Daniel, who provided the skutterudite samples and performed the structural characterizations, in the group of Prof.

Manfred Albrecht¹, Technische Universt¨at Chemnitz. The thin films of skuttuerudites under investigation consists of: thin films of CoSb₃ with various film thicknesses, thin

1Present address: Institute of Physics, University of Augsburg, Germany

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Chapter 1.Picosecond Ultrasonics

films of Yb_xCo₄Sb₁₂with different filling fractionx, and thin films of Fe_yCo1−ySb₃ with different substitution level y. Chapter 4 and 5 present the generation and detection of coherent acoustic phonons (CAPs) in cubic GaN (supported films and free-standing membranes) and free-standing GaAs membranes, respectively. The cubic GaN samples are provided by the group of Prof. Anthony J. Kent, University of Nottingham, UK; the theoretical analysis in Chapter 4 and 5 is supported by Prof. V. Gusev at Universit´e du Maine, France.

1.2 Fundamentals of Continuum Mechanics

In this section, the fundamentals of continuum mechanics and mechanics of elastic waves are reviewed [17–21].

1.2.1 Infinitesimal Deformation and Strain Tensor

When the wavelength λ of an elastic wave is longer than 10⁻⁶ cm or the frequency is below 10¹¹or 10¹²Hz, a solid (or a crystal) can be viewed as a homogeneous continuous medium. In this section the elastic properties of a solid are reviewed [17–19].

Figure 1.2: Infinitesimal deformation of a body: att0the body is stress-free while at t the body is deformed.

Figure 1.2 illustrates a body in the stress-free configuration at time t0 and the deformed state at a later time t. Consider two points at t₀, P(t₀) and Q(t₀), after applying an external disturbance, such as a force, a temperature change, etc., the body is deformed and at time t the two points move to P(t) and Q(t). Here only infinitesimal strain is treated, thus Hooke’s law holds, i.e. the strain in an elastic solid is directly proportional

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to the corresponding stress. As seen in Fig. 1.2, the point P(t₀) in the body at r₁ undergoes a displacementu so that it arrives at the position r2, P(t):

r2 =r1+u(r1, t), (1.1)

where u(r, t) is the displacement field, which is a vector-valued field. The point Q(t₀) atr1+dr1 arrives at r2+dr2, Q(t):

r2+dr2 =r1+dr1+u(r1+dr1, t). (1.2) Substituting Eq.(1.1) in Eq.(1.2), we obtain

dr₂=dr₁+u(r₁+dr₁, t)−u(r₁, t)

=dr₁+∇u·dr₁ = (I+∇u)·dr₁

=Fdr1, (1.3)

where∇uis the second-order tensor known as displacement gradient andF(F=I+∇u) the deformation gradient. F describes the relationship of dr₁(before the deformation) to dr2(after the deformation), i.e. the relationship of the body linePQat t0 and t. The matrix of ∇uand F with respect to rectangular Cartesian coordinates are

∇u=







∂ux

∂x

∂uy

∂x

∂uz

∂x

∂ux

∂y

∂uy

∂y

∂uz

∂y

∂ux

∂z

∂uy

∂z

∂uz

∂z







; F=







1 +^∂u_∂x^x ^∂u_∂x^y ^∂u_∂x^z

∂ux

∂y 1 +^∂u_∂y^y ^∂u_∂y^z

∂ux

∂z

∂uy

∂z 1 +^∂u_∂z^z





 .

Taking the dot product of Eq.(1.3) with itself,

dr₂·dr₂ = (Fdr₁)·Fdr₁ =dr₁F^T(Fdr₁)

=dr₁·(F^TF)·dr₁. (1.4)

⇒ dr²₂ =dr1·C·dr1, (1.5)

where

C=F^TF= (I+∇u)^T(I+∇u)

=I+∇u+ (∇u)^T+ (∇u)^T(∇u)

=I+ 2η^∗. (1.6)

and

η^∗ = 1

2[∇u+ (∇u)^T+ (∇u)^T(∇u)].

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The tensor Cis the right-Cauchy-Green deformation tensor. If C=I, then dr₂² =dr²₁. Therefore,Ccorresponds to a rigid body motion (translation and/or rotation). Eq.(1.6) shows that the tensor η^∗ characterizes the change of lengths in the continuum due to the displacement of the material points. Tensorη^∗ is a finite deformation tensor that is known as the Lagrange strain tensor.

For an infinitesimal deformation, the components of the displacement vector as well as their partial derivatives are very small, so that the absolute value of every component of (∇u)^T(∇u) is a small quantity of higher order than those of the components of (∇u).

In this case C ≈ I+ 2η, where η = ¹₂[∇u+ (∇u)^T] = symmetric part of (∇u). The tensorη is the infinitesimal strain tensor and in Cartesian coordinates:

η_ij = 1 2(∂ui

∂j + ∂uj

∂i ), i, j=x, y, z.

The tensor of strain describes the local strain at every point in an elastic body. Here we define six dimensionless coefficientse_αβ that completely describe the strain,

exx= ∂ux

∂x ; eyy = ∂u_y

∂y ; ezz = ∂uz

∂z ; exy = ∂u_x

∂y +∂u_y

∂x; eyz = ∂u_y

∂z +∂u_z

∂y ; ezx= ∂u_z

∂x +∂u_x

∂z . 1.2.2 Tensor of Stress and Elasticity

1.2.2.1 Stress Tensor

Figure 1.3: Traction vector on an oblique plane (left) and stress tensor cube (right).

Consider a body in equilibrium under external forces. For a small area ∆A inside the body the resultant force is ∆F. As shown on the left side of Fig. 1.3, n is the normal

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vector of ∆A and Tⁿ is the traction vector defined by:

Tⁿ(r,n) = lim

∆A→0

∆F

∆A.

The traction vector Tⁿ depends on both the position r and the normal vector n. It has two components: the normal stress Tⁿ_n, which is perpendicular to ∆A, and the shear stress Tⁿ_S, which is parallel to ∆A. In the Cartesian coordinate system, where

∆F= ∆F_xi+ ∆F_yj+ ∆F_zk,i,j, andkare the unit vectors along the axis, the traction vectorTⁿ also has 3 components and each of them can be further resolved into 3 stress components. Therefore we have 9 components to describe the elasticity property at a point inside the body (∆A → 0). This point property is better viewed using an infinitesimal cube, which is depicted on the right side of Fig. 1.3. The stress components σ_ij is

σ_ij = lim

∆Ai→0

∆F_j

∆Ai

, i, j =x, y, z.

where i indicates the plane on which the force acts and j denotes the direction of the force. σxx,σyy and σzz are called normal stress while the others are shear stress. So the traction vectors along the coordinates are

Tⁿ(r,i) =σxxi+σxyj+σxzk, Tⁿ(r,j) =σyxi+σyyj+σyzk, Tⁿ(r,k) =σzxi+σzyj+σzzk.

The stress tensor σ is given as

σ=







σxx σxy σxz

σyx σyy σyz

σzx σzy σzz





 .

On the left side of Fig. 1.3, the traction vector Tⁿ on an oblique plane with normal vectorn in the Cartesian coordinate system, where n=n_xi+n_yj+n_zk, is

Tⁿ=n_xTⁿ(r,i) +n_yTⁿ(r,j) + +n_zTⁿ(r,k)

=σ_xxn_x+σ_yxn_y+σ_zxn_z)i+σ_xyn_x+σ_yyn_y +σ_zyn_z)j +σ_xzn_x+σ_yzn_y+σ_zzn_z)k.

⇒Tⁿ_i =σ_jin_j.

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This is the traction-stress relation. Now let’s take a look at an elemental cube with a volume dxdydz in equilibrium, the force equilibrium and moment equilibrium require the resultant force and moment on the cube to be zero. The external force is still

∆F= ∆Fxi+ ∆Fyj+ ∆Fzk, therefore we have

∂σji

∂j +Fi = 0, (1.7)

σij =σji.

1.2.2.2 Tensor of Elasticity

For each small piece of the material, we assume Hooke’s law holds, that is, the stress is proportional to the strain. The stress tensor component σij is defined as the ith component of the force across a unit area perpendicular to thej-axis. Hooke’s law states that each stress component σ_ij is linearly related to each of the strain components η_kl. Since σ and η each have 9 components, there are 81 possible coefficients, C_ijkl, that describe the elastic properties of the material. They are constants if the material itself is homogeneous. So

σ_ij =X

k,l

C_ijkl·η_kl, (1.8)

where Cijkl are the elastic stiffness constants or moduli of elasticity, which have the dimensions of [force]/[area]. All these constants form a fourth-rank tensor named the tensor of elasticity. Since the stress tensor is symmetric, the strain tensor is also symmetric, i.e. σij =σji, ηij =ηji, which implies that C_ijkl = C_jikl = C_ijlk. This reduces the number of elastic constants from 81 to 36, and Eq.(1.8) can be written as:

σ_xx = C_xxxxe_xx+ C_xxyye_yy+ C_xxzze_zz+ C_xxyze_yz+ C_xxzxe_zx+ C_xxxye_xy; σyy = Cyyxxexx+ Cyyyyeyy+ Cyyzzezz+ Cyyyzeyz+ Cyyzxezx+ Cyyxyexy; σzz = Czzxxexx+ Czzyyeyy+ Czzzzezz+ Czzyzeyz+ Czzzxezx+ Czzxyexy; σyz = Cyzxxexx+ Cyzyyeyy+ Cyzzzezz+ Cyzyzeyz+ Cyzzxezx+ Cyzxyexy; σzx= Czxxxexx+ Czxyyeyy+ Czxzzezz+ Czxyzeyz+ Czxzxezx+ Czxxyexy; σxy = Cxyxxexx+ Cxyyyeyy+ Cxyzzezz+ Cxyyzeyz+ Cxyzxezx+ Cxyxyexy. Apply Voigt notation (xx= 1; yy= 2; zz= 3; yz= 4; zx= 5; xy = 6.), we have:

σ_xx = C₁₁e_xx+ C₁₂e_yy+ C₁₃e_zz+ C₁₄e_yz+ C₁₅e_zx+ C₁₆e_xy; σ_yy = C₂₁e_xx+ C₂₂e_yy+ C₂₃e_zz+ C₂₄e_yz+ C₂₅e_zx+ C₂₆e_xy; σzz = C31exx+ C32eyy+ C33ezz+ C34eyz+ C35ezx+ C36exy; σyz= C41exx+ C42eyy+ C43ezz+ C44eyz+ C45ezx+ C46exy;

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σ_zx= C₅₁e_xx+ C₅₂e_yy+ C₅₃e_zz+ C₅₄e_yz+ C₅₅e_zx+ C₅₆e_xy; σ_xy = C₆₁e_xx+ C₆₂e_yy+ C₆₃e_zz+ C₆₄e_yz+ C₆₅e_zx+ C₆₆e_xy.

Based on the fact that the tensor of elasticity symmetrically connects the two tensors, we can also get Cijkl = Cklij. This further reduces the number of elastic constants to 21. Omit the same constants, the array of the elastic stiffness constants can be written in the following matrix,

C=







C11 C12 C13 C14 C15 C16

C₂₂ C₂₃ C₂₄ C₂₅ C₂₆ C33 C34 C35 C36

C₄₄ C₄₅ C₄₆ C55 C56

C₆₆







. (1.9)

Since a cubic structure is symmetric about its three four-fold rotation axes, if we rotate the cubic structure about x-axis, x,y and z change tox,z and −y. Thus,

1→1; 2→3; 3→2; 4→ −4; 5→ −6; 6→5.

According to this transformation, the matrix of elastic constants in Eq.(1.9) changes to

C=







C11 C13 C12 C−14 C−16 C−15

C₃₃ C₃₂ C−34 C−36 C₃₅ C22 C−24 C−26 C25

C₄₄ C₄₆ C−45

C₆₆ C−56

C55







(1.10)

Compare Eq.(1.9) and Eq.(1.10), we get

C12= C13; C22= C33; C23= C32; C55= C66. and all the other constants equal zero. So

C=







C₁₁ C₁₂ C₁₂ 0 0 0

C₂₂ C₂₃ 0 0 0

C22 0 0 0

C₄₄ 0 0

C66 0 C₅₅







. (1.11)

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Likewise, we apply this to they- andz- axes,x, the array of the elastic stiffness constants can be written as:

C=







C₁₁ C₁₂ C₁₂ 0 0 0

C12 C11 C12 0 0 0

C₁₂ C₁₂ C₁₁ 0 0 0

0 0 0 C₄₄ 0 0

0 0 0 0 C44 0

0 0 0 0 0 C₄₄





 .

and

σ_xx = C₁₁e_xx+ C₁₂e_yy+ C₁₂e_zz; σ_yy = C₁₂e_xx+ C₁₁e_yy+ C₁₂e_zz; σ_zz = C₁₂e_xx+ C₁₂e_yy+ C₁₁e_zz; σyz = C44eyz;

σzx= C44ezx; σxy = C44exy. There are only 3 independent constants,

C₁₁= σ_xx exx

, longtitudinal compression (Young’s modulus);

C₁₂= σ_xx eyy

, transverse expansion;

C44= σyz

e_yz, shear modulus.

In isotropic elastic solids, the stress-strain relation can be written in terms of two independent elastic constantsλand µ:

σ_ij =λ(η_xx+η_yy+η_zz)δ_ij + 2µη_ij. (1.12) The elastic constant λ is called Lam´e’s constant while µ is referred to as the shear modulus or modulus of rigidity. Here we have C11 = λ+ 2µ, C12 = λ, C44 = µ.

Substituting Eq.(1.12) into Eq.(1.7), we obtain the Navier’s or Lam´e’s constant equation:

µ∇²ui+ (λ+µ)∂

∂i(∂ux

∂x +∂u_y

∂y +∂uz

∂z ) +Fi= 0, or

µ∇²u+ (λ+µ)∇(∇ ·u) +F= 0. (1.13)

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1.2.3 Plane Elastic Waves in Cubic Crystals

1.2.3.1 Plane Elastic Waves

A plane harmonic wave which propagates with a phase velocity ccan be written as:

u=Ade^{ik(n·r−ct)}, (1.14)

where ris the position vector, k the wavenumber and ω=kc the circular frequency,n anddare unit vectors that define the directions of propagation and motion, respectively.

Thusn·rdenotes a plane normal to the unit vectornandudescribes a plane wave that is normal to n and propagates with velocity c. In the previous sections we discussed the elastic properties of a body under any external force in static equilibrium, in the dynamic cases no body force exists, Eq.(1.7) and Eq.(1.13) become:

∂σji

∂j =ρ∂²ui

∂t² , (1.15)

µ∇²u+ (λ+µ)∇(∇ ·u) =ρ∂²u

∂t², (1.16)

whereρ is the density. Substituting Eq.(1.14) into Eq.(1.16), we obtain [21]:

(µ−ρc²)d+ (λ+µ)(n·d)n= 0.

This equation is valid only if either d=±n orn·d= 0 is satisfied. When d=±n we havec=c_l=p

(λ+ 2µ)/ρ, where the motion is parallel to the direction of propagation, this is called a longitudinal wave (also dilatational wave or a P-wave); whenn·d= 0 we have c=ct=p

µ/ρwhere the motion is normal to the direction of propagation, this is a transverse wave (also rotational wave or a S-wave). If we choose the wave travelling in the xz-plane, which contains n, the motion can be either in the xz-plane or normal to the xz-plane. These transverse waves are named SV- or SH- waves, respectively. Also we have the ratio of c_l and c_t:

cl

ct

=κ= s

λ+ 2µ µ .

1.2.3.2 Reflection and Transmission of Elastic Waves

In this section, we will discuss the reflection and transmission of longitudinal plane elastic waves at free surface and the interface of two elastic layers [21]. In the former case there is only incident and reflected waves, while in the latter case there is also refracted or transmitted waves. Here we focus on the two dimensional in-plane motions and both reflected waves and refracted waves have longitudinal and SV wave components.

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Reflection at Free Surface Figure 1.4 depicts the longitudinal wave incident from material a onto the free surface z = 0 in the xz-plane. The incident and the reflected waves are described by:

u⁽ⁿ⁾=Ad⁽ⁿ⁾e^ikⁿ⁽ⁿ⁽ⁿ⁾^·r−cⁿ^t), (1.17) where n = 0, n = 1, and n = 2 represent the incident longitudinal wave (n⁽⁰⁾ = d⁽⁰⁾;c₀ =c_l), the reflected longitudinal wave (n⁽¹⁾ =d⁽¹⁾;c₁ =c_l), and the reflected SV wave (n⁽²⁾·d⁽²⁾ = 0; c2 =ct), respectively. From Fig. 1.4 we can see that

n⁽⁰⁾= sinθ₀i_x+ cosθ₀i_z; n⁽¹⁾ = sinθ₁i_x−cosθ₁i_z; n⁽²⁾= sinθ₂i_x−cosθ₂i_z. Stress components for the waves can be obtained by substituting Eq.(1.17) into Eq.(1.12), and together with Eq.(1.17) atz= 0 we could summarize as follows:

u⁽ⁿ⁾_x =A_nd⁽ⁿ⁾_x eîkⁿ⁽ⁿ⁽ⁿ⁾^x ^x−cⁿ^t), u⁽ⁿ⁾_z =A_nd⁽ⁿ⁾_z eîkⁿ⁽ⁿ⁽ⁿ⁾^x ^x−cⁿ^t), σ_zz⁽ⁿ⁾=ik_n[λd⁽ⁿ⁾_x n⁽ⁿ⁾_x + (λ+ 2µ)d⁽ⁿ⁾_z n⁽ⁿ⁾_z ]A_neîkⁿ⁽ⁿ⁽ⁿ⁾^x ^x−cⁿ^t), σ_zx⁽ⁿ⁾=iknµ[d⁽ⁿ⁾_x n⁽ⁿ⁾_z +d⁽ⁿ⁾_z n⁽ⁿ⁾_x ]Aneîkⁿ⁽ⁿ⁽ⁿ⁾^x ^x−cⁿ^t).

Because of the boundary conditions at the free surface z = 0, i.e. the traction is zero, we have

σzz =σ_zz⁽⁰⁾+σ_zz⁽¹⁾+σ_zz⁽²⁾

=ik0(λ+ 2µcos²θ0)A0e^ik⁰^(sinθ⁰^x−c^l^t)+ik1(λ+ 2µcos²θ1)A1e^ik¹^(sinθ¹^x−c^l^t)

−2ik2µsinθ2cosθ2A2e^ik²^(sin^θ²^x−c^t^t)= 0, σzx=σ_zx⁽⁰⁾+σ_zx⁽¹⁾+σ_zx⁽²⁾

= 2ik0µsinθ0cosθ0A0eîk⁰^(sinθ⁰^x−c^l^t)−2ik1µsinθ1cosθ1A1eîk¹^(sinθ¹^x−c^l^t) +ik₂µ(sin²θ₂−cos²θ₂)A₂eîk²^(sin^θ²^x−c^t^t)= 0.

Since the two equations are valid for all values ofxandt, the three exponential functions in the equations must be the same, from this we get

k₀ =k₁; k₂ k0

= c_l ct

=κ; θ₀ =θ₁; sinθ₀=κsinθ₂. Therefore we obtain the amplitude ratios as follows:

A1

A₀ = sin 2θ0sin 2θ2−κ²cos²2θ2

sin 2θ₀sin 2θ₂+κ²cos²2θ₂, A₂

A₀ = 2κsin 2θ₀cos 2θ₂ sin 2θ₀sin 2θ₂+κ²cos²2θ₂.

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It is easy to getθ₁= 0, A₂/A₀ = 0, andA₁/A₀ =−1 for normal incidenceθ₀ = 0, where the incident longitudinal wave reflects also as longitudinal wave.

n⁽⁰⁾ n⁽¹⁾

n⁽²⁾ θ₁ θ₀ θ₂

x z

a

Figure 1.4: Incident, reflected (longitudinal and SV) waves at a free surface.

Reflection and Transmission at the Interface Figure 1.5 illustrates the reflection and refraction of an incident longitudinal wave at the interface of two elastic layersaand b. Heren= 3 andn= 4 represent the refracted longitudinal wave (n⁽³⁾=d⁽³⁾;c3 =c^b_l) and the refracted SV wave (n⁽⁴⁾·d⁽⁴⁾= 0; c₄=c^b_t.). Additionally,

n⁽³⁾ = sinθ₃i_x+ cosθ₃i_z, n⁽⁴⁾ = sinθ₄i_x+ cosθ₄i_z.

For the perfect contact the displacements and the stresses atz= 0 should be continuous, u⁽⁰⁾_x +u⁽¹⁾_x +u⁽²⁾_x =u⁽³⁾_x +u⁽⁴⁾_x ; u⁽⁰⁾_z +u⁽¹⁾_z +u⁽²⁾_z =u⁽³⁾_z +u⁽⁴⁾_z ,

σ_zx⁽⁰⁾+σ_zx⁽¹⁾+σ_zx⁽²⁾=σ⁽³⁾_zx +σ⁽⁴⁾_zx; σ_zz⁽⁰⁾+σ_zz⁽¹⁾+σ_zz⁽²⁾=σ⁽³⁾_zz +σ⁽⁴⁾_zz.

Again for normal incidenceθ₀ = 0, the SV waves vanish, i.e. A₂=A₄ = 0. In this case the amplitude ratios, A1/A0 and A3/A0, are also known as the reflection coefficient r and the transmission coefficientt,

r= Z_b−Z_a Z_b+Z_a, t= 2Z_b

Z_b+Z_a.

where Z_a=ρc_l and Z_b =ρ^bc^b_l are the acoustic impedance of the layera and b, respectively. ρ and ρ^b are the density of material a and b, respectively. If Za > Zb, a phase change of π occurs upon reflection from the interface. For Z_b = 0,r =−1, t= 0, which corresponds to the free surface. For Z_a =Z_b,r = 0, t= 1, which means that the pulse is completely transmitted.

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n⁽⁰⁾ n⁽¹⁾

n⁽²⁾ θ₁ θ₀ θ₂

x z

_θ _n⁽³⁾

3

a b

θ₄ n⁽⁴⁾

Figure 1.5: Reflection and refraction of an incident longitudinal wave at the interface of two elastic layersaandb.

1.2.3.3 Thickness-Modes in Free-Standing Membranes

In this section, the vibrations of thin films and free-standing membranes are reviewed [22]. The free-standing membrane is bound by two planesz=±z₀. Atz=±z₀ the two free surfaces are traction-free, i.e. σ_zz = σ_zy = σ_zx = 0. Additionally, in our analysis we are only concerned with the z-direction, i.e. uz = uz(z, t) and σzz = 0. Thus the stress-strain relation and the equation of motion are as follows:

σ_zz = C₁₁e_zz = (λ+ 2µ)∂u_z

∂z , (1.18)

C₁₁∂²uz

∂z² =ρ∂²uz

∂t² . (1.19)

For plane harmonic waves with the time-factor e^iωt, the equation of motion becomes

∂²u_z

∂z² +ρω² C11

uz = ∂²u_z

∂z² +β²uz = 0.

The solution to this equation isu_z=Asin(βz)+Bcos(βz), whereAandB are constants.

From the boundary condition at z=±z₀, we have

∂u_z

∂z |z=±z₀ =βAcosβz0±βBsinβz0 = 0.

The solutions are β =mπ/(2z₀) when A = 0, wherem = 2,4,6,· · · and β =mπ/(2z₀) when B = 0, wherem= 1,3,5,· · ·. The displacementuz becomes:

uz =Asin(mπ 2z0

z), m= 1,3,5,· · ·, uz =Bcos(mπ

2z0

z), m= 2,4,6,· · ·.

The displacement profiles of the modesm= 1,2,3,and 4 att= 0 are depicted in Fig. 1.6, where for the odd modes the vibrations are symmetric with respect to the middle plane

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z = 0, while the even modes are asymmetric with respect to the middle plane z = 0.

The frequency of the modes is f = mv/2(2z0) = mv/2d, where d is the thickness of the membrane. These modes are referred to as thickness modes, longitudinal confined acoustic phonon modes, or dilatational modes.

Figure 1.6: The displacement profile of the thickness modes in a free-standing membrane.

1.3 Coherent Phonons in Solids

In an unperturbed crystalline solid material, atoms are always in harmonic motion with a random phase. Because of the chemical bonds between them, the atoms are connect with each other, and if one atom is displaced from its equilibrium position, a force will be exerted on their neighbours which causes the collective atomic motions known as phonon that propagates through the solid. An ultrashort light pulse on a femtosecond scale incident onto a crystal can initiate such collective atomic motions, i.e. coherent phonons, that is, all the atoms are in phase with the each other, and these coherent phonons can interfere with each other in a wave-mechanics sense. There are two types of coherent phonons: optical and acoustic. Coherent optical phonons are standing-waves of in-phase atomic oscillations, where adjacent atoms oscillate against each other in the unit cell, while coherent acoustic phonons are ballistic wave packets of compressive/tensile stress.

The propagation of elastic waves in a medium is normally described by a dispersion relationω=ω(k), whereω andkare the frequency and wave vector of the propagating wave, respectively. In the Debye approximation, also known as continuum or long- wavelength approximation, the dispersion relation is simply ω = vk, where v is the sound velocity in the medium.

1.3.1 Opto-Acoustic Generation of Coherent Phonons

Ultrashort laser pulses can excite and detect high-frequency coherent phonons in the GHz to THz frequency range. For an ultrashort pulse the spectrum of the photogenerated

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acoustic strain pulse, ˜η(ω), is related to the spectrum of the pulse ˜f(ω) byK_OA(ω) [23]:

˜

η(ω) =K_OA(ω) ˜f(ω)∼K_OA(ω)

whereK_OA(ω) is the spectral transformation function of opto-acoustic (OA) conversion, which determines the effectiveness of different frequencies ω in ˜f(ω) to be transformed into acoustic frequencies. In order to convert the highest frequencies from the excitation spectrum ˜f(ω) into acoustic frequencies, the pump beam should be chosen so that the imaginary partk_g⁰⁰of its optical wave numberkg =k_g⁰+ik⁰⁰_g is as high as possible, i.e. the shortest absorption depth α⁻¹_g = (2k_g⁰⁰)⁻¹, where the index g represents the generation process andk_gis the wave number of the pump beam. Assuming no supersonic transport of energy either, the spectrum of the efficiently generated acoustic frequencies can extend up to the frequencies at which the acoustic wave number equals the absorption depth of the pump, i.e. ka^g(ω) = ωa^g/va = αg = 2k_g⁰⁰, where a denotes the acoustic waves and v_a the sound velocity of acoustic wave, thus the highest frequency ωa^g = 2v_ak_g⁰⁰. Furthermore the corresponding shortest duration of the photogenerated acoustic pulses is τa^g = (ω^ga)⁻¹ =α⁻¹_g /v_a, which is the time needed for the acoustic pulse to propagate across the absorption depth of α⁻¹_g . Generally the mechanisms considered in the OA conversion are thermoelasticity (TE), deformation potential (DP), inverse piezoelectric process, and electrostriction [2]. In this section, only the mechanisms in non-piezoelectric materials, i.e. TE and DP, are reviewed.

1.3.1.1 Thermoelasticity

The generation of thermal stress after the absorption of laser pulse by the TE mechanism is reviewed in this section [1, 9, 20]. For linear isotropic elastic solids, from the stress- strain relation Eq.(1.12), the strain can be written in terms of stress:

η_ij = 1

2µ[σ_ij− λ

3λ+ 2µ(σ_xx+σ_yy+σ_zz)]δ_ij

= 1 +ν

Y σ_ij− ν

Y(η_xx+η_yy+η_zz)δ_ij,

where Y = µ(3λ+ 2µ)/(λ+µ) is the Young’s modulus and ν = λ/2(λ+µ) is the Poisson’s ratio. So Eq.(1.12) becomes:

σ_ij = Y

1 +ν[η_ij + ν

1−2ν(η_xx+η_yy+η_zz)δ_ij].

When the temperature of a homogeneous and isotropic material increases uniformly, it will only increase in size, without shape change. Here the strain-stress and stress-strain

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relationships are

η_ij = 1 +ν

Y σ_ij − ν

Y(σ_xx+σ_yy+σ_zz)δ_ij+α∆T δ_ij, σij = Y

1 +ν[ηij+ ν

1−2ν(ηxx+ηyy+ηzz)δij]− Y

1−2να∆T δij, (1.20) whereαis the coefficient of linear thermal expansion and ∆Tthe increase in temperature of the solid. In addition, the bulk modulus, shear modulus, and Lam´e modulus of an elastic solid are defined as follows:

B= Y

3(1−2ν), µ= Y

2(1 +ν), λ= νY

(1 +ν)(1−2ν).

When the laser pulse is absorbed at the free surface of a homogeneous isotropic material, the temperature rise ∆T is a function of position and time. This sets up a thermal stress and a strain pulse is launched and propagates away from the surface. Generally, the area A illuminated by the laser beam is assumed large compared to the absorption depth of lightζ, hence the stress only depends onz, the only motion is parallel tozand the only non-zero component of the elastic strain tensor isηzz. Therefore we can write Eq.(1.20) as:

σzz = 31−ν

1 +νBηzz−3Bα∆T(z), (1.21)

where ∆T(z, t) = (1−R)Qe⁻^z^ζ/(ACζ) for t > 0, (R be the optical intensity reflection coefficient, Q the incident optical pulse energy, C the heat capacity per unit volume) and σ^{T E} =−3Bα∆T(z) is the isotropic thermal stress. Also we have:

ρ∂²uz

∂t² = ∂σzz

∂z , η_zz = ∂u_z

∂z .

With the initial condition of zero strain everywhere and the stressσ_zz is always zero at the free surface z= 0, the solution is

η_zz(z, t) = (1−R) Qα AζC

1 +ν

1−ν[e⁻^z^ζ(1−1

2e⁻^vt^ζ )−1 2e⁻

|z−vt|

ζ sgn(z−vt)], (1.22)

The strain described by Eq.(1.22) is shown in Fig. 1.7. As can be seen, the strain consists of two parts: a time-independent strain in the region near z = 0 because of thermal expansion and a pulse which propagates away from the free surface at the speed of the longitudinal sound velocity. This travelling strain pulse comes from the second term of the strain. Its shape is independent of time and its width is of the order of 2ζ.

Here we assume the time tis much less than d/v.

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Figure 1.7: The spatial dependence of the elastic strain at different times.

1.3.1.2 Deformation Potential

In a general case the photo-generated stress in a solid induced by an ultrashort light pulse can be written as [9]:

σij =X

k

δne(k)∂E_k

∂ηij

+X

k

δnph(k)~∂ω_k

∂ηij

,

whereδn_e(k) andδn_ph(k) are the changes in the electron and phonon distribution func- tionsne(k) and np(k), respectively, Ek is the energy of an electron of a wave vectork, ω_k is the frequency of an phonon of a wave vector k and ∂E_k/∂η_ij is the deformation potential. In the case of semiconductor the absorption of the light pulse produces electrons and holes, and as the electrons and holes relax to the band edge thermal phonons of energy (E −E_g) are produced for each photon of energy E. Therefore the stress generation depends not only on the temperature change but also on the excited carriers, and the stress generation related to the electronic distribution change caused by the carriers is referred to as the deformation potential (DP) mechanism [2]. The electronic and phononic contribution to the photo-excited stress in semiconductor are as follows [9]:

σ_ij^e =σ_ij^DP =n_e∂Eg

∂η_ij =−BdEg

dP δ_ijn_e, σ_ij^ph=−3Bα

C (E−E_g)δ_ijn_e,

where n_e is the total number of electron. The electron-hole and phonon contributions are normally of the same order of magnitude, but may have different signs. Generally

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the electrons and phonons are in local thermal equilibrium the stress σ_ij reduces to

−3Bα∆T(z).

1.3.2 Acousto-Optic Detection of Coherent Phonons

Likewise, the spectral transformation function of acousto-optic (AO) conversionK_AO(ω) determines the effectiveness of different acoustic frequencies that can be transformed into the spectrum of the reflected probe intensity variation in the detection region of the probe beam [23]. The highest frequency in the spectrum is determined byk^d_a(ω) = ω_a^d/va = 2kd = 2(k_d⁰ +ik⁰⁰_d), where d represents the detection process and ka is the wave number of the probe beam. Accordingly the characteristic frequency ω_d is also complex, i.e. ωd = 2va(k_d⁰ +ik_d⁰⁰) =ω_d⁰ +ω⁰⁰_d. There are two characteristic time scales, one is τ_d⁰ = (ω_d⁰)⁻¹ = (2v_ak_d⁰)⁻¹, which corresponds to the time of sound propagation across the wavelength of the probe in the material, while the other is τ_d⁰⁰ = (ω_d⁰⁰)⁻¹ = (2vak_d⁰⁰)⁻¹, which corresponds to the time needed for the acoustic pulse to travel through the absorption depth of the probe. In order to have a broad frequency bandwidth of the AO detection, the largest wavenumber of the probe is required.

Figure 1.8: Illustration of a polarized mono-chromatic EM wave incident on a material with a strain pulse atz=z⁰.

Consider a plane electromagnetic (EM) wave travelling through the vacuum in the +z- direction, as shown in Fig. 1.8, the EM wave incident on the surface at normal incidence and the wave equation is

E⁽ⁱ⁾(z, t) =E⁽⁰⁾e^i(k⁰^z−ωt),

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where ω is the optical angular frequency and k₀ = ω/c the free-space wave number.

When the EM wave travels across the air/medium interface at z = 0 into a medium with a complex refractive index ˜n= n+iκ, wheren is the refractive index and κ the extinction coefficient, the reflected and transmitted plane EM wave atz= 0 is expressed respectively as

E^(r)(z, t) =r0E⁽⁰⁾e^i(−k⁰^z−ωt),

E^(t)(z, t) =t0E⁽⁰⁾e^i(˜^kz−ωt)=t0E⁽⁰⁾e^i(k⁰^nz−ωt)^˜ e^−k⁰^κz,

where ˜k= ˜nω/c, the reflection coefficientr0 = (k0−k)/(k˜ 0+˜k) = (1−n−iκ)/(1+n+iκ), and the transmission coefficients t0 = 2k0/(k0+ ˜k) = 2/(1 +n+iκ). As can be seen, the electric field decays exponentially as the wave travels in the medium. For a strain pulse at position z=z⁰, the strain-induced variation of the relative dielectric constant ε_r (˜n = √

˜

ε_r) in terms of a spatial δ-function is: ∆˜ε_r(z) = F δ(z−z⁰), ∆˜ε_r(z) ε˜_r. So the wave equation in the medium involving the perturbation induced by the strain becomes

∂²E^(t)

∂z² =−k₀²[˜εr+ ∆˜εr(z)]E^(t).

As depicted in Fig. 1.8, the electric field incident on the strain pulse is defined asEⁱ = E₀e^i[˜^k(z−z⁰^)−ωt], where t₀E⁽⁰⁾ = E₀e⁻ⁱ^kz^˜ ⁰. By integrating this equation, we obtain the reflection coefficient of the electric field at the perturbation [1]:

r⁰ = iF k₀² 2˜k .

The reflected wave at the perturbation across the interface atz= 0, therefore the total reflected wave is

E^r=rE^(0)e^−i(k⁰^z−ωt) = (r₀+t₀r⁰t⁰₀e^−2k⁰^κz⁰)E⁽⁰⁾e^−i(k⁰^z−ωt),

where t⁰₀ = 2k/(k0+ ˜k) = 2(n+iκ)/(1 +n+iκ) is the transmission coefficient from the medium to the vacuum at z = 0. For a general distribution ∆˜ε_r(z), the reflection coefficientr becomes

r=r0+ik₀² 2kt0t⁰₀

Z ∞ 0

∆˜εr(z⁰)e^2ikz⁰dz⁰ =r0+dr,

where dr is the change in the reflection coefficient. The change in relative dielectric constant by the propagating strain is ∆˜εr(z, t) = 2˜n∆˜n = 2(n+iκ)(dn/dηzz + idκ/dη_zz)η_zz(z, t). Thus the relative change in the reflection coefficient is

dr(t)

r = r−r₀

r = 4ik₀n˜ 1−n˜²( dn

dη +i dκ dη )

Z ∞

ηzz(z⁰, t)e^2ik⁰^nz^˜ ⁰dz⁰. (1.23)

Ultrafast Dynamics of Coherent Phonons in Thin Films and Free-Standing Membranes