Pis'ma v ZhETF, vol.86, iss.9, pp.687-690
© 2007 November 10
Determination of Landau's Fermi-liquid parameters in Si-MOSFET
systems
A. Gold, V. T. Dolgopolov+ Centre d'Elaboration de Matériaux et d'Etudes Structurales (CEMES-CNRS), 31055 Toulouse, France + Institute of Solid State Physics, 142432 Chernogolovka, Moscow District, Russia
Submitted 6 September 2007 Resubmitted 27 September 2007
We analyze experimental data in order to evaluate Landau's Fermi-liquid parameters. By using row data of recent Shubnikov -de Haas measurements we derive, as function of the electron density ns, results for the compressibility mass of the charged two-dimensional electron gas. The compressibility mass is nearly equal to the transport mass even in the density region where the transport mass has the tendency to diverge. We conclude that Landau's Fermi-liquid parameter Fq(iis) is nearly independent of electron density and near to zero. This result is derived for silicon (100) and silicon (111) surfaces. We also obtain the dependence of Fi(ns), determining the transport mass, and of F§(ns), determining the spin-susceptibility.
PACS: 71.10.Ay, 72.10.-d, 73.20.-r
Recently, it was found in experiment that in Si-MOSFETs with (100) or (111) orientation the electron effective mass (the transport mass or cyclotron mass) to* drastically increases with decreasing electron density with a tendency to diverge at a definite electron concentration [1-3]. The critical electron density depends on the electron-electron interaction strength, so that the dependence m*(rs)/nib is universal in both silicon systems [3]. Here toj, is the bulk effective mass and rs is the dimensionless Wigner-Seitz radius. Simultaneously, in both systems a decrease of the Dingle temperature is observed by decreasing of the electron concentration [2, 3]. On the first glance the last observation seems to be very puzzling, because it means an increase of the single-particle relaxation time with decreasing density. In lowering the electron density disorder effects should become more important due to less screening and the single-particle relaxation time should decrease. We conclude that the observed behavior of the Dingle temperature is in contradiction with expectations obtained within the simplest theoretical consideration [4-6].
In a number of publications during last years [7-9] it was shown that despite of a strong increase of the effective mass the electron system can be described in the frame of the Fermi-liquid theory proposed by Landau [10]. In the two-dimensional case the effective transport mass in this theory is defined through the dimensionless parameter F* (a Landau Fermi-liquid parameter)
nib vpfTib 2
(1)
The possibility of a divergence of the effective mass can be easily seen if one uses in equation (1) the initial parameter of the theory f* = —r^jy F*. Here v(0) is the density of states defined with the mass toj. With f* the equation for the effective (transport) mass is written as
TO
mb
Ho )/i
(2)
The transport effective mass, corresponding to equations (1), (2), is not unique in Landaus Fermi-liquid theory. Another effective mass mcom, determining the thermodynamic density of states, the compressibility (com), and the screening behavior is defined by
_ ims2h2 _
^com — ^ —
9v
TO
Fi
I + IF*
where k represents the compressibility, ns is the electron density, and gv is the number of valleys. The mass TOCOm is a function of two macroscopic parameters F' and Fq, and can, in principle, have a very different dependence on the electron density in comparison to to*. We stress that the density of states is proportional to the mass mcom and that this mass also is responsible for screening effects. It defines the screening length in the long-range limit.
The aim of the present paper is to explain qualitatively the unusual behavior of the single-particle relaxation time (or sometimes called quantum scattering time) as a function of electron density [2, 3] and to extract from the available experimental data the dependences of TOCOm and Fq on electron density.
The single-particle relaxation time describes the amplitude of Shubnikov - de Haas (SdH) oscillations and the Dingle temperature via UbTd = ft/2itTs. Note that the effective transport mass m* is given by the energy difference of the Landau ladder seen in SdH oscillations. Therefore, we argue that one can determine two Landau Fermi-liquid parameters by measuring SdH oscillations as function of temperature.
First of all we would like to explain why we use the single particle relaxation time rs, and not the transport scattering time , where the compressibility mass also is an important parameter. The transport scattering time [11] is very sensitive to multiple scattering processes. Experimental data in both silicon systems are taken in the vicinity of the metal-insulator transition [12], where multiple scattering processes play a crucial role. The single-particle relaxation time is defined by density of state modulations and is much less sensitive to such multiple scattering processes [6]. For e j?rs > H one can write rs as [6]
(4)
2m*
f
J o
e(q) sjAk2F - q
with e(q < 2k f) = 1 + qs(q)Fc(q)/q as the screening function, the ^-dependent screening number qs(q)/qs = = 9v(mCOm/inb)(l - G(q)) and qs = gs/a*B, where a*B is the effective Bohr radius. Fc(q) is the form factor for the Coulomb interaction due to the finite extension of the electron gas [4]. (\U(q)\2) is the random potential due to disorder, and G(q) describes the local field corrections. G(q) takes into account exchange and correlation effects beyond the mean-field theory (the random-phase approximation) and reduces screening effects [10]. Impurity scattering with impurities at the Si/Si02 interface and interface-roughness scattering are important for the two-dimensional electron gas on silicon surfaces [4]. The analyzed Si(100) and Si(lll) data for rs are in the validity range of equation (4), see also figure 10 in [6].
There are different contributions to the single-particle relaxation time in Eq.(4). The increase of the effective mass m* with decreasing electron density in the numerator decreases rs for decreasing electron concentration. Similar influence has the effective quasi-particle interaction, because G{2kp) [13, 14] tends to unity as the electron density decreases. About the compressibility mass TOcom nothing is really known from experiment. From numerical calculations of Ref. ([9]) we conclude that the mass mcom should strongly increase near the critical electron density, the density where the transport mass diverges.
In the integral of Eq.(4) at qs 2kp we can neglect the unity in e(q), because the second term exceeds the
unity significantly, and deduce that m*/rs oc m*2/m201J1. Hence, the single-particle relaxation time is roughly proportional to the square of mcom and the effect of the mass increase, according to the experiment, prevails both other effects described above.
In Fig.l the ratio m*/mcom is shown versus electron
ns (10 cm )
Fig.l. Different mass ratios versus electron density for silicon (100): the ratios m*/mcom and m* /m6 extracted from experiment [2] are shown.The long arrow indicates the value of the MIT
density for experimental results obtained for Si(100) [2].We found this ratio within the following procedure. In SdH experiment m* is measured together with the single-particle relaxation time. At low electron density (m*¡Ts)1/2 is proportional to m*/mcom if the coefficient defined from Eq. (4) is slowly dependent on electron concentration. We tested this assumption and found that the variation of this coefficient is not more than 10%. Taking {m*¡Ts)1/2 oc m*/mcom equal to 1 at relatively large density, where the density dependence of m* is negligible, we can conclude that m* = mcom- Our numerical results indicate that m* = mcom holds for all densities. This would means that the
parameter Fq is
independent of density and near to zero. Note the strong density dependence of m* /mj [1], which is also shown in Fig.l.
Experimental results for Si(lll) are more complicate to analyze. MOSFET with (111) orientation are more disordered than the Si(100) samples and the effective mass increase is shifted to higher electron concentration. It means that the contribution from surface roughness scattering is more important. Nevertheless the same procedure as described above gives very similar result. In Fig.2 we show the mass ratio m*/mcom versus electron density for experimental results obtained for Si(lll) [3]. At large density the ratio was fixed to 1. A weak density
Determination of Landau's Fermi-liquid parameters in Si-MOSFET systems
689
ns (1012 cm 2)
Fig.2. Different mass ratios versus electron density for sili-con(lll): the ratios m*/mcom and m*/mt extracted from experiment [3] are shown
dependence is found. But again we can conclude that the essential density dependence of the mass mcom can be expressed by m*. We conclude that 0 < FJ < 0.3. It is not clear weather this is due to the higher disorder in the (111) sample compared to the (100) sample or related to degeneracy effects in Si(lll).
From existing experimental data one can extract two other Landau liquid parameters. Using Eqs. (1), (2) and mass values from SdH measurements we find Ff versus electron density. Our results are shown in Fig.3. For
o «
T3
I
II
hJ
a
£ " p.
- 1
a 1
I o
Si(111)
_i_i_i_i_i_
12 -2 ns (10 cm )
-F"
Si (100)
«11 -2. ns (10 cm )
Fig.3. Landaus Fermi-liquid parameter F[ for the effective transport mass and Fg for the spin susceptibility for Si(100) as deduced from experimental data [1] and [15], respectively. In the inset we show F( for the effective transport mass for Si(lll) as deduced from experimental data [3].The long arrow indicates the value of the MIT
both silicon surfaces F{ exhibits a strong tendency to diverge at low electron density.
Another Fermi liquid parameter Fg exists, defi
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