Transport measurements in quantum Hall systems

Quantum Matter Seminar

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Shafayat Hossain

Outline

• Hall effect measurements
• 2D electrons in perpendicular magnetic fields: Probing many-body phases via electrical transport
• Geometric resonance technique to probe the Fermi sea of composite fermions
• Measurements of spin and valley susceptibilities

Outline

• Hall effect measurements
• 2D electrons in perpendicular magnetic fields: Probing many-body phases via electrical transport
• Geometric resonance technique to probe the Fermi sea of composite fermions
• Measurements of spin and valley susceptibilities

Hall effect

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I

B

B

The sign and the density of carriers can be determined from the slope.

Edwin Hall (1879)

+

Standard sample geometry: Hall bar

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In 2D

{

In 2D

Van der Pauw geometry

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In 2D

7

magnet

Inner Vacuum chamber

Sorb

1K pot

Insert

Sample space

1K pot

magnet

still

Mixing chamber

Heat exchangers

Phase boundary

8

Dilution Refrigerator

Source: Wikipedia

Hall effect

9

I

B

B

The sign and the density of carriers can be determined from the slope.

Edwin Hall (1879)

+

Outline

• Hall effect measurements
• 2D electrons in perpendicular magnetic fields: Probing many-body phases via electrical transport
• Geometric resonance technique to probe the Fermi sea of composite fermions
• Measurements of spin and valley susceptibilities

2D electrons in a perpendicular magnetic field

11

12

2D electrons in a perpendicular magnetic field

Landau level filling factor:

ν = (n/B)(h/e)

K. von Klitzing, PRL (1980)

R_xx and R_xy

13

y

Edge states

Current only at the edges

B

Davies “The Physics of Low Dimensional Semiconductors”

E

Fractional quantum Hall effect (FQHE)

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Tsui, PRL (1982)

Shabani, PRL (2009)

Ground states in N = 0 Landau level (LL)

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Shabani, PRL (2009)

• Fractional quantum Hall effect (FQHE) at odd-denominator fillings (Tsui, 1982)
• Composite fermions at even-denominator fillings

n = 1.8 × 10¹¹ cm^-2

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Composite fermions (CFs)

Electron-flux CFs

Electrons

ν = 1/2

Jain, PRL (1989), Halperin, PRB (1993)

E

R^*_C

B^*_┴

real space

reciprocal space

K^*_F

B^*_┴

FQHE and composite fermions

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Jain, Composite Fermions (Cambridge University Press, New York, 2007)

FQHE of electrons = IQHE of CFs

Λ-levels

​

​

​

​

Half-filled (lowest) Landau level

Q. What is the ground state at this half-filled flatband?

Ground state at a half-filled flatband

• Predicted and observed in many systems. (Wu, PRL 2007; Tsai, NJP 2015, Si, Science 2010; Cao, Nature 2018.)

Correlated (Mott) insulator (Mott, 1937).

Ground state at half-filled (lowest) Landau level

1. Metallic at 𝛎 = ½!

2. A non-Fermi liquid of interacting electrons:

Theory: Halperin, PRB 1993, Nayak, Nucl. Phys. 1994; Jain, 2007; Shao, PRL 2015.

Experiment: Jiang, PRB 1989; Willett, PRL 1990.

Composite fermions (CFs) to the rescue

Jain, PRL (1989); Halperin, PRB (1993).

Ground states in N > 0 LL

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N = 1 LL:

• FQHE at even-denominator fillings ν = 5/2 (& 7/2) Willett, PRL (1987)
• Possible non-Abelian charge excitations (Moore, Read, 1991)

Ground states in N > 0 LL

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Theory:

Fogler, 1996

Fradkin, 1999

Experiment:

Lilly, 1999

Du, 1999

N = 1 LL:

• FQHE at even-denominator fillings ν = 5/2 (& 7/2) Willett, PRL (1987)
• Possible non-Abelian charge excitations (Moore, Read, 1991)

N > 1 LL :

​

• Nematic/Stripe phase

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Landau level filling factor:

ν = (n/B)(h/e)

2D electrons in a perpendicular magnetic field

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Landau level filling factor:

ν = (n/B)(h/e)

2D electrons in a perpendicular magnetic field

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2D electrons in a perpendicular magnetic field

Landau level filling factor:

ν = (n/B)(h/e)

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So many states in one (magnetic field) sweep!�

Shayegan, arXiv (2005)

ν = number of occupied Landau levels

Fractional quantum Hall effect (FQHE)

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Shabani, PRL (2009)

Composite fermion (CF) Fermi sea

• How to probe them via transport?

Outline

• Hall effect measurements
• 2D electrons in perpendicular magnetic fields: Probing many-body phases via electrical transport
• Geometric resonance technique to probe the Fermi sea of composite fermions
• Measurements of spin and valley susceptibilities

30

Probing CF Fermi sea

Condition for geometric resonance:

Minimum in 𝛒_xx at B^*

Weiss, EPL 1989; Winkler, PRL 1989; Beenakker, PRL 1989; Gerhardts, PRL 1989; Skuras, APL 1997; Kamburov, PRB 2012

<1%

n

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Willett, PRL 1993, Smet, PRL 1999

Probing CF Fermi sea

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Measurements of CFs’ spin polarization: Technique

Willett, PRL 1993, Smet, PRL 1999

Spin polarization

Fully polarized Fermi sea

Partially polarized Fermi sea

Spin polarization

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Geometric resonance/Weiss oscillation for electrons in zero magnetic field

Theory : C. W. J. Beenakker, Phys. Rev. Lett. 62, 2020 (1989).

Experiment : D. Weiss et al.,Europhys. Lett. 8, 179 (1989).

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• Periodic in 1/B.
• Frequency gives the area of Fermi Surface

Fermi sea area from SdH oscillations

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1. Measuring the Fermi contour shape

Simultaneously measure along two perpendicular directions

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1. Splitting of the Fermi contour in geometric resonance

B_|| along k_x

Mueed, PRL (2015)

2. CFs in the lowest Landau level

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MS Hossain, et al., PRB 100, 041112(R) (2019).

3. Comparison with the Dirac theory

Dirac theory of CFs:

Son, PRX 2015; Geraedts, Science 2016; Wang, PRX 2017; Cheung, PRB 2017; Pan, Nat. Phys. 2017; Geraedts, PRL 2018.

Density of CFs

Excellent agreement!

MS Hossain, et al., PRL, in press (2020).

4. Do CFs obey the Luttinger theorem?

MS Hossain, et al., PRL, in press (2020).

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Bloch

Ferromagnet

​

​

​

​

​

Ferromagnet

(expected)

​

​

​

​

​

​

​

Partial

Polarization

​

​

​

​

​

​

• Bloch ferromagnetism observed for CFs near 𝛖 = 1/2 at very low densities!

MS Hossain, et al., Nature physics, in press (2020).

5. Bloch ferromagnetism of CFs

What about other Landau levels?

Composite fermions in the lowest Landau level

​

Is there any composite fermion Fermi sea in higher Landau levels

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CFs in N = 1 Landau level

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MS Hossain, et al., PRL 120, 256601 (2018).

CF Fermi sea is the precursor of 5/2 state!

Summary

Transport measurements can probe quantum Hall effect, nematic phases, and Fermi surface of composite fermions in two-dimensional electron systems.

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