a r X i v :0807.2521v 1 [c o n d -m a t .m t r l -s c i ] 16 J u l 2008
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Fundamentals and Applications of Iso-tope Effect in Modern Technology.V.G.Plekhanov.
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Fonoriton Science Lab.,Garon Ltd.,P.O.Box 2632,Tallinn,13802,ESTONIA <e-mail > Different crystals (miconductors and insulators)with varying isotopic composition have been recently grown.I discuss here the effect of isotopic mass and isotopic disorder on the properties (vibrational,elas-tic,thermal and optical)of different crystals.The main applications of the stable isotopes are included lf-diffusion,neutron transmutative doping (NTD)of different miconductors,optical fibers,isotope-bad quantum computers,etc.Becau of space lim-itations this discussion will not exhaustive.I hope however,to give sufficient references to published work so that the interest reader can easily find the primary literature sources to this rapidly expanding field of solid state physics.Phonons,excitons,isotope-mixed crystals,
lar materials,quantum information,isotope-bad quantum computers.
It is well-known that the prence of randomly dis-tributed impurities in a crystal can give ri to signi
f-icant variations of its mechanical,electrical,thermal,and optical properties with respect to tho of the pure solid.All the properties are,more or less,directly related to the structure of the manifold of phonon states and any variation induced in this struc-ture by the prence of the impurities,will produce
a corresponding alteration of the physical properties of the material.Of particular interest is the ca in which the impurity species is of the same chemical nature,but with a different he ca of isotopic impurities.The mechanisms by which the impurities (isotopes)pertur
b the phonon distribu-tion will depend on the mass difference between the host and guest species [1-3].Phonons are the crys-tal excitations most directly related to the isotopic
mass.In monatomic crystals (like C,Ge,Si.,etc.),and within the harmonic approximation,all phonon
frequencies scale like the square root of the average
isotopic mass.Namely,this feature can be ud for the nondestructive isotopic characterization investi-gated materials.The isotopic effect can be classified into two categories:1)The first type is caud by the variation of the phonon frequencies with the average
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isotopic mass.To this type belongs the isotope effect
in superconductors,which plays an important role in the arch for the mechanism of high T c supercon-ductivity ([4]).The effect of changing the atomic mass M is to change the phonon frequencies ωaccording to:
ω= M ,(1)
where αis a force constant characteristic of the phonon under consideration.The change in atomic mass implies,at low temperatures (e below),a change in the average atomic displacement for each phonon mode.In the ca of one atom per primitive cell the mean squared phonon amplitude u 2 is given by [1;2]: u 2 = ¯h
2
4M 1/2α1/2
[1+2n B (ω)] ,(2)where n B (ω)is the Bo -Einstein statistical fac-tor,ωis the frequency of a given phonon and ...
reprents an average over all phonon modes.The average in r.h.s.of (2)is often simplified by tak-ing the value inside ... at an average frequency ωD which usually turns out to be clo to the Debye fre-quency.We should distinguish between the low tem-perature (¯
h ω>>k B T)and the high temperature (¯h ω<<k B T)limits and e:(¯h ω>>k B T), u 2 =¯
h 2M ω2
∼T in-
dependent of M (3).
Using Eq.(1)we can find from last equations that u 2
,the zero-point vibrational amplitude,is propor-tional to M −1/2at low temperatures:it thus decrea with increasing M and vanishes for M −→∞.For high T,however,we find that u 2 is independent of M and linear in T (details e [3]and references therein).
Another type of isotope effects is produced by the 1
isotopic massfluctuations about the average mass
M .Thefluctuations perturb the translational in-
variance of a crystal and lift,at least in part,k-vec-
tor conrvation.The most striking effect of this type
is obrved in the thermal conductivity which has a
maximum at a temperature T M<<ΘD(hereΘD is
Debye temperature,T M=80K for diamond,T M=
20K for silicon(e,also Figs.64-66in[3]).Reduc-
tion of the concentration of13C from the standard1%
(against99%of12C)by a factor of ten increas the
thermal conductivity of diamond by about a factor of
two,a fact that leads to amplifications in situations
where a large amount of generated heat has to be
豫湘桂会战driven as substrates for high power elec-
tronic devices[5]).As is well-known this maximum
reprents the transition from boundary scattering to
the phonon unklapp scattering regime and its value
K m is determined by the isotopicfluctuation param-
eter g(mass variance):
g= M2
2 )1/2
(5).
(for more details e Ref.[12]).The density
of states in diamond is asymmetric with respect to
ωLT O,causing asymmetry in the shape of the scatter-
ing line[7].This asymmetry also leads to the asym-
metric concentration dependence of the half-width of
the scattering line.As was shown early([3]
and references therein),in the ca of a weak poten-
tial of isotopic scattering of phonons,their lf-energy
ε(ω)does not depend on q(-phonon quasiimpuls).
This is precily the situation obrved for C and Ge.
Indeed,if we express the massfluctuation∆M/M
is the mean mass of all isotopes)in the form of the
广东好玩的地方variation of the phonon band width∆ω0=12cm−1
at q=0and compare it with the width of the band
of optical phonons in Ge equals to≈100cm−1,we
will e that the variations very small.Under this
conditions the localization of optical phonons in Ge
is naturally,abnt,and as obrved in experiment,
they stay delocalized(e below,however opposite
ca in LiH x D1−x crystals).Moreover,direct mea-
surements of the phonon lifetime in Ge show that,in
the ca of anharmonic decay,it is two orders of mag-
nitude shorter than the lifetime that is due to the ad-
2
ditional scattering by τanharm=τdisord·10−2[13].Therefore,the contribution of anharmonic-ity to the half-width of thefirst-order light scattering line in Ge is two orders of magnitude greater than that caud by the isotopic disorder in crystal lattice. In conclusion of this part of our report we should mention that analogous structure offirst-order RS and their dependence on isotope composition has by now been obrved many times,not only in elemen-tary Si andα-Sn,but also in compound CuCl,CuBr, ZnSe,GaN miconductors(details e Ref.[3]).
In Fig.2(curve1)the spectrum of cond-order RS of light in pure LiD crystal is shown[7].In spite of the fact,according to the nomogram of exciton states [14],the crystal studied should be considered to be pure,its RS spectrum contains a clear high-frequency peak around1850cm−1.The obrved peak does not have an analogue in RS of pure LiH(Fig.2,curve4) and has already been obrved earlier in the cond-order RS and has been interpreted(e[7]and ref-erences therein)as a local vibration of the hydrogen in LiD crystals.Further we note that as the con-centration grows further(x>0.15)one obrves in the spectra a decreasing intensity in the maximum of2LO(Γ)phonons in LiD crystal with a simulta-neous growth in intensity of the highest frequency peak in mixed LiH x D1−x crystals(Fig.2,curve3). The origin of the last one is in the renormalization of LO(Γ)vibrations in mixed
crystals[7].Compar-ison of the structure of RS spectra(curves1and2 in Fig.2)allows us,therefore,to conclude that in the concentration range of0.1<x<0.45the RS spectra simultaneously contain peaks of the LO(Γ) phonon of pure LiD and the LO(Γ)phonon of the mixed LiH x D1−x crystal.Thus,the cond-order RS spectra of LiH x D1−x crystals have one-and two-mode character for LO(Γ)phonons,and also contain a con-tribution from the local excitation at small values of x.Moreover,we should add that an additional struc-ture in RS spectra on the short-side of the2LO(Γ) peak(e Fig.21in Ref.[7])was obrved relatively ago in mixed LiH x D1−x crystals and,very recently, in isotopically mixed crystals of diamond,germanium andα-Sn(details e[3,11]).The effects caud by isotopic disorder in the crystal lattice of isotopi-cally mixed crystals[3].The obrvation of two-mode behavior of the LO(Γ)phonons in RS spectra of LiH x D1−x crystals contradicts the prediction of the CPA[15],according to which the width W of optical vibration band should be smaller than the frequency shift(∆)of transver optical phonon.However,as was shown early([7]and references therein) in LiH x D1−x mixed crystals,the rever inequality is W>|∆|.According[16],this discrep-ancy between experimental results and theory bad on CPA[15]is mainly explained by the strong po-tential of scattering of phonons,caud by a large change in the mass upon substitution of deuterium for hydrogen.Once more reason of the discrepancy between theory and results of the experiment may be connected with not taking into acc
ount in the-ory the change of the force-constant at the isotope substitution of the smaller in size D by H ion.We should stress once more that among the various pos-sible isotope substitution,by far the most important in vibrational spectroscopy is the substitution of hy-drogen by deuterium.As is well-known,in the limit of the Born-Oppenheimer approximation the force-constant calculated at the minimum of the total en-ergy depends upon the electronic structure and not upon the mass of the atoms.It is usually assumed that the theoretical values of the phonon frequen-cies depend upon the force-constants determined at the minimum of the adiabatic potential energy sur-face.This leads to a theoretical ratioω(H)/ω(D)of the phonon frequencies that always exceed the ex-perimental data.Very often anharmonicity has been propod to be responsible for lower value of this ra-tio.In isotope effect two different species of the same atom will have different vibrational frequencies only becau of the difference in isotopic mass.The ra-tio p of the optical phonon frequencies for LiH and LiD crystals is given in harmonic approximation by: p=ω(H)M(LiD)2(6) while the experimental value(which includes an-harmonic effects)is1.396÷1.288(e Table in Ref.
[17]).In this Table there are the experimental and theoretical values of p according to formula(6),as well as the deviationδ=P T heory-p exp
∼f(ln[∂E
∂M ).(7)
From the results of Fig.3,it can be concluded that only hydrogen compounds(and its isotope ana-log-deuterium)need to take into account the force-
constant changes in isotope effect.It is also en that for miconductor compounds(on Fig.3-points, which is below of Ox line)the isotope effect has only
the changes of the isotope mass(details e[3,7]). The dependence of the band gap energy on isotopic
composition(via mechanism of electron-phonon in-teraction)has already been obrved for insulators (Fig.4)and lowest(indirect-direct)gap of dif-
ferent miconductors([3]and references therein). It has been shown to result primarily from the ef-
fect of the average isotopic mass on the electron-phonon interaction,with a smaller contribution from the change in lattice constant.It was thefirst pa-
per[19]where the exciton binding energy E B was found to depend on the isotopic composition.It was shown further that this change in E B was attributed
to the exciton-phonon interaction(originally with LO phonons)(e,also[3]).At prent time such depen-
西红柿牛腩汤dence of E B∼f(x)(x-isotope concentration)was found for different bound excitons in miconductors [20-21].The simplest approximation,in which crys-
tals of mixed isotopic composition are treated as crys-tals of identical atoms having the average isotopic mass is referred to as virtual crystal approximation
(VCA)[15].Going beyond the VCA,in isotopically mixed crystals one would also expect localfluctua-tions in the band-gap energy from statisticalfluctu-ations in local isotopic composition within some ef-fective volume,such as that of an exciton( Fig.2of Ref.[18]).Using the least-squares method it was found the empirical dependence of ln ∂E g
∂M =6.105(lnE g)2-7.870(lnE g)+0.565.
(8)
From thisfigure it can be concluded also that the small variation of the nuclear mass(miconductors) caus the small changes in E g also.When the nu-
clear mass increas it caus the large changes in E g(C,LiH,CsH,etc.)(details,e[18,3]).
Detail analyze the process of lf-diffusion in iso-tope pure materials and hetero-structures was done in[5].Interest in diffusion in solids is as old as metal-lurgy or ceramics,but the scientific study of the phe-nomenon may probably be dated some sixth-ven decades ago.As is well-known,the measured diffu-sion coefficients depends on the chemistry and struc-ture of the sample on which it is measured.In cited paper[5]it was shown to u the stable isotopes for the study of diffusion process in different micon-ducting structures(bulk,hetero-structures etc.).
Chapter6indicated book[5]describes the new reactor technology-neutron transmutative doping (NTD).Capture of thermal neutrons by isotope nu-clei followed by nuclear decay produces new elements, resulting in a very number of possibilities for isotope lective doping of solids.The importance of NTD technology for studies of the miconductor doping as well as metal-insulator transitions and neutral im-purity scattering process is underlined.The low-temperature mobility of free carriers in miconduc-tors is mainly determined by ionized-and neutral-impurity scattering.The ionized-impurity scattering mechanism has been extensively studied([5] and references therein),and various aspects of this process are now quite well understood.Scattering by neutral impurities is much less than by ionized ,its contribution is significant only in crystals with l什么是电动势
ow compensation and at very low temperatures where most of the free carriers are frozen on the im-purity sites.The availability of highly enriched iso-topes of Ge which can be purified to residual dopant levels<1012cm−3has provided thefirst opportu-nity to measure neutral impurity scattering over a wide temperature range.In paper[22]three Ge iso-topes transmute into shallow acceptors(Ga),shallow donors(As)and double donors(Se)(e also above): 70
32
Ge+n→7132Ge EC(t
1/2
=11.2days)→7132Ga+νe, 74
32
Ge+n→7532Geβ−(t
1/2
=82.2min)→7532As+β−+¯νe,
76
32
Ge+n→7732Geβ−(t
1/2
=11.3h)→β−+¯νe+ 77
32
Asβ−(t
1/2
=38.8h)→7732Se+β−+¯νe.(9) The isotopes72Ge and73Ge are transmuted into
4
the stable73Ge and74Ge respectively.Controlling the ratio of70Ge and74Ge in bulk Ge crystals al-lo
wsfine tuning of the majority-as well as the minor-ity carrier concentration.Currently,this is the best method to vary the free-carrier concentration inde-pendently from compensation ratio.As oppod to other doping methods,NTD yields a very homoge-neous,perfectly random distribution of the dopants down to the atomic levels[5].Thus isotopically con-trolled crystals offer a unique possibility to study sys-tematically the scattering mechanism of the charge carriers in miconductors.Extensive Hall-effect and resistivity measurements from room temperature down to4.2K yielded very accurate free-carrier con-centrations and mobilities as a function of tempera-ture and doping level were done in paper[5].Itoh et al.[22]have performed temperature-dependent Hall measurements on four different p-type and two-different n-type Ge crystals(Fig.6).Fig.6shows the relative strength of the scattering from the ionized and the neutral impurities.There is only a relatively small temperature region in which the scattering from the neutral impurities dominates.This range extends to higher temperatures as the free-carrier concentra-tion is incread.The calculated”transition temper-atures”above which the ionized impurities are the main scattering centres compare very well with ex-perimental results of Itoh et al[22](e also Fig.6.31 in Ref.[5]).In order to demonstrate the importance of the homogeneous dopant distribution,Itoh et al. have performed the same study on samples cut from Ge:Ga crystals grown by the conventional Czochral-ski method,where Ga impurities were introduced to Ge melt during the crystal growth.The authors obrved deviations o
f the measured mobility from the theoretical calculations,which are most likely due to inhomogeneous Ga impurity distributions in melt-doped Ge.Only the u of NTD miconductors with randomly distributed dopants allows for an accurate test of the neutral impurity-scattering models(de-tails,e[5]).
Another application of isotope pure and isotope mixed crystals that will be discusd here is related to the possibility of using an isotopically mixed medium (e.g.LiH x D1−x or12C x13C1−x)as an oscillator of coherent radiation in the ultraviolet spectral range.
To achieve this,the u of indirect electron transi-tions involving,say,LO phonons was planned[23]. The detection of LO phonon replicas of free-exciton luminescence in wide-gap insulators attracted con-siderable attention to the crystals([10;23]). At the same time it is allowed one to po a question about the possibility of obtaining stimulated emission in UV(VUV)region(4-6eV)of the spectrum,where no solid state sources for coherent radiation exist yet. In thefirst place this related to the emitters working on the transitions of the intrinsic electronic excitation (exciton).The last one provides the high energetical yield of the coherent emission per unit volume of the substance.
In this part we will discuss the investigation re-sults of the influence of the excitation light density on t
he resonant condary emission spectra of the free-exciton in the wide-gap insulator LiH x D1−x (LiH1−x F x)crystals.The cubic LiH crystals are typ-ical wide-gap ionic insulator with E g=4.992eV [10]with relatively weak exciton-phonon interac-tion however:E B/¯hωLO=0.29where E B and¯hωLO are exciton binding energy and longitudinal optical phonon’s energy,respectively.Besides it might be pointed out that the analogous relation for CdS,di-amond and NaI is0.73;0.45and12.7,respectively .In the inrt of Fig.7depicts the luminescence of 1LO and2LO phonon replicas in LiH crystals.An increa in the density of the exciting light caus a burst of the radiation energy in the long-wave wing of the emission of the1LO and2LO repetitions(e Fig.7)at a rate is higher for the1LO replica line [23].A detailed dependence of the luminescence in-tensity and the shape of the2LO phonon replica line are prented in Fig.7.The further investigations have shown[5]that with the increa of the excitation light intensity at the beginning a certain narrowing can be obrved,followed by widening of the line of 2LO phonon replica with a simultaneous appearance of a characteristics,probably mode structure(e Fig.
8.11in Ref.[5]).From this Fig.it can be en that the coupling between longwavelength lumines-cence intensity and excitation light intensity is not only linear,but,in fact,of a threshold character as in ca of other crystals.A proximity of the exciton parameters of LiH and CdS(ZnO)crystals allowed to 5
carry out the interpretation of the density effects in
LiH on the analogy with the miconducting com-pounds.Coming from this in the paper[23]it was shown that for the obrved experimental picture on LiH crystals to suppo the exciton-phonon mecha-nism of light generation[5]is enough the excitons density about1015cm−3.This is reasonable value,if the high quality of the resonator mirrow-the crystal cleavage”in situ”and relatively large exciton radius (r=40˚A[10])is taken into account.To this light mechanism generation must be also promoting a large value of the LO phonon energy(¯hωLO=140meV). Owing to this the radiative transition is being real-ized in the spectral region with a small value of the absorption coefficient,and thus with a small loss in resonator(details e[5]).
In conclusion of this ction we should underlined that if the obrvable mode structure is really caud by the lar generation it may be smoothly tuned in the region of energies 4.5± 5.1eV owing to smooth transition of the line emission energy in the LiH x D1−x(LiH x F1−x;LiD x F1−x)mixed crystals as well as in the range5.35-5.10eV in12C x13C1−x mixed crystals(e also[10]).
Concluding our report we should be paid your at-tention to the reports of Professors Schoven,Westo
n, Wendt as well as Dr.Chai of our conference which are devoted in thefirst step of radioactive isotope applications.
Figure Captions.肠粉好吃吗
Fig.1.a)First-order Raman spectra of12C13x C1−x diamonds with different isotope compositions.The labels A,B,C,D,E and F correspond to x=0.989;
0.90;0.60;0.50;0.30and0.01respectively.The in-tensity is normalized at each peak(after[8]);b)First-order Raman scattering spectra in Ge with different isotope contents(after[13]).
Fig.2.Second-order Raman spectra of LiH x D1−x crystals at room temperature:(1);(2);(3)and(4)x =0;0.42;0.76and1,respectively(after[7]).
Fig. 3.The dependence of ln(δ%)∼f(ln[∂E
∂M ∼f(lnE g): points are experimental date and continuous line-calculation on the formula(8)(after[18]).
Fig.6.Temperature dependence of the carrier mo-bility of a)p-type and b)n-type NTD Ge crystals.
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c)Temperature dependence of relative contributions to the mobility.Note that the mobility is dominated by neutral impurity scattering below20K(70Ge:Ga ♯2crystal)(after[22]).
Fig.7.The dependence of the intensity in the maximum(1)and on the long-wavelength side(2)of 2LO replica emission line of LiH crystals on the exci-tation light intensity.In inrt:luminescence spectra of free excitons in LiH crystals in the region of the emission lines of1LO and2LO phonon repetitions at 4.2K for low(1)and high(2)density of excitations of4.99eV photons(after[23]).
References.
1.I.M.Lifshitz,Physics of Real Crystals and Disordered Systems,Selected Works(Eds.M.I. Kaganov,A.M.Kovich,Science,Moscow,1987)(in Russian).
2. A.A.Maradudin,E.W.Montroll,G.H.Weiss and I.P.Ipatova,Theory of Lattice Dynamics in the Harmonic Approximation,Solid State Physics,Vol.3, (Eds.F.Seitz,D.Turnbull and H.Ehrenreich,Aca-demic,New York,1971).
3.V.G.Plekhanov,Elementary Excitations in Isotope-Mixed Crystals,Physics Reports,410[1-3] 1(2005).
4.J.P.Franck,in:Physical Properties of High T c Superconductors(ed.D.M.Ginsberg,Vol.4.,World Scientific,Singapore,1984)p.189.
5.For a review,e,V.G.Plekhanov,Applications of the Isotopic Effect in Solids,Springer,Berlin-Hei-delberg,2004.
6.See,for example,R.Berman,Thermal Con-duction of Solids(Clarendon Press,Oxford,1976); T.M.Tritt,Thermal Conductivity(Springer,Berlin-Heidelberg,2005).
7.V.G.Plekhanov,Isotopic Effects in Lattice Dynamics,Physics-Uspekhi(Moscow)46[7]689 (2003).
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