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Size effect on quantum magnetic and thermo-magnetic oscillations in the non-spin domain phase
Physica B 502 (2016) 160–165
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Size effect on quantum magnetic and thermo-magnetic oscillations in the non-spin domain phase L.A. Bakaleinikov a,b,n, A. Gordon b a b
A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation Department of Exact Sciences, Faculty of Natural Sciences, University of Haifa, Oranim Campus, Tivon 36006, Israel
art ic l e i nf o
a b s t r a c t
Article history: Received 2 June 2016 Received in revised form 16 August 2016 Accepted 1 September 2016 Available online 3 September 2016
Magnetic and thermo-magnetic (magneto-caloric) oscillations are studied in quantizing magnetic fields in slabs under conditions of the existence of non-spin (Condon) domains. Size effects on the magnetization oscillations in thin samples are calculated in the domain phase. Computations are carried out in the center of the period of the magnetization and temperature oscillations, taking into account the sample size. Phase diagrams, describing diamagnetic phase transitions and formation of Condon domains, are presented in finite size silver and quasi-two-dimensional organic conductors (2D) samples. & 2016 Elsevier B.V. All rights reserved.
Keywords: Electron gas in quantizing magnetic fields De Haas-van Alphen oscillations Thermo-magnetic oscillations Condon domains Diamagnetic phase transitions Size effects
1. Introduction In the presence of high magnetic fields and sufficiently low temperatures metallic samples may split into two Condon domains under conditions of the de Haas-van Alphen effect (dHvA) [1–3]. These domains are of non-spin nature. This instability of an electron gas occurs as “diamagnetic phase transition” [2]. This phase transition from the homogeneous state to the inhomogeneous phase may take place in metallic single crystals of very high quality. Condon domains were detected in silver [3], beryllium [4] and white tin [5] by means of nuclear magnetic resonance (NMR) and muon spin-rotation spectroscopy ( μSR ) respectively. Formation of Condon domains and occurrence of diamagnetic phase transitions were studied in [6–9]. A direct observation of Condon domains in silver has also been produced by Hall probes [10]. A hysteresis loop in the dHvA effect due Condon domains has been observed and investigated in [11]. The transformation occurs in each cycle of dHvA oscillations when the reduced amplitude of oscillations approaches unity. A series of phase transitions leading to domain splitting takes place at discrete values of the external magnetic field. Static and dynamic effects close to diamagnetic phase transitions have been considered in [12,13]. n Corresponding author at: A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation. E-mail addresses: [email protected], [email protected] (L.A. Bakaleinikov).
http://dx.doi.org/10.1016/j.physb.2016.09.001 0921-4526/& 2016 Elsevier B.V. All rights reserved.
The phase transition is similar to the diamagnetic phase transition. In recent years there has been renewed interest in the problem of strong interactions between conduction electrons leading to diamagnetic phase transitions in experiment [14] and astrophysics [15]. All the theoretical and experimental researches on diamagnetic phase transitions and Condon domains have been studied in large samples and have been investigated on a macroscopic scale. Static mesoscopic properties of metals undergoing diamagnetic phase transitions or containing Condon domains have not been considered. We study the size effect in diamagnetic 3D and 2D metals by considering the three oscillations phenomena: 1) magnetization oscillations, 2) magnetic induction bifurcation of NMR and μSR lines due to Condon domains, 3) thermo-magnetic or magnetocaloric oscillations close to the instability point of 3D and 2D metals caused by diamagnetic phase transitions.
2. Model and results The oscillator part of the density of the thermodynamic potential can be written by using the Lifshitz-Kosevich formula within the first harmonic approximation [1]:
Ω=
⎤ 1 ⎡ 1 a cos( b) + a2sin2( b)⎥, 2⎢ ⎣ ⎦ 2 4π k
(1)
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
b = k(B − H0) = k⎡⎣ hex + 4π ( 1 − n)M ⎤⎦ , B is the magnetic induction, H0 is the magnetic field inside the material, k =
2πF
, F is the 2
H0
fundamental oscillation frequency, n is the demagnetization factor, hex = Hex − H0 , is the small increment of the applied magnetic field. M is the oscillatory part of the magnetization. In this approximation, the magnetization is found from the implicit equation of state [1]:
4πkM = a sin⎡⎣ k hex + 4π ( 1 − n)M ⎤⎦,
(
where a = 4π
)
( )
∂M ∂B B = H 0
(2)
is the reduced amplitude of magnetization
oscillations, i.e., the ratio between the amplitude of dHvA oscillations and their period. If the magnetic interaction is strong enough, a state of lower density of the thermodynamic potential can be achieved over part of an oscillation cycle by the sample splitting into domains of different values of magnetization. Since the demagnetization factor n is not a well-defined quantity, we omit it in Eq. (2). The demagnetization effect will be further included. Close to the phase transition temperature (which is found from the equation a ¼ 1 in the case of infinite specimen) we can present the density of the thermodynamic potential as an expansion in powers of the magnetization [8]:
8 3 2 4 π k aM . 3
Ω = 2π ( 1 − a)M2 +
(3)
We arrived at the density of the thermodynamic potential of the Landau theory-type:
Ω=−
A 2 B 4 M + M , 2 4
A = 4π (a − 1),
B=
32 3 2 aπ k . 3
(4)
According to [1], the temperature and magnetic field dependence of the amplitude a is:
a( T , H , TD) = a 0( H )
λT exp( −λTD), sinh( λT )
(5)
2
2π k m c
λ ≡ eℏBH c , mc is the cyclotron mass, c is the velocity of light, kB is the Boltzmann constant, e is the absolute value of the electron charge, ℏ =
h , 2π
h is the Planck constant, a0(H ) =
( )
3
Hmax 2 H
is de-
termined by Hmax which is the largest field at which the domain phase exists. In the following discussion we will analyze the effect of the sample size on the magnetization and temperature oscillations in the domain phase. The geometry to be considered is that of a slab of infinite planar extent, and of finite thickness L. We consider the strip domain structure. The rectangular, ideally uniaxial slab with the magnetization direction perpendicular to its thickness breaks up into domains of alternating slices of up magnetization and down magnetization with a domain width D. The demagnetization energy per unit volume is reduced to 142 ζ ( 3) M02D/L , where ζ is the
161
magnetic field and we are in the center of the cycle so that the up and down domain are equally wide. We suppose that L 4 D. The width of the domain walls d is very small compared with the distance between them, including the inhomogeneous term 1 ∂M K ( ∂x )2 [17] into (4) , (K is the positive coefficient, proportional to 2
rc2, where rc is the cyclotron radius) we determine the equilibrium magnetization in the domain center for a given thickness L by minimizing the density of the thermodynamic potential with respect to M0 and D. Minimizing the density of the thermodynamic potential with respect to the magnetization in the domain center and domain width
∂Ω ∂M0
= 0and
∂Ω ∂D
= 0and using σ =
2
4 KM0 3 d
[18],
rc2
where d is the domain wall thickness and K = 4 , we derive the equilibrium value of M0 for the given slab thickness L and domain width D:
M0 =
1 1⎡ 1 ⎛ 3 ⎞2 ⎜ ⎟ ( a − 1) 2 ⎢ 1 − ⎝ ⎠ 2π k 2a ⎣
D = 0.3rc
Lc =
1
Lc ⎤ 2 ⎥ , L ⎦
L , d
(8)
(9)
rc2 2
9d( a − 1)
, (10)
In the limiting case of a very thick specimen ( L → ∞), the diamagnetic phase transition occurs at a = 1. In the approximation, in which Eq. (8) is derived, the obtained result implies a shift of the diamagnetic phase transition compared to the infinite sample case and a shift of the temperature-magnetic field phase diagram. As films become thinner, the demagnetization becomes important, lowering both the phase transition point and the magnetization. The order parameter of the diamagnetic phase transition M0 , the magnetization in the paramagnetic domain, is equal to zero when L = Lc , where Lc is the minimal slab thickness below which the Condon domain phase disappears. Equation a = 1 describes the temperature-magnetic field phase diagram determining the locus of points showing the series of the phase-transition temperatures in an infinite sample. The envelope of these points serves as the diamagnetic phase transition boundary. However, following from Eq. (8), the domain phase appears in the slab not at a ¼1 but at
a=1+
rc 3 dL
(11)
in the film of thickness L . Thus, there is the shift of the phase transition point as a result of the size effect. This means that the diamagnetic phase transition in the finite size specimen exhibits a new behavior, which is different from that known from the literature [1–3].
π
ζ - function [16] and M0 is the magnetization in the center of the domain. The breakup creates domain walls with a surface energys. The contribution of the domain-wall energy and the total demagnetization-energy of a multiple-domain plate to the density of the thermodynamic potential is then
⎛L ⎞1 14 D ζ ( 3)M02 + σ ⎜ − 1⎟ . ⎝D ⎠L L π2
(6)
Thus, the density of the thermodynamic potential per unit volume of the slab is
Ω=−
⎛L ⎞1 A 2 B 4 14 D M0 + M0 + 2 ζ ( 3)M02 + σ ⎜ − 1⎟ . ⎝ ⎠L 2 4 L D π
(7)
To simplify the treatment we suppose the slab normal to the
2.1. Size effect on magnetization oscillations Temperature-magnetic field-slab thickness diagrams showing the regions of existence of the disordered, no domain phase, and the ordered, domain phase, are presented in Fig. 1 for TD = 0.1 K , L¼ 2 μm and L ¼20 μm in silver. It should be noted that the size effect exists only at high magnetic fields for which the domain wall width and the cyclotron radius are smaller than the domain width and the sample thickness: d, rc < D < L . For this reason, we present only the relevant part of the phase diagram corresponding to high magnetic fields on the right side of the phase diagram. The magnetization as a function of the slab thickness is presented in Fig. 2 in silver for T¼ 2.8 K. H¼35 T, TD ¼0.1 K. It is seen that the magnetization tends to zero while the
162
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Fig. 1. Phase diagrams, temperature (in K ) - magnetic field (in Tesla), for two specimens of silver of thickness L ¼2 μm and L ¼ 20 μm and TD ¼ 0.1 K. The calculated curve is the locus of phase transitions from the homogeneous phase outside the “bell” to the domain phase inside the “bell”. The curve separating the two phases r is determined by a = 1 + c . 3 dL
Fig. 3. The temperature dependence of the magnetization M0 at H¼ 35 T and thickness L ¼2 μm and L ¼ 20 μm. This result indicates that it tends to zero at T = Tc .
Measurements in silver [3] and aluminum [19] were made in the center of the period of dHvA oscillations or near it because the intensities of the lines due to the two Condon domains are equal or almost equal. The calculated temperature dependences of the magnetic induction bifurcation in a 3D electron gas describe well the temperature dependence of the magnetic induction splitting in silver [20] and aluminum [21]. This effect is also considered in the case of quasi-two-dimensional metals [22]. The magnetic induction splitting due to Condon domains ΔB is equal to the difference between the paramagnetic part of the magnetization 4πM0 > 0and the diamagnetic part −4πM0 < 0, i.e., ΔB = 4π ΔM = 8πM0 , where the magnetization in the domain center M0 is determined by Eq. (8). Thus, the magnetic induction bifurcation is given: 1 1⎡ 4 ⎛ 3 ⎞2 ⎟ ( a − 1) 2 ⎢ 1 − ΔB = ⎜ ⎠ ⎝ k 2a ⎣
Fig. 2. Dependence of the magnetization M0 (given in A/m ) in the paramagnetic Condon domain ( M0 40) on the thickness (given in μm) for the silver specimen at TD ¼ 0.1 K.
thickness tends to the minimal value of the slab thickness Lс . The domain wall used in Figs. 1–3 is the parameter of the theory and is equal to 0.2 μm. Fig. 3 demonstrates the temperature dependence of the magnetization. It can be seen that M0 tends to zero when the temperature tends to the phase transition temperature, Tc , determined by Eq. (11). Following from the phase diagram the value Tc is equal to 2.9 K at H = 35 T, L = 2 μm and 3.4 K at H¼ 35 T, L¼ 20μm. Since widths of domain walls have not been measured in such small samples, we take them to be equal to the order of magnitude of the cyclotron radius. 2.2. Size effect on the splitting of NMR and μ SR lines due to condon domains The utilization of local probes is an important tool of the investigation of the non-spin magnetism. As is known, NMR [3] and μ SR [4] can directly measure internal magnetic fields in the domain phase within a high accuracy. In NMR and μSR experiments the domain structure is revealed by the appearance of a doublet corresponding to two domain magnetic inductions [3–5].
1
Lc ⎤ 2 ⎥ . L ⎦
(12)
The magnetic induction bifurcation ΔB due to Condon domains is presented in Fig. 4 as a function of the slab thickness in silver for T¼ 2.8 K, H ¼35 T, TD ¼ 0.1 K. The induction splitting tends to zero while the thickness tends to the minimal value of the slab thickness at which the domain phase disappears. Close to the phase transition the magnetization in the paramagnetic domain M0 is given by 1
1
M0 ≈
1 ⎛ L − L ⎞2 1 ⎛ 3 ⎞2 c ⎜ ⎟ ( a − 1) 2 ⎜ ⎟ , ⎝ L ⎠ 4π k ⎝ a ⎠
M0 ≈
⎤2 1 ⎡ 3 ⎤ 2 ⎡ ⎛ da ⎞ ⎢ ⎥ ⎢ −⎜ ⎟|T = Tc ( Tc − T )⎥ , 2πk ⎢⎣ 2a( Tc ) ⎥⎦ ⎣ ⎝ dT ⎠ ⎦
1
(13)
1
(14)
or 1
M0 ≈
1 1 ⎛ 3λ ⎞ 2 ⎜ ⎟ [Λ( λTc )( Tc − T )]2 , 2π k ⎝ 2 ⎠
Λ( λTc ) = cth( λTc ) −
1 λTc
(14a)
and the magnetic induction splitting re-
lated to the existence of two Condon domains is given by
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Fig. 4. Dependence of the magnetic induction splitting ΔB (given in G) on the thickness (given in μm) for the silver specimen at TD ¼0.1 K.
Fig. 5. Amplitude of temperature oscillations ΔT as a function of the specimen thickness in silver at TD = 0.1 K , T ¼ 2.8 K and H¼ 35 T.
1
1
ΔB ≈
1 ⎛ L − L ⎞2 2 ⎛ 3 ⎞2 c ⎜ ⎟ ( a − 1) 2 ⎜ ⎟ , ⎝ ⎠ ⎝ L ⎠ k a
(15)
In Fig. 5, we present the amplitude of the temperature oscillations ΔT as a function of the specimen thickness in silver at the temperature T ¼2.8 K, magnetic field H¼ 35 T and C = 633
ΔB ≈
1 ⎤2 ⎡
1 ⎤2
⎛ da ⎞ 4⎡ 3 ⎢ ⎥ ⎢ −⎜ ⎟|T = T ( Tc − T )⎥ , k ⎢⎣ 2a( Tc ) ⎥⎦ ⎣ ⎝ dT ⎠ c ⎦
(16)
1
ΔB ≈
1 4 ⎛ 3λ ⎞ 2 ⎡ ⎜ ⎟ ⎣ Λ( λTc )( Tc − T )⎤⎦ 2 , k⎝ 2 ⎠
(16a) da dT
where Tc is the phase transition temperature, < 0 since a decreases with increasing T . Thus, the magnetization in the domain M0 and the induction splitting ΔB reflecting the behavior of the order parameter of the phase transition approach zero when temperature and the sample thickness tend to their limiting values 1
according to the formula ( Xc − X ) 2 , where Xc = Tc or Xc = Lc , X = T or X = L . This dependence is characteristic of second order phase transitions. The calculated critical indices are typical of the Landau theory of phase transitions. 2.3. Size effect on thermo-magnetic or magneto-caloric oscillations close to diamagnetic phase transitions The thermo-magnetic oscillations present the known magnetocaloric effect. The amplitude of the temperature oscillations of a sample ΔT is given by [1]
ΔT = −
T C
163
⎛ dM0 ⎞ ⎟ dH, dT ⎠H
∫ ⎜⎝
(17)
where C is the specific heat and the integration is produced over the dHvA period for the given magnetic field. The considered temperature changes are carried out at adiabatic changes of the applied magnetic field. The influence of the magnetic interaction on thermo-magnetic oscillations has been considered in [1]. However, this effect has not been studied close to the phase transition and in finite size samples. We present the amplitude of temperature oscillations in vicinity of the phase transition and its size effect. Under these conditions, the expected effects should be much more significant than those calculated and measured in [1].
J K ⋅ m3
[23].
The amplitude of temperature oscillations increases, while the sample thickness decreases. It diverges at the L = Lc . We can interpret this effect as follows. The rate of the temperature change of the order parameter of the phase transition, i.e., the magnetization in the paramagnetic domain, increases with the slab thickness decreasing. This means that the entropy and, in fact, the intensity of magnetic disorder increase, since the entropy is proportional to the derivative of the magnetization with respect to temperature. For this reason, the entropy growth leads to increasing the amplitude of temperature oscillations for lower slab thicknesses tending to infinity close to the critical sample thickness Lc . Size effects presented in Figs. 1–5 take place only at high magnetic fields corresponding to the right side of the phase diagram.
3. Quasi-two-dimensional organic metals Condon domains and diamagnetic phase transitions have not been observed in quasi 2D metals. The long-range nature of magnetic interactions between orbital magnetic moments of conduction electrons in metals under quantizing magnetic fields, typical for the Landau second phase transition theory, justifies the usage of the mean-field approach for the description of diamagnetic phase transitions in cases of 2D and 3D electron gases. For this reason, the results obtained in Chapter II should be similar to those for the 2D electron gas. The only difference between the two cases is the equation for the reduced amplitude of magnetic oscillations. The reduced amplitude of oscillations a( T , H , TD) is presented according to [9,22]:
a( T , H , TD) = 4π
2 ⎛ Ce ⎛ F ⎞ 2λ T ⎞ ⎜ ⎟ exp( −λTD)⎜ 1 − 2 ⎟, ⎝ ⎠ ⎝ ηc H π ⎠
(18)
164
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Tc =
ℏωc ⎛ H2 ⎞ ⎜⎜ 1 − 2 ⎟⎟, 4kB ⎝ H0 ⎠
H0 = F
4π C e exp( −λTD), ηc
⎛ e2 ⎞ Ce ≡ ⎜ ⎟/πme c 2, ⎝ c* ⎠ ηc =
mc , me
(19)
me is the electron mass, ωc =
eH mc c
is the cyclotron fre-
quency, kB is the Boltzmann constant, c* is the lattice constant in direction perpendicular to the conducting plane,
2π c*
is the effective
height of the cylinder modeling the Fermi surface in the 2D conductor. r Using a = 1 + c , we calculate phase diagrams for nearly 2D 3 dL
organic conductors of the type
( ET )2X with
X = Cu( NCS ) , 2
KHg ( SCN ) , I3,AuBr2, IBr2, etc., where ET stands for bis (ethelene4
dithio) – tetrathiafulvalene (or BEDT-TTF) [24]. The fundamental Fig. 8. The dependence of the magnetic induction splitting ΔB (given in G) on the thickness (given in μm) for the κ(BEDT − TTF )2I3 specimen at T ¼ 0.09 K, H¼ 5 T, TD ¼0.1 K. 0
frequency is F = 3.8⋅103T [25], ηc = 2.7 [24], c* = 8.5A [24], TD = 0.1 K . The phase diagrams are shown in Fig. 6. The thickness of the domain wall is equal to 0.35 μm . Using Eqs. (8), (18) and (19), we obtain the magnetization in the paramagnetic Condon domain as a function of the slab thickness L in μm for T ¼0.09 K, H¼ 5 T, TD ¼0.1 K (see Fig. 7). The magnetic induction bifurcation ΔB due to Condon domains is presented in Fig. 8 as a function of the slab thickness in μm for T¼ 0.09 K, H¼ 5 T, TD ¼0.1 K (Eq. (12)). It is seen that the induction splitting tends to zero while the thickness tends to the minimal value of the slab thickness Lc at which the domain phase disappears. We have not calculated the amplitude of temperature oscillations in quasi-2D organic conductors since results of measurements of the specific heat at such low temperatures are unknown. Fig. 6. Phase diagrams, temperature (in K ) – magnetic field (in Tesla), for two specimens of κ(BEDT − TTF )2I3 of thickness L ¼4 μm and L ¼ 40 μm and TD = 0.1 K .
Fig. 7. Dependence of the magnetization M0 (given in A/m ) in the paramagnetic Condon domain ( M0 40) on the thickness (given in μm ) for κ(BEDT − TTF )2I3specimen at T ¼0.09 K, H¼ 5 T, TD ¼0.1 K.
4. Conclusions We have analyzed diamagnetic phase transitions in samples of finite thickness (thin film). The geometry considered is that of a slab of infinite planar extent and of finite thickness. We have examined the stripe-like domain structure with diamagnetic phase transitions of the second order. We have shown that the critical sample size for the diamagnetic phase transition exists, i.e., there is the film thickness, below which the order parameter of the diamagnetic phase transition, the magnetization in the paramagnetic domain, is zero. We have presented phase diagrams (temperature-magnetic field) exhibiting the domain phase and giving the curve separating the ordered and disordered phases in 3D silver and quasi 2D organic metals. We have calculated the thickness dependence of the magnetization in the paramagnetic domain in 2D and 3D metals. We have obtained a shift of the phase transition temperature and the temperature-magnetic field phase diagram as a result of the size effect. We have computed the size effect on the thermo-magnetic (magneto-caloric) oscillations in silver. We have shown that the amplitude of thermo-magnetic oscillations grows at lower slab thicknesses diverging at minimal value Lc . We have discovered that the critical indices for the temperature and sample thickness of the magnetization in the paramagnetic domain are typical of the Landau mean-field theory of second phase transitions.
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
References [1] D. Shoenberg, Magnetic Oscillations in Metals, Cambridge University Press, Cambridge, England, 1984. [2] A. Privorotskii, Thermodynamic Theory of Domain Structures, Wiley, New York, and Israel University Press, Jerusalem, 1976. [3] J.H. Condon, R.E. Walstedt, Phys. Rev. Lett. 21 (1968) 612. [4] G. Solt, C. Baines, V.S. Egorov, D. Herlach, E. Krasnoperov, U. Zimmermann, Phys. Rev. Lett. 76 (1996) 2575. [5] G. Solt, V.S. Egorov, C. Baines, D. Herlach, U. Zimmermann, Phys. Rev. B62 (2000) R11933. [6] A. Gordon, I.D. Vagner, P. Wyder, Phys. Rev. B41 (1990) 658. [7] B. Grushko, A. Gordon, I.D. Vagner, P. Wyder, Phys. Rev. B45 (1992) 3119. [8] A. Gordon, M.A. Itskovsky, P. Wyder, Phys. Rev. B55 (1997) 812. [9] A. Gordon, M.A. Itskovsky, I.D. Vagner, P. Wyder, Phys. Rev. Lett. 81 (1998) 2787. [10] R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, W. Joss, Phys. Rev. Lett. 95 (2005) 267209. [11] R.B.G. Kramer, V.S. Egorov, A.G.M. Jansen, W. Joss, Phys. Rev. Lett. 95 (2005)
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187204. [12] L. Bakaleinikov, A. Gordon, Physica B407 (2012) 3408. [13] L. Bakaleinikov, A. Gordon, J. Magn. Magn. Mater. 333 (2013) 124. [14] R.B.G. Kramer, V.S. Egorov, V.A. Gasparov, A.G.M. Jansen, W. Joss, Low Temp. Phys., 37, 2011, p. 39. [15] Z. Wang, G. Lu, C. Zhu, W. Huo, Astrophys. J. 773 (2013) 160. [16] M. Prutton, Thin Ferromagnetic Films, Butterworths, London, 1964. [17] L.D. Landau, E.M. Lifshitz, Statistical Physics, Pergamon Press, London, 1958. [18] T. Mitsui, J. Furuichi, Phys. Rev., 90, 1953, p. 193; Phys. Rev., 95, 1954, p. 558. [19] G. Solt, V.S. Egorov, C. Baines, D. Herlach, U. Zimmermann, Physica B326 (2003) 536. [20] A. Gordon, M.A. Itskovsky, P. Wyder, Phys. Rev. B59 (1999) 10864. [21] A. Gordon, N. Logoboy, W. Joss, Physica B353 (2004) 296. [22] L. Bakaleinikov, A. Gordon, J. Magn. Magn. Mater. 368 (2014) 281. [23] D.L. Martin, Phys. Rev. 141 (1966) 576. [24] J. Wosnitza, Int. J. Modern Phys., B7, 1993, p. 2707. [25] E. Balthes, D. Schweitzer, I. Heinen, W. Strunz, E. Steep, A.G.M. Jansen, P. Wyder, Physica C317–318 (1999) 108.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Size effect on quantum magnetic and thermo-magnetic oscillations in the non-spin domain phase L.A. Bakaleinikov a,b,n, A. Gordon b a b
A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation Department of Exact Sciences, Faculty of Natural Sciences, University of Haifa, Oranim Campus, Tivon 36006, Israel
art ic l e i nf o
a b s t r a c t
Article history: Received 2 June 2016 Received in revised form 16 August 2016 Accepted 1 September 2016 Available online 3 September 2016
Magnetic and thermo-magnetic (magneto-caloric) oscillations are studied in quantizing magnetic fields in slabs under conditions of the existence of non-spin (Condon) domains. Size effects on the magnetization oscillations in thin samples are calculated in the domain phase. Computations are carried out in the center of the period of the magnetization and temperature oscillations, taking into account the sample size. Phase diagrams, describing diamagnetic phase transitions and formation of Condon domains, are presented in finite size silver and quasi-two-dimensional organic conductors (2D) samples. & 2016 Elsevier B.V. All rights reserved.
Keywords: Electron gas in quantizing magnetic fields De Haas-van Alphen oscillations Thermo-magnetic oscillations Condon domains Diamagnetic phase transitions Size effects
1. Introduction In the presence of high magnetic fields and sufficiently low temperatures metallic samples may split into two Condon domains under conditions of the de Haas-van Alphen effect (dHvA) [1–3]. These domains are of non-spin nature. This instability of an electron gas occurs as “diamagnetic phase transition” [2]. This phase transition from the homogeneous state to the inhomogeneous phase may take place in metallic single crystals of very high quality. Condon domains were detected in silver [3], beryllium [4] and white tin [5] by means of nuclear magnetic resonance (NMR) and muon spin-rotation spectroscopy ( μSR ) respectively. Formation of Condon domains and occurrence of diamagnetic phase transitions were studied in [6–9]. A direct observation of Condon domains in silver has also been produced by Hall probes [10]. A hysteresis loop in the dHvA effect due Condon domains has been observed and investigated in [11]. The transformation occurs in each cycle of dHvA oscillations when the reduced amplitude of oscillations approaches unity. A series of phase transitions leading to domain splitting takes place at discrete values of the external magnetic field. Static and dynamic effects close to diamagnetic phase transitions have been considered in [12,13]. n Corresponding author at: A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, 194021, Russian Federation. E-mail addresses: [email protected], [email protected] (L.A. Bakaleinikov).
http://dx.doi.org/10.1016/j.physb.2016.09.001 0921-4526/& 2016 Elsevier B.V. All rights reserved.
The phase transition is similar to the diamagnetic phase transition. In recent years there has been renewed interest in the problem of strong interactions between conduction electrons leading to diamagnetic phase transitions in experiment [14] and astrophysics [15]. All the theoretical and experimental researches on diamagnetic phase transitions and Condon domains have been studied in large samples and have been investigated on a macroscopic scale. Static mesoscopic properties of metals undergoing diamagnetic phase transitions or containing Condon domains have not been considered. We study the size effect in diamagnetic 3D and 2D metals by considering the three oscillations phenomena: 1) magnetization oscillations, 2) magnetic induction bifurcation of NMR and μSR lines due to Condon domains, 3) thermo-magnetic or magnetocaloric oscillations close to the instability point of 3D and 2D metals caused by diamagnetic phase transitions.
2. Model and results The oscillator part of the density of the thermodynamic potential can be written by using the Lifshitz-Kosevich formula within the first harmonic approximation [1]:
Ω=
⎤ 1 ⎡ 1 a cos( b) + a2sin2( b)⎥, 2⎢ ⎣ ⎦ 2 4π k
(1)
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
b = k(B − H0) = k⎡⎣ hex + 4π ( 1 − n)M ⎤⎦ , B is the magnetic induction, H0 is the magnetic field inside the material, k =
2πF
, F is the 2
H0
fundamental oscillation frequency, n is the demagnetization factor, hex = Hex − H0 , is the small increment of the applied magnetic field. M is the oscillatory part of the magnetization. In this approximation, the magnetization is found from the implicit equation of state [1]:
4πkM = a sin⎡⎣ k hex + 4π ( 1 − n)M ⎤⎦,
(
where a = 4π
)
( )
∂M ∂B B = H 0
(2)
is the reduced amplitude of magnetization
oscillations, i.e., the ratio between the amplitude of dHvA oscillations and their period. If the magnetic interaction is strong enough, a state of lower density of the thermodynamic potential can be achieved over part of an oscillation cycle by the sample splitting into domains of different values of magnetization. Since the demagnetization factor n is not a well-defined quantity, we omit it in Eq. (2). The demagnetization effect will be further included. Close to the phase transition temperature (which is found from the equation a ¼ 1 in the case of infinite specimen) we can present the density of the thermodynamic potential as an expansion in powers of the magnetization [8]:
8 3 2 4 π k aM . 3
Ω = 2π ( 1 − a)M2 +
(3)
We arrived at the density of the thermodynamic potential of the Landau theory-type:
Ω=−
A 2 B 4 M + M , 2 4
A = 4π (a − 1),
B=
32 3 2 aπ k . 3
(4)
According to [1], the temperature and magnetic field dependence of the amplitude a is:
a( T , H , TD) = a 0( H )
λT exp( −λTD), sinh( λT )
(5)
2
2π k m c
λ ≡ eℏBH c , mc is the cyclotron mass, c is the velocity of light, kB is the Boltzmann constant, e is the absolute value of the electron charge, ℏ =
h , 2π
h is the Planck constant, a0(H ) =
( )
3
Hmax 2 H
is de-
termined by Hmax which is the largest field at which the domain phase exists. In the following discussion we will analyze the effect of the sample size on the magnetization and temperature oscillations in the domain phase. The geometry to be considered is that of a slab of infinite planar extent, and of finite thickness L. We consider the strip domain structure. The rectangular, ideally uniaxial slab with the magnetization direction perpendicular to its thickness breaks up into domains of alternating slices of up magnetization and down magnetization with a domain width D. The demagnetization energy per unit volume is reduced to 142 ζ ( 3) M02D/L , where ζ is the
161
magnetic field and we are in the center of the cycle so that the up and down domain are equally wide. We suppose that L 4 D. The width of the domain walls d is very small compared with the distance between them, including the inhomogeneous term 1 ∂M K ( ∂x )2 [17] into (4) , (K is the positive coefficient, proportional to 2
rc2, where rc is the cyclotron radius) we determine the equilibrium magnetization in the domain center for a given thickness L by minimizing the density of the thermodynamic potential with respect to M0 and D. Minimizing the density of the thermodynamic potential with respect to the magnetization in the domain center and domain width
∂Ω ∂M0
= 0and
∂Ω ∂D
= 0and using σ =
2
4 KM0 3 d
[18],
rc2
where d is the domain wall thickness and K = 4 , we derive the equilibrium value of M0 for the given slab thickness L and domain width D:
M0 =
1 1⎡ 1 ⎛ 3 ⎞2 ⎜ ⎟ ( a − 1) 2 ⎢ 1 − ⎝ ⎠ 2π k 2a ⎣
D = 0.3rc
Lc =
1
Lc ⎤ 2 ⎥ , L ⎦
L , d
(8)
(9)
rc2 2
9d( a − 1)
, (10)
In the limiting case of a very thick specimen ( L → ∞), the diamagnetic phase transition occurs at a = 1. In the approximation, in which Eq. (8) is derived, the obtained result implies a shift of the diamagnetic phase transition compared to the infinite sample case and a shift of the temperature-magnetic field phase diagram. As films become thinner, the demagnetization becomes important, lowering both the phase transition point and the magnetization. The order parameter of the diamagnetic phase transition M0 , the magnetization in the paramagnetic domain, is equal to zero when L = Lc , where Lc is the minimal slab thickness below which the Condon domain phase disappears. Equation a = 1 describes the temperature-magnetic field phase diagram determining the locus of points showing the series of the phase-transition temperatures in an infinite sample. The envelope of these points serves as the diamagnetic phase transition boundary. However, following from Eq. (8), the domain phase appears in the slab not at a ¼1 but at
a=1+
rc 3 dL
(11)
in the film of thickness L . Thus, there is the shift of the phase transition point as a result of the size effect. This means that the diamagnetic phase transition in the finite size specimen exhibits a new behavior, which is different from that known from the literature [1–3].
π
ζ - function [16] and M0 is the magnetization in the center of the domain. The breakup creates domain walls with a surface energys. The contribution of the domain-wall energy and the total demagnetization-energy of a multiple-domain plate to the density of the thermodynamic potential is then
⎛L ⎞1 14 D ζ ( 3)M02 + σ ⎜ − 1⎟ . ⎝D ⎠L L π2
(6)
Thus, the density of the thermodynamic potential per unit volume of the slab is
Ω=−
⎛L ⎞1 A 2 B 4 14 D M0 + M0 + 2 ζ ( 3)M02 + σ ⎜ − 1⎟ . ⎝ ⎠L 2 4 L D π
(7)
To simplify the treatment we suppose the slab normal to the
2.1. Size effect on magnetization oscillations Temperature-magnetic field-slab thickness diagrams showing the regions of existence of the disordered, no domain phase, and the ordered, domain phase, are presented in Fig. 1 for TD = 0.1 K , L¼ 2 μm and L ¼20 μm in silver. It should be noted that the size effect exists only at high magnetic fields for which the domain wall width and the cyclotron radius are smaller than the domain width and the sample thickness: d, rc < D < L . For this reason, we present only the relevant part of the phase diagram corresponding to high magnetic fields on the right side of the phase diagram. The magnetization as a function of the slab thickness is presented in Fig. 2 in silver for T¼ 2.8 K. H¼35 T, TD ¼0.1 K. It is seen that the magnetization tends to zero while the
162
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Fig. 1. Phase diagrams, temperature (in K ) - magnetic field (in Tesla), for two specimens of silver of thickness L ¼2 μm and L ¼ 20 μm and TD ¼ 0.1 K. The calculated curve is the locus of phase transitions from the homogeneous phase outside the “bell” to the domain phase inside the “bell”. The curve separating the two phases r is determined by a = 1 + c . 3 dL
Fig. 3. The temperature dependence of the magnetization M0 at H¼ 35 T and thickness L ¼2 μm and L ¼ 20 μm. This result indicates that it tends to zero at T = Tc .
Measurements in silver [3] and aluminum [19] were made in the center of the period of dHvA oscillations or near it because the intensities of the lines due to the two Condon domains are equal or almost equal. The calculated temperature dependences of the magnetic induction bifurcation in a 3D electron gas describe well the temperature dependence of the magnetic induction splitting in silver [20] and aluminum [21]. This effect is also considered in the case of quasi-two-dimensional metals [22]. The magnetic induction splitting due to Condon domains ΔB is equal to the difference between the paramagnetic part of the magnetization 4πM0 > 0and the diamagnetic part −4πM0 < 0, i.e., ΔB = 4π ΔM = 8πM0 , where the magnetization in the domain center M0 is determined by Eq. (8). Thus, the magnetic induction bifurcation is given: 1 1⎡ 4 ⎛ 3 ⎞2 ⎟ ( a − 1) 2 ⎢ 1 − ΔB = ⎜ ⎠ ⎝ k 2a ⎣
Fig. 2. Dependence of the magnetization M0 (given in A/m ) in the paramagnetic Condon domain ( M0 40) on the thickness (given in μm) for the silver specimen at TD ¼ 0.1 K.
thickness tends to the minimal value of the slab thickness Lс . The domain wall used in Figs. 1–3 is the parameter of the theory and is equal to 0.2 μm. Fig. 3 demonstrates the temperature dependence of the magnetization. It can be seen that M0 tends to zero when the temperature tends to the phase transition temperature, Tc , determined by Eq. (11). Following from the phase diagram the value Tc is equal to 2.9 K at H = 35 T, L = 2 μm and 3.4 K at H¼ 35 T, L¼ 20μm. Since widths of domain walls have not been measured in such small samples, we take them to be equal to the order of magnitude of the cyclotron radius. 2.2. Size effect on the splitting of NMR and μ SR lines due to condon domains The utilization of local probes is an important tool of the investigation of the non-spin magnetism. As is known, NMR [3] and μ SR [4] can directly measure internal magnetic fields in the domain phase within a high accuracy. In NMR and μSR experiments the domain structure is revealed by the appearance of a doublet corresponding to two domain magnetic inductions [3–5].
1
Lc ⎤ 2 ⎥ . L ⎦
(12)
The magnetic induction bifurcation ΔB due to Condon domains is presented in Fig. 4 as a function of the slab thickness in silver for T¼ 2.8 K, H ¼35 T, TD ¼ 0.1 K. The induction splitting tends to zero while the thickness tends to the minimal value of the slab thickness at which the domain phase disappears. Close to the phase transition the magnetization in the paramagnetic domain M0 is given by 1
1
M0 ≈
1 ⎛ L − L ⎞2 1 ⎛ 3 ⎞2 c ⎜ ⎟ ( a − 1) 2 ⎜ ⎟ , ⎝ L ⎠ 4π k ⎝ a ⎠
M0 ≈
⎤2 1 ⎡ 3 ⎤ 2 ⎡ ⎛ da ⎞ ⎢ ⎥ ⎢ −⎜ ⎟|T = Tc ( Tc − T )⎥ , 2πk ⎢⎣ 2a( Tc ) ⎥⎦ ⎣ ⎝ dT ⎠ ⎦
1
(13)
1
(14)
or 1
M0 ≈
1 1 ⎛ 3λ ⎞ 2 ⎜ ⎟ [Λ( λTc )( Tc − T )]2 , 2π k ⎝ 2 ⎠
Λ( λTc ) = cth( λTc ) −
1 λTc
(14a)
and the magnetic induction splitting re-
lated to the existence of two Condon domains is given by
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Fig. 4. Dependence of the magnetic induction splitting ΔB (given in G) on the thickness (given in μm) for the silver specimen at TD ¼0.1 K.
Fig. 5. Amplitude of temperature oscillations ΔT as a function of the specimen thickness in silver at TD = 0.1 K , T ¼ 2.8 K and H¼ 35 T.
1
1
ΔB ≈
1 ⎛ L − L ⎞2 2 ⎛ 3 ⎞2 c ⎜ ⎟ ( a − 1) 2 ⎜ ⎟ , ⎝ ⎠ ⎝ L ⎠ k a
(15)
In Fig. 5, we present the amplitude of the temperature oscillations ΔT as a function of the specimen thickness in silver at the temperature T ¼2.8 K, magnetic field H¼ 35 T and C = 633
ΔB ≈
1 ⎤2 ⎡
1 ⎤2
⎛ da ⎞ 4⎡ 3 ⎢ ⎥ ⎢ −⎜ ⎟|T = T ( Tc − T )⎥ , k ⎢⎣ 2a( Tc ) ⎥⎦ ⎣ ⎝ dT ⎠ c ⎦
(16)
1
ΔB ≈
1 4 ⎛ 3λ ⎞ 2 ⎡ ⎜ ⎟ ⎣ Λ( λTc )( Tc − T )⎤⎦ 2 , k⎝ 2 ⎠
(16a) da dT
where Tc is the phase transition temperature, < 0 since a decreases with increasing T . Thus, the magnetization in the domain M0 and the induction splitting ΔB reflecting the behavior of the order parameter of the phase transition approach zero when temperature and the sample thickness tend to their limiting values 1
according to the formula ( Xc − X ) 2 , where Xc = Tc or Xc = Lc , X = T or X = L . This dependence is characteristic of second order phase transitions. The calculated critical indices are typical of the Landau theory of phase transitions. 2.3. Size effect on thermo-magnetic or magneto-caloric oscillations close to diamagnetic phase transitions The thermo-magnetic oscillations present the known magnetocaloric effect. The amplitude of the temperature oscillations of a sample ΔT is given by [1]
ΔT = −
T C
163
⎛ dM0 ⎞ ⎟ dH, dT ⎠H
∫ ⎜⎝
(17)
where C is the specific heat and the integration is produced over the dHvA period for the given magnetic field. The considered temperature changes are carried out at adiabatic changes of the applied magnetic field. The influence of the magnetic interaction on thermo-magnetic oscillations has been considered in [1]. However, this effect has not been studied close to the phase transition and in finite size samples. We present the amplitude of temperature oscillations in vicinity of the phase transition and its size effect. Under these conditions, the expected effects should be much more significant than those calculated and measured in [1].
J K ⋅ m3
[23].
The amplitude of temperature oscillations increases, while the sample thickness decreases. It diverges at the L = Lc . We can interpret this effect as follows. The rate of the temperature change of the order parameter of the phase transition, i.e., the magnetization in the paramagnetic domain, increases with the slab thickness decreasing. This means that the entropy and, in fact, the intensity of magnetic disorder increase, since the entropy is proportional to the derivative of the magnetization with respect to temperature. For this reason, the entropy growth leads to increasing the amplitude of temperature oscillations for lower slab thicknesses tending to infinity close to the critical sample thickness Lc . Size effects presented in Figs. 1–5 take place only at high magnetic fields corresponding to the right side of the phase diagram.
3. Quasi-two-dimensional organic metals Condon domains and diamagnetic phase transitions have not been observed in quasi 2D metals. The long-range nature of magnetic interactions between orbital magnetic moments of conduction electrons in metals under quantizing magnetic fields, typical for the Landau second phase transition theory, justifies the usage of the mean-field approach for the description of diamagnetic phase transitions in cases of 2D and 3D electron gases. For this reason, the results obtained in Chapter II should be similar to those for the 2D electron gas. The only difference between the two cases is the equation for the reduced amplitude of magnetic oscillations. The reduced amplitude of oscillations a( T , H , TD) is presented according to [9,22]:
a( T , H , TD) = 4π
2 ⎛ Ce ⎛ F ⎞ 2λ T ⎞ ⎜ ⎟ exp( −λTD)⎜ 1 − 2 ⎟, ⎝ ⎠ ⎝ ηc H π ⎠
(18)
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L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
Tc =
ℏωc ⎛ H2 ⎞ ⎜⎜ 1 − 2 ⎟⎟, 4kB ⎝ H0 ⎠
H0 = F
4π C e exp( −λTD), ηc
⎛ e2 ⎞ Ce ≡ ⎜ ⎟/πme c 2, ⎝ c* ⎠ ηc =
mc , me
(19)
me is the electron mass, ωc =
eH mc c
is the cyclotron fre-
quency, kB is the Boltzmann constant, c* is the lattice constant in direction perpendicular to the conducting plane,
2π c*
is the effective
height of the cylinder modeling the Fermi surface in the 2D conductor. r Using a = 1 + c , we calculate phase diagrams for nearly 2D 3 dL
organic conductors of the type
( ET )2X with
X = Cu( NCS ) , 2
KHg ( SCN ) , I3,AuBr2, IBr2, etc., where ET stands for bis (ethelene4
dithio) – tetrathiafulvalene (or BEDT-TTF) [24]. The fundamental Fig. 8. The dependence of the magnetic induction splitting ΔB (given in G) on the thickness (given in μm) for the κ(BEDT − TTF )2I3 specimen at T ¼ 0.09 K, H¼ 5 T, TD ¼0.1 K. 0
frequency is F = 3.8⋅103T [25], ηc = 2.7 [24], c* = 8.5A [24], TD = 0.1 K . The phase diagrams are shown in Fig. 6. The thickness of the domain wall is equal to 0.35 μm . Using Eqs. (8), (18) and (19), we obtain the magnetization in the paramagnetic Condon domain as a function of the slab thickness L in μm for T ¼0.09 K, H¼ 5 T, TD ¼0.1 K (see Fig. 7). The magnetic induction bifurcation ΔB due to Condon domains is presented in Fig. 8 as a function of the slab thickness in μm for T¼ 0.09 K, H¼ 5 T, TD ¼0.1 K (Eq. (12)). It is seen that the induction splitting tends to zero while the thickness tends to the minimal value of the slab thickness Lc at which the domain phase disappears. We have not calculated the amplitude of temperature oscillations in quasi-2D organic conductors since results of measurements of the specific heat at such low temperatures are unknown. Fig. 6. Phase diagrams, temperature (in K ) – magnetic field (in Tesla), for two specimens of κ(BEDT − TTF )2I3 of thickness L ¼4 μm and L ¼ 40 μm and TD = 0.1 K .
Fig. 7. Dependence of the magnetization M0 (given in A/m ) in the paramagnetic Condon domain ( M0 40) on the thickness (given in μm ) for κ(BEDT − TTF )2I3specimen at T ¼0.09 K, H¼ 5 T, TD ¼0.1 K.
4. Conclusions We have analyzed diamagnetic phase transitions in samples of finite thickness (thin film). The geometry considered is that of a slab of infinite planar extent and of finite thickness. We have examined the stripe-like domain structure with diamagnetic phase transitions of the second order. We have shown that the critical sample size for the diamagnetic phase transition exists, i.e., there is the film thickness, below which the order parameter of the diamagnetic phase transition, the magnetization in the paramagnetic domain, is zero. We have presented phase diagrams (temperature-magnetic field) exhibiting the domain phase and giving the curve separating the ordered and disordered phases in 3D silver and quasi 2D organic metals. We have calculated the thickness dependence of the magnetization in the paramagnetic domain in 2D and 3D metals. We have obtained a shift of the phase transition temperature and the temperature-magnetic field phase diagram as a result of the size effect. We have computed the size effect on the thermo-magnetic (magneto-caloric) oscillations in silver. We have shown that the amplitude of thermo-magnetic oscillations grows at lower slab thicknesses diverging at minimal value Lc . We have discovered that the critical indices for the temperature and sample thickness of the magnetization in the paramagnetic domain are typical of the Landau mean-field theory of second phase transitions.
L.A. Bakaleinikov, A. Gordon / Physica B 502 (2016) 160–165
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