Thermal Motion in a 2D Structure - Physics of Important Developments That Predestined Graphene

Chapter 2 Physics of Important Developments That Predestined Graphene

2.2 Thermal Motion in a 2D Structure

The Lorentz force provides a current (Fig. 2.9), j=υρ, an electric field E=υB c/ and σxy, a Hall conductance:

σ ρ

xy J

E c

= = B ; (2.27)

where c = speed of light.

In real heterostructures ρ is fixed by doping and the value of the gate voltage is not precisely quantized. The above-mentioned imperfection in the quantum Hall experiment manifests itself in the Hall plateau formation. The Hall conductance in the plateau is equal to a multiplication of e²/h (the parallel resistance is zero in a plateau). The explanation is that the new state adiabatically transformed into a filled Landau with the exception of excitation with partial charges⁸. It is argued that a perfect system is invariant along the x-direction and σxy=P Bc ; ρ is not quantized charge density. The high probability of the electron cloud in the z- direction makes the electron fully-quantized in a f(z) bound state in a potential well narrower than the de Broglie wavelength for the electron, λ =h

p. The de Broglie wavelength is the wavelength, λ, associated with a particle and connected with its momentum, p.

Adding to the theory by von Klitzing, it has been argued⁹ that a magnetic field creates at least several delocalized states. Summarizing the above discussion, we can see that electrons on the surface of liquid helium crystallize forming a lattice that exists immediately next to the liquid helium phase. In semiconductor layers, with an applied high magnetic field, electrons confined between the layers exhibit the qualities of a 2D state. Graphene that as a 2D structure has the same quantum Hall effect thus remains (in this aspect) a typical 2D crystal. Although graphene is generally a good conductor, at temperatures close to zero, it exhibits an exponentially decreasing conductance with increasing sample size¹⁰. Prospective field-effect transistors should be capable of current densities of 10¹² A/m².

24 Graphene for Defense and Security the 3D case was provided by Landau and Lifshitz¹². A large crystal may be described by a vector:

   

R na mb lc= + + ; (2.29) where n, m, l are integers.

At a finite temperature (2.29) is rather idealistic, not taking into account crystal inner structure displacement. However, the Fourier decomposition of the displacement may be calculated as:

 

u uke^{ik r}

∑

k ^•^{; (2.30)}

The probability of fluctuation is evaluated by its free energy change (cost), ΔF : P∝exp(−∆F k T/ B ); (2.31) ΔF, the potential energy cost:

∆F V u uik k k k

kil

lk i l x y z

= ^/²

∑

^{ } ^ϕ^,^{( , , );}^(2.32)

where ϕi l,( , , ) are the components of the displacement; V is the entire body k k kx y z

volume connected with the displacement u. The displacement is analogous to the  mechanical displacement of a spring with K, a spring constant in the classical mechanics expression W=1Kx

2 ² (spring force work). The expectation value of the displacement vectors:

u ujk lk T V A n kil il

∗ = ^/

∑

^{( )/ ;}^ ²^(2.33)

where T is temperature and Ail depends on n, the direction of the vector  k n k k  ( = / ), the value of u ujk lk∗ is found from the wave vector k. The mean-square displacement vector may be calculated:

  

u2 =T d k

∫

^[ 3 ^/(2^π^{) ] ( )/}3 A n kll 2=T dkd

∫

^[ ^σ^/(2^π^{) ] ( );}3 A nll ^(2.34)

where σ= component of area;

If k is not large, the integral converges at k = 0 and the integral is linear. It means that the mean square fluctuation displacement is a finite quantity and does not depend on the size of the system.

Next, the authors (Landau and Lifshitz) approach the 2D problem:

u uik lk∗ =T dk dk

∫

^[ x y^/(²^π^{) ] ( )/ ;}² A n kil ^ ²^(2.35)

The integral equals a constant times ln(k) and dk dkx y=2πkdk . k is proportional to 1/L and 1/a where L is the size of the system and a = lattice constant. Thus, the integral diverges as ln(k) (logarithmically):

u² =const T.× ln( / );L a (2.36) Summarizing the above argument, the absolute thermal motion of a point in a 2D system with size L is proportional to T ln(L/a), where T = temperature¹². This loga- rithmic divergence is rather slow and even for the large size L, the displacement remains comparatively small. In such a case, the system has properties of a solid state crystal at low temperatures. The actual temperature is relevant to the at least Debye temperature which is comparatively high for graphene as it is for graphite.

More specifically, at low temperature, the displacement u r ( ) of atoms is considered slowly changing at the distance close to the lattice constant distance a. Then, the variation in density and the local value of the displacement vector are considered¹². Let ρ0( )r be the density distribution at T = 0. Then, the density may be expressed:

ρ( )r =ρ0[r u r  − ( )]; (2.37) The correlation function describes the relation between fluctuations at different locations:

ρ ρ(r r 1 (2 = ρ0 1[  r u r− ( )] [1 ρ0 2r u r  − ( )] ;2 (2.38) The expansion in a Fourier series:

ρ0 ρ ρ

( )r AV be ;

= + ib r

≠

∑

• ^(2.39)

where b are vectors in the 2D reciprocal lattice. Inserting (2.42) into (2.41) yields:

ρb²exp[ib r r  ⋅ −(1 2)] exp[− ⋅ib u u  ( 1− 2)] ; (2.40) where u1=u r ( );1

The right-hand side of (2.40) can be expressed by the expectation value of the reciprocal lattice parameters:

exp[− ⋅ib u u  ( − )] =exp( /− b bi il);

1 2 1 2 1χ (2.41)

where χil( )r =T dk dk

∫

[ x y/(²π) ](² ¹−cosk r A n k ⋅ ) ( )/ ;il  ²^(2.42)

26 Graphene for Defense and Security Eq. (2.42) converges for small values of k since the upper part of the fraction of the integrand. The maximum value of k may be written as:

kmax=T hc/( ); (2.43)

where c = velocity of sound, h= Plank’s constant.

Then,

χ_il( )r =T/πA_ilAVln(k rmax ); (2.44) The result for the correlation function with a fictitious temperature T`:

ρ( ) ( )r ρ r ρAV cos(b r r )/ T T^/ ;

1 2 − 2 = ⋅ ^′ (2.45)

where 

b is a reciprocal lattice vector corresponding to the maximum value of r^{T T}^/ ^′. For practical purposes, in the two-dimensional lattice, we can consider the motion at close locations to be correlated at low temperatures (although as r goes to infinity, so the correlation goes to zero). The temperature T` was defined¹²:

T′ =4πmc b k²/( ² b); (2.46) where m is an 2D crystal atom’s mass.

In the 3D lattice, the correlation approaches a constant but in a liquid, the correlation diminishes exponentially. The nearest-neighbor correlation is highly correlated. The question of thermal stability: at finite T’ the correlations of the nearest are finite for any L determined the melting temperature of the 2D lattice¹³. It is stated the existence of a Lindemann criterion of 2D melting. The criterion, Υ_M^C ≡ [ (u R a  + )−u R ( )] /a

0 2 2 was found independent of the nature of a 2D classical crystal, for 2D dipole and 2D Lennard-Jones crystals γ_M^C =0 12. and that for a 2D electron crystal γMC =0 10. . Using a differential criterion:

γM= u r a    ( + −) u r( ) / ;² a² (2.47)

In Fig. 2.10 the simulated curves of the Lindemann criterion are presented. At T ~ 4900 K graphene melts. The lowest curve is received for three neighbors, the middle curve – for 12 neighbors and the upper curve was built for 9 nearest neighbors.

The next question is whether high temperature deformations may be large enough to destroy a 2D crystal (e.g. to melt it). A thermal displacement exceeding a lattice constant, if it is local, does not imply melting since melting presumes a long-range order destruction. The experiment with graphene annealing at T = 2300 K (with the melting point at about T = 3900 K) shows local order defects but not melting¹⁴.

The melting temperature dependence on lattice defects as predicted by Zakharchenko, et al¹⁵ caused by clustering defects and octagon for motion leads to carbon chains. It is possible that melting occurs as the atomic lattice bends. Thus, unit cells that are located some distance from the source of thermal fluctuations of a large amplitude are no longer positioned according to the vector R na mb lc= +  + . Although, the cells at a distance from the source may be thermally dislocated, the local cell position remains stable. A one- square- meter graphene sample, for example, at room temperature will have a root mean square (rms) amplitude approximately 2 mm and the oscillation period of about 270 hours. Such a sample is not a crystal by defini- tion since the unit cells in this sample move (oscillate) over distances exceeding their lattice constants. Three-dimensional semiconductor samples (such as Si, for example) will not have such oscillations and remain perfectly crystalline. The melting effect, therefore, is a pure matter of a two-dimensional system.

1500 2000 2500 3000 3500 4000 4500 5000 0

0.02 0.04 0.06 0.08 0.1 0.12

T, K 0.14

0.16

FIGURE 2.10 Dependence of the Lindemann criterion vs temperature¹⁵

28 Graphene for Defense and Security

Dalam dokumen Graphene for Defense and Security (Halaman 36-41)