**2. Theoretical perspectives**

## **2.1 Place, place attachment and environmental stewardship**

Place concept is an increasingly contested concept across disciplines including positivistic scholars, social constructivism and sychometrics. In a review of place literature Low and Altman [3], established that as space is an integrating concept, there is no systematic theory of place and scholars have increasingly echoed luck of conceptual coherence in place research. The concept of place broadly connote to the subjective experience of embodied human experience in the material world [4]. Morgan argues further that place concept is paradoxical, with meaning that is not readily understood, but difficult to define. While understanding place is a step towards comprehending place attachment, this chapter is about place attachment and its implicature to environmental stewardship in peri-urban areas. The next paragraphs of this Section therefore narrows down to the conception of place attachment and its interlinkages to environmental stewardship behavior.

Place attachement is defined as an attitude of bonding (trust and belonging), mutual concerns and shared values with other members of one's group or locale' [5]. "…. Place attachment may contribute to the formation, maintanance and preservation of the identity of a person, group, or culture. And it may also be that place attachment plays a role in fostering individual, group and cultural selfesteem, self-worth, and self-pride …." [3]. Place attachment is also assumed to be of importance to the community since it facilitates engagement in local affairs [6]. Studies variously justify that place attachment motivates indiviaduals to contribute

#### *Community Attachment and Environmental Stewardship: A Peri-Urban Perspective DOI: http://dx.doi.org/10.5772/intechopen.99495*

to civic activity on behalf of one's residential location, in the form of sustainable behavior [7, 8], ecological behaviors [9] or reactions to encroachment of one's territory [10, 11]. On this basis, place attachment serves both the individual and large community [6]. Place attachment has also been correlated to community resilience building processes [12] and fostering local friendship [13].

## **2.2 Peri-urban in the context of environmental stewardship**

Situating the peri-urban concept is largely a daunting endeavor. This owes to its increasingly contested milieu by both academia and development practitioners [14, 15]. This Section reviews the conceptual contestation of peri-urban not in the sense of engaging in the debate of its contestations but for understanding its complex nature towards realizing environmental stewardship. The remainder of this Section therefore provide the contested environment and deployed discussion of the peri-urban concept, thereby its linkage to environmental stewardship and to the broad landscape planning discourse.

Broadly, the peri-urban concept is variously and increasingly viewed and conceptualized across both the geographical and disciplinary niches. Arguments behind the multiple conceptions of the peri-urban include lack of scientific definition Forsyth [15], diversity of engaged disciplinary orientations Thuo [14], the difficulties linked to the delimiting the spatial extent of this dynamic region Brook [16] and the equivocation of the concept itself. Notably, scholars increasingly argue that rural, periurban and urban environments operate as a system rather than independent entities [17, 18]. At the same time, peri-urban area is increasingly claimed to constitute the intersection point between urban and rural areas [19, 20]. In nutshell, there are growing yet converging understandings within academia on the lived reality of the diverse and context-laden definitions of the peri-urban concept [21]. Another converging understanding linked to peri-urban conception is the co-existence of urban and rural features within cities and beyond their limits [17, 21]. Notably, inspired by the aforehinted converging understandings, this chapter adopts the conception of peri-urban as defined by Mngumi [22], i.e. peri-urban is a city transitional zone, amalgamating both urban and rural landscape functions and features. In addition to the conceptual definition, the empirical materials discussed herein are drawn from the Pugu and Kazimzumbwi forest reserves at the peri-urban of Dar es Salaam city. Dar es Salaam is among the most rapidly growing cities in Africa and currently the business capital of Tanzania. Similar to other peri-urban areas in the Global South [23], natural resources and/ecosystem services in these peri-urban forest reserves are increasingly claimed to deteriorate [22].

Literature has increasingly established that the quantity and quality of natural resources are deteriorating in peri-urban areas, and particularly so in Sub-Saharan Africa [23, 24]. One explanation behind the decline of natural resources in periurban areas is the ongoing urbanization which has increasingly resulted in the expansion of the built environment on ecologically sensitive land, especially in periurban areas [25–27]. Onother explanation of the growing degradation of natural resources in peri-urban areas in Sub-Saharan Africa is the rapid urbanization due to poverty [28, 29]. The poverty aspect which is largely prevelant in the Global South context renders it difficult for the community members to inculcate the environmental stewardship habit. This is due to the fact that their livelihoods are intricately connected to the resources in their vicinity and so despite having strong natural attachment to the environment, this is compromised by livelihood hardships and as a result leading to low stewardship to the environment. Notably, there is a host of other contributing factors to the decline of natural resources at the peri-urban areas of Pugu and Kazimzumbwi forest reserves. These include climate change, the

increase of anthropogenic activities and the expansion of the city's built area to the peri-urban areas [22].


Early attempts assumed a three-temperature (3T) phenomenological model with rate equations for electron, phonon, and spin temperatures (energies). Using a spin temperature is quite unsatisfactory, considering that the spin system is not in equilibrium on the femtosecond timescale. If the spin is instead coupled to a two-temperature (2T) model for phonon and electron temperatures, we can obtain some improvement. Such a model deals with an energy flow concept, interpreting the ultrafast demagnetization as a non-coherent, "thermal" process. The energy goes from photon to electron and then to the spin system, without the need for stating any underlying quantum mechanism behind the spin flip. Among the candidates for underlying quantum mechanisms are Elliott-Yafet (EY) electronphonon scattering, EY electron scattering with impurities or other electrons and the electron-electron inelastic exchange scattering. The EY model for phonon scattering seems to

Further progress in the field of phenomenological models (Atxitia & Chubykalo-Fesenko, 2010) has resulted in three central branches: 1) Langevin dynamics based on the Landau-Lifshitz-Gilbert (LLG) equation and classical Heisenberg Hamiltonian for localized atomic spin moments, 2) the Landau-Lifshitz-Bloch (LLB) micromagnetics model, and 3) the

The LLB model includes the dynamics governed by both the atomistic LLG model and the M3TM model, with both classical and quantum versions. The LLB equation for a quantum spin *S* is based on the density matrix approach. For simple quantum spin systems modeled a s a two level system with spin-up and spin-down bands (i.e. *S* = ±1/2) it has been shown to be equivalent to the M3TM model (Atxitia & Chubykalo-Fesenko, 2010). The LLB equation for classical spins is equivalent to an ensemble of exchange-coupled atomistic spins modeled

Quantum mechanisms responsible for the ultrafast demagnetization in the LLB model are described by the coupling parameter λ, which defines the rate of the spin flip. Relations

$$\begin{aligned} \mathbf{W}\_{\mathbf{q}}^{\mathrm{r}} \left( \mathbf{t}\_{1}, \boldsymbol{\alpha} \right) &= \int \mathbf{t} \mathbf{t}\_{2} \mathbf{W}\_{\mathbf{q}}^{\mathrm{r}} \left( \mathbf{t}\_{1}, \mathbf{t}\_{2} \right) \mathbf{e}^{\mathrm{i}\boldsymbol{\alpha} \left( \mathbf{t}\_{1} - \mathbf{t}\_{2} \right)}, \\ \boldsymbol{\omega}\_{\mathbf{q}}^{-1} \left( \mathbf{t}\_{\mathrm{D}}, \boldsymbol{\alpha} \right) &= \mathbf{W}\_{\mathbf{q}}^{\mathrm{r}} \left( \mathbf{t}\_{\mathrm{D}}, \boldsymbol{\alpha} \right) / \mathbf{V}\_{\mathbf{q}} \end{aligned} \tag{24}$$

with *Vq* denoting the strength of the bare Coulomb interactions. Huber et al. (Huber et al., 2005b) investigated the formation of hybrid modes in intrinsic InP using a 12-fs Ti:sapphire laser pump with a single-cycle THz probe at 28 THz, delayed with the time tD . Real-time evolution of the probe THz electric field was directly measured by means of the ultrabroadband electro-optic sampling technique with sampling delay time s t . The measured data allowed access to both real and imaginary parts of q D (t , ) in a 7 - 60

THz window.

Delaying the probe with respect to the pump resulted in characteristic trailing oscillations long after the original probe signal had died out. The trailing oscillations represent a change in the probe signal which was monitored by varying the sampling delay time s t . Increasing the delay between pump and probe resulted in more trailing oscillation cycles. These oscillation cycles are manifestations of a quantum beating between the newly developing upper and lower hybrid modes and in Fig. 3.

The hybrid resonances can be followed by a Fourier transformation with respect to s t of the change in the real-time probe signal for various pump-probe delays tD , thereby extracting the inverse dielectric function 1 q d (t , ) in the long wavelength limit *q* = **0**. For an unexcited semiconductor only the bare LO phonon resonance is found, which vanishes after photoexcitation and then two new hybrid modes arise after a while. By varying the carrier densities the position of the peaks in of Im(1 / ) *q***0** can be followed. As a reference, TO phonons in InP oscillate at TO / 2 9.7 THz, and LO phonons at LO 10.3 THz. In the time-domain, it was demonstrated both experimentally and theoretically that the time for establishment of fully developed hybrid modes depends on the carrier density as H 1.6 / 2 . This means that the buildup takes shorter time in denser systems, c.f. Fig 3. Typical values of H for InP would be ~ 140 fs at a carrier density of 18 3 10 cm .

#### **3.2.6 Magnetic systems**

Diluted ferromagnetic semiconductors are currently explored for use in information processing and storage devices. By introducing Mn atoms in GaAs or other III-V semiconductors a new, ferromagnetic material is obtained in which fast spin manipulation seems possible. Other candidate material classes investigated in the literature are chrome spinels, manganese oxides, transition metals, rare earths, pyrochlore, EuO, and EuS.

 Spin manipulation is a basic ingredient in spintronics, spin-photonics and quantum computation applications. Carrier density induced ferromagnetic effects can either be controlled by light or current via electrical contacts and gates. Magnetic field pulses and spin-currents can control the spin on the picosecond timescale, but for femtosecond manipulation, ultrashort laser pulses will have to be introduced. For the investigation of candidate materials a particular experimental method is used, ultrafast pump-probe magneto-optical spectroscopy. Here the pump optical pulse creates carrier populations 158 Femtosecond–Scale Optics

Huber et al. (Huber et al., 2005b) investigated the formation of hybrid modes in intrinsic InP using a 12-fs Ti:sapphire laser pump with a single-cycle THz probe at 28 THz, delayed with the time tD . Real-time evolution of the probe THz electric field was directly measured by means of the ultrabroadband electro-optic sampling technique with sampling delay time s t .

Delaying the probe with respect to the pump resulted in characteristic trailing oscillations long after the original probe signal had died out. The trailing oscillations represent a change in the probe signal which was monitored by varying the sampling delay time s t . Increasing the delay between pump and probe resulted in more trailing oscillation cycles. These oscillation cycles are manifestations of a quantum beating between the newly developing

 in Fig. 3. The hybrid resonances can be followed by a Fourier transformation with respect to s t of the change in the real-time probe signal for various pump-probe delays tD , thereby extracting

semiconductor only the bare LO phonon resonance is found, which vanishes after photoexcitation and then two new hybrid modes arise after a while. By varying the carrier

time-domain, it was demonstrated both experimentally and theoretically that the time for establishment of fully developed hybrid modes depends on the carrier density as

TO / 2 9.7 THz, and LO phonons at

. This means that the buildup takes shorter time in denser systems, c.f. Fig 3.

H for InP would be ~ 140 fs at a carrier density of 18 3 10 cm .

Diluted ferromagnetic semiconductors are currently explored for use in information processing and storage devices. By introducing Mn atoms in GaAs or other III-V semiconductors a new, ferromagnetic material is obtained in which fast spin manipulation seems possible. Other candidate material classes investigated in the literature are chrome

 Spin manipulation is a basic ingredient in spintronics, spin-photonics and quantum computation applications. Carrier density induced ferromagnetic effects can either be controlled by light or current via electrical contacts and gates. Magnetic field pulses and spin-currents can control the spin on the picosecond timescale, but for femtosecond manipulation, ultrashort laser pulses will have to be introduced. For the investigation of candidate materials a particular experimental method is used, ultrafast pump-probe magneto-optical spectroscopy. Here the pump optical pulse creates carrier populations

spinels, manganese oxides, transition metals, rare earths, pyrochlore, EuO, and EuS.

r r itt 1 2 12

W t , dt W t ,t e ,

1 2

in the long wavelength limit *q* = **0**. For an unexcited

*q***0** can be followed. As a reference, TO

LO 10.3 THz. In the

(24)

in a 7 - 60

 q D (t , )

*q qq*

t , W t , /V

1 r D D

The measured data allowed access to both real and imaginary parts of

 and

q d (t , )

>

densities the position of the peaks in of Im(1 / )

with *Vq* denoting the strength of the bare Coulomb interactions.

THz window.

upper and lower hybrid modes

the inverse dielectric function 1

phonons in InP oscillate at

**3.2.6 Magnetic systems**

 H 1.6 / 2

Typical values of

1

t

*q q*

whose subsequent interactions starts the magnetization dynamics which can be followed as a function of time via Faraday or Kerr rotation.

Natural timescales for phonon mediated spin relaxation and magnetization precession lie in the picosecond range. This was considered to be the ultimate limit for the attainable speed of magnetization switching prior to investigations with fs pulsed lasers. Then ultrafast demagnetization with the aid of fs lasers was demonstrated and this opened up new possibilities for high-speed magnetic devices. We shall first consider the metals because a large body of results has been obtained here and the corresponding theory has matured considerably in the last few years.

The ultrafast demagnetization process seen in transition metals such as Ni, Co, Fe (delocalized magnetism of *d*-electrons) and the rare-earth metal Gd (magnetism of *f*  electrons), is related to three characteristic timescales: 1) a femtosecond demagnetisation with timescale *τE* , 2) a picosecond magnetization recovery with timescale *τM*, and 3) a hundred picosecond-nanosecond magnetization precession.

Finding a satisfactory theory from first principles of the ultrafast physics going on under 1) is not straightforward. Instead, it seems at the moment more profitable to start with simpler models in order to describe the influences of the different subsystems (charge carriers, spin, photons, and phonons). For example, it is known that spin-orbit interactions are important, but modeling of all three temporal regimes on the same footing is currently not possible using quantum theory.

Early attempts assumed a three-temperature (3T) phenomenological model with rate equations for electron, phonon, and spin temperatures (energies). Using a spin temperature is quite unsatisfactory, considering that the spin system is not in equilibrium on the femtosecond timescale. If the spin is instead coupled to a two-temperature (2T) model for phonon and electron temperatures, we can obtain some improvement. Such a model deals with an energy flow concept, interpreting the ultrafast demagnetization as a non-coherent, "thermal" process. The energy goes from photon to electron and then to the spin system, without the need for stating any underlying quantum mechanism behind the spin flip.

Among the candidates for underlying quantum mechanisms are Elliott-Yafet (EY) electronphonon scattering, EY electron scattering with impurities or other electrons and the electron-electron inelastic exchange scattering. The EY model for phonon scattering seems to be a strong candidate in many of the metals.

Further progress in the field of phenomenological models (Atxitia & Chubykalo-Fesenko, 2010) has resulted in three central branches: 1) Langevin dynamics based on the Landau-Lifshitz-Gilbert (LLG) equation and classical Heisenberg Hamiltonian for localized atomic spin moments, 2) the Landau-Lifshitz-Bloch (LLB) micromagnetics model, and 3) the Koopmans's magnetization dynamics model (M3TM).

The LLB model includes the dynamics governed by both the atomistic LLG model and the M3TM model, with both classical and quantum versions. The LLB equation for a quantum spin *S* is based on the density matrix approach. For simple quantum spin systems modeled a s a two level system with spin-up and spin-down bands (i.e. *S* = ±1/2) it has been shown to be equivalent to the M3TM model (Atxitia & Chubykalo-Fesenko, 2010). The LLB equation for classical spins is equivalent to an ensemble of exchange-coupled atomistic spins modeled by LLG equations.

Quantum mechanisms responsible for the ultrafast demagnetization in the LLB model are described by the coupling parameter λ, which defines the rate of the spin flip. Relations

Time-Resolved Laser Spectroscopy of

free electron-hole pair or go via exciton recombination.

temperature *Tc* and free-carrier density *Neh*.

excitons and free carriers.

the populations.

Semiconductors - Physical Processes and Methods of Analysis 161

we embark on a study of the carrier cooling and hot phonon effects, we shall investigate another type of condensation. As the photoexcited carriers start to cool, the free carrier electron-hole pairs begin to condensate into excitons. Both excitions and free electron-hole pairs have rather long lifetimes, often in the µs range. Recombination can happen within a

The Saha equation (Kaindl et al., 2009, Suzuki & Shimano 2011), gives the equilibrium ratio between the density of excitons *Nex* and free carriers (i.e. when these two populations are

> 2 E eh B c 2 k T

where *μ* and *E0* are the exciton reduced mass and the binding energy, respectively. For a given total *e*-*h* pair density *N* = *Neh* + *Nex*, it yields the relationship between carrier

Free electron hole pair populations obtain thermal equilibrium with excitons within 2 ps. If we consider long timescales of some ps to hundreds of ps after the pump pulse with no external disturbance, this could be considered an instant equilibrium. The Saha equation implies that hot carriers have a low exciton density while cold carriers contain a high exciton density. However, the above equation only represents an exclusive equilibrium between free electron-hole pairs and excitons. As we saw above the carriers will still not be in equilibrium with the phonon system within a 2 ps timescale. What can happen under the carrier cooling phase is a thermal re-ionization of excitons by the phonons to create freeelectron-hole pairs. The result will be a slow formation time of excitons on the order of several hundreds of ps when lattice and excitons eventually reach an equilibrium. We can consider the temporal region of carrier cooling to be free from recombination lifetime effects. After the cooling phase a final equilibrium of all subsystems is reached by recombination of

This effect has been investigated in Si by Suzuki et al. (Suzuki & Shimano 2011) using optical pump and THz probe measurements. They evaluated the time-dependent fraction of free carriers and excitons. The waveforms of the THz pulse transmitted after the sample with and without the optical pump were recorded by electro-optic sampling. The complex transmittance change was then obtained by Fourier transformation, thereby yielding a photoinduced change in the complex dielectric function. To estimate the free *e*-*h* pair density at each delay time, the data were fitted with the free-carrier based Drude model in the low photon energy region (2.0–7.8 meV), where the Drude component dominates the spectrum. The Drude component decreases with increasing delay time, and an exciton component (Kaindl et al., 2009) emerged as a peak at ∼ 10-12 meV, an energy that corresponds to the 1*s*-2*p* transition of excitons in Si. Thus the formation dynamics of excitons was revealed through the observation of the 1*s*-2*p* transition at ∼ 3 THz. Contrary to a free-carrier behavior, the exciton density, as indicated by the 1*s*-2*p* absorption at 12 meV, gradually increased after the photoexcitation and reached a constant value after 400 ps for an initial lattice temperature of 30 K. Long after this period, recombination eventually will influence

As we have just seen, the interaction between the hot free electron-hole plasma and the phonons constitutes a very important basis for the formation of a stable exciton

0 B c

(25)

3

e

reciprocally thermalized) in the Boltzmann limit. For bulk materials it reads,

N 2kT 1

ex

N 4

between such parameters in the phenomenological models are a key to understand whether or not a given mechanism can act on more than one of the three characteristic timescales of the demagnetization process mentioned above.

The LLB equation can handle large spatial scales, which is a prerequisite for modeling multiscale magnetization dynamics. Coupling the spin to the electron temperature from the 2T model, we automatically include a purely carrier-based spin flip process. If both electron and phonon temperatures are coupled to the spin dynamics we automatically include the carrier-phonon scattering induced Elliott-Yafet mechanism. Coupled to the 2T model, the LLB equation has recently been shown to describe correctly all three stages of the ultrafast demagnetization processes: the sub-picosecond demagnetization, the picosecond magnetization recovery and the nanosecond magnetization precession (Atxitia & Chubykalo-Fesenko, 2010).

As regards semiconductors, similar magnetization effects have been demonstrated (Kapetanakis et al., 2009). Using density matrix equations of motion a nonequilibrium theory of ultrafast magnetization reorientation in ferromagnetic (Ga,Mn)As was presented. Interactions and coherent nonlinear optical effects were treated in a similar way as the Semiconductor Bloch equations. Both resonant and non-resonant photoexcitation were included, taking into account the relevant bands. A femtosecond collective spin tilt was induced by nonlinear, near-ultraviolet (3 eV), coherent photoexcitation with linearly polarized light. The magnetization was initiated by the interaction of non-thermal itinerant carriers (holes) with local Mn spins. The dynamics generated a subsequent uniform magnetic precession. This was interpreted as a possibility of non-thermal magnetization control by tuning the laser frequency and polarization direction.

## **3.3 Incoherent ultrafast regime**

We shall now consider ultrafast phenomena after coherence has been completely destroyed by scattering. This is the particular branch of scattering dominated carrier relaxation dynamics which is representative for traditional micreoelectronics devices.

When electrons are photoexcited high into the conduction band, scattering processes are initiated and the initial configuration of excited charge carriers in the bands starts to change. First the charge carriers thermalize among themselves within 1 ps due to carriercarrier scattering. Thermalization means that the charge carriers obtain a common temperature which is different from the lattice temperature. Then the cooling of the carriers towards the lattice temperature starts, and this can take tens to hundreds of picoseconds depending on the amount of photoexcited carriers and how high above the band gap the electrons were excited. Screening of the carrier-phonon scattering mechanisms at high photoexcitation densities can reduce the energy transfer rate to the lattice. Considerable feedback from the lattice to the carriers are related to bottleneck effects when the most active phonon types receive too much energy in a short time. These hot phonons obtain much higher temperatures than the less active phonons, which tend to have temperatures very close to the initial lattice temperature. Usually the energy of the pump pulse is very small in these kinds of experiments, so the hot phonons are essentially due to poor energy distribution between different phonon modes.  After the cooling has resulted in equilibration of carrier and lattice temperatures, a final 'condensation' process where electrons and holes recombine completes the photoexcitation cycle. Before 160 Femtosecond–Scale Optics

between such parameters in the phenomenological models are a key to understand whether or not a given mechanism can act on more than one of the three characteristic timescales of

The LLB equation can handle large spatial scales, which is a prerequisite for modeling multiscale magnetization dynamics. Coupling the spin to the electron temperature from the 2T model, we automatically include a purely carrier-based spin flip process. If both electron and phonon temperatures are coupled to the spin dynamics we automatically include the carrier-phonon scattering induced Elliott-Yafet mechanism. Coupled to the 2T model, the LLB equation has recently been shown to describe correctly all three stages of the ultrafast demagnetization processes: the sub-picosecond demagnetization, the picosecond magnetization recovery and the nanosecond magnetization precession (Atxitia &

As regards semiconductors, similar magnetization effects have been demonstrated (Kapetanakis et al., 2009). Using density matrix equations of motion a nonequilibrium theory of ultrafast magnetization reorientation in ferromagnetic (Ga,Mn)As was presented. Interactions and coherent nonlinear optical effects were treated in a similar way as the Semiconductor Bloch equations. Both resonant and non-resonant photoexcitation were included, taking into account the relevant bands. A femtosecond collective spin tilt was induced by nonlinear, near-ultraviolet (3 eV), coherent photoexcitation with linearly polarized light. The magnetization was initiated by the interaction of non-thermal itinerant carriers (holes) with local Mn spins. The dynamics generated a subsequent uniform magnetic precession. This was interpreted as a possibility of non-thermal magnetization

We shall now consider ultrafast phenomena after coherence has been completely destroyed by scattering. This is the particular branch of scattering dominated carrier relaxation

When electrons are photoexcited high into the conduction band, scattering processes are initiated and the initial configuration of excited charge carriers in the bands starts to change. First the charge carriers thermalize among themselves within 1 ps due to carriercarrier scattering. Thermalization means that the charge carriers obtain a common temperature which is different from the lattice temperature. Then the cooling of the carriers towards the lattice temperature starts, and this can take tens to hundreds of picoseconds depending on the amount of photoexcited carriers and how high above the band gap the electrons were excited. Screening of the carrier-phonon scattering mechanisms at high photoexcitation densities can reduce the energy transfer rate to the lattice. Considerable feedback from the lattice to the carriers are related to bottleneck effects when the most active phonon types receive too much energy in a short time. These hot phonons obtain much higher temperatures than the less active phonons, which tend to have temperatures very close to the initial lattice temperature. Usually the energy of the pump pulse is very small in these kinds of experiments, so the hot phonons are essentially due to poor energy distribution between different phonon modes.  After the cooling has resulted in equilibration of carrier and lattice temperatures, a final 'condensation' process where electrons and holes recombine completes the photoexcitation cycle. Before

control by tuning the laser frequency and polarization direction.

dynamics which is representative for traditional micreoelectronics devices.

the demagnetization process mentioned above.

Chubykalo-Fesenko, 2010).

**3.3 Incoherent ultrafast regime**

we embark on a study of the carrier cooling and hot phonon effects, we shall investigate another type of condensation. As the photoexcited carriers start to cool, the free carrier electron-hole pairs begin to condensate into excitons. Both excitions and free electron-hole pairs have rather long lifetimes, often in the µs range. Recombination can happen within a free electron-hole pair or go via exciton recombination.

The Saha equation (Kaindl et al., 2009, Suzuki & Shimano 2011), gives the equilibrium ratio between the density of excitons *Nex* and free carriers (i.e. when these two populations are reciprocally thermalized) in the Boltzmann limit. For bulk materials it reads,

$$\frac{\text{N}\_{\text{eh}}^2}{\text{N}\_{\text{ex}}} = \frac{1}{4} \left( \frac{2 \,\mu\text{k}\_{\text{B}}\text{T}\_{\text{c}}}{\hbar \pi} \right)^{\frac{3}{2}} \text{ e}^{\left( -\text{E}\_{\text{b}} \text{\AA}\_{\text{B}} \text{T}\_{\text{c}} \right)} \tag{25}$$

where *μ* and *E0* are the exciton reduced mass and the binding energy, respectively. For a given total *e*-*h* pair density *N* = *Neh* + *Nex*, it yields the relationship between carrier temperature *Tc* and free-carrier density *Neh*.

Free electron hole pair populations obtain thermal equilibrium with excitons within 2 ps. If we consider long timescales of some ps to hundreds of ps after the pump pulse with no external disturbance, this could be considered an instant equilibrium. The Saha equation implies that hot carriers have a low exciton density while cold carriers contain a high exciton density. However, the above equation only represents an exclusive equilibrium between free electron-hole pairs and excitons. As we saw above the carriers will still not be in equilibrium with the phonon system within a 2 ps timescale. What can happen under the carrier cooling phase is a thermal re-ionization of excitons by the phonons to create freeelectron-hole pairs. The result will be a slow formation time of excitons on the order of several hundreds of ps when lattice and excitons eventually reach an equilibrium. We can consider the temporal region of carrier cooling to be free from recombination lifetime effects. After the cooling phase a final equilibrium of all subsystems is reached by recombination of excitons and free carriers.

This effect has been investigated in Si by Suzuki et al. (Suzuki & Shimano 2011) using optical pump and THz probe measurements. They evaluated the time-dependent fraction of free carriers and excitons. The waveforms of the THz pulse transmitted after the sample with and without the optical pump were recorded by electro-optic sampling. The complex transmittance change was then obtained by Fourier transformation, thereby yielding a photoinduced change in the complex dielectric function. To estimate the free *e*-*h* pair density at each delay time, the data were fitted with the free-carrier based Drude model in the low photon energy region (2.0–7.8 meV), where the Drude component dominates the spectrum. The Drude component decreases with increasing delay time, and an exciton component (Kaindl et al., 2009) emerged as a peak at ∼ 10-12 meV, an energy that corresponds to the 1*s*-2*p* transition of excitons in Si. Thus the formation dynamics of excitons was revealed through the observation of the 1*s*-2*p* transition at ∼ 3 THz. Contrary to a free-carrier behavior, the exciton density, as indicated by the 1*s*-2*p* absorption at 12 meV, gradually increased after the photoexcitation and reached a constant value after 400 ps for an initial lattice temperature of 30 K. Long after this period, recombination eventually will influence the populations.

As we have just seen, the interaction between the hot free electron-hole plasma and the phonons constitutes a very important basis for the formation of a stable exciton

Time-Resolved Laser Spectroscopy of

"hot", as illustrated in Fig. 5.

1

2

3

4

N

5

6

Semiconductors - Physical Processes and Methods of Analysis 163

not large. Electrons in MCT will mainly interact with small wavevector longitudinal optical phonons, since electrons occupy a very narrow band in *k*-space with small electron wave vectors. Thus, only longitudinal optical phonons with small wavevectors will become really

> 0.5 ps 2 ps 10 ps 20 ps 30 ps

0,0 5,0x105 1,0x106 1,5x106 2,0x106

Fig. 5. Longitudinal phonon occupation at select times in simulations of coupled electron

Longitudinal optical phonon modes with larger wavevectors have a phonon occupancy

Transversal phonons do not become hot either, as we can see from Fig. 6. Holes receive less of the laser energy and occupy a much broader range of wave vectors and will therefore interact with a much broader range of phonon wave vectors. Because the phonons that are hot have small wave vectors these phonons will not interact as efficiently with the broad-

At high laser fluences carrier densities will increase. Then the interaction with polar optical phonons tends to be weakened by carrier induced screening, and transversal phonons becomes a more important relaxation route. Another complication at high carrier densities and laser fluences is the occurrence of plasmons or longitudinal hybrid mode oscillations. An individual carrier can also be scattered by plasmons in elemental semiconductors such

In polar materials however, such as the compound semiconductors GaAs and MCT the plasmon mode and the longitudinal optical mode combine via their respective long range electrical fields, as we saw earlier. The combined modes replace the plasmon and polar optical mode as scattering agents of the charge carriers, and can be simulated by the same

and hole ensembles at 16 3 *n* 10 cm , with hot phonons accounted for.

corresponding more closely to the lattice temperature.

wavevector hole ensemble as one could expect.

as Ge, Si etc., but not by hybrid modes.

method as we have outlined here.

Phonon wavenumber (cm-1)

population. In (Suzuki & Shimano 2011), cooling of the hot carriers was studied using a set of cooling rate expressions. A very flexible, but more computationally heavy alternative for studying this particular cooling dynamics would be to use a particle-based Monte Carlo (MC) simulation. The flexibility lies in the fact that very few assumptions have to be made about the system at hand, so that not only special cases can be considered. In the following we present some of our recent MC results in order to demonstrate the physics behind the dissipation of short laser pulses. In this case we shall investigate a low-gap semiconductor, Hg0.72Cd0.28Te (MCT). As shown in Fig. 2, laser energy can be exchanged between carriers and phonons in several ways before finally escaping through the phonon distributions. Plots of the rate of energy exchange for Hg0.72Cd0.28Te due to the interactions are shown in Fig. 4, for coupled MC simulations with hot phonons and exact screening.

Longitudinal optical phonons constitute the most important route for energy relaxation when carrier densities are low. Almost all the energy relaxed out of the system goes by this route, through polar optical interaction with high energy electrons. Relaxation through transversal phonons occurs via the non-polar deformation potential interaction and is usually less important.

Fig. 4. Energy transfer rates of the different mechanisms at a carrier density of 16 3 *n* 10 cm . Hot phonons are simulated and the screening length is calculated from the simulated distributions.

We also see from the figures that the green graph representing the hole energy loss rate to longitudinal optical phonons goes slightly negative after 5 ps. This means that some energy is flowing into the hole ensemble indirectly via the mainly electron-heated hot longitudinal optical phonons. This can help energy relaxation and even out the temperature difference between electrons and holes. For the particular case investigated here however, the flow is 162 Femtosecond–Scale Optics

population. In (Suzuki & Shimano 2011), cooling of the hot carriers was studied using a set of cooling rate expressions. A very flexible, but more computationally heavy alternative for studying this particular cooling dynamics would be to use a particle-based Monte Carlo (MC) simulation. The flexibility lies in the fact that very few assumptions have to be made about the system at hand, so that not only special cases can be considered. In the following we present some of our recent MC results in order to demonstrate the physics behind the dissipation of short laser pulses. In this case we shall investigate a low-gap semiconductor, Hg0.72Cd0.28Te (MCT). As shown in Fig. 2, laser energy can be exchanged between carriers and phonons in several ways before finally escaping through the phonon distributions. Plots of the rate of energy exchange for Hg0.72Cd0.28Te due to the interactions are shown in Fig. 4, for coupled MC simulations

Longitudinal optical phonons constitute the most important route for energy relaxation when carrier densities are low. Almost all the energy relaxed out of the system goes by this route, through polar optical interaction with high energy electrons. Relaxation through transversal phonons occurs via the non-polar deformation potential interaction and is

0 5 10 15 20 25 30

Time (ps)

Fig. 4. Energy transfer rates of the different mechanisms at a carrier density of 16 3 *n* 10 cm . Hot phonons are simulated and the screening length is calculated from the simulated

We also see from the figures that the green graph representing the hole energy loss rate to longitudinal optical phonons goes slightly negative after 5 ps. This means that some energy is flowing into the hole ensemble indirectly via the mainly electron-heated hot longitudinal optical phonons. This can help energy relaxation and even out the temperature difference between electrons and holes. For the particular case investigated here however, the flow is

e to L-phon via PO h to L-phon via PO h to L-phon via nonPO h to T-phon via nonPO Out of the system via L-phon Out of the system via T-phon Into e ensemble via C-C Into h ensemble via C-C

with hot phonons and exact screening.

0

1x1011

2x1011

3x1011

4x1011

5x1011

6x1011

usually less important.

 Energy loss rate [eV/(fs cm-3

distributions.

)]

not large. Electrons in MCT will mainly interact with small wavevector longitudinal optical phonons, since electrons occupy a very narrow band in *k*-space with small electron wave vectors. Thus, only longitudinal optical phonons with small wavevectors will become really "hot", as illustrated in Fig. 5.

Fig. 5. Longitudinal phonon occupation at select times in simulations of coupled electron and hole ensembles at 16 3 *n* 10 cm , with hot phonons accounted for.

Longitudinal optical phonon modes with larger wavevectors have a phonon occupancy corresponding more closely to the lattice temperature.

Transversal phonons do not become hot either, as we can see from Fig. 6. Holes receive less of the laser energy and occupy a much broader range of wave vectors and will therefore interact with a much broader range of phonon wave vectors. Because the phonons that are hot have small wave vectors these phonons will not interact as efficiently with the broadwavevector hole ensemble as one could expect.

At high laser fluences carrier densities will increase. Then the interaction with polar optical phonons tends to be weakened by carrier induced screening, and transversal phonons becomes a more important relaxation route. Another complication at high carrier densities and laser fluences is the occurrence of plasmons or longitudinal hybrid mode oscillations. An individual carrier can also be scattered by plasmons in elemental semiconductors such as Ge, Si etc., but not by hybrid modes.

In polar materials however, such as the compound semiconductors GaAs and MCT the plasmon mode and the longitudinal optical mode combine via their respective long range electrical fields, as we saw earlier. The combined modes replace the plasmon and polar optical mode as scattering agents of the charge carriers, and can be simulated by the same method as we have outlined here.

Time-Resolved Laser Spectroscopy of

**4. Conclusions**

**5. References**

195

Semiconductors - Physical Processes and Methods of Analysis 165

results can be incorporated in laser irradation simulations. In addition, the data must also take into account the possibility of different carrier and lattice temperatures, a feature which

As we have seen, no single post evaluation tool or method will suffice to cover all pulselength ranges, and not all tools are suited for a given experiment. Combining time resolved laser experiments with satisfactory analysis of the results can therefore become a demanding task. When evaluating potential analysis tools, we will take the view that it is vital to have access to flexible methods, even if we see that this often comes at the expense of heavy numerics. It is true that the analysis of e.g. a semiconductor device by simple analytical models often benefits the physical understanding. But in the field of laser spectroscopy many processes contribute in unpredictable ways, and to have some of these ways sporadically blocked by a priori assumptions or limitations within the method of analysis can become a frustrating experience. Some of the methods reviewed in this chapter introduce substantial simplifications, and should be used, but not without focusing on their limitations and assuring the access to more flexible tools. As regards the future prospects of laser spectroscopy, it is clear that the techniques developed in this field will play a crucial role in achieving quantum coherent control of information bearing variables such as spin as

we have seldom seen considered for cases involving low-intensity laser irradiation.

well as continuing to be a central method for device and material characterization.

Ashcroft, N.W. & Mermin, N.D. (1976). *Solid State Physics*, Holt-Saunders Japan LTD, Tokyo Atxitia U. & Chubykalo-Fesenko O. (2010). Ultra-fast magnetization rates within the Landau-Lifshitz-Bloch model. *ArXiv:1054v1 [cond-mat.mtr-sci]* (23 Nov 2010*)* Axt, V.M. & Stahl, A. (1994*).* A Dynamics-Controlled Truncation Scheme for the Hierarchy

Butov, L.V.; Lai, C.W.; Ivanov, A. L., Gossard, A.C. & Chemla, D.S. (2002). Experimental approaches for exciton Bose-Einstein condensation. *Nature*, Vol.417, pp. 47 Blanchard, F.; Sharma, G.; Razzari, L.; Ropagnol, X.; Bandulet, H.C.; Vidal, F.; Morandotti,

*Selected Topics in Quantum Electronics*, Vol.17, No.1, January/February 2011 Chang, E; Bussi, G.; Ruini, A & Molinari, E. (2004). Excitons in Carbon Nanotubes: An *Ab*

Christensen, N.E.; Svane, A.; Cardona, M.; Chantis, A.; Laskowski, N.R.; van Schilfgaarde,

Combescot, M.; Betbeder-Matibet, O. & Leuenberger, M.N. (2009). Analytical approach to

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M. & Kotani, T. (2011). Calculations of quasi-particle spectra of semiconductors

Fig. 6. Transversal phonon occupation at select times in simulations of coupled electron and hole ensembles at 16 3 *n* 10 cm , with hot phonons accounted for.

## **3.4 Long pulse regime**

We shall devote some space (albeit small) also to this case, because of all the non-linear optical effects that appear. Using infrared (IR) wavelengths, intraband free carrier (FCA) and inter-valence band (IVA) absorption between e.g. the light and heavy hole bands can be studied, see Fig. 1. Although the total number of carriers generated in the conduction band can be comparable to that of a short pulse experiment, the majority of the carriers are well thermalized at temperatures which lie close to the overall lattice temperature. Nevertheless, differences between carrier and lattice temperature smaller than 100 K can have a major impact on processes occurring near the band gap, according to (Storebo et al., 2010). For example, it was found that a slightly elevated carrier temperature can strongly reduce the Pauli-induced saturation or "bleaching" of one-photon absorption across the band gap, or affect other long-timescale parameters, like carrier lifetimes and impact ionization rates. The difficulties within the modeling in this regime lies not only in the solution of many coupled differential equations, but also to a large degree on severe inaccuracies in recombination rates and TPA, FCA, IVA absorption coefficients which has now lasted for decades. The discrepancies have persisted because it has been experimentally very difficult to isolate and quantify all the simultaneous absorption and recombination effects. Theoretical estimates could be used to eliminate the uncertainties, but the methods used in the past have proven much too primitive for this task. With the advent of *ab-initio* band structure numerical codes it will become possible to obtain accurate estimates of each process, and examples of progress in this direction will become available in the near future. An important point which we would like to emphasize in this context is that even if the data is obtained numerically, it is crucial for the usefulness of the data that analytical expressions are extracted, so that the results can be incorporated in laser irradation simulations. In addition, the data must also take into account the possibility of different carrier and lattice temperatures, a feature which we have seldom seen considered for cases involving low-intensity laser irradiation.

## **4. Conclusions**

164 Femtosecond–Scale Optics

 0.5 ps 2.5 ps 5 ps 10 ps 20 ps

0,0 5,0x106 1,0x107 1,5x107 2,0x107

)

Phonon wavenumber (cm-1

Fig. 6. Transversal phonon occupation at select times in simulations of coupled electron and

We shall devote some space (albeit small) also to this case, because of all the non-linear optical effects that appear. Using infrared (IR) wavelengths, intraband free carrier (FCA) and inter-valence band (IVA) absorption between e.g. the light and heavy hole bands can be studied, see Fig. 1. Although the total number of carriers generated in the conduction band can be comparable to that of a short pulse experiment, the majority of the carriers are well thermalized at temperatures which lie close to the overall lattice temperature. Nevertheless, differences between carrier and lattice temperature smaller than 100 K can have a major impact on processes occurring near the band gap, according to (Storebo et al., 2010). For example, it was found that a slightly elevated carrier temperature can strongly reduce the Pauli-induced saturation or "bleaching" of one-photon absorption across the band gap, or affect other long-timescale parameters, like carrier lifetimes and impact ionization rates. The difficulties within the modeling in this regime lies not only in the solution of many coupled differential equations, but also to a large degree on severe inaccuracies in recombination rates and TPA, FCA, IVA absorption coefficients which has now lasted for decades. The discrepancies have persisted because it has been experimentally very difficult to isolate and quantify all the simultaneous absorption and recombination effects. Theoretical estimates could be used to eliminate the uncertainties, but the methods used in the past have proven much too primitive for this task. With the advent of *ab-initio* band structure numerical codes it will become possible to obtain accurate estimates of each process, and examples of progress in this direction will become available in the near future. An important point which we would like to emphasize in this context is that even if the data is obtained numerically, it is crucial for the usefulness of the data that analytical expressions are extracted, so that the

hole ensembles at 16 3 *n* 10 cm , with hot phonons accounted for.

1,11150

**3.4 Long pulse regime**

1,11152

1,11154

1,11156

1,11158

N

1,11160

1,11162

1,11164

As we have seen, no single post evaluation tool or method will suffice to cover all pulselength ranges, and not all tools are suited for a given experiment. Combining time resolved laser experiments with satisfactory analysis of the results can therefore become a demanding task. When evaluating potential analysis tools, we will take the view that it is vital to have access to flexible methods, even if we see that this often comes at the expense of heavy numerics. It is true that the analysis of e.g. a semiconductor device by simple analytical models often benefits the physical understanding. But in the field of laser spectroscopy many processes contribute in unpredictable ways, and to have some of these ways sporadically blocked by a priori assumptions or limitations within the method of analysis can become a frustrating experience. Some of the methods reviewed in this chapter introduce substantial simplifications, and should be used, but not without focusing on their limitations and assuring the access to more flexible tools. As regards the future prospects of laser spectroscopy, it is clear that the techniques developed in this field will play a crucial role in achieving quantum coherent control of information bearing variables such as spin as well as continuing to be a central method for device and material characterization.

## **5. References**

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Time-Resolved Laser Spectroscopy of

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**0**

**7**

*México*

**Lasers in Atomic Collisions, Cold Plasma**

In the more than 50 years since its conception, the laser and its variety of applications has covered most of the scientific and technological areas of science, from fundamental physics to biophysics and medicine. Probably, the scientific area that has benefited the most is the same area from where the laser originally came: Atomic, Molecular, and Optical (AMO) Physics. The purpose of this chapter is to present, through a selection of examples, how the applications of the laser in atomic collisions, cold plasmas and cold atom physics is still opening new areas of research. The examples chosen are a sample of what is the current research at the Instituto de Ciencias Nucleares (Nuclear Science Institute) and the Instituto de Ciencias Físicas (Physics Science Institute) at the Universidad Nacional Autónoma de México. We hope that, in choosing these examples, the reader will have the opportunity to get a wide

The chapter is separated in three sections. Section 2 deals with applications of numerical calculations related to laser assisted collisions, in particular the chapter deals with Finite Difference methods to the calculation of charge transfer cross sections. It shows the bits and pieces of how this numerical technique can be implemented. Section 3 shows an example of a calculation for experimental work with cold atoms. It discusses the technique of stimulated Raman adiabatic passage to produce highly excited Rydberg states. Finally, Sec. 4 shows the reader applications of the Optogalvanic Effect, and in particular to the new emergence of applications that are coming about due to the availability of solid state sources in the UV, and mid infrared. The applications of this technique go from simple basic physics research, to

The study of ion-atom and ion-molecule interactions in the presence of an intense laser is a new and exciting field in atomic physics. Laser intensities up to *I* = 3.5 × 1016 W/cm2 that correspond to the atomic unit of electric field *E* = 5.1 × 109 V/cm are so intense that compete with the Columbic interaction force in the control of the electron dynamics. As a result, atoms and molecules in the presence of intense laser fields exhibit new properties and behavior which can be studied by means of multiphotonic processes. Those properties generate new behaviors of matter in the presence of intense lasers, with applications that go

view of three main areas of research in applications of lasers to AMO physics.

National Security, Biophysics and Medical applications.

**2. Laser assisted charge transfer collisions**

**1. Introduction**

R. Cabrera-Trujillo, J. Jiménez-Mier and A. M. Juárez

**and Cold Atom Physics**

*Universidad Nacional Autónoma de México*