week ending 14 JUNE 2013
PHYSICAL REVIEW LETTERS
PRL 110, 246603 (2013)
How to Directly Measure a Kondo Cloud’s Length Jinhong Park,1 S.-S. B. Lee,1 Yuval Oreg,2 and H.-S. Sim1,* 1
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea 2 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel (Received 26 October 2012; published 14 June 2013)
We propose a method to directly measure, by electrical means, the Kondo screening cloud formed by an Anderson impurity coupled to semi-infinite quantum wires, on which an electrostatic gate voltage is applied at distance L from the impurity. We show that the Kondo cloud, and hence the Kondo temperature and the electron conductance through the impurity, are affected by the gate voltage, as L decreases below the Kondo cloud length. Based on this behavior, the cloud length can be experimentally identified by changing L with a keyboard type of gate voltage or tuning the coupling strength between the impurity and the wires. DOI: 10.1103/PhysRevLett.110.246603
PACS numbers: 72.15.Qm, 72.10.Fk, 73.63.Kv
Introduction.—The Kondo effect is a central many-body problem of condensed matter physics [1,2]. It involves a spin singlet, formed by the spin-spin interaction between a magnetic impurity and surrounding conduction electrons. A deeper understanding of the effect has been achieved by using a quantum dot, that hosts a magnetic impurity spin, under systematic control [3–6]. Although the Kondo effect is well known, its spatial features still remain to be addressed. The Kondo spin singlet is formed below the energy scale of Kondo temperature TK . This implies that the singlet is spatially formed over a conduction-electron region of length scale K ¼ @vF =ðkB TK Þ; when TK 1 K and the Fermi velocity vF 105 –106 m=s, K 1 m. The region is called the Kondo screening cloud. There have been several proposals [7–19] for ways to detect the cloud. Despite the proposals, there has been no conclusive measurement supporting the existence of the Kondo cloud [20,21]. The difficulty to detect the cloud arises because it is a spin cloud showing quantum fluctuations with zero averaged spin. The cloud manifests itself in the spin-spin correlation [7–10] between the impurity and the conduction electrons. However, it requires measurements of spin dynamics of time scale @=ðkB TK Þ. STM studies probing local density of states may be useful for detecting the cloud [12–14,22]. Recent STM measurements [23–26] show the Kondo effect in the region away from a magnetic impurity, whose spatial extension is, however, much shorter than K . Another direction is to study a magnetic impurity in a finite-size system [15–19,27,28]. Because the cloud cannot extend beyond the finite size, the Kondo effect is strongly affected, and suppressed when the system is shorter than K . There has been no conclusive experimental detection of K in this direction [28]. In this Letter, we propose a new way of detecting the Kondo cloud, based on the intuition that a change of conduction electrons inside the cloud will affect the Kondo effect. We consider a Kondo impurity formed in a 0031-9007=13=110(24)=246603(5)
quantum dot coupled to two semi-infinite ballistic quantum wires with electron tunneling amplitude tWD (see Fig. 1). Electrostatic gate voltages Vg are applied to the wires (or to only one wire) at distance L from the dot, modifying indirectly the local density of states ðÞ of conduction electrons nearby the dot (Fig. 2). We find that Vg does not affect the cloud, when L K . However, when L K , the cloud, hence the Kondo temperature TK and electron conductance G through the dot, are sensitive to Vg . The crossover between the two regimes occurs at L K . By measuring G or TK with varying L or tWD (Figs. 3 and 4), one can detect the crossover and K . We use the poor man’s scaling [29], numerical renormalization group study (NRG) [30–32], and Fermi liquid theory [33]. The setup.—We describe the wires by the tight-binding Hamiltonian with sites j’s in wire i ¼ l, r, HW ¼
1 X X X
½0 nij þ ðtcyij ciðjþ1Þ þ H:c:Þ
i¼l;r j¼1 ¼";#
eVg
1 X X
X
nij ;
(1)
i¼l;r j¼Nþ1 ¼";#
FIG. 1 (color online). Setup for detecting the Kondo cloud. A quantum dot located at x ¼ 0 hosts a magnetic impurity spin. It couples to quantum wires along x^ axis, with electron tunneling amplitude tWD . Gate voltage Vg is applied at distance L from the dot (in jxj > L). The Kondo effect becomes sensitive to Vg , as L decreases below the cloud length.
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Ó 2013 American Physical Society
0.2
H ¼ HD þ HW þ HT ’ J s~ S~ þ V
0.15
(2)
FIG. 2 (color online). Hybridization function ðÞ ¼ 2t2WD ðÞ in units of t for infinite (blue dashed curve) and finite L (green solid curve); we choose 0 =2t ¼ 0:931, tWD =2t ¼ 0:2, and eVg =2t ¼ 0:125. For finite L, ðÞ has resonances with spacing @vF =L; F is chosen to be located at a resonance center (red vertical dashed line). When L is so large that Vg is applied outside the Kondo cloud, the Kondo temperature TK ¼ TK1 is determined by 1 and independent of Vg . On the other hand, TK depends on Vg and L, when L & the cloud size K . When L K , TK is determined by 0 ðF Þ.
P Here, the Kondo impurity spin, S~ ¼ ;0 dy ~ 0 d0 =2, couples with the spin of the neighboring conduction P electrons, s~ ¼ ;0 ðcyl1 þcyr1 Þ~ 0 ðcl10 þcr10 Þ=2, with strength J ¼ 2t2WD ½1=ðd F Þ þ 1=ðU þ d F Þ. The second term describes the potential scattering with strength V ¼ t2WD ½1=ðd F Þ 1=ðU þ d F Þ=2. Local density of states.—We show how the gate voltage Vg changes the local density of states ðÞ at sites j ¼ 1 of the wires. We calculate ðÞ, by matching the singleparticle wave functions of HW between j ¼ N and N þ 1, ðÞ ¼
cyij
where creates an electron with spin and energy 0 at site j in wire i, nij cyij cij , and t is the hopping energy. The last term describes the gate voltage Vg applied in jxj > L ¼ Na, where a is the lattice spacing. Vg ðxÞ changes at x ¼ L abruptly over the length shorter than the Fermi wavelength. The dot Hamiltonian HD is modeled by the P Anderson impurity [34], HD ¼ ¼";# d dy d þ Und" nd# , where dy creates an electron with energy d and spin in the dot, nd dy d , and U is the electron repulsive interP P action. HT ¼ tWD i ¼";# ðcyi1 d þ H:c:Þ describes electron tunneling between the wire and the dot. The dot is occupied by a single electron in the Coulomb blockade regime of d < F and ðF Þ d þ F , U þ d F , where ðÞ ¼ 2jtWD j2 ðÞ is the hybridization function between the dot and the wires, ðÞ is the local density of states at energy in the neighboring sites j ¼ 1 of the dot, and F is the Fermi energy. In this regime, the total Hamiltonian of the setup becomes [2,6,35]
0
(a) 1 0.5
-0.2 1
2
L/
K
3
0
L/
K
=0.009
L/
K
=0.10
L/
K
=1.0
L/
K
L/
K
=13 =
0.01
sin½kðN þ 1Þa sinðkNaÞ
(b)
(3)
NRG Fermi Liquid
10
0.5
eff
-4
1
10
T/TK 0
1.0E-4
0.01
T/
1
0 0
0.1
1
L/
constant
0.4
eff
FIG. 3 (color online). Case A under the resonance condition of kF ¼ kF;n . In this case, one changes L with keeping tWD (hence TK1 and K1 ) constant. (a) Kondo temperature TK as a function of L, obtained by the poor man’s scaling (blue solid curve) and NRG (green dashed curve). The two approaches show qualitatively the same overall behavior that TK drastically changes for L & K1 , while TK TK1 for L * K1 ; their discrepancy in L * K1 is discussed in the text. (b) NRG result of the temperature T dependence of conductance G for various values of L=K1 . We choose 1 =2t ¼ 0:28, 0 =2t ¼ 0:925, eVg =2t ¼ 0:125, and U=2t ¼ 3:6.
:
-3
2
0.2
eVg t
k and q are the wave vectors in jxj < L and jxj > L, respectively, satisfying ¼ 0 2t cosðkaÞ ¼ 0 eVg 2t cosðqaÞ. ðÞ ¼ 2t2WD ðÞ shows resonances with level spacing @vF =L; see Fig. 2. The oscillation amplitude of ðÞ is proportional to Vg . The value of ðF Þ of the L ! 1 limit is denoted as 1 , which equals the average of ðÞ around F for finite L. We sketch how the change of ðÞ by Vg affects the Kondo effect in different regimes of L. For L K1 (i.e., TK1 ), the Kondo temperature is determined by 1 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TK1 1 U=2 exp½ðd F ÞðU þ d F Þ=ð21 UÞ, and the cloud size is K1 ¼ @vF =ðkB TK1 Þ. In this regime, the average 1 of many resonances of ðÞ around F determines the Kondo effect, insensitively to Vg . On the other hand, for L K1 (i.e., TK1 ), the resonance of ðÞ located at F , namely 0 ðF Þ, determines the Kondo effect, resulting in TK ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TK0 0 U=2 exp½ðd F ÞðUþd F Þ=ð20 UÞ and
G [2e /h]
0.4
G [2e2/h]
(b) 1 NRG result Poorman's scaling
0.6
at½sin2 ðkaÞ þ
Teff
(a) 0.8
sinðqaÞsin2 ðkaÞ
ln (TK / TK )
0.05
ln(TK /TK )
X ni1 þ HW : i;
0.1
0
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PHYSICAL REVIEW LETTERS
PRL 110, 246603 (2013)
2 L/ K
4
10
K
FIG. 4 (color online). Case B. In this case, one changes tWD (hence K ) with keeping L constant. (a) The NRG result of GðTÞ for different values of tWD ; we show the values of K instead of tWD . (b) Teff ðL=K Þ [defined in Eq. (8)], obtained from the two different approaches of the Fermi liquid theory and the NRG. Inset: The NRG result of TK ðL=K1 Þ exhibits the same behavior as Fig. 3(a). We choose L ¼ 100a, 0 =2t ¼ 0:931, eVg =2t ¼ 0:125, and U=2t ¼ 3:6. F is chosen to be located at a center of a resonance of ðÞ, for simplicity.
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PHYSICAL REVIEW LETTERS
K ¼ @vF =ðkB TK0 Þ. In this case, Vg affects conduction electrons within the cloud, and modifies TK . We will discuss how TK changes between TK0 and TK1 as a function of L=K in the two possible situations, case A where one changes L with keeping tWD constant, and case B where tWD changes and L remains constant. Kondo temperature.—We compute TK , using the poor man’s scaling and the NRG [32]. In the poor man’s scaling, the renormalization of J ! R RD0 J þ J 2 ð D D0 þ D ÞdðÞ=jj is performed with reducing the energy bandwidth of the wire R from D0 to D, and stopped at the bandwidth where J 2 dðÞ=jj is comparable with J. The final bandwidth provides TK , eVg cos½kF ð2L þ aÞ TK 2L Ci ln ’ ; (4) K TK1 2tsin2 ðkF aÞ eVg 2L Ci (5) ’ for kF ¼ kF;n ; 2 K 2tsin ðkF aÞ R 0 0 0 where CiðyÞ y 1 dy ðcosy Þ=y , kF;n ¼ 2n=ð2L þ aÞ, and n is an integer. Equation (4) is obtained by putting k ! kF kðF Þ, q ! qðF Þ, L a, vF ð1=@Þð@=@kÞjkF ¼ ð2at=@Þ sinðkF aÞ, jeVg j 2tsin2 ðkF aÞ, and the linearization of ’ F þ @vF ðk kF Þ into Eq. (3); k ! kF and q ! qðF Þ are valid within the small energy scale of TK . We remark that K depends on L and Vg in Eq. (4). In Eq. (4), the term Cið2L=K Þ gives the information on the cloud, while another L dependence of the 2kF oscillation appears because resonance centers in ðÞ shift across F as L changes. One can focus on the former. In case A, where one changes L, one can reduce the effect of the 2kF oscillation, by considering the situation that F is located near the bottom of an energy band where the 2kF term slowly oscillates, or by considering the resonance condition of kF ¼ kF;n where the 2kF term provides the maximum value; see Eq. (5). The resonance condition can be achieved at each value of L, by tuning an additional gate voltage applied to the entire region of the wires (not shown in Fig. 1) with monitoring the conductance through the dot. On the other hand, in case B, where one changes tWD with keeping L constant, the term cos½kF ð2L þ aÞ is constant, and hence, can be ignored. For case A under the resonance situation of kF ¼ kF;n , the poor man’s scaling in Eq. (5) is plotted as a function of L=K1 (rather than L=K ) in Fig. 3(a). As expected, TK stays at TK1 for K1 & L, drastically changes around L ¼ K1 , and approaches to TK0 for L K . In addition, the oscillation of lnðTK =TK1 Þ sincð2L=K1 Þ appears for L * K1 ; we put CiðxÞ sincx ¼ ðsinxÞ=x and K ’ K1 (valid for x ¼ 2L=K1 * 1) into Eq. (4). The oscillation originates from the average effect of ðÞ within TK1 , and becomes suppressed for longer L as more ( TK1 =) resonances appear within TK1 . On the other hand, for L K1 , we find lnðTK =TK1 Þ / lnðL=K Þ, using
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CiðxÞ lnðxÞ þ 0:577 for x 1. The above behavior of TK ðL=K1 Þ reveals the Kondo cloud. Conductance.—We compute the temperature T dependence of electron conductance G between the wires through the dot, using the NRG [17,30–32]. We will discuss how to extract K1 from GðTÞ in cases A and B. We continue to discuss case A under the resonance condition of kF ¼ kF;n . In Fig. 3(b), we plot GðTÞ for different L’s. It is customary [36] to get an estimate for TK from the temperature at which GðTÞ equals the half of the zero-temperature conductance GðT ¼ 0; L ! 1Þ of the L ! 1 case. GðTÞ shows the behavior distinct between L * K1 and L & K1 . For L * K1 , GðTÞ equals GðT ¼ 0; L ! 1Þ=2 at almost the same temperature, implying that TK equals TK1 independent of L. On the other hand, for L & K1 , GðTÞ shows that TK changes toward TK0 as L decreases. This NRG result agrees with the poor man’s scaling; see Fig. 3(a). In this way, one directly measures K1 . There is a discrepancy between the two curves in Fig. 3(a). In the NRG case, lnðTK =TK1 Þ decreases only monotonically for L * K , without showing the behavior sincð2L=K1 Þ of the poor man’s scaling. The discrepancy may come from the known limitation that the logarithmic discretization scheme of NRG does not perfectly capture the behavior of high-energy states (higher than in our case). On the other hand, for L K1 , where low-energy states mainly contribute to the Kondo effect, the NRG shows the same behavior of lnðTK =TK1 Þ lnðL=K Þ as the poor man’s scaling. Next, we discuss case B where one changes tWD with keeping L constant (hence, ¼ @vF =L is constant). Figure 4 shows the NRG result of GðTÞ for different K ’s. We obtain TK from the high-temperature behavior of GðTÞ in the same way as above, by choosing the temperature at which GðTÞ ¼ GðT ¼ 0; L ! 1Þ=2. The result of TK agrees with case A; see the inset of Fig. 4(b). Below, we suggest another way to see the cloud from the lowtemperature behavior of GðTÞ. Note that as T changes across , GðTÞ can show a jump due to the resonance structure of ðÞ, as shown for L=K ¼ 26 in Fig. 4(a). We describe the regime of T TK , , using the fixedpoint Hamiltonian of the Fermi liquid theory [6,33,37], X k þ k0 X 1 þ p cyk ck0 Hlow ’ k cyk ck ðF Þ kk0 2TK k X y 1 þ c ck " cy ck # ; (6) TK 2 ðF Þ k1 k2 k3 k4 k1 " 2 k3 # 4 where cyk creates an electron with momentum k, spin , and energy k . p is the phase shift by the potential scattering, which occurs as the particle-hole symmetry is broken. Although depends on , we take for simplicity ðF Þ in Eq. (6) as a crude approximation. The second term of Eq. (6) describes elastic scattering of electrons by the
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PHYSICAL REVIEW LETTERS
Kondo singlet, with scattering phase shift ðÞ ¼ =2 þ ð F Þ=TK þ p . The third term shows repulsive interactions that break the Kondo singlet, and contributes to inelastic T matrix tin as ðF ÞImtin ðÞ ¼ ½ð F Þ2 þ 2 T 2 =ð2TK2 Þ. By combining ðÞ and Imtin ðÞ, we obtain [6] 2e2 1 þ cosð2p Þ 2 T 2 GðTÞ ¼ 1 2 ; (7) 2 h Teff 2 cosð2p Þ 1 2 þ ; 2 2 Teff 3ð þ Þ ½1 þ cosð2p ÞTK2
(8)
where and are constants depending on Vg and kF but independent of L. p is obtained by comparing GðT ¼ 0Þ with Eq. (7). For nonzero p , GðT ¼ 0Þ deviates from the unitary-limit value of 2e2 =h. Note that when is independent of and the particle-hole symmetry is preserved, p ¼ 0 and ¼ 0, hence, Teff ! TK . Teff is obtained by comparing GðTÞ with Eq. (7) in experiments or in the NRG, while computed from Eq. (8) in the Fermi liquid theory. We plot Teff ðL=K Þ in Fig. 4(b), showing good agreement between the NRG and the Fermi liquid theory; their quantitative discrepancy may come from our approximation in the Fermi liquid theory. The dependence of Teff on L=K or L=K1 is useful for identifying K in case B, since is constant so that Teff ðL=K Þ directly provides the information of TK ðL=K Þ; see Eq. (8). For L * K , Teff is almost constant, implying that TK and K are independent of L. For L & K , Teff (hence, TK ) depends on L=K . The crossover occurs around K ’ L. Discussion.—Our proposal may be within experimental reach. Case A, where L varies, may be achieved with keyboard-type gate voltages, while one tunes tWD by a gate in case B. A good candidate for our proposal may be a carbon nanotube, where TK 1 K and K 1 m [38]. For both the cases, a (single-mode or multimode) wire whose Fermi level F lies near the bottom Eb [van Hove singularity (VHS)] of one of the energy bands is useful to achieve a conclusive evidence of the cloud; our results of Eq. (4) and NRG are applicable to this regime, since they are obtained, taking into account the energy dependence of . In this regime, ðÞ is sensitive to Vg ; hence, it may not be difficult to obtain a sizable difference between TK1 and TK0 by Vg ð L (not suppressed even for L K ) and can be only weakly (perturbatively) modified, allowing direct detection of K and the spatial structure of the cloud. We thank I. Affleck, G. Finkelstein, L. Glazman, D. Goldharber-Gordon, S. Ilani, A. K. Mitchell, and S. Tarucha for useful discussions, and Minchul Lee for advice on NRG calculations. We acknowledge support by NRF (Grant No. 2011-0022955; H.-S. S.), and by BSF and Minerva Grants (Y. O.). H.-S. S. thanks J. Moore and UC Berkeley, where this Letter was written, for hospitality.
*[email protected] [1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964). [2] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, England, 1993). [3] D. Goldhaber-Gordon, H. Shtrikman, D. Mahalu, D. Abusch-Magder, and U. Meirav, Nature (London) 391, 156 (1998). [4] S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, 540 (1998). [5] L. Kouwenhoven and L. I. Glazman, Phys. World 14, 33 (2001). [6] L. I. Glazman and M. Pustilnik, in Nanophysics: Coherence and Transport, edited by H. Bouchiat et al. (Elsevier, New York, 2005), pp. 427–478. [7] J. E. Gubernatis, J. E. Hirsch, and D. J. Scalapino, Phys. Rev. B 35, 8478 (1987). [8] V. Barzykin and I. Affleck, Phys. Rev. Lett. 76, 4959 (1996); Phys. Rev. B 57, 432 (1998). [9] L. Borda, Phys. Rev. B 75, 041307 (2007). [10] A. Holzner, I. P. McCulloch, U. Schollwo¨ck, J. von Delft, and F. Heidrich-Meisner, Phys. Rev. B 80, 205114 (2009). [11] E. S. Sørensen and I. Affleck, Phys. Rev. B 53, 9153 (1996). [12] I. Affleck, L. Borda, and H. Saleur, Phys. Rev. B 77, 180404 (2008). [13] C. A. Bu¨sser, G. B. Martins, L. Costa Ribeiro, E. Vernek, E. V. Anda, and E. Dagotto, Phys. Rev. B 81, 045111 (2010). [14] A. K. Mitchell, M. Becker, and R. Bulla, Phys. Rev. B 84, 115120 (2011). [15] I. Affleck and P. Simon, Phys. Rev. Lett. 86, 2854 (2001). [16] R. Yoshii and M. Eto, Phys. Rev. B 83, 165310 (2011). [17] P. S. Cornaglia and C. A. Balseiro, Phys. Rev. B 66, 115303 (2002); Phys. Rev. Lett. 90, 216801 (2003). [18] P. Simon and I. Affleck, Phys. Rev. Lett. 89, 206602 (2002); Phys. Rev. B 68, 115304 (2003).
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[19] T. Hand, J. Kroha, and H. Monien, Phys. Rev. Lett. 97, 136604 (2006). [20] For a review, see I. Affleck, in Perspectives of Mesoscopic Physics (World Scientific, Singapore, 2010), pp. 1–44. [21] J. P. Boyce and C. P. Slichter, Phys. Rev. Lett. 32, 61 (1974); Phys. Rev. B: Solid State 13, 379 (1976). [22] G. Bergmann, Phys. Rev. B 77, 104401 (2008). [23] V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie, and N. S. Wingreen, Science 280, 567 (1998). [24] H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature (London) 403, 512 (2000). [25] H. Pru¨ser, M. Wenderoth, P. E. Dargel, A. Weismann, R. Peters, T. Pruschke, and R. G. Ulbrich, Nat. Phys. 7, 203 (2011). [26] Y.-S. Fu et al., Phys. Rev. Lett. 99, 256601 (2007). [27] W. B. Thimm, J. Kroha, and J. von Delft, Phys. Rev. Lett. 82, 2143 (1999). [28] Yu. Bomze, I. Borzenets, H. Mebrahtu, A. Makarovski, H. U. Baranger, and G. Finkelstein, Phys. Rev. B 82, 161411 (2010). [29] P. W. Anderson, J. Phys. C 3, 2436 (1970).
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[30] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975); H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980). [31] R. Bulla, T. A. Costi, and T. Pruschke, Rev. Mod. Phys. 80, 395 (2008). [32] We use the full density matrix NRG method developed by A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99, 076402 (2007). We use the NRG discretization parameter of ¼ 2 and keep 300 states at each iteration. [33] P. Nozie`res, J. Low Temp. Phys. 17, 31 (1974); J. Phys. (Paris) 39, 1117 (1978). [34] P. W. Anderson, Phys. Rev. 124, 41 (1961). [35] J. R. Schrieffer and P. A. Wolff, Phys. Rev. 149, 491 (1966). [36] D. Goldhaber-Gordon, J. Go¨res, M. Kastner, H. Shtrikman, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998). [37] I. Affleck and A. W. W. Ludwig, Phys. Rev. B 48, 7297 (1993). [38] J. Nygard, D. H. Cobden, and P. E. Lindelof, Nature (London) 408, 342 (2000).
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PRL 110, 246603 (2013)
How to Directly Measure a Kondo Cloud’s Length Jinhong Park,1 S.-S. B. Lee,1 Yuval Oreg,2 and H.-S. Sim1,* 1
Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea 2 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel (Received 26 October 2012; published 14 June 2013)
We propose a method to directly measure, by electrical means, the Kondo screening cloud formed by an Anderson impurity coupled to semi-infinite quantum wires, on which an electrostatic gate voltage is applied at distance L from the impurity. We show that the Kondo cloud, and hence the Kondo temperature and the electron conductance through the impurity, are affected by the gate voltage, as L decreases below the Kondo cloud length. Based on this behavior, the cloud length can be experimentally identified by changing L with a keyboard type of gate voltage or tuning the coupling strength between the impurity and the wires. DOI: 10.1103/PhysRevLett.110.246603
PACS numbers: 72.15.Qm, 72.10.Fk, 73.63.Kv
Introduction.—The Kondo effect is a central many-body problem of condensed matter physics [1,2]. It involves a spin singlet, formed by the spin-spin interaction between a magnetic impurity and surrounding conduction electrons. A deeper understanding of the effect has been achieved by using a quantum dot, that hosts a magnetic impurity spin, under systematic control [3–6]. Although the Kondo effect is well known, its spatial features still remain to be addressed. The Kondo spin singlet is formed below the energy scale of Kondo temperature TK . This implies that the singlet is spatially formed over a conduction-electron region of length scale K ¼ @vF =ðkB TK Þ; when TK 1 K and the Fermi velocity vF 105 –106 m=s, K 1 m. The region is called the Kondo screening cloud. There have been several proposals [7–19] for ways to detect the cloud. Despite the proposals, there has been no conclusive measurement supporting the existence of the Kondo cloud [20,21]. The difficulty to detect the cloud arises because it is a spin cloud showing quantum fluctuations with zero averaged spin. The cloud manifests itself in the spin-spin correlation [7–10] between the impurity and the conduction electrons. However, it requires measurements of spin dynamics of time scale @=ðkB TK Þ. STM studies probing local density of states may be useful for detecting the cloud [12–14,22]. Recent STM measurements [23–26] show the Kondo effect in the region away from a magnetic impurity, whose spatial extension is, however, much shorter than K . Another direction is to study a magnetic impurity in a finite-size system [15–19,27,28]. Because the cloud cannot extend beyond the finite size, the Kondo effect is strongly affected, and suppressed when the system is shorter than K . There has been no conclusive experimental detection of K in this direction [28]. In this Letter, we propose a new way of detecting the Kondo cloud, based on the intuition that a change of conduction electrons inside the cloud will affect the Kondo effect. We consider a Kondo impurity formed in a 0031-9007=13=110(24)=246603(5)
quantum dot coupled to two semi-infinite ballistic quantum wires with electron tunneling amplitude tWD (see Fig. 1). Electrostatic gate voltages Vg are applied to the wires (or to only one wire) at distance L from the dot, modifying indirectly the local density of states ðÞ of conduction electrons nearby the dot (Fig. 2). We find that Vg does not affect the cloud, when L K . However, when L K , the cloud, hence the Kondo temperature TK and electron conductance G through the dot, are sensitive to Vg . The crossover between the two regimes occurs at L K . By measuring G or TK with varying L or tWD (Figs. 3 and 4), one can detect the crossover and K . We use the poor man’s scaling [29], numerical renormalization group study (NRG) [30–32], and Fermi liquid theory [33]. The setup.—We describe the wires by the tight-binding Hamiltonian with sites j’s in wire i ¼ l, r, HW ¼
1 X X X
½0 nij þ ðtcyij ciðjþ1Þ þ H:c:Þ
i¼l;r j¼1 ¼";#
eVg
1 X X
X
nij ;
(1)
i¼l;r j¼Nþ1 ¼";#
FIG. 1 (color online). Setup for detecting the Kondo cloud. A quantum dot located at x ¼ 0 hosts a magnetic impurity spin. It couples to quantum wires along x^ axis, with electron tunneling amplitude tWD . Gate voltage Vg is applied at distance L from the dot (in jxj > L). The Kondo effect becomes sensitive to Vg , as L decreases below the cloud length.
246603-1
Ó 2013 American Physical Society
0.2
H ¼ HD þ HW þ HT ’ J s~ S~ þ V
0.15
(2)
FIG. 2 (color online). Hybridization function ðÞ ¼ 2t2WD ðÞ in units of t for infinite (blue dashed curve) and finite L (green solid curve); we choose 0 =2t ¼ 0:931, tWD =2t ¼ 0:2, and eVg =2t ¼ 0:125. For finite L, ðÞ has resonances with spacing @vF =L; F is chosen to be located at a resonance center (red vertical dashed line). When L is so large that Vg is applied outside the Kondo cloud, the Kondo temperature TK ¼ TK1 is determined by 1 and independent of Vg . On the other hand, TK depends on Vg and L, when L & the cloud size K . When L K , TK is determined by 0 ðF Þ.
P Here, the Kondo impurity spin, S~ ¼ ;0 dy ~ 0 d0 =2, couples with the spin of the neighboring conduction P electrons, s~ ¼ ;0 ðcyl1 þcyr1 Þ~ 0 ðcl10 þcr10 Þ=2, with strength J ¼ 2t2WD ½1=ðd F Þ þ 1=ðU þ d F Þ. The second term describes the potential scattering with strength V ¼ t2WD ½1=ðd F Þ 1=ðU þ d F Þ=2. Local density of states.—We show how the gate voltage Vg changes the local density of states ðÞ at sites j ¼ 1 of the wires. We calculate ðÞ, by matching the singleparticle wave functions of HW between j ¼ N and N þ 1, ðÞ ¼
cyij
where creates an electron with spin and energy 0 at site j in wire i, nij cyij cij , and t is the hopping energy. The last term describes the gate voltage Vg applied in jxj > L ¼ Na, where a is the lattice spacing. Vg ðxÞ changes at x ¼ L abruptly over the length shorter than the Fermi wavelength. The dot Hamiltonian HD is modeled by the P Anderson impurity [34], HD ¼ ¼";# d dy d þ Und" nd# , where dy creates an electron with energy d and spin in the dot, nd dy d , and U is the electron repulsive interP P action. HT ¼ tWD i ¼";# ðcyi1 d þ H:c:Þ describes electron tunneling between the wire and the dot. The dot is occupied by a single electron in the Coulomb blockade regime of d < F and ðF Þ d þ F , U þ d F , where ðÞ ¼ 2jtWD j2 ðÞ is the hybridization function between the dot and the wires, ðÞ is the local density of states at energy in the neighboring sites j ¼ 1 of the dot, and F is the Fermi energy. In this regime, the total Hamiltonian of the setup becomes [2,6,35]
0
(a) 1 0.5
-0.2 1
2
L/
K
3
0
L/
K
=0.009
L/
K
=0.10
L/
K
=1.0
L/
K
L/
K
=13 =
0.01
sin½kðN þ 1Þa sinðkNaÞ
(b)
(3)
NRG Fermi Liquid
10
0.5
eff
-4
1
10
T/TK 0
1.0E-4
0.01
T/
1
0 0
0.1
1
L/
constant
0.4
eff
FIG. 3 (color online). Case A under the resonance condition of kF ¼ kF;n . In this case, one changes L with keeping tWD (hence TK1 and K1 ) constant. (a) Kondo temperature TK as a function of L, obtained by the poor man’s scaling (blue solid curve) and NRG (green dashed curve). The two approaches show qualitatively the same overall behavior that TK drastically changes for L & K1 , while TK TK1 for L * K1 ; their discrepancy in L * K1 is discussed in the text. (b) NRG result of the temperature T dependence of conductance G for various values of L=K1 . We choose 1 =2t ¼ 0:28, 0 =2t ¼ 0:925, eVg =2t ¼ 0:125, and U=2t ¼ 3:6.
:
-3
2
0.2
eVg t
k and q are the wave vectors in jxj < L and jxj > L, respectively, satisfying ¼ 0 2t cosðkaÞ ¼ 0 eVg 2t cosðqaÞ. ðÞ ¼ 2t2WD ðÞ shows resonances with level spacing @vF =L; see Fig. 2. The oscillation amplitude of ðÞ is proportional to Vg . The value of ðF Þ of the L ! 1 limit is denoted as 1 , which equals the average of ðÞ around F for finite L. We sketch how the change of ðÞ by Vg affects the Kondo effect in different regimes of L. For L K1 (i.e., TK1 ), the Kondo temperature is determined by 1 as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TK1 1 U=2 exp½ðd F ÞðU þ d F Þ=ð21 UÞ, and the cloud size is K1 ¼ @vF =ðkB TK1 Þ. In this regime, the average 1 of many resonances of ðÞ around F determines the Kondo effect, insensitively to Vg . On the other hand, for L K1 (i.e., TK1 ), the resonance of ðÞ located at F , namely 0 ðF Þ, determines the Kondo effect, resulting in TK ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TK0 0 U=2 exp½ðd F ÞðUþd F Þ=ð20 UÞ and
G [2e /h]
0.4
G [2e2/h]
(b) 1 NRG result Poorman's scaling
0.6
at½sin2 ðkaÞ þ
Teff
(a) 0.8
sinðqaÞsin2 ðkaÞ
ln (TK / TK )
0.05
ln(TK /TK )
X ni1 þ HW : i;
0.1
0
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2 L/ K
4
10
K
FIG. 4 (color online). Case B. In this case, one changes tWD (hence K ) with keeping L constant. (a) The NRG result of GðTÞ for different values of tWD ; we show the values of K instead of tWD . (b) Teff ðL=K Þ [defined in Eq. (8)], obtained from the two different approaches of the Fermi liquid theory and the NRG. Inset: The NRG result of TK ðL=K1 Þ exhibits the same behavior as Fig. 3(a). We choose L ¼ 100a, 0 =2t ¼ 0:931, eVg =2t ¼ 0:125, and U=2t ¼ 3:6. F is chosen to be located at a center of a resonance of ðÞ, for simplicity.
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K ¼ @vF =ðkB TK0 Þ. In this case, Vg affects conduction electrons within the cloud, and modifies TK . We will discuss how TK changes between TK0 and TK1 as a function of L=K in the two possible situations, case A where one changes L with keeping tWD constant, and case B where tWD changes and L remains constant. Kondo temperature.—We compute TK , using the poor man’s scaling and the NRG [32]. In the poor man’s scaling, the renormalization of J ! R RD0 J þ J 2 ð D D0 þ D ÞdðÞ=jj is performed with reducing the energy bandwidth of the wire R from D0 to D, and stopped at the bandwidth where J 2 dðÞ=jj is comparable with J. The final bandwidth provides TK , eVg cos½kF ð2L þ aÞ TK 2L Ci ln ’ ; (4) K TK1 2tsin2 ðkF aÞ eVg 2L Ci (5) ’ for kF ¼ kF;n ; 2 K 2tsin ðkF aÞ R 0 0 0 where CiðyÞ y 1 dy ðcosy Þ=y , kF;n ¼ 2n=ð2L þ aÞ, and n is an integer. Equation (4) is obtained by putting k ! kF kðF Þ, q ! qðF Þ, L a, vF ð1=@Þð@=@kÞjkF ¼ ð2at=@Þ sinðkF aÞ, jeVg j 2tsin2 ðkF aÞ, and the linearization of ’ F þ @vF ðk kF Þ into Eq. (3); k ! kF and q ! qðF Þ are valid within the small energy scale of TK . We remark that K depends on L and Vg in Eq. (4). In Eq. (4), the term Cið2L=K Þ gives the information on the cloud, while another L dependence of the 2kF oscillation appears because resonance centers in ðÞ shift across F as L changes. One can focus on the former. In case A, where one changes L, one can reduce the effect of the 2kF oscillation, by considering the situation that F is located near the bottom of an energy band where the 2kF term slowly oscillates, or by considering the resonance condition of kF ¼ kF;n where the 2kF term provides the maximum value; see Eq. (5). The resonance condition can be achieved at each value of L, by tuning an additional gate voltage applied to the entire region of the wires (not shown in Fig. 1) with monitoring the conductance through the dot. On the other hand, in case B, where one changes tWD with keeping L constant, the term cos½kF ð2L þ aÞ is constant, and hence, can be ignored. For case A under the resonance situation of kF ¼ kF;n , the poor man’s scaling in Eq. (5) is plotted as a function of L=K1 (rather than L=K ) in Fig. 3(a). As expected, TK stays at TK1 for K1 & L, drastically changes around L ¼ K1 , and approaches to TK0 for L K . In addition, the oscillation of lnðTK =TK1 Þ sincð2L=K1 Þ appears for L * K1 ; we put CiðxÞ sincx ¼ ðsinxÞ=x and K ’ K1 (valid for x ¼ 2L=K1 * 1) into Eq. (4). The oscillation originates from the average effect of ðÞ within TK1 , and becomes suppressed for longer L as more ( TK1 =) resonances appear within TK1 . On the other hand, for L K1 , we find lnðTK =TK1 Þ / lnðL=K Þ, using
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CiðxÞ lnðxÞ þ 0:577 for x 1. The above behavior of TK ðL=K1 Þ reveals the Kondo cloud. Conductance.—We compute the temperature T dependence of electron conductance G between the wires through the dot, using the NRG [17,30–32]. We will discuss how to extract K1 from GðTÞ in cases A and B. We continue to discuss case A under the resonance condition of kF ¼ kF;n . In Fig. 3(b), we plot GðTÞ for different L’s. It is customary [36] to get an estimate for TK from the temperature at which GðTÞ equals the half of the zero-temperature conductance GðT ¼ 0; L ! 1Þ of the L ! 1 case. GðTÞ shows the behavior distinct between L * K1 and L & K1 . For L * K1 , GðTÞ equals GðT ¼ 0; L ! 1Þ=2 at almost the same temperature, implying that TK equals TK1 independent of L. On the other hand, for L & K1 , GðTÞ shows that TK changes toward TK0 as L decreases. This NRG result agrees with the poor man’s scaling; see Fig. 3(a). In this way, one directly measures K1 . There is a discrepancy between the two curves in Fig. 3(a). In the NRG case, lnðTK =TK1 Þ decreases only monotonically for L * K , without showing the behavior sincð2L=K1 Þ of the poor man’s scaling. The discrepancy may come from the known limitation that the logarithmic discretization scheme of NRG does not perfectly capture the behavior of high-energy states (higher than in our case). On the other hand, for L K1 , where low-energy states mainly contribute to the Kondo effect, the NRG shows the same behavior of lnðTK =TK1 Þ lnðL=K Þ as the poor man’s scaling. Next, we discuss case B where one changes tWD with keeping L constant (hence, ¼ @vF =L is constant). Figure 4 shows the NRG result of GðTÞ for different K ’s. We obtain TK from the high-temperature behavior of GðTÞ in the same way as above, by choosing the temperature at which GðTÞ ¼ GðT ¼ 0; L ! 1Þ=2. The result of TK agrees with case A; see the inset of Fig. 4(b). Below, we suggest another way to see the cloud from the lowtemperature behavior of GðTÞ. Note that as T changes across , GðTÞ can show a jump due to the resonance structure of ðÞ, as shown for L=K ¼ 26 in Fig. 4(a). We describe the regime of T TK , , using the fixedpoint Hamiltonian of the Fermi liquid theory [6,33,37], X k þ k0 X 1 þ p cyk ck0 Hlow ’ k cyk ck ðF Þ kk0 2TK k X y 1 þ c ck " cy ck # ; (6) TK 2 ðF Þ k1 k2 k3 k4 k1 " 2 k3 # 4 where cyk creates an electron with momentum k, spin , and energy k . p is the phase shift by the potential scattering, which occurs as the particle-hole symmetry is broken. Although depends on , we take for simplicity ðF Þ in Eq. (6) as a crude approximation. The second term of Eq. (6) describes elastic scattering of electrons by the
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Kondo singlet, with scattering phase shift ðÞ ¼ =2 þ ð F Þ=TK þ p . The third term shows repulsive interactions that break the Kondo singlet, and contributes to inelastic T matrix tin as ðF ÞImtin ðÞ ¼ ½ð F Þ2 þ 2 T 2 =ð2TK2 Þ. By combining ðÞ and Imtin ðÞ, we obtain [6] 2e2 1 þ cosð2p Þ 2 T 2 GðTÞ ¼ 1 2 ; (7) 2 h Teff 2 cosð2p Þ 1 2 þ ; 2 2 Teff 3ð þ Þ ½1 þ cosð2p ÞTK2
(8)
where and are constants depending on Vg and kF but independent of L. p is obtained by comparing GðT ¼ 0Þ with Eq. (7). For nonzero p , GðT ¼ 0Þ deviates from the unitary-limit value of 2e2 =h. Note that when is independent of and the particle-hole symmetry is preserved, p ¼ 0 and ¼ 0, hence, Teff ! TK . Teff is obtained by comparing GðTÞ with Eq. (7) in experiments or in the NRG, while computed from Eq. (8) in the Fermi liquid theory. We plot Teff ðL=K Þ in Fig. 4(b), showing good agreement between the NRG and the Fermi liquid theory; their quantitative discrepancy may come from our approximation in the Fermi liquid theory. The dependence of Teff on L=K or L=K1 is useful for identifying K in case B, since is constant so that Teff ðL=K Þ directly provides the information of TK ðL=K Þ; see Eq. (8). For L * K , Teff is almost constant, implying that TK and K are independent of L. For L & K , Teff (hence, TK ) depends on L=K . The crossover occurs around K ’ L. Discussion.—Our proposal may be within experimental reach. Case A, where L varies, may be achieved with keyboard-type gate voltages, while one tunes tWD by a gate in case B. A good candidate for our proposal may be a carbon nanotube, where TK 1 K and K 1 m [38]. For both the cases, a (single-mode or multimode) wire whose Fermi level F lies near the bottom Eb [van Hove singularity (VHS)] of one of the energy bands is useful to achieve a conclusive evidence of the cloud; our results of Eq. (4) and NRG are applicable to this regime, since they are obtained, taking into account the energy dependence of . In this regime, ðÞ is sensitive to Vg ; hence, it may not be difficult to obtain a sizable difference between TK1 and TK0 by Vg ð L (not suppressed even for L K ) and can be only weakly (perturbatively) modified, allowing direct detection of K and the spatial structure of the cloud. We thank I. Affleck, G. Finkelstein, L. Glazman, D. Goldharber-Gordon, S. Ilani, A. K. Mitchell, and S. Tarucha for useful discussions, and Minchul Lee for advice on NRG calculations. We acknowledge support by NRF (Grant No. 2011-0022955; H.-S. S.), and by BSF and Minerva Grants (Y. O.). H.-S. S. thanks J. Moore and UC Berkeley, where this Letter was written, for hospitality.
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