PHYSICAL REVIEW B 15 APRIL 1997-IVOLUME 55, NUMBER 15

Thermally assisted tunneling of hydrogen in silicon: A path-integral Monte Carlo study

Carlos P. HerreroInstituto de Ciencia de Materiales, Consejo Superior de Investigaciones Cientı´ficas (C.S.I.C.), Campus de Cantoblanco,

28049 Madrid, Spain~Received 13 December 1996!

Quantum transition-state theory, based on the path-integral formalism, has been applied to study the jumprate of atomic hydrogen and deuterium in crystalline silicon. This technique provides a methodology to studythe influence of vibrational mode quantization and quantum tunneling on the impurity jump rate. The atomicinteractions were modeled by effective potentials, fitted to earlierab initio pseudopotential calculations. Siliconnuclei were treated as quantum particles up to second-nearest neighbors of the impurity. The hydrogen jumprate follows an Arrhenius law, describable with classical transition-state theory, at temperatures higher than100 K. At ;80 K, a change in the slope of the Arrhenius plot is obtained for hydrogen, as expected for theonset of a diffusion regime controlled by phonon-assisted tunneling of the impurity. For deuterium, no changeof slope is observed in the studied temperature range~down to 40 K!. @S0163-1829~97!01616-0#

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Hydrogen diffusion in crystalline semiconductors hbeen studied for many years, and in the last decade this thas received much attention after the discovery that atohydrogen can passivate a large number of extended anddefects in these materials.1–3 The classical permeation experiments by Van Wieringen and Warmoltz4 ~VWW! at hightemperatures (T.1000 K! gave for hydrogen a diffusion coefficient in crystalline silicon described by an Arrhenius laD(T)5D0exp(2EA /kBT), with an activation energyEA50.48 eV. In recent years, several experimental techniqhave been applied to obtain the hydrogen diffusivity at lowtemperatures, and a certain controversy appeared in theerature about the validity of extrapolating the VWW eqution to the diffusion of hydrogen at room temperature2,3

Moreover, it is now well accepted that at low temperatu(T,70 K! hydrogen can tunnel quantum mechanically btween different minimum-energy sites in hydrogen-impurcomplexes in semiconductors.5–7

From a theoretical point of view, several research grohave calculated diffusion barriers for hydrogen by using dferent methods~see Ref. 8 for a recent review!. In particular,Van de Walle et al.9 performed pseudopotential-densitfunctional calculations on a supercell Si32H, and found thatthe lowest-energy position for hydrogen is the bond-cente~BC! site, halfway between adjacent Si atoms, which rebackwards by about 0.4 Å. For the~adiabatic! diffusion bar-rier between nearest BC sites, these authors obtained a vof about 0.2 eV. Most of the calculations reported in tliterature agree in the lowest-energy site for hydrogen, thaalso supported by electron paramagnetic resonance exments, showing the presence of hydrogen in the so-caAA9 defect10 with axial symmetry around the@111# crystalaxis, as expected for a BC site.

Hydrogen diffusion in silicon has also been studiedmolecular dynamics simulations,11–13 which gave diffusioncoefficients that compare well with the VWW results at hitemperatures. However, in these simulations the atomicclei are assumed to be classical particles and thus quaneffects like zero-point motion are not taken into accouThese effects can be important for light impurities like h

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drogen, especially at low temperatures. In fact, ChengStavola7 found recently that the reorientation rate of B-complexes in silicon is enhanced by thermally activated tneling atT,70 K. In this case, hydrogen hops from BC sito BC site around the boron atom, in a way similar to thexpected for H diffusion in pure silicon. Since these jumare associated with large changes in the relaxation ofadjacent host atoms, coherent tunneling between BC sdoes not take place. However, as indicated by Stoneham14,15

the jump probability can be enhanced by phonons, as therfluctuations can yield a coincidence geometry, for whichwould have the same energy when associated with eisite. This mechanism has been shown to be responsiblehydrogen tunneling in metals at temperatures lower thanK.16

In this paper, we present a calculation of the jump ratehydrogen, based on the quantum transition-state theory,17–19

by means of path-integral Monte Carlo~PIMC! simulations.The results indicate that quantum effects appear at abouK for hydrogen diffusion. A similar calculation for deuterium shows that this atom presents a nearly classical beior down to 40 K.

For our system ofP11 quantum particles (P silicon nu-clei plus one impurity!, the partition functionZ at tempera-ture T can be expressed as a path integral in the followway:20

Z5E expS 21

\S@R~u!# DDR~u!, ~1!

whereu is a parameter with dimensions of time andR is avector in a 3(P11)-dimensional space, the componentswhich are the Cartesian coordinates of the nucR5(r0 , . . . ,rP). The pathsR(u) satisfy the cyclic conditionR(0)5R(b\) with b51/(kBT), and the functionalS@R(u)# is given by

S@R~u!#5E0

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mpr p2~u!1V@R~u!# D du, ~2!

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with mp the mass of nucleusp, and r p the derivative ofr pwith respect to the ‘‘time’’ coordinateu. In the following,r0 will refer to the impurity coordinate, andr p(p51, . . . ,P) to the coordinates of the Si nuclei. Since ocalculations are performed within the Born-Oppenheimerproximation, we employ a potential-energy surfaceV(R) forthe nuclei coordinates, as described below. For our calctions we have takenP511, i.e., we consider as quantuparticles the silicon nuclei up to next-nearest neighborsthe impurity. Farther Si nuclei are kept fixed in their relaxpositions.

The formalism employed here to calculate the transitrate of the impurity is based on a generalization of the csical transition-state theory, according to which the jump ris controlled by the probability distribution of th‘‘centroid’’: 17–19,21

P~x0!5Z21E d~x02 r0!expS 21

\S@R~u!# DDR~u!, ~3!

where the centroidr0 of a given pathr0(u) is defined as

r051

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r0~u!du. ~4!

Note that the restriction on the centroid to be atx0 in Eq. ~3!affects only the impurity coordinates, and those of thenuclei are free with the only conditionr p(0)5r p(b\) (p51, . . . ,P). The path integral in Eq.~3! has been evaluateby a discretization of the cyclic pathsR(u) into N points(R1 ,R2 , . . . ,RN). To assure the right convergence of tpath integrals in Eqs.~2! and ~3!, we take N as atemperature-dependent parameter, withN;10 000/T K 21

~see Refs. 21 and 22 for details on the discretization produre!.

In this formalism, the transition rate constant is given

k5 12 CP~x0* !, ~5!

whereC is a weakly temperature-dependent factor, givenC5 v/L, with v the thermal velocityv5A2/(pbm0). L is alength of the order of the distance beween a minimuenergy site and the saddle point of the energy surface,has been taken to be half the distance between two neighing BC sites.x0* is the position of the saddle point, which iour case is the point calledC* in Fig. 1. The centroid prob-ability P(x0* ) can be written as the integral

FIG. 1. Schematic diagram showing the integration path forhydrogen coordinate, employed to calculate the centroid probabP(x0* ) in Eq. ~6!.

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wheref(x0) is the mean force acting on the quantum imprity for the centroid positionr05x0:

f~x0!52^¹V~r0!& r 05x0. ~7!

In Eq. ~6!, x0m indicates the position of a BC site~BC1 in Fig.

1!. The forcef(x0) has been calculated in our Monte Carapproach for 11 points along the path indicated in Fig. 1.each centroid position, all the other degrees of freedom waveraged to obtain the mean force at a given temperatFor each point in the integration path of Eq.~6!, we gener-ated 53104 quantum paths for the calculation of the esemble average properties, and 5000 quantum paths fortem equilibration.

The Si-Si interaction has been modeled by the StillingWeber potential,23 that gives results for crystalline silicon igood agreement with those derived from experiment.24 TheSi-H interaction has been described by a three-body potedeveloped to reproduce the energy surface obtained byde Walleet al.9 ~see Refs. 22 and 25 for details on our prametrized potential!. With this interaction potential, the BCsite is the absolute energy minimum for H, with a backwarelaxation of the nearest-neighbor Si atoms of 0.3 Å. In tminimum-energy configuration, we find a potential energy–1.39 eV with respect to the pure host supercell. This enecan be separated into Si-Si and Si-H interactions, andobtain a Si-H energyV~Si-H!5–2.83 eV, along with a relax-ation energy of the latticeDV~Si-Si!51.44 eV.

In Fig. 2 we present the jump rate for hydrogen~opensquares! and deuterium~black squares! obtained from theMonte Carlo simulations by applying Eq.~5!. The dotted lineis an extrapolation of our high-temperature results (T.300

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FIG. 2. Rate for impurity jumps from BC to BC, as derivefrom the PIMC calculations. White squares, hydrogen; blasquares, deuterium. The dotted line is an extrapolation of the htemperature results (T.300 K! obtained for hydrogen in the PIMCsimulations. The dashed line is a fit to the low-temperature res(T,70 K! for H.

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K!, and gives an activation energy of 0.23 eV, close toenergy barrier of 0.27 eV obtained from the relaxed atconfigurations~classical nuclei! atT50. Although the resultsobtained for deuterium are close to the Arrhenius plotpected for a classical behavior of the impurity, the jump rfound for hydrogen is clearly higher than the classical exptation at temperatures lower than;80 K. Note that the errorbars of the calculated points, due to the statistical error inMonte Carlo sampling and to the numerical integration ofmean force in Eq.~6!, are of the order of the symbol sizethe lowest temperatures shown in Fig. 1, and decreaseincreasing temperature. The dashed line in Fig. 2 cosponds to an activation energy of 0.19 eV, slightly lowthan that found for the high-temperature regime.

It is interesting to analyze the probability distribution fthe impurity with the path centroid fixed at the saddle poC* . This gives information on the delocalization of thquantum particle in the two potential wells correspondingtwo adjacent BC sites. In Fig. 3 we display the projectionthe probability density for the impurity, along the@110# di-rection shown in Fig. 1. The impurity distribution broadewith decreasing temperature, and more important, its shchanges from a single maximum around the C* site at hightemperatures, to a bimodal distribution with a distancetween maxima that increases as temperature goes dowthe formalism employed here, such a splitting of the proability density for the impurity into two well-defined peaksa fingerprint of the appearance of quantum tunneling.26 Thisqualitative change in the distribution of the quantum pathstemperature decreases corresponds to a transition from sclassical motion over the effective barrier to quantum tunning through the barrier. The crossover temperature betwboth regimes (T;80 K! and the activation energies obtainehere for H are similar to those found experimentally for tjump rate of hydrogen in B-H complexes in silicon.7 We notethat the quantum character of the nearby host atoms is

FIG. 3. Probability density for hydrogen obtained in the PIMsimulations with the path centroid of hydrogen fixed at the sadpointC* . Continuous line, 200 K; dashed line, 70 K; dotted line,K. The appearance of two well-defined maxima in the probabidensity atT540 K is an indication of impurity tunneling.

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portant for impurity tunneling in the point defect studiehere. In fact, similar PIMC simulations atT540 K with afixed path centroid for H, and in which the Si nuclei aconsidered as classical particles, give a probability denfor hydrogen with only a maximum around the siteC* .Probability-density profiles qualitatively similar to those prsented in Fig. 3, have been found in path-integral studiehydrogen on metal surfaces. In particular, hydrogen is fouto undergo incoherent tunneling on Ni~001! at temperatureslower than 40 K.26

The effect of the impurity mass on the quantum delocization is important, especially at low temperatures, as canseen in Fig. 4, where the dashed and continuous lines cospond to projections of the impurity density on the@110# axisat 40 K. The deuterium distribution is that typical of a clasical particle, in agreement with the fact that, down to 40we do not obtain any clear indication of an enhancementhe jump rate due to quantum effects.

The slope of the Arrhenius plots presented in Fig. 2lower than the activation energy found by Van Wieringand Warmoltz,4 and corresponds basically to the energy brier for diffusion obtained by Van de Walle and co-workerAt this point, it is worthwhile mentioning that Blo¨chl, Vande Walle, and Pantelides,27 using the sameab-initio methodas that employed in Ref. 9, found an activation energyabout 0.5 eV, in surprisingly good agreement with the VWresults at high temperatures.4 Those calculations, which tookinto account entropic corrections at finite temperatures,difficult to reconcile with those presented here for the qutum impurities, including all contributions to the free enerin a direct way. It is not yet clear why such a correctionnecessary for the diffusion of H in pure silicon, and not fthe jump rate of H around boron~the calculated adiabaticbarrier coincides with the experimental activation energythe jump rate7,28,29! since the mechanisms and energy baers are assumed to be similar in both cases.

leFIG. 4. Probability density for hydrogen and deuterium found

PIMC simulations at 40 K with the path centroid of hydrogen fixat the saddle pointC* . Continuous line, hydrogen; dashed lindeuterium.

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In summary, path-integral Monte Carlo simulations prvide us with a good tool to study quantum effects onjump rate of hydrogen in semiconductors. Thermally acvated quantum tunneling is possible, in spite of the lahost-atom relaxations involved in the diffusion proceSimulations similar to those presented here can give insinto the mechanism responsible for the non-Arrhenius re

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entation kinetics for acceptor-hydrogen complexes in seconductors.

This work was supported by CICYT~Spain! under Con-tract No. PB93-1254. R. Ramı´rez, J.C. Noya, and E. Artachare thanked for inspiring discussions and for critically reaing the manuscript.

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