pump probe

Coherent Phonons in topological insulators and ferroelectrics

Pascal Ruello

Ins6tut des Molécules et Matériaux du Mans, UMR 6283 CNRS-Le Mans Université, FRANCE

h ν

pump probe

Acoustic phonons

1

1

Photo-induced coherent acoustic phonons in phase change material GeTe

R. Gu ⁽¹⁾ , T. Perrault ⁽¹⁾ , A. Levchuk ⁽¹⁾ , M. Weis ⁽¹⁾ , Z. Cheng ⁽²⁾ , H. Bhaskaran ⁽²⁾ , G. Vaudel ⁽¹⁾ , V. Gusev ⁽³⁾ , V. Juvé ⁽¹⁾ , N. Chigarev ⁽³⁾ ,S. Raetz ⁽³⁾ , A. Bulou ⁽¹⁾ , P. Ruello ⁽¹⁾

(1) Institut des Molécules et Matériaux du Mans, UMR CNRS 6283, Le Mans Université, 72085 Le Mans, France.

(2) Department of Materials, Oxford University, United Kingdom.

(3) Laboratoire d’Acoustique, UMR CNRS 6613, Le Mans Université, 72085 Le Mans, France

Interest: Chalcogenide glass, diatomic semi-conductor, phase change memory devices, ferroelectricity

Objective: Understanding the electron & phonon dynamics at short time and space scale

Sample Germanium-Tellurium: E _gap = 0.8eV(amorphous)

= 0.2eV(crystalline) x _pump = 142nm(amorphous) = 20nm(crystalline)

Pump-probe experimental setup: P _pump (830nm) = 5mW(1.5eV)

P _probe (586.1nm) = 300mW(2.2eV)

Laser duration t = 200fs (FWHM)

S.Kohara and al. Appl. Phys. Lett. 89,201910 (2006)

UP-DOWN project (18-CE09-0026-04)

SANTA project (18-CE24-0018-03)

LIA IM-LED

France-Japan

Ultrafast phenomena in condensed ma2er and picosecond acous4cs : people

V. Gusev S. Raetz N. Chigarev

cnrs cnrs cnrs cnrs

2

B. Arnaud V. Besse F. Calvayrac R. Busselez

Acknowledgments !

Ph.D.

G. Vaudel V. Juvé V. Temnov

Mariusz Lejman (Former PhD Now Post-doc CEA Lydil Paris)

R. Gu A. Levchuk M. Weis (now Post- doc LOA

Ec. Polytechnique)

Collaborators

3

B. Dkhil, I. C. Infante P. Gemeiner, C. Paillard

G. Nataf, M. Guennou J. Kreisel

M. Viret

B. Wilk, K. Balin, R.

Rapacz, J. Szade Topological insulators

Ferroelectrics, mul6ferroics

T. Marou6an S. Matzen S. Gable P. Lecoeur

C. Paillard, L. Bellaiche

-1 Photogenera6on of phonon with ultrashort light pulse

-2 Op6cal and acous6c phonons genera6on in ultra-thin film of topological insulators (Bi ₂ Te ₃ ).

-3 Acous6cs phonon genera6on and detec6on in ferroelectric materials (BiFeO3, LiNbO3)

-4 concluding remarks and perspec6ves OUTLINE

4

Driving la=ce dynamics with ultrashort light pulses

z

Phonon genera6on

Newton law

x h ν

pump probe

:

;<6

;= = = =

=

=

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC

<, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

Ti ₂ O ₃

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC <, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

A gradient of the energy deposited by the pump ac6on (δU (J.m ^-3 )) generates a force (N.m ^-3 ) that drives the atoms mo6on (ρ= mass density (kg.m ^-3 )

! ^! !

!! ^! + Ω ^! ! = 1

!

! (!")

!" = !/!

This approach works for both op6cal and acous6c phonons.

:

;<6

;= = = =

=

=

Q is the normal coordinate (see the previous slide) and describes one possible phonon mode over the dispersion curve (here this simple equa6on does not take into account the dispersion - k dependence)

! = −!"#$!

z

Op6cal phonon genera6on

Newton law

x h ν

pump probe

Op6cal phonon = strain field

restricted to the unit cell

Raman process

(Merlin Sol. Stat. Comm 1997)

Pergamon Solid State Communications, Voi. 102, No. 2-3, pp. 207-220, 1997

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00

PII: S01D8-1098(96)00721.1

GENERATING COHERENT THz PHONONS WITH LIGHT PULSES R. Merlin

The Harrison M. Randall Laboratory of Physics and Center for Ultrafast Optical Science, The University of Michigan, Ann Arbor, MI 48109, U.S.A.

We present an overview of experimental work and discuss the theory of femtosecond-pulse generation of coherent optical phonons. Recent developments and differences between solids and molecular systems are emphasized. Theoretically, we focus on Raman mechanisms for the coherent driving force providing a comprehensive approach for the description of both below-gap and resonant excitation. The survey covers also polaritons and folded acoustic phonons in superlattices as well as a comparison with spontaneous Raman scattering. © 1997 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION AND HISTORICAL ASPECTS Following recent advances in femtosecond laser tech- nology, several groups demonstrated that the propagation of light pulses in solids is accompanied by intense THz lattice vibrations showing a high degree of spatial and temporal coherence [1-21]. The availability of coherent

optical phonons at such frequencies has led to a variety of suggestions for applications and experiments involving, in particular, time-domain spectroscopy of phonons [1-21], conversion of mechanical into coherent electro- magnetic energy [14] and intriguing proposals bearing on photon control of the ionic motion [5, 22, 23]. This article provides an overview of recent developments in these areas and the theory of phonon generation.

The most common experiment on coherent phonons involves two laser pulses obtained by splitting a single fs-pulse. As shown in Fig. 1, the stronger pump pulse creates a vibrational wave which perturbs the weaker

probe pulse that follows behind. Here, the signal of interest is the transmitted or reflected intensity of the probe beam as a function of the time-delay, as measured by the relative distance between the two pulses. In its simplest form, the generation of coherent modes and their effect on the probe beam rely on the modulation of the refractive index n by the ion motion [24, 25]. To first order in the phonon fieM Q, we have ~n(r, t) = 2r~x/n -~

2r[ax/OQ]Q(r, t)/n (x is the electronic susceptibility).

The resulting change in electromagnetic energy density is

~U = ~XlEo(r, 012 1 / 8X ~

= -~-~)Q(r,t)lEo(r,t)l 2, (1)

where E 0 is the magnitude of the pump electric field.

Since ~U ~ Q, equation (1) gives a force density F acting on Q which is proportional to the electric field intensity.

In the following, F will be referred to as the driving force. If we ignore phonon dispersion, the equation of

motion for the lattice field is

dt 2 2 \ 8Q/IE0(r, t)l 2 = F(r, t). (2) Using the Green's function method, the solution is

t

I sin [f~(t - O].F(r ' r) dr. (3) Q(r, t) = fl

This indicates that the lattice motion is controlled by Fourier components of F at ~- fl and, accordingly, that the coupling is large only if the duration of the pulse is small compared with fl-1. The properties of the coherent field can be studied by monitoring the scattering of the probe pulse by the time- and space-varying refractive index associated with Q(~n oc Q). If we consider the fact that the susceptibility is not a scalar but a tensor, the expression for the force becomes F = Euv(XuvEuEv)/2 R

where Eu denotes a component of the optical pump field, X~v "-~ OxJOQ is the nonlinear Raman and Xuv is the linear susceptibility [24, 25]. It is remarkable (although not surprising) that the mechanism responsible for the driving force is also the source of inelastic light scattering since the Raman cross-section is proportional to IXuRvl 2 [26]. Notice that in the derivation of equation (3) we neglected the depletion of the pump field (which 207

Pergamon Solid State Communications, Voi. 102, No. 2-3, pp. 207-220, 1997

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098/97 $17.00+.00

PII: S01D8-1098(96)00721.1

GENERATING COHERENT THz PHONONS WITH LIGHT PULSES R. Merlin

The Harrison M. Randall Laboratory of Physics and Center for Ultrafast Optical Science, The University of Michigan, Ann Arbor, MI 48109, U.S.A.

We present an overview of experimental work and discuss the theory of femtosecond-pulse generation of coherent optical phonons. Recent developments and differences between solids and molecular systems are emphasized. Theoretically, we focus on Raman mechanisms for the coherent driving force providing a comprehensive approach for the description of both below-gap and resonant excitation. The survey covers also polaritons and folded acoustic phonons in superlattices as well as a comparison with spontaneous Raman scattering. © 1997 Elsevier Science Ltd. All rights reserved

1. INTRODUCTION AND HISTORICAL ASPECTS Following recent advances in femtosecond laser tech- nology, several groups demonstrated that the propagation of light pulses in solids is accompanied by intense THz lattice vibrations showing a high degree of spatial and temporal coherence [1-21]. The availability of coherent

optical phonons at such frequencies has led to a variety of suggestions for applications and experiments involving, in particular, time-domain spectroscopy of phonons [1-21], conversion of mechanical into coherent electro- magnetic energy [14] and intriguing proposals bearing on photon control of the ionic motion [5, 22, 23]. This article provides an overview of recent developments in these areas and the theory of phonon generation.

The most common experiment on coherent phonons involves two laser pulses obtained by splitting a single fs-pulse. As shown in Fig. 1, the stronger pump pulse creates a vibrational wave which perturbs the weaker

probe pulse that follows behind. Here, the signal of interest is the transmitted or reflected intensity of the probe beam as a function of the time-delay, as measured by the relative distance between the two pulses. In its simplest form, the generation of coherent modes and their effect on the probe beam rely on the modulation of the refractive index n by the ion motion [24, 25]. To first order in the phonon fieM Q, we have ~n(r, t) = 2r~x/n -~

2r[ax/OQ]Q(r, t)/n (x is the electronic susceptibility).

The resulting change in electromagnetic energy density is

~U = ~XlEo(r, 012 1 / 8X ~

= -~-~)Q(r,t)lEo(r,t)l 2, (1)

where E 0 is the magnitude of the pump electric field.

Since ~U ~ Q, equation (1) gives a force density F acting on Q which is proportional to the electric field intensity.

In the following, F will be referred to as the driving force. If we ignore phonon dispersion, the equation of

motion for the lattice field is

dt 2 2 \ 8Q/IE0(r, t)l 2 = F(r, t). (2) Using the Green's function method, the solution is

t

I sin [f~(t - O].F(r ' r) dr. (3) Q(r, t) = fl

This indicates that the lattice motion is controlled by Fourier components of F at ~- fl and, accordingly, that the coupling is large only if the duration of the pulse is small compared with fl-1. The properties of the coherent field can be studied by monitoring the scattering of the probe pulse by the time- and space-varying refractive index associated with Q(~n oc Q). If we consider the fact that the susceptibility is not a scalar but a tensor, the expression for the force becomes F = Euv(XuvEuEv)/2 R

where Eu denotes a component of the optical pump field, X~v "-~ OxJOQ is the nonlinear Raman and Xuv is the linear susceptibility [24, 25]. It is remarkable (although not surprising) that the mechanism responsible for the driving force is also the source of inelastic light scattering since the Raman cross-section is proportional to IXuRvl 2 [26]. Notice that in the derivation of equation (3) we neglected the depletion of the pump field (which 207

:

;<6

;= = = =

=

=

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC

<, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

Ti ₂ O ₃

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC <, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

Electromagne6c density of energy

The electric field of the light polarizes the electronic cloud, so that so the lagce is distorted

! = 1

2 !! ^!

Force (N.m ^-3 ) F=d(δU)/dQ (gradient of pressure)

z

Op6cal phonon genera6on

Newton law

x h ν

pump probe

Op6cal phonon = strain field

restricted to the unit cell

Electronic excita6on (deforma6on poten6al), - Zeigler et al Phys Rev B 1992

:

;<6

;= = = =

=

=

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC

<, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

Ti ₂ O ₃

In this model, the electronic system is excited by the laser pump and the new electronic distribu6on (n(t)) forces the lagce to have a new ini6al

coordinate Q ₀ which depend on this electronic

distribu6on.

z

Op6cal phonon genera6on

Newton law

x h ν

pump probe

:

;<6

;= = = =

=

=

216 GENERATING COHERENT TI-Iz PHONONS WITH LIGHT PULSES Vol. 102, No. 2-3 (a) 2

CC

<, 0

% -1 -2

sotropic ,

• . onisotropic

Time Dolay (ps)

o -2

v-

-4 -6

(b) 1.0

"O

¢l N e=

E O.S

0

0.0 2

... isotropic ,,"~ A 1 anisotropiq ::

!,

3

Fmquoney ffHz)

E"TO

x~.__

Fig. 7. Reflectivity modulation due to coherent phonons in Te obtained with pulses at 2 eV. (a) Time-domain data. The trace labeled "isotropic" shows the A1 mode; results do not depend on the angle between the polarizations of the pump and probe beam which are perpendicular to the c-axis. The "anisotropic" E-sym- metry oscillations were gained from the difference between two orthogonaUy polarized probe components (see Raman tensors in Table 1). 0a) Fourier transform intensity. After Dekorsy et al. [14].

two-band situations. This correlation strongly indicates that TSRS is the underlying mechanism for the driving force. We remark that Eg-osciUations were not observed in the early experiments at hco 0 ~- 2 eV that led to the DECP interpretation [5], where the A 1g-coherence was found to exhibit the now characteristic cos(ri0-behavior [57]. In this regard, it is of interest to consider the fits to exp - (rt)sin (rit + ¢) in (a) indicating that the dynamics of the modes at 1.52 eV are neither purely displacive (I,Pl = lr/2) nor impulsive (~ = 0) [55]. This, and the A lg-results at 2 eV are consistent with the Raman model in that the latter excitation energy favors displacive behavior for it is much closer to the E~-maximum (see Section 2.2).

The case of Te in Fig. 7 serves to further amplify the discussion on the correlation between TSRS and RS. The results show A r and E-type oscillations obeying selection rules identical to those for Sb (see Table 1), as expected for a Raman process [14]. Unlike Sb, how- ever, the comparison between time-domain amplitudes and RS intensities reveals a large discrepancy [14] in

0.95 0 . ~ ~

(a)

0.85

0.0 7.1 I--

I I I

0.5 1.0 1.5 2.0

~ ^{6.9 -}

6.7

~ .6 (b)

6.5

o,o 0.s L0 t.s 2.0

Time Delay (ps)

Fig. 8. (a) Pump-probe data for Ti203 showing coherent Alg modes. The pulses are centered at 2eV. (b) Dependence of the oscillation frequency on the delay.

After Cheng et al. [9].

that, relative to the Al-phonon, TO-signals are much larger in RS (however, notice that the time-domain and RS data referred to in [14] were obtained at different laser energies). Since Te can be described by two-band contributions [58], does this contradict TSRS predic- tions? The answer is: not necessarily. There is no inversion symmetry in Te and Raman E-modes are also infrared-active. As the oscillations are due to the TO component of the phonon, TSRS and RS spectra are not required to correlate on the grounds that there is a second X(2)-mechanism operating on TO-phonons (see Section 2.3). This, and the fact that TO-modes are at all observed in a geometry which is tantamount to forward scattering follow from non-conservation of momentum as the penetration depth is only ~- 500 ,~.

The reader may have noticed in the examples presented so far that there were no departures from harmonic behavior, other than for the conventional anharmonic decay. An exception to this are the results shown in Fig. 8 for the Alg-mode of Ti203 [9]. At high excitation intensities, the oscillation frequency exhibits a large redshift at small delays returning to ground-state values after relatively few cycles. Notice the rather large reflectivity modulation from where the vibrational ampli- tude can be inferred. An estimate gives as much as 0.04-0.07 ,g, for the Ti-Ti displacement (interestingly, Ti203 exhibits a transition into the metallic state involving a slightly larger change, ~- 0.1 ,~, in the lattice

Ti ₂ O ₃

Whatever the model, the first necessary condi6on to

generate coherent op6cal phonon is to excite the solid with a laser pulse dura6on shorter than the op6cal phonon

period, else, « interferences » will make the genera6on

process vanishing

9

! ^! !

!! ^! + Ω ^! ! = !/!

If Q(t=0)=0 and dQ/dt) _t=0 =V

then Q=(V/Ω)sin(Ωt)

V(impulsive velocity)

If Q(t=0)=Q ₀ and dQ/dt) _t=0 =0

then Q=Q ₀ cos(Ωt)

Solving the oscillator equa6on with different ini6al condi6ons

Q ₀ (ini6al

displacement) Q ₀

(like impulsive Raman process)

(like displacive excita6on – DECP

– deforma6on poten6al)

z

Electron-hole / photon / acous6c phonon coupling

Thermoelasticity B : bulk modulus,

β : thermal expansion coefficient, Δ T : lattice temperature

σ _therm. = − 3B β Δ T (t, x )

Piezoelectricity :

e : piezoelectric constant, E : electric field

σ _piezo = eE (t, x )

Newton law

10

σ _electro = 1

2 ρ ^∂ ε

∂ ρ ^E

2 (t, x) Electrostriction :

ε : dielectric constant, ρ : mass density E : electric field

σ _magneto = 1

2 ρ ^∂ µ

∂ ρ ^H

2 (t, x) Light Magnetic field striction : µ : magnetic permeability H : magnetic field

Deformation potential :

N : photocarriers concentration, d _eh : deformation potential parameter

σ _{e− ph} = − d _eh N (t, x )

x h ν

light

Efficient in semiconductors

Efficient in metals

Efficient in piezoelectric materials

Driven mechanism in transparent solids like dielectrics (σ _electro >σ _magneto )

Origin of photoinduced stress σ

Coupled effects (magnetostric6ve magne6c materials, ferroelectrics,

+ …)

- Yan et al, J. Chem. Phys. 83 (11) 1985

- Thomsen et al, Phys. Rev. B, 34, 4129 (1986) -Gusev, Karabutov, Laser optoacous,cs AIP 1993 NY -Ruello & Gusev, Ultrasonics 56, 21-35 (2015)

:

;<6

;= = = =

=

=

P _eq = − ∂ E _electrons

∂V + ∂ E _phonons

∂V

#

$ % &

' (

N

Total energy of a solid

P = − Δn _e ∂E (  k )

∂ η ⁺

k 

∑ ^{Δn ph  ∂ ω (}

k  )

∂ η

q 

% ∑

&

' (

) *

Pressure at the thermodynamic equilibrium

Electronic pressure (deforma6on poten6al)

η

Δ n _e , Δ n _ph

Strain

Concentra6on of Photoexcited electrons and phonons

with P _eq = − ∂E

∂ V

#

$ % &

' (

N

E = E _electrons + E _phonons + ...

Phonon pressure

(thermoelas6c effect) Light-induced pressure when light excita6on leads to electron

and phonon energies changes

Light-induced pressure

If we consider only e and ph

(H.J. Maris, in: W.P. Mason, R.N. Thurston

(Eds.), Physical Acous6cs, vol. 7, Academic,

New York, 1971).

12

Lagce anharmonicity (thermoelas6city)

Transfer of the electronic energy towards the phonon subsystem

Laser induced pressure : phonons (thermoelas6city)

σ _TE. = − 3B β Δ T _L (t, x ) ^{(C L} lagce heat capacity, ϒ _L : gruneisen coefficient, B: bulk modulus, β thermal expansion coefficient T _L lagce temperature)

P _phonon = − Δ n _ph  ∂ ω ^{( k  )}

∂ η

q 

∑

12

:

B

:

4

*

"

.20=

.20

,

*

)B B

?0 )B .L?0 *

) . 0

_:

+

*

:

_:

:

(=

₊ (

?0 )B .L?0 *

) . 0 A0)) B

)

C06)B . 0

Lagce energy

σ _TE = − P _phonon (conven6on)

σ _TE = − Δ n _ph  ω _k γ _k = − γ _L ^C _L Δ T _L

k

∑

P = − Δ n _e ∂ E(  k )

∂ η ⁺

k 

∑ ^{Δ n ph  ∂ ω (}

k  )

∂ η

q 

% ∑

&

' (

) *

13

P = − Δ n _e ∂ E (  k )

∂ η ⁺

k 

∑ ^{Δ n ph  ∂ ω (}

k  )

∂ η

q 

% ∑

&

' (

) *

σ _electron = − 2 3

E _electrons V

The internal pressure of quantum free electrons (Sommerfeld model)

η

Δn _e , Δn _ph

Strain

Concentra6on of Photoexcited electrons and phonons

with

Light-induced pressure : electrons

*

)B B

?0 )B .L?0 *

) . 0

_:

+

*

:

_:

:

(=

₊ (

?0 )B .L?0 *

) . 0 A0)) B

)

C06)B . 0

σ _electrons = Δ n _e ∂ E (  k )

∂ η

k 

∑

Light excita6on modifies the internal electronic energy

Electronic pressure in metals (deforma6on poten6al)

P _eq = 2 3

E _electrons V E _electrons =  ² k ²

2m g(  k )d

0 k _F

∫ ^{k  =} ₈ ^V _π ^{3 sin θ d θ d ϕ }

2 k ²

2m k ² dk

θ ∫∫ , ϕ ^{= V }

2 k _F ⁵ 10m π ² Electronic pressure within the Sommerfeld model

P _eq = − ∂E _electrons

∂ V

#

$ % &

' (

N

N = 4

3 π ^k _F ³ × V

8 π ^{3 × 2}

Number of electrons in the Fermi sphere

E _electrons ( N, V ) =

V  ² × ( N 3 π ² V ) ^5/3

10m π ² The internal pressure of quantum free electrons depends on its energy E

14

(T=0K)

Fermi sphere

Modifica6on of the electronic pressure by laser excita6on

ΔP ≈ 2 3

ΔE _electrons

V ≈ γ _e ^C _e ^ΔT _e σ _e = − γ _e ^C _e Δ T _e

Absorbed laser energy by electrons = ΔE _electrons

σ Conven6on External applied stress

Compression σ > 0 σ < 0 Expansion

C _e ΔT _e = ΔE _electrons / V

( )

15

γ _e = 2 / 3

( )

Orders

ΔP ≈ N × 1 eV

V ≈ 10 ²⁸ × 10 ⁻¹⁹ = 1 GPa

Grüneisen coeff of electrons

P _eq = 2 3

E _electrons

V

Deforma6on poten6al in SC

16

Deforma4on poten4al : different views -  modifica6on of the electrons distribu6on

(i.e. modifica6on of the microscopic electrosta6c ca6on-electron interac6on) -   “Photo-doping” of empty orbitals

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Varia6on of the lagce volume (V) associated to a modifica6on of the electronic energy U

δ ^V

V = δ ^U d _eh

d _eh = deforma6on poten6al parameter

d _eh =10eV for GaAs (Brillouin zone center) in Yu&Cardona, Fundamentals of SC, Springer d _eh =35eV for Bi ₂ Te ₃ (Huang et al PRB 2008)

“Molecular view” : 2 possible situa6ons

Ruello & Gusev, Ultrasonics 56, 21-35 (2015)

(Review )

17

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Δn _e (k ) E _k

∂E _k

∂ η

- Energy of the level k

-Modifica6on (due to laser ac6on) of the electron popula6on in the level k -Deforma6on poten6al parameter (η=strain) Deforma6on poten6al stress : general expression

(H.J. Maris, in: W.P. Mason, R.N. Thurston (Eds.), Physical Acous6cs, vol. 7, Academic, New York, 1971).

“Molecular view” : 2 possible situa6ons

Ruello & Gusev, Ultrasonics 56, 21-35 (2015) (Review )

Deforma6on poten6al in SC

σ _electron = − Δ n _e ∂ E _k

∂ η

k

∑

Long life6me of hot carriers in SC = long life6me of deforma6on poten6al stress

18

riers. For ⌦ _pp ⇥835 nm ⇤⇧ _pp ⇥1.485 eV⌃ the nonequilibrium electrons are photoexcited with E _ex ^e ⇥ E _th ⌃ 8 meV close to the thermal energy while the initial average energy of the holes is reduced of ⇥10 meV compared to that for ⌦ _pp ⇥810 nm ⇤E _ex ^h ⇥ E _th ⌅ 24 meV⌃. However, the transient transmissiv- ity reaches its equilibrium value on a much shorter time scale as shown in Fig. 3 consistent with the smaller initial excess energy of the electrons. ⁴⁰ In contrast, when holes are photo- excited with an initial energy close to E _th ⇤i.e., ⌦ _pp ⇥780 nm,

⇧ _pp ⇥1.59 eV⌃ a slowly decaying signal is observed with a larger amplitude than for ⌦ _pp ⇥810 nm ⇤Fig. 3⌃, consistent with higher energy injection into the electron system ⇤E _ex ^e

⇥ E _th ⌃ 89 meV⌃. This large dependence of the ↵ T / T tempo- ral behavior on the initial excess energy of the electrons con- firms that the measured slow thermalization can be ascribed to hot-electron cooling and that hole thermalization dynam- ics plays a minor role in the measured transient response.

When electrons are photoexcited with a significant excess energy relative to E _th ⇤i.e., ⌦ _pp ⇥780 or 810 nm⌃, ↵ T/ T slowly reaches a plateau, indicating that the electrons are still hotter than the lattice for times as long as 8 ps ⇤Fig. 3⌃. This slow thermalization is evidenced in Fig. 4 where the differ- ence, DT ⇧ ( ↵ T / T) ⌅ ( ↵ T/ T) _QE , between ↵ T / T and its quasiequilibrium ⇤QE⌃ value (↵ T / T ) _QE has been plotted on a logarithmic scale as a function of the probe time delay for

⌦ _pr ⇥810 nm ( ↵ T / T ) _QE , which corresponds to carrier-lattice thermal equilibrium, has been measured for t ⇧15 ps . The amplitude of DT is related to the excess occupancy of the probed electron states f _e ( T _e )⌅ f _e ⇤295 K⌃ and hence for small system perturbations is proportional to the excess tem- perature ↵ T _e of the electron gas when it is internally ther- malized ⇤i.e., for t ⌥500 fs⌃. The DT decay thus directly reflects the electron cooling dynamics.

A slow DT decay is observed for t ⇤1 ps with an almost monoexponential behavior for t ⇤3 ps with a time constant of

e ⇥1.9 ps comparable with the LO-phonon dephasing time in GaAs at room temperature ⇤⇥2.1 ps⌃. This slow thermal- ization, with a time constant close to that observed for cold electron heating, ⁵ is in good agreement with that predicted by the rate equation model ⇤3⌃. It has to be noted that for

short time delay ⇤t ⌅3 ps⌃ a faster decay time is observed because of nonequilibrium LO-phonon momentum space dif- fusion internal thermalization of the LO-phonon system

⇤Fig. 2⌃ which is equivalent to increasing C _ph with time in our simple model.

The DT amplitudes measured for different pump wave- lengths are directly comparable for the same probe wave- length and photoexcited carrier density (↵ T / T ) _QE is thus constant permitting us to compare the electron excess tem- perature for different excitation conditions. As the pump wavelength decreases from 810 to 780 nm, the long-time delay DT amplitude ⇤and thus ↵ T _e ⌃ is measured to increase by a factor of 2.1 ⇤Fig. 4⌃. As the heat capacity of the coupled electron and phonon system is almost unchanged, this ↵ T _e rise simply reflects the increase of the energy in- jected into the system by more than a factor of 2 ⇤the average excess energy ↵ E _e of the photoexcited electrons increases from 43 to 89 meV⌃. The measured decay times for t ⇤3 ps are, however, identical ⇤ _e ⇧1.9⇥0.1 ps⌃ as they are mainly determined by the LO-phonon anharmonic decay and thus independent of the excess energy stored in the electron and LO-phonon coupled systems ⇤3⌃.

The experimental results are in very good agreement with the transient transmission changes calculated from the com- puted carrier distributions using _LO ⇧2.1 ps Fig. 4⇤b⌃ . The computed ↵ T/ T neglecting the hot-phonon effect i.e., ne- glecting modifications of n _LO by imposing n _LO ( q) ⇧ n _LO ⁰ in the simulations shows a fast decay, which is only weakly sensitive to the screening model ⇤dotted line in Fig. 5⌃. Our calculations show that for intermediate time delays

⇤1⌅t ⌅3 ps⌃, the electron thermalization dynamics is dominated by momentum space diffusion of the nonequilib- rium LO phonons ⇤Fig. 2⌃. As the electron gas strongly in- teracts with the small q _LO phonons ⇤0.7 10 ⁶ ⌅q _LO ⌅2 10 ⁶

FIG. 3. Measured transient differential transmission ↵ T/T in GaAs at 295 K for a probe wavelength of 810 nm and a carrier density of 1 10 ¹⁷ cm ^⌅3 . The pump wavelength is 810 nm ⇤full line⌃, 780 nm ⇤dotted line⌃, and 835 nm ⇤dashed line⌃.

FIG. 4. Measured ⇤a⌃ and calculated ⇤b⌃ transient transmission difference DT ⇧ ↵ T/T⌅ ( ↵T /T) _QE on a logarithmic scale (↵ T/T ) _QE is the long delay ↵T /T measured for t ⇧ 15 ps . The carrier density is 1 10 ¹⁷ cm ^⌅3 and the pump wavelength 810 nm

⇤full line⌃ and 780 nm ⇤dotted line⌃. The probe wavelength is 810 nm.

14 490 P. LANGOT et al. 54

Langot et al PRB 1996

*

)B B

?0 )B .L?0 *

) . 0 _:

+

*

:

_:

:

(=

₊ (

?0 )B .L?0 *

) . 0 A0)) B

)

C06)B . 0

Intraband process (electron- phonon relaxa6on ~1ps)

Interband process (electron recombina6on ) ~1ns)

GaAs

N : concentra6on of out-of equilibrium electron-hole pairs

B = bulk modulus

deforma6on poten6al parameter

∂E _g

∂ P σ _electron = Δn _e ∂E _k

∂ η ^{= N}

∂ E _g

∂ η ^{= − NB}

∂ E _g

∂ P

k

∑ ^{= −(d e + d h )N}

d _e , d _h

19

- Yan et al, J. Chem. Phys. 83 (11) 1985

- Thomsen et al, Phys. Rev. B, 34, 4129 (1986)

- Gusev, Karabutov, Laser optoacous,cs AIP 1993 NY - Ruello & Gusev, Ultrasonics 56, 21-35 (2015)

Electron-hole / photon / acous6c phonon coupling Electron-hole / photon / acous6c phonon coupling

à Lecture of Vitali Gusev

Full theore6cal descrip6on of microscopic mechanisms

Some references