Response of Si and InSb to ultrafast laser pulsesdtraian/laser-si-insb.pdf · ductors Si and InSb, an orthogonal tight-binding model was used. A nonorthogonal, density-func-tional-based

Response of Si and InSb to ultrafast laser pulses

Traian Dumitrica1; 2, Andrea Burzo1, Yusheng Dou1; 3, and Roland E. Allen*; 1

1 Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station,TX 77843, USA

2 Department of Mechanical Engineering and Materials Science, and Center for Nanoscale Scienceand Technology, Rice University, Houston, TX 77251, USA

3 Department of Chemistry and Center for Ultrafast Laser Applications, Princeton University,Princeton, NJ 08544, USA

Received 1 March 2004, revised 1 May 2004, accepted 31 May 2004Published online 19 July 2004

PACS 78.20.Bh, 78.47.+p

We present simulations of the response of Si and InSb to femtosecond-scale laser pulses of variousintensities. In agreement with the experiments by various groups on various materials, there is a non-thermal phase transition for each of these semiconductors above a threshold intensity. Our simulationsemploy semiclassical electron-radiation-ion dynamics (SERID), a technique which is briefly describedin the text. We also introduce a new addition to the technique, which provides a simple treatment ofthe correction due to motion of the atomic-orbital basis functions. We find that this correction is smallin the present context, but it may be substantial in situations with more rapid atomic motion. Ourexpression for this correction is remarkably simple to employ because it amounts to nothing more thana generalized Peierls substitution.

# 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

The interaction of matter with ultrafast and ultra-intense laser pulses is a current frontier of science[1–25]. New discoveries often result from the ability to explore a new regime. Here one is exploringboth extremely short time scales (below one hundred femtoseconds) and extremely high intensities(above one terawatt per square centimeter). The usual approximations of theoretical physics and chem-istry break down under these conditions, and both electrons and atoms exhibit new kinds of behavior.The experimental techniques to achieve these conditions are relatively new, and so is the theoreticalapproach outlined below.

A first-principles formulation of our method has been presented and employed elsewhere [10]. How-ever, we find that a tight-binding representation is preferable for practical calculations: (1) Since the timestep is of order 30 attoseconds, and the system may contain many atoms, the method must be computation-ally fast. (2) A tight-binding representation involves chemically-meaningful basis states which are loca-lized on the atoms, and which have the same symmetries as atomic orbitals. One can then immediatelyinterpret the results using intuitive ideas based on ground-state and excited-state chemistry [26–28].

Our method is called semiclassical electron-radiation-ion dynamics (SERID), because the nuclear(or ion-core) motion and the radiation field are both treated classically, while the quantum dynamicsof the electrons is treated via the time-dependent Schr€odinger equation. A pleasant feature of thissemiclassical approach is the fact that it effectively includes n-photon and n-phonon processes corre-

phys. stat. sol. (b) 241, No. 10, 2331–2342 (2004) / DOI 10.1002/pssb.200404934


* Corresponding author: e-mail: [email protected], Phone: +01 979 845 4341, Fax: +01 979 845 2590

sponding to absorption and stimulated emission [11, 12]. The method is applicable to general nonadia-batic processes, including interactions with an intense radiation field. The vector potential Aðx; tÞ forthe field is included in the electronic Hamiltonian H through a time-dependent Peierls substitution.The time-dependent Schr€odinger equation is solved with an algorithm that conserves probability andsatisfies the Pauli exclusion principle. Finally, the atomic forces are obtained from a generalized Hell-mann-Feynman theorem, which may also be interpreted as a generalized Ehrenfest theorem.

The value of simulations is that one can monitor both those properties which are experimentallyaccessible and those which are not. In the present context, the experiments focus principally on opti-cal properties such as the dielectric function E wð Þ, which probes the electronic structure, and thesecond-order nonlinear susceptibilty c 2ð Þ wð Þ, which probes the symmetry of the material. In our theo-retical studies of GaAs presented elsewhere [5, 6, 9], we have calculated both E wð Þ and c 2ð Þ wð Þ, andfound good agreement with the experiments.

In addition, however, the simulations reveal the behavior of other properties as functions of time, inthe few hundred femtoseconds following application of a laser pulse, and their dependence on theparameters of the pulse, including duration, intensity, and photon energy. In Fig. 1, for example, weshow the time evolution of the average displacement of a Si atom from its initial position following alaser pulse with a duration of 70 femtoseconds (full width at half maximum), a photon energy of1:95 eV, and various intensities of the laser radiation field. This same figure shows what fraction ofthe electrons intially in the valence band are promoted to the excited states of the conduction band,for various intensities. It can be seen that the lattice is destabilized for amplitudes A0 which are great-er than 2:00 G cm, or alternatively when more than about 10% of the electrons are promoted toconduction-band states. This can be interpreted as a very substantial weakening of the tetrahedralbonding, since in a simple picture the valence band and conduction band are respectively composed ofbonding and antibonding states. Further results for Si are shown in Figs. 2–4.

2 Method

The equations of SERID are essentially a time-dependent Schr€odinger equation for the electrons

i�h@Y j

@t¼ S�1 �H �Y j ð2:1Þ

2332 T. Dumitrica et al.: Ultrafast laser pulses, InSb and Si

Fig. 1 Left panel: Average displacement of a Si atom from its initial position as a function of time, for variouslaser radiation intensities. The amplitude of the vector potential A is given in G cm. As shown in Ref. 5, anamplitude A0 ¼ 1:00 G cm corresponds to a fluence of 0:815 kJ/m2, and the fluence is proportional to the squareof the amplitude for a given pulse shape and duration. In all the simulations for Si and InSb described here, thepump pulse had a FWHM duration of 70 femtoseconds with a photon energy �hw equal to 1:95 eV. The lattice isclearly destabilized for the four highest intensities, corresponding to amplitudes greater than A0 ¼ 2:00 G cm.Right panel: Time evolution of the occupancy of the excited states (i.e., conduction band) in Si, again for variousvalues of A0. It can be seen that the lattice is destabilized when more than about 10% of the electrons arepromoted to excited states.


together with a Newton’s equation for the atoms (or, more precisely, ion cores):

M‘d2X‘a

dt2¼ � 1

2

PjY y

j �@H@X‘a

� i�h@S@X‘a

@

@t

� ��Y j þ h:c:� @Urep

@X‘a: ð2:2Þ

Here S is the overlap matrix and Urep is a summation over repulsive potentials which model the ion-ion repulsion (together with the negative of the electron–electron repulsion, which is doubly countedin the one-electron Hamiltonian). Both of the above equations follow from extremalization of thetight-binding version of the Lagrangian for particles treated in a time-dependent self-consistent-fieldapproximation [3, 4]. We can adopt the point of view that each electron is labeled by j and has itsown time-dependent state vector Y j. If there are N tight-binding basis functions in the system, Y j isN-dimensional, and the time-dependent Hamiltonian H is N � N.

A proper first-principles treatment of nonequilibrium problems (including many-body effects) wouldemploy methods like those of Martin and Schwinger [29], Kadanoff and Baym [30], or Keldysh [31],with a self-energy that is even more complicated than that for equilibrium or quasiequilibrium pro-blems [32]. In the present context, however, it is a reasonable approximation to adopt a time-depen-dent self-consistent field picture.

phys. stat. sol. (b) 241, No. 10 (2004) / www.pss-b.com 2333

0200

400time (fs)

0 1 2 3 4 5 6

distance (Å)

02468

10

Number of Neighbors

0200

400time (fs)

0 1 2 3 4 5 6

distance (Å)

02468

10

Number of Neighbors

Fig. 2 Time evolution of the pair correlation function in Si for a field strength of A0 ¼ 1:00 G cm (left) and2:45 G cm (right). At t ¼ 0, there are peaks corresponding to first, second, etc. neighbors. For the lower intensity,this structure is only broadened after a few hundred femtoseconds. For the higher intensity, however, it is essen-tially lost at 500 fs, indicating a change from the original tetrahedral bonding.

-8

-6

-4

-2

0

2

4

6

0 50 100 150 200 250 300 350 400 450

Ene

rgy

(eV

)

time (fs)

-8

-6

-4

-2

0

2

4

6

8

0 50 100 150 200 250 300 350 400 450

Ene

rgy

(eV

)

time (fs)

Fig. 3 Electronic energy eigenvalues for Si at the ð2p=aÞð1=4; 1=4; 1=4Þ point as a function of time, withA0 ¼ 1:00 G cm (left) and 2:45 G cm (right). The band gap at this particular k-point is larger than the fundamen-tal band gap, but at the higher intensity it has collapsed to zero, demonstrating that the material is now metallicrather than semiconducting. The rapid oscillations during application of the pulse are due to the Peierls factor in(2.3).


First-principles molecular dynamics (which was introduced just after tight-binding molecular dy-namics [33, 34]) is ordinarily an accurate method, because the local density approximation for ex-change and correlation is quite good for total energies. However, the direct application of density-functional methods is not ordinarily suitable for the kind of problems addressed here, which involveexcited states and nonadiabatic processes. The excited states are typically too low for semiconductors,and the simulation of nonadiabatic processes requires a time step of about 30 attoseconds. It is there-fore prohibitively expensive to treat large systems in a true first-principles simulation.

For these reasons and others, SERID appears to be the preferred method for simulations of theinteraction of light with matter. On the other hand, the results of Ref. [7] indicate that a density-functional-based approach is more accurate than one in which the parameters are fitted more naively.The approach of Sankey, Demkov, and coworkers [35–40] is also very useful and promising, since itpermits self-consistent calculations.

Equations (2.1) and (2.2) represent the nonorthogonal formulation of SERID. These same equationscan, however, be cast into an orthogonalized form. In the simulations reported here, for the semicon-ductors Si and InSb, an orthogonal tight-binding model was used. A nonorthogonal, density-func-tional-based model has been used in more recent simulations for the fullerenes and various organicmolecules [7, 8, 11, 12].

The electrons and ions are coupled in (2.1) and (2.2), because H is a function of the ion coordi-nates and the forces on the ions are influenced by the electronic states. We now need to couple theelectrons to the radiation field. (One can also easily couple the ions to the electromagnetic field, butthis is a minor effect if the field oscillates on a one femtosecond time scale, two orders of magnitudesmaller than the response time of the ions.) The most convenient way to introduce the field into theelectronic Hamiltonian is to employ a time-dependent Peierls substitution [41, 42]:

HabðX � X0Þ ¼ H0abðX � X0Þ exp iq

�hcA � ðX � X0Þ

� �ð2:3Þ


0

200

400time (fs)

1 2 3 4 5 6

Energy (eV)

Im ε

0

200

400time (fs)

1 2 3 4 5 6

Energy (eV)

Im ε

0

200

400time (fs)

1 2 3 4 5 6

Energy (eV)

Im ε

0

200

400time (fs)

1 2 3 4 5 6

Energy (eV)

Im ε

Fig. 4 Time evolution of the dielectric function of Si for four intensities: A0 ¼ 1:73; 2:00; 2:24, and 2:45 G cm. Inthis figure, Im EðwÞ is shown for a time interval of 450 fs, with the pulse applied between 50 and 190 fs. For the higherintensities, the loss of structural features in Im EðwÞ, and the emergence of a Drude-like peak at low frequencies,demonstrate a change from the original tetrahedral bonding, together with an onset of metallic behavior.


where q ¼ �e is the electron’s charge. This approach requires no additional parameters and is validfor strong time-dependent fields.

Solution of the ionic equations of motion (2.1) is essentially the same as in tight-binding moleculardynamics [33, 34] and the velocity Verlet method appears to be optimal. Solution of (2.1) requiresmore care, since a naive algorithm for this first order equation fails to conserve probability. In earlierwork we followed a standard prescription. More recently a still better method has been introduced byTorralva in Ref. [7]. In this approach, the first-order term in a Dyson-like series for the time evolutionoperator U t þ Dt; tð Þ is written in unitary form:

Uðt þ Dt; tÞ ¼ 1þ i2�h

ðtþDt

t

dt0 H t0ð Þ

0@ 1A�1

1� i2�h

ðtþDt

t

dt0 H t0ð Þ

0@ 1A: ð2:4Þ

After evaluating each element of Uðt þ Dt; tÞ with Simpson’s rule (for example), one then obtains theelectron states from

Y j t þ Dtð Þ ¼ Uðt þ Dt; tÞ �Y j tð Þ : ð2:5Þ

With this algorithm, unitarity (i.e., orthonormality of the one-electron states Y j) is preserved to themachine accuracy of better than 10�12.

3 Results

In experiments by several groups [13–22], the time evolution of the dielectric function EðwÞ and thesecond-order nonlinear susceptibility cð2Þ has been measured in GaAs and Si, using a probe pulse witha photon energy of about 2:2 eV after excitation with an intense pump pulse at about 1:9 eV. Theobservations indicate that the response of a semiconductor to an ultrafast laser pulse, with a durationof 100 femtoseconds or less, is fundamentally different from its response to a pulse with a duration of1 picosecond or more. Whereas the longer pulses appear to produce ordinary heating of the sample byphonon emission, there is convincing evidence that ultrafast pulses induce a structural transition bydirectly destabilizing the atomic bonds.

Using the method outlined above, we have previously performed calculations for the electronic andstructural response of GaAs to ultra-intense and ultrashort laser pulses [5, 6, 9]. Here we report simi-lar calculations for Si and InSb. The time dependence of the electronic states and ionic positions wascalculated as described above, and the imaginary part of the dielectric function was obtained from theformula [42]

Im EðwÞ / 1w2

Pn;m;k

½ fnðkÞ � fmðkÞ� pnmðkÞ � pmnðkÞ dðw� wmnðkÞÞ : ð3:1Þ

The tight-binding model of Vogl et al. was employed [28], together with Harrison’s r�2 scaling for theinteratomic matrix elements [26]. We used a nonstandard repulsive potential with the formuðrÞ ¼ a=r4 þ b=r6 þ g=r8, which is a generalized form of Harrison’s r�4 scaling. The three para-meters a, b, and g were fitted to the experimental values of the cohesive energy, interatomic spacing,and bulk modulus – properties associated with the zeroth, first, and second derivative of the totalenergy. A cubical cell containing eight atoms was used for the present simulations.

Representative results for Si are shown in Figs. 1–4. As indicated in the figure captions, the timeevolution of the atomic displacements, the dielectric function, and the pair correlation function demon-strate that the threshold intensity is above 2.0 G cm. Also, a laser pulse with this intensity promotessomewhat above 10% of the electrons to excited states. One can clearly observe the destabilization ofthe covalent bonding at high intensities as electrons are excited across the band gap [43–46], frombonding to antibonding states.

The power of simulations is that one can calculate various properties which show in detail how theelectrons and ions respond. These include the population of excited states, the time-dependent band



structure, and the atomic pair-correlation function. Typical results are shown in Figs. 1–7. For exam-ple, the eigenvalues enðkÞ at the special point k ¼ ð1=4; 1=4; 1=4Þð2p=aÞ are plotted as functions oftime for Si in Fig. 3. Notice that the band gap at this point (which is larger than the fundamental bandgap at ð0; 0; 0Þ) has completely closed up for the higher intensity shown (A0 ¼ 2:45 G cm) because ofthe large atomic displacements associated with lattice destabilization.

4 Correction for motion of atomic-orbital basis functions

In this section we introduce an addition to the technique which enables us to eliminate one of theapproximations in the simple version of SERID employed above. In this version we have not allowedfor the fact that the time dependence of an electron wavefunction results from both the time depen-dence of the coefficients Y j x; tð Þ and that of the atomic-orbital basis functions fa x� Xð Þ, whichmove with the atomic nuclei rather than being static and thus time-independent.


0 100 200 300 400 500 600 700 800 900time (fs)

0

1

2

3

4

5

disp

lace

men

t (A

ngst

rom

s)

InSb average displacement

A=3.0A=2.82A=2.45A=2.23A=2.0A=1.58A=1.41A=1.28A=1.0A =0.7

0 100 200 300 400 500 600 700 800 900time (fs)

0.0

0.1

0.2

0.3

0.4

Occupancy

A=2.82A=2.45A=2.23A=2.0A=1.58A=1.41A=1.0

Fig. 5 (online colour at: www.pss-b.com) Left panel: Average distance moved by an atom in InSb, as a functionof time for various pulse intensities. Right panel: Percentage of valence electrons in InSb promoted to excitedstates, as a function of time and for varying pulse intensities.

A = 1.5

0100

200300

400

time (fs)

0 1 2 3 4 5 6

energy (eV)

0100200300400500600700800

Im εA = 2.23

0100

200300

400

time (fs)

0 1 2 3 4 5 6

energy (eV)

0

200

400

600

800

1000

1200Im ε

Fig. 6 Time evolution of the dielectric function in InSb for a field strength corresponding to 1:50 G cm (left)and 2:23 G cm (right). For the higher intensity, the loss of structural features in Im EðwÞ, and the emergence of aDrude-like peak at low frequencies, demonstrate a change from the original tetrahedral bonding, together with anonset of metallic behavior. I.e., an amplitude of 2:23 G cm is above the threshold for a nonthermal phase transi-tion in this material.


Let us begin with the model action in the coordinate representation which yields both the quantumdynamics of the electrons and the classical dynamics of the nuclei (or ion cores) [3, 4]:

�SS ¼ðdt

12

ðd3x

Pj

Y�j x; tð Þ i�h

@

@t� bHH� �

Y j x; tð Þ þ h:c:þ 12

PM _XX2 � U

" #: ð4:1Þ

The action is represented by �SS to avoid confusion with the overlap matrix S. Also, the second summa-tion is over all the atomic nuclei (with masses M) and over the three components of each time-depen-dent nuclear position X. If an electron wavefunction Y j x; tð Þ is expanded in terms of atomic-orbitalbasis functions fa x� Xð Þ,

Y j x; tð Þ ¼PXa

Y j Xa; tð Þ fa x� Xð Þ ; ð4:2Þ

it changes during a time dt by an amount

dY j x; tð Þ ¼PXa

dY j Xa; tð Þ fa x� Xð Þ þY j Xa; tð Þ dfa x� Xð Þ� �

: ð4:3Þ

Then, as stated above, its time dependence has two contributions:

@Y j x; tð Þ@t

¼PXa

@Y j Xa; tð Þ@t

fa x� Xð Þ þY j Xa; tð Þ @fa x� Xð Þ@X

_XX� �

: ð4:4Þ

The second contribution, which results from the motion of the basis functions as they follow thenuclei with velocities _XX, is a correction to the simple version of our method that was employed above.To obtain an approximate approach for including this correction, let us first notice that

@fa x� Xð Þ@X

¼ � @fa x� Xð Þ@ x� Xð Þ ¼ � @fa x� Xð Þ

@x¼ �rfa x� Xð Þ ð4:5Þ

so that

i�h@Y j x; tð Þ

@t¼PXa

i�h@Y j Xa; tð Þ

@tfa x� Xð Þ þY j Xa; tð Þ _XXbppfa x� Xð Þ

� �ð4:6Þ


A = 1.5

0100

200300

400

time (fs)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

distance (Angstroms)

0

2

4

6

8

10

12

number of neighbors

A = 2.23

0100

200300

400

time (fs)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

distance (Angstroms)

0

2

4

6

8

10

12

number of neighbors

Fig. 7 Time evolution of the pair correlation function in InSb for a field strength coresponding to 1:50 G cm(left) and 2:23 G cm (right). At t ¼ 0, there are peaks corresponding to first, second, etc. neighbors. For the lowerintensity, this structure is only broadened after a few hundred femtoseconds. For the higher intensity, however, itis essentially lost at 500 fs, indicating a change from the original tetrahedral bonding.


where bpp ¼ �i�hr is the momentum operator. It follows thatðd3xY*j x; tð Þ i�h @

@tY j x; tð Þ ¼

PX0a0

PXa

Y*j X0a0; tð Þ i�h _YYj Xa; tð Þ S X0a0;Xað Þ�

þY*j X0a0; tð ÞY j Xa; tð Þ bpp X0a0;Xað Þ _XX� ; ð4:7Þ

where

S X0a0;Xað Þ ¼Ðd3xf�

a0 x� X0ð Þ fa x� Xð Þ ; ð4:8Þ

p X0a0;Xað Þ ¼Ðd3xf�

a0 x� X0ð Þ bppfa x� Xð Þ : ð4:9Þ

With inner products implied for N-dimensional vectors and matrices, but indicated explicitly with adot for 3-dimensional vectors, we now haveð

d3xY yj x; tð Þ i�h @

@tY j x; tð Þ ¼ Y y

j i�hS _YYj þY yj P � _XXY j : ð4:10Þ

Here P is the N � N matrix with components p X0a0;Xað Þ. (It is thus treated as both an N-dimensionalmatrix and a 3-dimensional vector.) But the action �SS also contains the Hermitian conjugate, so

�SS ¼ðdt

12

Pj

Y yj i�hS

@

@tþ P � _XX �H

� �Y j þ h:c:

� �þ 12

PM _XX2 � U

" #: ð4:11Þ

Now let Y 0j ¼ S1=2Y j (where S1=2 is Hermitian) and consider

i�hYyjS

@

@tY j � i�h

@

@tY j

� �ySY j

¼ i�hðS�1=2Y 0jÞy S

@

@tðS�1=2Y 0

jÞ � i�h@

@tðS�1=2Y 0

jÞ� �y

SðS�1=2Y 0jÞ

¼ i�hY 0yj S

�1=2S@S�1=2

@tY 0

j þ S�1=2@Y 0

j

@t

!� i�h

@S�1=2

@tY 0

j þ S�1=2@Y 0

j

@t

!y

SS�1=2Y 0j

¼ i�hYyjS

1=2 @S�1=2

@tY 0

j þ i�hY 0yj

@Y 0j

@t� i�hY 0y

j@S�1=2

@tS1=2Y 0

j � i�h@Y 0

j

@tY 0y

j :

In a representation that diagonalizes S1=2, S�1=2 and @S1=2=@t are also diagonal. Then S1=2 and@S1=2=@t commute, and the first and third terms cancel. After an integration by parts within �SS, thefourth term doubles the contribution of the second term. Then the above expression for the actionbecomes

�SS ¼ðdtPjY 0y

j i�h@

@tþ 1

2P0 � _XX þ 1

2_XX � P0 �H0

� �Y 0

j þ12

PM _XX2 � U

" #ð4:12Þ

where

H0 ¼ S�1=2HS�1=2 ; P0 ¼ S�1=2PS�1=2: ð4:13ÞLet us define an operator b_XX_XX by

b_XX_XXfa x� Xð Þ ¼ _XXfa x� Xð Þ ð4:14Þ

for any X and a, so that

f*a0 ðx� X0Þ b_XX_XXy ¼ f*a0 x� X0ð Þ _X0X0 ð4:15Þ



with the operator acting to the left in this second equation. (Recall thatÐd3xf*a0 ðx� X0Þ b_XX_XXy

faðx� XÞ ¼ ðÐd3xf*aðx� XÞ b_XX_XXfa0 ðx� X0ÞÞ* ð4:16Þ

with b_XX_XXfa0 x� X0ð Þ ¼ _XX0fa0 ðx� X0Þ.) We then haveÐd3xf*a0 ðx� X0Þ bpp � b_XX_XXfaðx� XÞ ¼ pðX0a0;XaÞ � _XX ð4:17ÞÐd3xf*a0 ðx� X0Þ b_XX_XXy � bppfaðx� XÞ ¼ _XX0 � pðX0a0;XaÞ : ð4:18Þ

If the fa x� Xð Þ are interpreted as a complete set of states (with ionized atomic states included), thecompleteness relation for nonorthogonal basis functionsP

X0a0

PXa

fa0 x0 � X0� �S�1 X

0a0;Xa

� �f*a x� Xð Þ ¼ d x� x0ð Þ ð4:19Þ

allows us to rewrite (4.17) and (4.18) as

PS�1 _XX ¼ P � _XX ð4:20Þ_XXyS�1P ¼ _XX � P ð4:21Þ

where _XX and _XXy are the N � N matrices with componentsÐd3xf*aðx� XÞ b_XX_XXfa0 ðx� X0Þ ¼ _XXSðXa;X0a0Þ ; ð4:22ÞÐd3xf*aðx� XÞ b_XX_XXy

fa0 ðx� X0Þ ¼ _XX0SðXa;X0a0Þ : ð4:23Þ

Then the action of (4.11) and (4.12) can be rewritten

�SS ¼ðdtPjY 0y

j i�h@

@tþ 1

2P0 � _XX0 þ 1

2_XX0y � P0 �H0

� �Y 0

j þ12

PM _XX2 � U

" #ð4:24Þ

with

_XX0 ¼ S�1=2 _XXS�1=2: ð4:25ÞLet

Heff ¼ H � 12 P � _XX þ _XX � PÞ�

ð4:26Þ

so that

H0eff ¼ H0 � 1

2 ðP0 � _XX0 þ 1

2_XX0y � P0Þ : ð4:27Þ

Let us now change notation by removing the primes in the new orthogonalized basis:

�SS ¼ðdtPjYy

j i�h@

@t� Heff

� �Y j þ

12

PM _XX2 � U

" #: ð4:28Þ

From either (4.26) or (4.27), the operator form of Heff is

bHHeff ¼ bHH � 12ðbpp � b_XX_XX þ b_XX_XX � bppÞ ð4:29Þ

¼ 12m

bpp� qcbAA 2

� 12ðbpp � b_XX_XX þ b_XX_XX � bppÞ þ U ð4:30Þ

¼ 12m

bpp� qcbAA� mb_XX_XX 2

þU � 12mb_XX_XX2 þ q

2cðA � b_XX_XX þ b_XX_XX � AÞ : ð4:31Þ



Since M _XX2=2 is already small, and M is about 104m, m _XX2=2 is very small. Let us additionally neglectthe terms with A � _XX and _XX � A, since A and _XX are separately important in different time regimes. Wethen obtain the remarkably simple form

bHHeff � 12m

ðbpp� qcbAA� mb_XX_XXÞ2 þ U ; ð4:32Þ

which is exactly the same as for the coupling of electrons to the radiation field, except that

qcbAA ! q

cbAAþ mb_XX_XX : ð4:33Þ

I.e., we can now treat both the radiation field and the ‘‘ion velocity field” with a generalized Peierlssubstitution:

HeffðXa;X0a0Þ ¼ expi�h

qcAðXÞ þ m _XX

h i� X

� �H0ðXa;X0a0Þ exp � i

�hqcAðX0Þ þ m _XX0

h i� X0

� �:

ð4:34ÞThen everything goes through just as before, with

�SS ¼ðdtPjY y

j i�h@

@t�Heff

� �Y j þ

12

PM _XX2 � U

" #: ð4:35Þ

As in Refs. [3] and [4], but with qA=c ! qA=cþ m _XX, the requirement d�SS ¼ 0 for arbitrary varia-tions dY y

j and dX leads to

i�h@Y j

@t¼ Heff �Y j ð4:36Þ

and

Md2Xdt2

¼ �PjY y

j �@Heff

@X�Y j �

@U@X

ð4:37Þ

where X is any nuclear coordinate.We have in fact used these modified equations in calculations for organic molecules [11, 12], and

we find that the effect is small for nuclear velocities that are relevant in the present context. I.e., forthe simulations of the present paper, and for those of Refs. [5–9], the correction considered in thissection is much less important than our other approximations.

We close this section with a comment regarding the main limiting aproximation of our SERIDtechnique, or of any method which treats the nuclear motion classically. If we had started with aquantum description of the nuclei, we would have obtained the operator equation

Md2 bXXdt2

¼ �PjYy

j � @Heff

@ bXX � Y j �@U

@ bXX ð4:38Þ

from the Heisenberg equations of motion for the nuclear position and momentum operators. Thisimmediately gives Ehrenfest’s theorem

Md2h bXXidt2

¼ �PjY y

j �@Heff

@ bXX� �

�Y j �@U

@ bXX� �

: ð4:39Þ

As long as these expectation values are dominated by a single set of similar nuclear trajectories, oursemiclassical approximation is valid. However, if there are several competing processes with compar-able probabilities (or if one is interested in processes which have low probability), it is not physicallymeaningful to follow the average trajectory represented by (4.39). Fortunately, there are many interest-ing situations in which the nuclei do behave classically on the relevant time scale, making a semiclas-



sical approach useful for understanding the most important features in the evolution of the nuclearpositions and electronic states.

5 Conclusion

In this paper we have presented two sets of results:For the semiconductors Si and InSb, we have studied the detailed response to ultrafast laser pulses.

Various properties were monitored as functions of time, including the atomic motion, the pair correla-tion function, the occupancy of the excited states, the band structure, and the dielectric function. Forboth semiconductors, there is clearly a nonthermal transition to a disordered and metallic state on atime scale of a few hundred femtoseconds.

We also have introduced a new method for including the correction due to motion of the atomic-orbital basis functions. In the present context, the resulting correction is found to be relatively small,but we anticipate that it may be substantial in situations with more rapid atomic motion. Our expres-sion is remarkably simple to employ because it amounts to nothing more than the generalized Peierlssubstitution (4.34).

We also introduce a new addition to the technique, which provides a simple treatment of the correc-tion due to motion of the atomic-orbital basis functions.

Acknowledgements This work was supported by DARPA and the Robert A. Welch Foundation.

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Response of Si and InSb to ultrafast laser pulsesdtraian/laser-si-insb.pdf · ductors Si and InSb, an orthogonal tight-binding model was used. A nonorthogonal, density-func-tional-based