On the Mott formula for the thermopower of non-interacting electrons in quantum point contacts

Transcription

1 INSTITUTE OF PHYSICSPUBLISHING J.Pys.: Condens. Matter 17 (25) JOURNAL OFPHYSICS: CONDENSED MATTER doi:1.188/ /17/25/14 On te Mott formula for te termopower of non-interacting electrons in quantum point contacts Anders Matias Lunde and Karsten Flensberg Niels Bor Institute, University of Copenagen, DK-21 Copenagen, Denmark Received 12 April 25 Publised 1 June 25 Online at stacks.iop.org/jpyscm/17/3879 Abstract We calculate te linear response termopower S of a quantum point contact using te Landauer formula and terefore assume non-interacting electrons. Te purpose of te paper is to compare analytically and numerically te linear termopower S of non-interacting electrons to te low-temperature approximation, S (1) = (π 2 /3e)kB 2 T µ[ln G(µ, T = )], and te so-called Mott expression, S M = (π 2 /3e)kB 2 T µ[ln G(µ, T )], were G(µ, T ) is te (temperature-dependent)conductance. Tis comparison is important, since te Mott formula is often used to detect deviations from single-particle beaviour in te termopower of a point contact. 1. Introduction Anarrow constriction in for example a two-dimensional electron gas makes a small cannel between two electron reservoirs. Tis constriction is called a quantum point contact [1]. Te widt of te cannel can be controlled by a gate voltage, and by applying a small bias te penomenon of quantized conductance as a functionofte widt (i.e. gate voltage) is observed at low temperatures [2]. Tis quantization is due to te wave nature of te electronic transport troug te sort ballistic point contact. Experimentally [3 7], it is also possible to eat up one of te sides of te point contact, tereby producing a temperature difference T across te contact, wic in turn gives an electric current (and a eat current) toug te point contact. By applying a bias V in te opposite direction to te temperature difference T,tetwocontributions to te electric current I can be made to cancel, wic defines te termopower S as V S = lim T T. (1) I= For a quantum point contact, te termopower as a function of gate voltage as a peak every time te conductance plateau canges from one subband of te transverse quantization to te next [5, 8] /5/ $3. 25 IOP Publising Ltd Printed in te UK 3879

2 388 AMLunde and K Flensberg In order to compare experiment and teory fortetermopower of a point contact, te so-called Mott formula, S M Vg [ln G(V g, T )], (2) is often a valuable tool, because by differentiating te experimentally found conductance G(V g, T ) wit respect to te gate voltage V g one can see if tere is more information in te termopower tat in te conductance. Tis additional information could for example be many-body effects [7], since S M is an approximation to te single-particle termopower. Note tat tis approximationisindependent of te specific form of te transmission T (ε) troug te point contact. It is te purpose of tis paper to determine te validity of te Mott approximation S M,andtereby decide if it is really deviations from single-particle beaviour te experiments [6, 7, 9]reveal or rater artefacts of tis approximation. 2. Termopower from te Landauer formula For te sake of completeness, we begin by deriving te single-particle termopower formula in linear response to te applied bias V and temperature difference T.Te current toug a ballistic point contact is found from te Landauer formula [1,p111, equation (7.3)]: I = 2e dε T (ε)[ fl (ε) f R (ε)], (3) were T (ε) is te transmission and fi (ε) is te Fermi function for te rigt/left (i = R, L) lead. Te Landauer formula assumes non-interacting electrons and terefore so will te derived termopower formula. Wen a small bias V = (µ L µ R )/( e) and temperature difference T = T L T R are applied, we can expand te distribution functions around µ, T as ( T /T 1and ev µ): fi (ε) f (ε) ε f (ε)(µ µ i ) (ε µ) ε f (ε) T T i, (4) T were f (ε) is te Fermi function wit te equilibrium cemical potential µ and temperature T and i = L, R. To obtain te termopower equation (1) weinsertte distribution functions in equation (3), set it equal to zero and obtain S(µ, T ) = 1 et wic is our exact single-particle formula. dε T (ε)(ε µ)[ ε f (ε)], (5) dε T (ε)[ ε f (ε)] 3. Approximations to te termopower and teir validity 3.1. Te low-temperature (first-order) approximation For T = weave ε f (ε) = δ(ε µ), sotenumerator in equation (5) iszero, i.e. S(µ, T = ) =. For temperatures k B T muc lower tan te scale of variation of T (ε) and k B T µ, wecan expand T (ε) around µ to first order (i.e. a Sommerfeld expansion) to obtain S (1) (µ, T ) = π 2 k B 3 e k 1 T (µ) BT = π 2 k B T (µ) ε 3 e k 1 G(µ, T = ) BT, (6) G(µ, T = ) µ were G(µ, T = ) is te conductance for zero temperature, i.e. G(µ, T = ) = 2e2 T (µ).

3 On te Mott formula for termopower of QPCs Te Mott approximation and analytical considerations of its validity Te Mott approximation 1 [6, 7]is S M (µ, T ) = π 2 k B 3 e k 1 G(µ, T ) BT, (7) G(µ, T ) µ were G(µ, T ) is te temperature-dependent conductance G(µ, T ) = 2e2 dε T (ε)[ ε f (ε)]. (8) Te form of S M stated in equation (2) assumes tat te cemical potential and gate voltage are linear dependent. Te Mott approximation to te single-particle termopower equation (5) and its range of validity are not so obvious compared to te approximation of te first-order Sommerfeld expansion equation (6). One way of comparing S from equation (5)andS M is to differentiate equation (8)toobtain (assuming tat T (ε) is independent of µ): S M (µ, T ) = π 2 k B 1 ( ) ε µ dε T (ε) tan [ ε f (ε)], (9) 3 e G(µ, T ) 2k B T i.e. by using te Mott formula we approximate (ε µ)/k B T in te integral by (π 2 /3) tan[(ε µ)/(2k B T )]. To compare S and S M in anoter way, we observe tat for low temperatures k B T µ te Mott approximation S M simplifies to te S (1) equation (6), because G(µ, T ) 2e2 T (µ) for T, i.e. S(µ, T ) = S (1) (µ, T ) = S M (µ, T ) for k B T/µ. Terefore, we compare S and S M by expanding bot quantities in orders of k B T and comparing order by order. Using 1 n T (µ) T (ε) = (ε µ) n, (1) n! ε n we can exactly rewrite equation (8): G = 2e2 1 n T (µ) dε(ε µ) n [ n! ε n ε f (ε)] = 2e2 were (y = (ε µ)/k B T ) 1 n! n T (µ) (k ε n B T ) n B n ( ) µ, (11) k B T ( ) µ y n B n dy k B T µ k B 4cos 2 (y/2) I y n n dy 4cos 2 (y/2) T for k B T µ, (12) were we note tat I 2n+1 = for all integer n. Numerically, it turns out tat B n (µ/k B T )/B n () forµ (1 + n)k B T as seen in figure 1. Te integral I n can be calculated, and te first values are I = 1, I 2 = π 2 3, I 4 = 7π 4 15, I 6 = 31π 6 21, I 8 = 127π 8, (13) Using te approximation equation (12)weget G(µ, T ) 2e2 1 2n T (µ) I (2n)! ε 2n 2n (k B T ) 2n. (14) 1 In te early works by Mott and co-workers [11, 12]itwasactually te first-order approximation equation (6)wic was referred to as te Mott formula.

4 3882 AMLunde and K Flensberg Figure 1. Left: te approximation in equation (12) is pictured for odd integer values of n from 1 (left) to 19 (rigt) in B n (µ/k B T ). Wenote tat B n (µ/k B T )/B n () forµ (1 + n)k B T. Rigt: te numerical values of te factors in te series expansions of te Mott approximation equation (15) (lower)and te exact linear single-particle series expansion equation (16) (upper). Tis leads to a Mott approximation to te termopower for low temperatures as S M (µ, T ) k [ ] B 1 2e 2 I 2 I 2n 2n+1 T (µ) (k e G(µ, T ) (2n)! ε 2n+1 B T ) 2n+1. (15) Writing te exact single-particle termopower S equation (5) byusingequation (1) andte approximation of low temperatures equation (12), we get S(µ, T ) k B e 1 2e 2 G(µ, T ) [ I 2n+2 2n+1 T (µ) (k (2n +1)! ε 2n+1 B T ) 2n+1 ]. (16) We see tat bot formulae only ave odd terms in k B T,andtefirst-order term is te same (wic is S (1) ). However, none of te iger-order terms are te same, and in figure 1(rigt) te different numerical factors of te two seriesexpansions are seen to beave very differently as te power of k B T grows: I 2n+2 (2n +1)! 4. n + π 2 3 and I 2 I 2n 6.58 for n 1. (17) (2n)! So te Mott approximation is better te smaller te temperature compared to µ, butit is not a bad approximation for moderate temperatures (i.e. k B T comparable to oter energy scales), as we sall see numerically. Note tat if te approximation equation (12)isnot valid, ten we ave all powers of k B T. 4. Comparison of te approximations to te exact single-particle termopower from numerical integration We need a specific model for te transmission to do a numerical comparison of S from equation (5) tos M and S (1). Using a armonic potential in te point contact, i.e. a saddle point potential, a transmission in te form of a Fermi function can be derived [13]: T (ε) = n max n=1 1 exp( nε ε ε s ) +1, (18)

5 On te Mott formula for termopower of QPCs 3883 Figure 2. Termopower S from numerical integration of equation (5) (black solid line), te Mott formula S M equation (7) (red dased line) and te first-order approximation S (1) equation (6)(green dotted line). From (a) to (f) te temperature is canged from te low-temperature regime k B T <ε s to k B T >ε s in small steps. Te smearing of te transmission ε s is kept constant, and note tat ε s, k B T ε and ε s, k B T ε F in all te graps. Te termopowers are all in units of k B /e, but note te different magnitudes of te termopower from (a) to (f). Te conductance G is sown (in arbitrary units) for comparison. (Tis figure is in colour only in te electronic version) were ε s is te smearing of te transmission between te steps and ε is te lengt of te steps (often called te subband spacing). In terms of te armonic potential V (x, y) = const mω 2 x x 2 /2+mω 2 y y2 /2, were x is along te cannel, we ave ε = ω y and ε s = ω x /(2π). Oterfunctional forms of T ave also been tested, but provided tey ave te same grapical structure (suc as for example a tan dependence) te same conclusions are obtained. Tree regimes of temperatures relevant to experiments are investigated numerically: k B T <ε s (figure 2(a)), k B T ε s (figures 2(b) (d)) and k B T >ε s (figures 2(e), (f)). (19)

6 3884 AMLunde and K Flensberg Te termopower S for te transmission model equation (18) isfound from numerical integration of equation (5) andcompared to te Mott approximation S M equation (7) and te first-order approximation S (1) equation (6). In all tree regimes, we ave a staircase conductance, so k B T ε,andg(µ, T ) is also sown in te figures (in arbitrary units) for comparison. Furtermore, µ = ε F is of order ε,soteapproximation k B T ε F used for example in equation (12)isindeed very good. Note tat all energiesin te figuresare given in units of te step lengt ε. Te information obtained from te numerical calculations is te following. Figures 2(a), (b) sow tat for k B T being te lowest energy scale bot approximations work very well, as expected from te analytical considerations. Wen te temperature becomes comparable to te smearing of te steps, k B T ε s,teculter Mott formula works well and is better tan te first-order approximation, as seen in figures 2(b) (d). For k B T bigger tan ε s, te Mott approximation still works quite well, wereas S (1) is no longer a good approximation. Te reason tat te Mott approximation works well is found in te similar terms in te analytic temperatureexpansions equations (15)and(16). Note tat as k B T increases bot S (1) and S M sow a tendency to overestimate S at te peaks and underestimate it at te valleys. In summary,we ave found tat te Mott approximation to te single-particle termopower is afairly good approximation provided te temperature is smaller tan te Fermi level, but k B T can be bot compatible and larger tan te smearing of te transmission ε s.however,to rule out any doubt one could use an experimental determination of T (ε) from te (very lowtemperature) conductance to find te single-particle termopower from equation (5), wic could peraps give an interesting comparison to te experimental result. Tereby one would obtain an even more convincing statement of deviations from single-particle beaviour in te termopower. Acknowledgment We would like to tank James T Nicolls for saring is experimental results wit us and for discussions of te termopower in point contacts in general. References [1] Houten H V and Beenakker C 1996 Pys. Today (July) 22 for an minor review [2] van Wees B J et al 1988 Pys. Rev. Lett [3] Molenkamp L W et al 199 Pys. Rev. Lett [4] Molenkamp L W et al 1992 Pys. Rev. Lett [5] van Houten H, Molenkamp L W, Beenakker C W J and Foxon C T 1992 Semicond. Sci. Tecnol. 7 B215 [6] Appleyard N J et al 1998 Pys. Rev. Lett [7] Appleyard N J et al 2 Pys. Rev. B (R) [8] Streda P 1989 J. Pys.: Condens. Matter [9] Proskuryakov Y Y et al 24 unpublised [1] Bruus H and Flensberg K 24 Many-body quantum teory in condensed matter pysics Oxford Graduate Texts 1st edn (New York: Oxford University Press) [11] Mott N F and Jones H 1936 Te Teory of te Properties of Metals and Alloys 1st edn (Oxford: Clarendon) [12] Cutler M and Mott N F 1969 Pys. Rev [13] Büttiker M 199 Pys. Rev. B (R)