Pis’ma v ZhETF, vol. 109, iss. 2, pp. 89 - 90

January 25

On the accuracy of conductance quantization in spin-Hall insulators

S. K. Konyzheva, E. S. Tikhonov, V. S. Khrapai¹⁾

Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Russa

Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia

Submitted 31 October 2018

Resubmitted 31 October 2018

Accepted 15

November 2018

DOI: 10.1134/S0370274X19020048

Transverse conductance in an ordinary quantum

in the absence of spin relaxation. Similarly, a non-

Hall effect is quantized with metrological accuracy.

additivity of the edge resistances is observable in a two-

In the effective Landauer-Büttiker description [1] this

terminal measurement. We bridge our results with the

is interpreted as a conductance quantization of one-

model of disordered contacts [10] and estimate the spin

dimensional (1D) edge channels. Such edge channels are

relaxation resistance contribution in HgTe-based QSH

protected by chirality thereby their four-terminal resis-

devices.

tance vanishes. By contrast, conventional 1D systems in-

To begin with, we develop an experimentally rele-

evitably suffer from contact effects [2] and even the best

vant model of a multi-terminal bar for QSH measure-

ones exhibit poor quantization [3] and non-vanishing

ments. In a typical experimental setup [6], a lithographic

four-terminal resistance [4].

gate covers the inner part of the mesa, excluding the

Somewhat intermediate case is realized in quantum

leads. All the leads are assumed identical and are repre-

spin-Hall (QSH) insulators [5, 6]. Here, the electric cur-

sented by regions of two-dimensional electron gas. The

rent is carried by a pair of 1D helical edge channels with

leads have finite resistance and interconnect helical edge

opposite spin and chirality, thereby the backscattering

channels with the ohmic contacts. The ohmic contacts

is easier than in the quantum Hall case and more diffi-

have negligible resistance and serve as macroscopic equi-

cult compared to the conventional 1D case. In spite of

librium reservoirs, connecting the device to external

the expected immunity to a non-magnetic disorder in a

electric circuit. We will consider the idealized case of bal-

phase-coherent helical edge channel [7], the mean-free

listic topologically protected edge states, such that the

path in experiments is relatively small and longer chan-

spin relaxation occurs only in the leads and the ohmic

nels behave as quasi-classical diffusive conductors [8, 9].

contacts. In addition, we assume that the edge channels

The shorter, ballistic, channels exhibit four-terminal re-

are perfectly coupled to the leads. This means that the

sistance which is poorly quantized and additive. Often

chemical potentials of the outgoing edge channels coin-

the resistance randomly drops below the quantum value

cide with those of the same-spin electrons in the leads

g^-10 = h/e² in local measurements [6] and below the ex-

nearby the bulk-edge transition point. All the leads are

pected fraction of g^-10 in non-local measurements. This

assumed to be quasi-1D, such that any dependence of

indicates that the measured signal is not the 1D con-

the chemical potentials within the cross-section of the

ductance and is influenced by contact effects. Backscat-

lead is neglected. The lead conductance is denoted g_L.

tering of helical electrons at a contact can be revealed

The spin relaxation in the leads is taken into account via

in transport [10] and noise measurements as well as in

the total spin relaxation conductance as G_s = g_s +g_L/4,

spin injection and photogalvanic experiments.

which takes into account two contributions, from a di-

In this work, we elaborate the role played by the

rect spin- relaxation (g_s term) and from an indirect re-

leads and ohmic contacts in resistance measurements

laxation via out-diffusion into the ohmic contact (g_L/4

in ballistic helical edge channels. A simple model of a

term).

phase-incoherent transport taking spin relaxation in the

Our main result is the expression for a four-terminal

leads and contacts into account is presented for a re-

resistance of the helical edge channel, determined as the

alistic experimental setup. We observe that the four-

ratio of the measured voltage drop between the ohmic

terminal resistance is always below g^-10 and vanishes

contacts on either side of the channel to the current in

the channel. Such resistanc[ reads:

]

G_s

R_4T =

(1)

¹⁾e-mail: dick@issp.ac.ru

g₀

G_s + g₀

Письма в ЖЭТФ том 109 вып. 1 - 2

2019

S. K. Konyzheva, E. S. Tikhonov, V. S. Khrapi

Equation (1) dictates that the measured R_4T is al-

edge resistances are additive in the experiments, as well

ways smaller than the quantum resistance g^-10. In the

as with numerous observations of R_4T below the quan-

limit of large lead resistance and negligible spin relax-

tum value g^-10 = h/e² in local measurements [6] and

ation (G_s → 0) the R_4T is suppressed down to zero,

below the expected fraction of g^-10 in non-local mea-

similar to zeroing of the longitudinal resistance in the

surements.

ordinary quantum Hall regime. Thus, apparently, the

In summary, we have shown how the spin relaxation

requirement of quantum phase-coherence in the leads

in the current/voltage leads affects the resistance mea-

raised in [7] is excessive as long as the spin relaxation is

surements of ballistic QSH helical edge channels in ex-

suppressed.

perimentally relevant geometries. Negligible relaxation

In the opposite limit G_s ≫ g₀ the spin relaxation is

results in a vanishing four-terminal resistance and non-

strong and we obtain a small correction to the resistance

additive edge resistances in a two-terminal setup even if

quantum R_4T ≈ g^-10 - G^-1s. It is interesting to compare

the quantum phase-coherence is not preserved, similar

with [10], which addresses a QSH measurement with dis-

to the case of ordinary quantum Hall effect. Available

ordered current/voltage probes. Relevant to our case,

experiments are in the opposite limit of strong spin re-

that calculation predicts R_4T = g^-10 (1 - D) / (1 + D),

laxation, which explains a poor quality of the resistance

where D is a reflection probability at a contact(formula

quantization as well as the edge resistances smaller than

(17) with all D the same [10]). This coincides with the

expected ballistic value.

result (1) given D = (1 + 2G_s/g₀)^-1, thereby bridging

We acknowledge valuable discussions with

the model of disordered probes [10] with the spin relax-

K.E. Nagaev, S.A. Tarasenko, Z.D. Kvon, S.U. Piatrusha

ation in the leads in typical experiments.

and M.L. Savchenko. This work was supported by the

We also calculate a two-terminal resistance between

Russian Science Foundation project # 16-42-01050.

the neighboring contacts in an N-terminal bar, with the

Full text of the paper is published in JETP Letters

following result:

journal. DOI: 10.1134/S0021364019020024

(N - 2) (G_s + 2g₀) G_s

R_2T =

(2)

g_L

2g₀

(NG_s + 2g₀) (G_s + g₀) 2g₀

1. M. Büttiker, Phys. Rev. B 38, 9375 (1988).

2. H.-L. Engquist and P. W. Anderson, Phys. Rev. B 24,

The first two terms in (2) are, respectively, the in-

1151 (1981).

evitable contribution of the lead resistance and the re-

3. A. Yacoby, H. L. Stormer, N. S. Wingreen, L. N. Pfeiffer,

sistance of two helical edges in parallel. The last term

K. W. Baldwin, and K. W. West, Phys. Rev. Lett. 77,

takes a non-additivity of the helical edge resistances into

4612 (1996).

account, given the spin relaxation is finite. In the limit

4. R. de Picciotto, H. L. Stormer, L. N. Pfeiffer,

G_s → 0, as well as for N = 2, this term vanishes and

K. W. Baldwin, and K. W. West, Nature

411,

we recover a result equivalent to the ordinary (spin-

(2001).

degenerate) quantum Hall effect. In the opposite limit

5. B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science

G_s ≫ g₀ the edge resistances become completely addi-

314, 1757 (2006).

tive and R_2T is a sum of the lead resistance and the

6. M. Konig, S. Wiedmann, C. Brune, A. Roth, H. Buh-

edge resistances g^-10 and (N - 1)g^-10 connected in par-

mann, L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Sci-

allel. In a multi-terminal bar with N → ∞ we have

ence 318, 766 (2007).

R_2T ≈ 2g^-1L + g^-10 + (2G_s)^-1, i.e., the first-order cor-

7. X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057

rection here is opposite in sign compared to the R_4T

(2011).

case.

8. G. M. Gusev, Z. D. Kvon, E. B. Olshanetsky,

Finally, we estimate the contribution of the spin re-

A. D. Levin, Y. Krupko, J. C. Portal, N.N. Mikhailov,

laxation resistance G^-1s in experiments, based on the

and S. A. Dvoretsky, Phys. Rev. B 89, 125305 (2014).

measurements of the weak anti-localization in HgTe

9. E. S. Tikhonov, D. V. Shovkun, V. S. Khrapai,

quantum wells. We obtain, roughly, G^-1s ∼ 100 Ω, such

Z. D. Kvon, N. N. Mikhailov, and S. A. Dvoretsky,

that the expected contribution of the spin relaxation re-

JETP Lett. 101, 708 (2015).

sistance to R_4T and R_2T is within a few percent. This

10. A. Mani and C. Benjamin, J. Phys. Cond. Matter 28,

estimate is consistent with the rule of thumb that the

145303 (2016).

Письма в ЖЭТФ том 109 вып. 1 - 2

2019