Metastability in the formation of Condon domaines
aa r X i v : . [ c ond - m a t . o t h e r] O c t Metastability in the formation of Condon domaines
Leonid Bakaleinikov
A. F. Ioffe Physico-Technical Institute, Russian Academy of Sciences,St. Petersburg, 194021, Russian Federation andDepartment of Exact Sciences, Faculty of Natural Sciences,University of Haifa, Campus Oranim, Tivon 36006, Israel
Alex Gordon
Department of Exact Sciences,Faculty of Natural Sciences, University of Haifa,Campus Oranim, Tivon 36006, Israel (Dated: September 11, 2018)Metastability effects in the formation of Condon non-spin magnetic domains are considered. Apossibility for the first-order phase transition occurrence in a three-dimensional electron gas isdescribed in the case of two-frequency de-Haas-van Alphen magnetization oscillations originatingfrom two extremal cross sections of the Fermi surface. The appearance of two additional domainsis shown in the metastable region in aluminum. The phase diagram temperature-magnetic fieldexhibits the presence of second-order and first- order phase transitions in the two-frequency case.
PACS numbers: 71.10 Ca; 71.70 Di; 75.60.- d; 74.62 – c; 75. 30. Kz;Keywords: Electron gas in quantizing magnetic fields; de Haas-van Alphen effect; diamagnetic phase transi-tions; Condon domains; metastability.
I. INTRODUCTION
Domain formation related to orbital magnetic mo-ments of conduction electrons and their “magnetic”interaction have been detected in silver, beryllium,white tin, indium, aluminum and lead by nuclear mag-netic resonance (NMR) and muon spin-rotation spec-troscopy (SR) . These domains were observed in thede Haas-van Alphen (dHvA) oscillations of the orbitalpart of magnetization. The reason for the stratificationof metallic samples into Condon non-spin domains is theinstability of an electron gas called “diamagnetic phasetransition” . This ordering occurs when the internal fieldin the sample is different from the external magnetic field(the Shoenberg effect) . Properties of Condon domainsin a two-dimensional electron gas have been consideredin . The direct observation of domains in silverhas been achieved by Hall probes . The hysteresis loopdue to domains has been observed and investigated inberyllium . Static, dynamic, size and tunneling effectsclose to phase transitions have been examined in . Inrecent years there has been renewed interest in the prob-lem of interactions between electrons leading to phasetransitions in new measurements of magnetization oscil-lations, spectroscopy and Hall probes experiments andastrophysics . Domain formation has been consideredwithin the outer crust, deeper part of the internal crustand core of magnetars .So far, Condon domains and diamagnetic phase transi-tions have been considered for the simplest Fermi surfacewith one extremal cross section. A more realistic consid-eration of two frequencies dHvA oscillations caused bytwo extremal cross sections of the Fermi surface has not carried out for the examination of Condon domains for-mation. Up to now these domains have been thoughtto be formed as a result of a temperature second-orderphase transition. We will show that magnetization os-cillations due to two extremal cross sections can lead tometastable phenomena in the domain formation. II. MODEL
The oscillatory part of the thermodynamic potentialdensity can be written by using the Lifshitz-Kosevich for-mula in the ‘first harmonic approximation’ :Ω = 14 πk (cid:20) a cos ( b ) + 12 a sin ( b ) (cid:21) . (1)Here b = k ( B − H ) = k [ h + 4 πM ] , B is the induction, H is the internal magnetic field, k = 2 πF/H , F is theoscillation frequency, h = H ex − H is the small incre-ment of the field ( H ex the external magnetic field). Inthe ‘first harmonic approximation’ the magnetization isfound from the equation of state :4 πkM = a sin [ k ( h + 4 πM )] , (2)where a = 4 πkA = 4 π ( ∂M/∂B ) B = H is the reducedamplitude of oscillations , and A is the amplitude of thefundamental oscillation. According to , the temperatureand field dependence of the amplitude is a ( T, T D , H ) = a ( H ) λT exp [ − λ ( H ) T D ] / sinh ( λT ) . (3)Here λ ≡ π k B m c c/ ( e ¯ hH ), mc is the cyclotron mass, k B is the Boltzmann constant, e is the absolute value ofthe electron charge, c is the light velocity, h is the Planckconstant, T D is the Dingle temperature. The limiting am-plitude a ( H ) is given by a ( H ) ≡ a ( λT → , , H ) =( H m /H ) / , where H m is the limiting field above whichthe diamagnetic phase transition does not occur at anytemperature. For temperature-magnetic field phase dia-grams giving the range of diamagnetic phase transitionsone can use the equation for the phase transition tem-perature making the expression (3) equal to unity at thephase transition point .We present the thermodynamic potential density Ω asfollows : Ω = 14 πk (cid:20) a cos ( kh + µ ) + 12 µ (cid:21) , (4)where µ = 4 πkM is the dimensionless magnetization.The case of h = 0 corresponds to the center of the dHvAperiod.Equations (1) and (4) present the approximated caseof the simplest Fermi surface with one extremal crosssection. In many cases the presentation of the Fermi sur-face with two extremal cross sections corresponds to thenature of metal better than the approximated equations(1) and (4). We consider the two extremal cross sectionscase following Ref. . Then the thermodynamic potentialdensity is given byΩ = 14 πk (cid:20) Ak cos ( µ ) + 12 µ (cid:21) +14 πk ′ (cid:20) A ′ k ′ cos ( µ ′ ) + 12 µ ′ (cid:21) , (5) µ ′ = 4 πk ′ M . Taking the amplitudes and frequenciesof dHvA magnetization oscillations due to the two ex-tremal cross sections of the Fermi surface A and A ′ as A ′ = αA, k ′ = βk, a = Ak, where k ′ = 2 πF ′ /H and F ′ is the second oscillation frequency, α and β are ratios be-tween oscillation amplitudes and frequencies respectively,we have the following equation for the thermodynamicpotential densityΩ = 14 πk (cid:20) a cos ( µ ) + αaβ cos ( βµ ) + µ (cid:21) . (6)The new equation for the thermodynamic potential den-sity gives a richer picture of the domain formation thanthat in the one-frequency case. In the two-frequency case,the instability of an electron gas does not occur at a = 1. III. RESULTS AND DISCUSSIONS
Condon domains were observed in aluminum by µ SRspectroscopy thereby validating results of the earlier work , in which the effect of the phase transition onhelicon waves was studied. We demonstrate here thethermodynamic potential density as a function of the di-mensionless magnetization µ at different temperaturesin aluminum in the two-frequency case (equations (3)and (6)). According to the known theory of diamagneticphase transitions at a < a = 0 . µ = 0. The start-ing point of the appearance of two minima is a = 0 . α = 0 . β = 2 .
5. Condon domains appear as a resultof a second order phase transition . One can obtain thesame result by expansion of the thermodynamic potentialdensity in powers of µ . µ = s a (1 + αβ ) − a (1 + αβ ) (7)It is seen from (7) that for α = 0 . β = 2 . a = 0 . µ is equal to zero. In Fig. 1 the ther-modynamic potential density is shown as a function of µ at various values of a . It is seen from Fig. 1 that there aretwo domains at a = 4 .
0, while at a = 8 . α = 9 . a = 11 .
8) the external domains are stable,while the internal domains are metastable. This meansthat a first-order phase transition occurs and metastableCondon domains are formed.By using equation (3) for aluminum we construct thephase diagram temperature-magnetic field in the two-frequency case when T D = 0 . K using ε F = 3 . eV and η c = m c / m e = 1 . , where m e is the electron mass(Fig.2). At a = 0.8 we obtain the curve of second-orderphase transitions. As is seen from Fig.2, the phase tran-sition occurs at T = 1 . K for H = 4 . T according tothe measurements . Starting from a = 7 . a = 9 .
138 we obtain thecurve of first-order phase transitions. At a = 11 .
95 thefour domains disappear. Condon domains in aluminumhave been observed by measurements of magnetizationoscillations of the relaxation rate of the muon precessionpolarization. According to the author’s interpretation,the line doublet indicating the appearance of two Con-don domains was not resolved by Fourier analysis due tolow resolution of µ SR spectrometer which was not ableto detect the expected splitting of induction lines. Wethink that the absence of the doublet splitting may bedue to the presence of numerous Condon domains takingplace in the metastability range. The doubling of Condondomains accompanied by metastable phenomena is char-acteristic of the two-frequency case. It can be detectedby improving the resolution of measurements of the µ SRspectrum. Within a slight deviation from the center ofmagnetization oscillations, second-order and first-orderdiamagnetic phase transitions occur. -4 -2 0 2 4-5051015 k a=4 a=8.2 a=9.138 a=11.8 FIG. 1. The dimensionless thermodynamic potential density4 πk Ω as a function of the dimensionless magnetization µ for α = 0 . β = 2 .
5: 1. a = 4 - two stable domains; 2. a = 8 - twometastable external domains and two stable internal domains;3. a = 9 .
138 - four domains have the same depth. The first-order phase transition occurs; 4. a = 11 . The first direct observation of Condon domains was de-tected in silver, in which magnetization oscillations aredue to belly and rosette oscillations . The authors ofRef. communicated that: “During the temperature de-pendence studies it was noted that there was no mag-netic induction splitting to two domains above 2 . K andthat the existence of a splitting at 2 . K depended onwhether the sample has been heated or cooled to thattemperature, consistent with a supercooling effect”. Themeasured temperature hysteresis which is approximatelyequal to 0 . K should be related to the obtained first-order phase transition. As is seen from the phase diagramfor aluminum, the temperature hysteresis is about 0 . K at H = 4 . T , whereas the observed temperature hystere-sis in silver is 0 . K , which is close to the calculated one.The appearance of an additional splitting of the magneticinduction lines in NMR and µ SR experiments should beexpected in the metastability range. The doubling of thenumber of Condon domains accompanied by metastablephenomena is characteristic of the two-frequency case. It can be detected by improving the resolution of NMR and µ SR measurements. However, it may be observed in thenarrow range of temperature and magnetic fields. By us-ing the temperature-magnetic field phase diagrams onecan find the metastability range and detect first-orderphase transitions.
IV. SUMMARY
Metastability effects in the formation of Condon non-spin magnetic domains have been considered. A possi-bility for the first-order phase transition occurrence in T ( K ) H (T)
Second-order phase transition First-order phase transition The lower limit of metastable states existence The upper limit of metastable states existence
FIG. 2. The phase diagram – temperature-magnetic field –in aluminum for T D = 0 . K . At a = 0 . a = 7 . a = 9 .
138 is a curveof first-order phase transitions. At a = 11 .
95 the four domainsdisappear and only two domains remain. a three-dimensional electron gas has been studied in thecase of two-frequency magnetization oscillations originat-ing from two extremal cross sections of the Fermi sur-face. The appearance of two additional domains has beenshown in the metastable region. The phase diagram ex-hibits the presence of second-order and first- order phasetransitions leading to formation of Condon domains inthe two-frequency case. D. Shoenberg,
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